WHAT’S THE STORY WITH STORY PROBLEMS? EXPLORING THE
RELATIONSHIP BETWEEN CONTEXTUAL MATHEMATICS TASKS, STUDENT
ENGAGEMENT, AND MOTIVATION TO LEARN MATHEMATICS IN MIDDLE
SCHOOL
By
Jamie L. W. Wernet

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics Education—Doctor of Philosophy
2015

ABSTRACT
WHAT’S THE STORY WITH STORY PROBLEMS? EXPLORING THE
RELATIONSHIP BETWEEN CONTEXTUAL MATHEMATICS TASKS, STUDENT
ENGAGEMENT, AND MOTIVATION TO LEARN MATHEMATICS IN MIDDLE
SCHOOL
By
Jamie L. W. Wernet
Contextual tasks, or tasks that include scenarios described at least in part with
nonmathematical language or pictures, are a long-standing part of mathematics education in the
United States. These tasks may have potential to promote student engagement and motivation to
learn mathematics by highlighting applications of mathematics to everyday matters and
generating interest in the content (e.g., van den Heuvel-Panhuizen, 2005). Yet, several scholars
have challenged the belief that contextual tasks can serve to motivate students and problematized
their role in mathematics curricula (e.g., Chazan, 2000; Gerofsky, 2004). Some theoretical and
empirical evidence exists to support both claims.
This study addresses a call for more research on how student motivation and engagement
in mathematics are influenced in specific learning situations, namely, working on contextual
tasks. Motivation describes a person’s choice, persistence, and performance when engaging in an
activity (Brophy, 2004), whereas engagement is active involvement in a learning activity (Helme
& Clarke, 2001) and the observable manifestation of motivation (Skinner, Kindermann, &
Furrer, 2008). The purpose of this multiple-case study was to consider the general questions, Do
contextual tasks have potential to engage students, and if so, under what circumstances?, and
How do students experience these tasks relative to their motivation to learn mathematics? In
particular, I considered enactment of tasks across lessons in two 7th-grade mathematics
classrooms. Through analyzing data from observations, lesson-specific teacher and student

surveys, and focus group interviews, I identified the most and least engaging lessons for
students, then characterized the tasks in these lessons as written and enacted.
I found that students were more likely to show high levels of engagement in contextual
tasks than noncontextual tasks. Their engagement in contextual tasks was related, however, to
the learning goals of the task, its placement in a unit, and the function of the context in problem
solving. In high-engagement lessons, the tasks tended toward open-ended tasks with contexts
central in solving the problem. I also found differences in the way students and teachers attended
to contextual features of tasks between the high- and low-engagement lessons. Students drew on
the context more in the high engagement lessons, and were more likely to connect the context to
the main mathematical ideas in the lesson. Teachers also paid more attention to contexts and in
more diverse ways across the high-engagement lessons.
I also drew on the data sources using expectancy-value theory to explore in depth how
students responded to individual tasks relative to their motivation to learn mathematics. Aspects
of tasks students attended to (including contexts) when reflecting on the value of mathematical
content and their experiences in lessons was related to their underlying motivation to learn.
Trends across groups of students, however, indicate that task contexts play little role in
promoting students’ valuing of mathematics or beliefs that they can be successful on a task.
Based on these findings, I argue that some contextual tasks engage students by eliciting
genuine interest in the context itself, providing entry into and support in solving the problem, and
anchoring the instruction to provide students a shared experience on which to develop their
understanding of the mathematical concepts. Yet, contextual tasks do not necessarily have the
same potential to motivate students to learn. I discuss implications for teachers, curriculum
design, and future research regarding the purpose and function of contextual tasks.

To Kevin, Penelope, Adrianna, Donald, and Edmund

iv

ACKNOWLEDGEMENTS

As I finish my degree, people like to ask, “How did you do it?” With four young kids at
home, it is a reasonable question with a simple answer—only by the gracious support and
encouragement of many people. I have much to be thankful for.
To my dissertation committee, Drs. Kristen Bieda (chair), Robert Floden, Sandra Crespo,
and Corey Drake, thank you for your guidance and advice that made this a better project in every
way. I extend my deepest appreciation to Kristen, my advisor, for coming alongside me from the
first phone call accepting me to the program to making comments and suggestions on this paper
down to the last draft. Thank you Bob for welcoming me on the Algebra Teaching Study team
and answering my countless questions with patience and wisdom and helped me develop as a
scholar and researcher. Corey and Sandra, thank you for your practical feedback and thoughtful
questions as I planned and reported on this project. I would also like to extend gratitude to
Elizabeth Phillips, Yvonne Grant, and Glenda Lappan for their advice and support throughout
my time at Michigan State.
And then there are all the people supporting me on the home front. To my husband,
Kevin—I would not and could not have finished this program without your friendship and
encouragement—and to Nellie, Annie, DJ, and Eddie, thank you for keeping me grounded and
balanced. Thank you to Mike and Linda Wacyk, John Wernet (who donated two laptops to my
graduate student career!), and Lisa Ward. I also would not have made it to this point without the
tangible and intangible help of all our siblings: Erin and Steve Stepek, Dave and Laurel Wacyk,
Emily and Mike Paski, Alex and Sarah Wernet, and Genny Ward-Wernet. Thank you Stephanie
Atchison for the loving care you provided for the kids over the years.

v

Along the way, I expected to meet and get to know new colleagues, but I did not expect
them to become such close friends! To my writing group, Jerilynn Lepak, Alex Musselman, and
Funda Gonulates, I so appreciate the hours you spent reading and commenting on every single
thing I have written over the last five years. But I am especially grateful for the laughs and tears
and stories (and Jerilynn—the hours and hours of driving time and conference hotels!) we have
shared. Thank you Marianna Levin and Julie Nurnberger-Haag for your thoughtful feedback on
my work and for your friendship.
Thank you to the Program in Mathematics Education at Michigan State University for
supporting me, financially and otherwise. Thank you to the College of Natural Science for
awarding me Dissertation Continuation and Completion Fellowships, making it possible to finish
in a timely manner. Finally, to the teachers and students who participated in this project, I extend
my gratitude for your hard work, openness, and for letting me be a part of this snapshot of your
lives.

vi

TABLE OF CONTENTS

LIST OF TABLES

xii

LIST OF FIGURES

xiii

KEY TO ABBREVIATIONS

xiv

CHAPTER 1: INTRODUCTION
A Study of Engagement, Motivation, and Contextual Tasks
Conceptualization of Key Terms
Purpose Statement
Situating the Study in Extant Literature
Final Note on the Importance of the Study
Overview of Forthcoming Chapters

1
2
2
5
5
8
8

CHAPTER 2: LITERATURE REVIEW
Contextual Tasks in the Mathematics Curriculum
Purposes of Contextual Tasks, Through History and Present
Connected Mathematics Project—An Example
Implementation of Contextual Tasks in Mathematics Classrooms
Student Motivation to Learn Mathematics and Engage in Mathematics
Tasks
Motivation to Learn Mathematics
Motivation to learn in both learned and adaptive
Learning goals are preferable to performance goals
A growth mindset is preferable to a fixed mindset
Instructional moves can promote motivation to learn
Motivation is situated with tasks as well as cumulative
Linking motivation and engagement
Empirical Research on Motivating Instructional Practices in
Mathematics
Linking Contextual Mathematics Tasks and Student Motivation
Theoretical Support for Contextual Tasks as Motivators
Theoretical Objections to Contextual Tasks as Motivators
Empirical Support for Contextual Tasks as Motivators
Empirical Challenges to Contextual Tasks as Motivators
Synthesis
Research Questions
Theoretical Foundation—Expectancy-Value Theory

24
25
27
29
34
36
37
38

CHAPTER 3: METHODS
Research Strategy
Settings and Participants

40
40
42

vii

10
10
10
13
15
17
17
18
18
19
19
20
20
22

Pine River Middle School
Mrs. Meyer
Students in Pine River participating class
Southpoint Junior High School
Ms. Pearson
Students in Southpoint participating class
Focus Students
Data Collection and Instruments
Classroom Observations
Tasks
Fieldnotes and video
Engagement protocol
Lesson-Specific Surveys by Teacher and Students
Focus Group Interviews
Beliefs and Values Survey
Note on Validity of Measures
Data Analysis
Ranking Lessons by Engagement
Students’ and teachers’ lesson-specific survey numerical
responses
Holistic engagement ratings
Counts of evidence of engagement
Notes on coding
Reliability coding
Comparing engagement results by data source
Identifying lessons with high and low engagement
Characterizing Lessons
Characterizing contexts as written
Characterizing contexts as enacted
Identifying patterns and commonalities in tasks
Coding Interview and Survey Data
Limitations and Delimitations of Research Design
CHAPTER 4: STUDENT ENGAGEMENT AND CONTEXTS AS WRITTEN
Description of Observed Lessons and Tasks
Lesson Overview
Identification of High- and Low-Engagement Lessons
Contexts in Highly Engaging Lessons
Lessons with Open-Ended Contextual Questions as the Core
Activity
Pine River Lesson 8: Solving the mystery of the teacher in
disguise
Pine River Lesson 9: Meeting the Wumps and impostors
Pine River Lesson 10: Mathematizing the Wumps and
impostors
Southpoint Lesson 4: Making sense of cola advertising

viii

44
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48
52
53
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56
57
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80
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86
88
90

Other Highly Engaging Contextual Tasks
Southpoint Lesson 2: Exploring similarity in the Wumps
noses and mouths
Southpoint Lesson 7: Analyzing Sascha’s bike trip
Trends and Themes in Contexts as Written in Highly Engaging
Lessons
Contexts in the Least Engaging Lessons
Pine River Lesson 11: Counting and Pricing Apples, Bananas, and
Eggs
Southpoint Lesson 8: What do Puffins, Airplanes, Enchiladas, and
Student Council Elections Have in Common?
Southpoint Lesson 10: Money Models
Trends and Themes in Contexts as Written and Enacted in Least
Engaging Lessons
Discussion: Engagement Potential of Contextual Tasks as Written
Meaningful Contexts Can Engage Students
Importance of Task Purpose and Centrality of the Context
Summary and Preliminary Argument
CHAPTER 5: STUDENT ENGAGEMENT AND CONTEXTS AS ENACTED
Student Engagement Related to Context
What Does Context-Related Engagement Look Like?
Commenting on context without connecting to mathematics
Using the context to make sense of and solve a problem
Connecting the context with the lesson’s core mathematical
concepts
Comparing Context-Related Engagement in High- and LowEngagement Lessons
Themes in Teachers’ Enactment of Tasks in High- and Low-Engagement
Lessons
Attention to Context by Type and Lesson Phase
Relationship Between Teachers’ and Students’ Enactment
Qualitative Differences in Teachers’ Enactment of Contextual
Tasks
Cases of Enactment of Contextual Tasks
Pine River Lesson 9: “Let Me Tell You, You Will Not Forget the
Wumps”
Southpoint Lesson 7: Using the Bike Context to Make Sense of
Unit Rates
Pine River Lesson 11: Inauthenticity of Apples and Bananas
Southpoint Lesson 8: Lots of Contexts, Little Engagement
Summary, and a Note on Comparisons Across Classrooms
Discussion: Multiple Roles of Contexts in Student Engagement
Contexts as Likeable and Interesting
Contexts Provide Entry Into and Support in Problem Solving
Contexts as Anchors for Instruction

ix

91
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107
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135
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138
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Considering Alternate Explanations and Other Factors Influencing
Engagement
Conclusion
CHAPTER 6: THE RELATIONSHIP BETWEEN MOTIVATION TO LEARN
MATHEMATICS AND STUDENTS’ RESPONSES TO CONTEXTUAL TASKS
Theoretical Background
Expectancy-Value Theory
Analytical Framework
Results Part 1: Trends in Focus Group Students’ Responses to Tasks
Contextual Tasks and Beliefs About the Value of Mathematical
Content
Generating ideas for future use of mathematical concepts
and skills
Motivation to learn and value-related responses to contexts
Low motivation group
Neutral and mixed motivation group
High motivation group
Money-related contexts and rational number operations
Student Responses to Tasks—Expectation of Success and Intrinsic
Value
Patterns in student responses to lessons by motivation group
Low motivation group
Neutral and mixed motivation group
High motivation group
Contextual tasks not related to individual high- and lowengagement lessons
Results Part 2: Illustrative Student Cases
Felix—I feel smart on weekends, because I'm not being told I'm
wrong!
Elijah—Math is My Friend!
Sophia—What if I Become a Famous App-Maker?
Discussion
Contexts, Value, and Application
Hidden applications
Differences related to motivation group
Money contexts as motivators
Contexts, Expectations of Success, and Intrinsic Value
Conclusion
CHAPTER 7: CONCLUSIONS AND IMPLICATIONS
Potential of Contextual Tasks to Motivate and Engage
Returning to the Research Questions
So What’s the Story on ‘Story Problems’?
Limitations and Directions for Future Research
Implications for Research and Practice

x

143
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189
192

Research
Curriculum Design
Instruction
Final Thoughts

192
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197

APPENDICES
APPENDIX A: SAMPLE LESSON MODIFIED BY MS. PEARSON
APPENDIX B: BELIEFS AND VALUES SURVEY – FALL
APPENDIX C: OBSERVATION PROTOCOL FOR STUDENT
ENGAGEMENT
APPENDIX D: LESSON-SPECIFIC STUDENT SURVEY
APPENDIX E: LESSON-SPECIFIC TEACHER SURVEY
APPENDIX F: FOCUS GROUP INTERVIEW PROTOCOL
APPENDIX G: LESSON ENGAGEMENT CODEBOOK
APPENDIX H: ENGAGEMENT ANALYSIS CHARTS FOR PINE
RIVER: MRS. MEYER’S CLASS
APPENDIX I: ENGAGEMENT ANALYSIS CHARTS FOR PINE
RIVER: MS. PEARSON’S CLASS
APPENDIX J: COMPARING AND SCALING 3.2 MODIFIED BY MS.
PEARSON
APPENDIX K: TABLE K1: SUMMARY OF OBSERVED LESSONS
REFERENCES

198
199
201
205
207
208
209
212
216
218
220
221
231

xi

LIST OF TABLES

Table 1 School Demographics

45

Table 2 Study Focus Students

50

Table 3 Map of Research Questions to Data Sources

59

Table 4 Correlation Matrix for Pine River Engagement Results

67

Table 5 Correlation Matrix for Southpoint Results

68

Table 6 Different Types of Context

72

Table 7 Lesson Engagement Results

83

Table 8 Summary of Features of Tasks in High-Engagement Lessons

93

Table 9 Summary of Features of Tasks in Low-Engagement Lessons

100

Table 10 Summary of Context-Related Engagement by Lesson Phase

112

Table 11 Summary of Students’ Context-Related Engagement by Types

119

Table 12 Summary of Teachers’ Attention to Context

121

Table 13 Analytic Framework for Students’ Motivation-Related Statements

150

Table 14 Belief in the Value of Mathematics Expressed in Focus Group Interviews

154

Table F1 Summary of Lessons Preceding Interviews

210

Table G1 Lesson Engagement Codebook for Evidence of Student Engagement
from Lesson Transcripts

212

Table K1 Summary of Observed Lessons

221

xii

LIST OF FIGURES

Figure 1. Diagram showing how certain actions by the teacher and students during
task enactment changed the authenticity and centrality of the contexts as written in
the curriculum (Wernet, 2001).

16

Figure 2. Problem statement and image from Stretching and Shrinking 1.1.

86

Figure 3. Problem statement and image from Stretching and Shrinking 2.1.

88

Figure 4. The problem as written in Stretching and Shrinking 2.1. Mrs. Meyer
focused on one core question: Who are the Wumps, and who are impostors?

89

Figure 5. Comparison statements from taste test results for Comparing and
Scaling 1.1.

90

Figure 6. Variables defined in task given to students.

96

Figure 7. Operationalized questions guiding analysis for research question 3. I
conceptualized the relationship between a student’s motivation to learn
mathematics and their responses to contextual tasks to be bidirectional, with
questions to guide each aspect of the relationship.

147

Figure 8. Questions to consider when designing or using contextual tasks to
engage students based on results from high- and low-engagement lessons.

186

Figure 9. Purposes of contextual tasks and questions to consider when using them
to serve these purposes. The first column follows from the findings of the study.
The second and third columns are italicized to show these are my suggested
questions for consideration based on extant literature.

188

Figure A1. Accompanying resource page distributed by Ms. Pearson.

200

Figure H1. Dotplots of student and teacher engagement ratings over lessons at Pine
River.

216

Figure H2. Scatterplot showing average weighted engagement counts per hour
versus the average focus student engagement rating at Pine River.

217

Figure I1. Dotplots of student and teacher engagement ratings over lessons at
Southpoint.

218

Figure I2. Scatterplot showing average weighted engagement counts per hour
versus the average focus student engagement rating at Southpoint.

219

xiii

KEY TO ABBREVIATIONS

CCSSI

Common Core State Standards Initiative

CMP

Connected Mathematics Project (Lappan et al., 2006, 2014)

NCTM

National Council of Teachers of Mathematics

xiv

CHAPTER 1
INTRODUCTION
Negative mathematics dispositions, in the form of math anxiety, avoiding challenge and
higher level mathematics, or believing one is not really a “math person”, are a widespread plague
in the United States. These beliefs influence students’ motivation to learn mathematics (or not),
and are influenced by classroom instruction (e.g., Brophy, 2004). Motivation to avoid
mathematics warrants a close consideration of mathematics instruction and curricula that can
promote student engagement and motivation to learn (Turner & Meyer, 2009). These issues are
in fact receiving attention; for example, the National Council of Teachers of Mathematics
(NCTM) recently published two books for teachers to address mathematics-related motivation,
engagement, and dispositions (Brahier & Speer, 2011; Middleton & Jansen, 2011).
Middleton and Jansen (2011) discussed several themes in literature on students’
motivation to learn mathematics. One is that although motivation to learn develops early and
stays relatively stable over time, student motivation can be influenced by instructional practices
and tasks. That is, a student’s level of motivation might be resistant to change, but is not innate,
and can be molded through localized experiences with mathematics in and out of the classroom.
This points to the importance of considering the potential for specific mathematics lessons or
tasks to engage and motivate students.
Adding contexts to mathematics problems is believed to have the potential to promote
student motivation, among other purposes (e.g., Chapman, 2006; Meyer, Dekker, & Querelle,
2001; Middleton & Jansen, 2011; Mitchell & Carbone, 2011; van den Heuvel-Panhuizen, 2005).
On one hand, several curricula and instructional research programs have been designed around
contextual tasks, either to meet recommendations by NCTM to put more emphasis on real-world

1

applications or to attempt an explicit connection between students’ everyday, cultural
mathematical activity and school mathematics. Their developers argue that these tasks can serve
to catch students’ interest and engage them in the mathematics along with supporting their
learning (e.g., Civil, 2002; Gutstein, 2003; Lappan & Phillips, 2009; Romberg & Shafer, 2003;
Silva, Moses, Rivers, & Johnson, 1990). On the other hand, several scholars have argued that
believing contextual tasks will motivate students is too simplistic. They have problematized the
use of contextual problems and called for a reconsideration of their role in the curriculum
(Boaler, 1993; Chazan, 2000; Gerofsky, 2004; Sullivan, Zevenbergen, & Mousley, 2003;
Verschaffel, Greer, & De Corte, 2000). Meanwhile, it seems the popular opinion of contextual
tasks is negative, as “the hated word problems” (Thomas in Thomas & Gerofsky, 1997, p. 21).
This debate suggests some complexity in the relationship between contextual tasks and
student motivation to learn mathematics. There is reasonable theoretical support for both views,
but there is a need for more empirical evidence to confirm or dispute the motivational potential
of various types of contextual tasks under various circumstances. Consequently, the purpose of
this multiple-case study was to gain a more nuanced understanding of the relationship between
the use of contextual tasks, engagement, and student motivation to learn in secondary
mathematics classrooms.
A Study of Engagement, Motivation, and Contextual Tasks
Conceptualization of Key Terms
I use the term contextual task or contextual problem to refer to any task or project that
includes some kind of realistic or imaginary scenario described at least in part by
nonmathematical language or a nonmathematical pictorial representation (Li, 2000). The term is
both inclusive and exclusive. It is inclusive because it covers the variety of other terms used in

2

the literature such as word problems, story problems, real-world or applied problems, and realworld connections. It is exclusive in the sense that it is limited to a particular subset of tasks
enacted in mathematics classrooms. “Context” can also refer to the physical classroom
environment; the social, cultural, or historical setting; or the placement of a task within in the
curriculum (i.e., where in the unit and course a topic is taught). With the definition used here,
contextualization is a feature of a task as written or designed.
Motivation is an umbrella construct used to describe a person’s choice, persistence, and
performance when engaging in an activity, particularly a goal-directed activity (Brophy, 2004). It
is related but not equivalent to constructs like values, beliefs, and attitudes. Motivation
encompasses the reasons underlying people’s behavior (Middleton & Jansen, 2011). To consider
the effects of contextual tasks on motivation, I focused on student motivation to learn, meaning a
“tendenc[y] to find academic activities meaningful and worthwhile and to try to get the intended
learning benefits from them” (Brophy, 2004, p. 16). This definition focuses on learning over
performance-based rewards like grades, making it different than extrinsic motivation. It is also a
different construct than intrinsic motivation, which relates to enjoyment of an activity.
Motivation to learn is in essence different than the motivation to do other things (e.g., sports,
games, playing music) because of the nature of school and its constraints (Brophy, 2004).
Motivation to learn, however, can coexist with and be supported by extrinsic and intrinsic
motivation.
The term engagement is sometimes used synonymously with motivation, but they are not
equivalent (Fredericks & McColskey, 2012; Middleton & Spanias, 1999). Basically, what
students actually do in the classroom may not accurately reflect their level of motivation to learn.
For example, a student might value learning mathematics and believe they can be successful (i.e.,

3

they are motivated to learn), but still sleep in class (i.e., be disengaged) because they stayed up
too late the night before. Or, students might express low motivation to learn, but engage richly in
academic tasks that are interesting to them. Motivation is the generally non-observable
mechanism underlying people’s behavior, whereas engagement is the observable manifestation
of motivation (Skinner, Kindermann, & Furrer, 2008) as active involvement in a learning activity
(Helme & Clarke, 2001). All students are motivated to do something, and that drives their level
of engagement—which is more prone to change with specific classroom situations. The
relationship between engagement and motivation is complicated, though, as various definitions
of engagement are inextricably related to other motivation constructs.
Most literature on engagement identifies interrelated components of school
engagement—including behavioral, emotional, and cognitive engagement—making it a
multidimensional concept (Fredericks, Blumenfeld, & Paris, 2004). In their review of literature
on academic engagement, Fredericks et al. (2004) describe behavioral engagement as relating to
participation in the classroom, adherence to rules, and absence of disruptive behaviors; that is,
being on task. Emotional engagement refers to student affect related to school activities,
including happiness, sadness, boredom, and excitement. Cognitive engagement, which is closely
linked to motivation to learn, refers more to the quality of behavioral engagement, or the level of
investment students make in learning content. It incorporates challenge seeking, hard work, and
self-regulation (Fredericks et al., 2004).
In this study, I focused on emotional and cognitive engagement and the interplay between
them. Behavioral engagement has primarily been captured at the school level (e.g., participation
in extracurricular activities, attendance). It is also difficult to conceptually and practically
separate behavioral engagement and the other types of engagement. Moreover, as will be

4

discussed, motivation theory suggests emotions (e.g., enjoyment, surprise) are an inherent part of
motivation. Thus, I plan to include emotional engagement in this study because although
motivation to learn is linked most closely with cognitive engagement, it seems emotional
engagement cannot be separated from student motivation to learn (Stipek et al., 1998). From this
point on, including in my research questions, when I use the term engagement I refer to both
cognitive and emotional engagement.
Along with engagement, I also aimed to explore broader motivational factors underlying
student behavior, such as the value students place on mathematical content. Accordingly, I refer
to both engagement and motivation throughout the following chapters, recognizing they are
different but related constructs. In the literature review, I explicate on research linking
motivation and engagement.
Purpose Statement
The purpose of this study was to gain a more nuanced understanding of the relationship
between the use of contextual tasks in mathematics classrooms and student engagement and
motivation to learn mathematics, which can inform curriculum development and instructional
decisions. More specifically, I aimed to identify lessons in which students exhibited high or low
levels of engagement, characterize the tasks in those lessons, and explore the motivational
foundations for student engagement in contextual tasks. The cases were classes of 7th-grade
students in which a variety of contextual tasks were a regular part of instruction. That is, students
worked on contextual tasks on their own, in small groups, or as a class multiple times per week.
Situating the Study in Extant Literature
Student motivation to learn mathematics is an important outcome to study. First, student
motivation and engagement has been empirically linked to student achievement in mathematics

5

(see, e.g., Cordova & Lepper, 1996; Singh, Granville, & Dika, 2002; Stipek et al., 1998), which
is the ultimate goal of school mathematics. Though the relationship between motivation and
engagement and student achievement is complex and, in some cases, nonlinear, many studies
link the quality of student engagement with positive gains in learning (Fredericks, Blumenfeld,
& Paris, 2004).
Second, some argue that students’ positive mathematical dispositions should be viewed
as a core intended outcome of mathematics education (National Research Council, 2001;
Brophy, 2008). That is, teachers and researchers should attend to “changes and acquisitions in
the motivational aspects of learning (i.e., certain content-related values, dispositions, and
appreciations), not just the knowledge and skill aspects (Brophy, 2008, p. 135). In part, this focus
serves to address and counteract widespread negative attitudes toward mathematics. Middleton
and Jansen (2011) describe this as a “motivational epidemic—motivation to avoid mathematics”
(p. 184). This response does not seem to generalize to other school subjects, encouraging the
study of mathematics and motivation in particular. “The longevity of math anxiety, both in
popular lore and research, indicate that the linkage of motivation and mathematics is a
compelling argument for considering how mathematics teaching, learning, and motivation are
negotiated in the classroom” (Turner & Meyer, 2009, p. 527).
Another reason for focusing on engagement and motivation to learn mathematics is that
meeting the calls for mathematics education reform such as those by NCTM (1989, 2000) and
the Standards for Mathematical Practice in the Common Core State Standards for Mathematics
(CCSSI, 2010) require students to be actively involved in the learning process. Unlike in lecturebased classrooms, teachers are dependent on students’ intellectual engagement in studentcentered classrooms based on inquiry, discussion, and reasoning, (Chazan, 2000). Thus, it is

6

worthwhile to consider curricular features and instructional practices that may influence student
engagement and motivation.
Taken together, literature suggests that interest and engagement are malleable. Tasks and
classroom instruction matter, which is where contextual tasks come in—they can serve to
promote or hinder student engagement and motivation to learn. An important feature of
contextual tasks is their ubiquitous nature—word problems have a presence in nearly all
mathematics curricula and are a familiar part of school mathematics (Gerofsky, 1996). This
feature makes them a potentially strong place to start in increasing student motivation and
engagement. They are also uniquely positioned to support engagement among students typically
underrepresented in mathematics, including students of color. Contexts that are personally
relevant can be developed to more actively and deliberately link content to students’ racial,
ethnic, and cultural experiences (Tate, 1996).
The claim that contextual tasks have the potential to promote student motivation seems to
run counter to the prevailing theme in school culture that students dread story problems. This
theme is implied in Thomas’s (1997) statement, “lots of people don’t call them ‘story problems.’
I refer to them usually as ‘the hated word problems’” (p. 21). Similarly, in her description of the
typical mathematics classroom, Wilson (2003) wrote, “Ample time is usually left for practicing
problems, and an audible sigh of relief is heard whenever word problems are not assigned” (p.
4). This suggests a need to take a closer look at the relationship between student motivation and
contextual tasks. Whereas the relationship between contextual tasks and problem solving
strategies, and between contextual tasks and student learning, has been the focus of recent
research (e.g., Walkington, Sherman, & Petrosino, 2012; Walkington, in press), the relationship
between contextual tasks and motivation has been inadequately investigated empirically.

7

Final Note on the Importance of the Study
As I finished the data analysis and wrote my final chapters, I had the opportunity to
return to the classroom to teach middle school mathematics. I am reminded daily of how salient
engagement is in a student-centered, problem-based classroom—little is learned or accomplished
without it. If contextual tasks have potential to support student engagement and motivation, it is
worth investigating. The immediacy of motivating students to learn mathematics typically falls
on teachers. Though it is apparent that no widely-used written mathematics curriculum can
incorporate contextual tasks that will be motivating for all students (e.g., Boaler, 1993), it is also
unlikely that teachers will be able to develop their own contexts that will be highly relevant for
all their students, due to the demands on teachers’ time and knowledge and the diversity existing
in a single classroom (Chazan, 2000). A clearer understanding of the types of contexts and
instructional practices that relate to student engagement and motivation could be valuable for
teachers as they select and enact tasks, and for curriculum designers as they design these tasks
and support teachers in using them.
Overview of Forthcoming Chapters
In the next chapter, I review literature on contextual tasks, motivation in mathematics
education, and theoretical and empirical links between contextual tasks and motivation. I close
the chapter with the research questions and a description of expectancy-value theory and how it
provided a theoretical framework for the study. Chapter 3 begins with the research strategy I
employed, then describes the participants in the two classrooms. I then outline how data was
obtained and analyzed from classroom observations, student and teacher surveys, and focus
group interviews to capture student engagement and motivation and to characterize contextual
tasks.

8

In Chapter 4, I describe the most and least engaging lessons and patterns in the
characteristics of the tasks (as written and as presented to students) in these lessons. Descriptive
statistics support the identification of noteworthy lessons and begin to explain the relationship
between the use of contextual mathematics tasks and student engagement in those lessons. In
Chapter 5, I focus on how teachers and students enacted the tasks, particularly the contextual
aspects of tasks, in the high- and low-engagement lessons. Chapter 6 describes in more detail the
relationships between task contexts and student motivation to learn. Drawing primarily on survey
and interview data, I consider how students responded to particular tasks and contexts, how their
engagement reflected (or not) their expressed motivation to learn mathematics, and how their
work with contextual tasks seemed to influence their motivational beliefs. I present cases of
students who expressed diverse views of mathematics and themselves as doers of mathematics
and discuss how they described their experiences in certain lessons. In Chapter 7, I conclude
with reflections on the posited answers to the research questions, suggestions for future studies
building on the results of this investigation, implications for researchers, and implications for
teaching and curriculum development that support student cognitive and emotional engagement
with contextual tasks.

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CHAPTER 2
LITERATURE REVIEW
With this study, I aimed to explore two main variables: contextual tasks, including their
implementation in classrooms, and student motivation to learn mathematics. In this section, I
start with an overview of the purposes contextual tasks serve, historically and presently. Next, I
discuss literature related to the implementation of contextual tasks. Then, I briefly synthesize
literature on motivation to learn mathematics (and clarify the relationship between motivation
and engagement). Finally, I consider theoretical and empirical links between contextual tasks and
motivation in order to highlight the need for the study.
Contextual Tasks in the Mathematics Curriculum
Purposes of Contextual Tasks, Through History and Present
Contextual tasks are a long-standing part of mathematics education, though their role in
the curriculum has changed over time. Gerofsky (1996) wrote, “the form and addressivity of
school mathematics word problems is recognizable nearly universally among most people who
have attended school mathematics classes” (p. 37). She argued that contextual tasks persist
perhaps in part simply because of tradition—teachers and parents did those problems, so they
teach them, and so on. But others claim contextual tasks have served multiple important purposes
in school mathematics, which I explicate in this section.
Texts including contextual tasks date back to at least 1850 B.C. across broadly diverse
cultures (Gerofsky, 1996; Verschaffel et al., 2000). For example, the Rhind papyrus is an
Egyptian mathematics “textbook” of sorts that dates written in approximately 1550 B.C.
(Trustees of the British Museum, n. d.). It contains 84 mathematics problems and tables of
values, including both non-contextual and contextual arithmetic problems. The contextual tasks

10

seem to represent practical problems faced by scribes, such as division of food for men and
livestock.
Historically, contextual tasks have served multiple purposes, to some extent reflecting
beliefs about mathematics and mathematics education at the time (Verschaffel et al., 2000). Like
in the Rhind papyrus, one purpose was to provide “training exercises in practical skills aimed as
[sic] specific classes of students” (p. 140) such as surveying, sales, and navigation. Tasks of this
nature were evident in mathematics texts as early as the 13th century, alongside non-contextual
tasks introducing central mathematics facts (e.g., Fibonacci’s Liber Abaci). Another historical
purpose for contextual tasks was to serves as puzzles or exercises in “recreational mathematics.”
These ancestors of traditional story problems tend to ignore realistic constraints, highlighting
instead some aspect of mathematical structure.
Since the early days of mathematics education in the United States, textbooks have
included a mix of contextual and non-contextual tasks, reflecting the fact that mathematics in
school served dual purposes: the “cultivation of reasoning power” and as an “instrument for
solving everyday problems (Kliebard & Franklin, 2003, p. 401). The relative balance between
the types of tasks has shifted over time, with larger societal changes that influenced the
demographics of the student population and the perceived role of mathematics in school, like
industrialization, immigration, and World War II (Kilpatrick & Izsák, 2008, Kliebard &
Franklin, 2003). During the New Math Era in the 1950s to the 1970s, there was a major shift in
school mathematics curricula toward logic, deductive reasoning, proof, and emphasis on
structure and sets (Fey & Graeber, 2003)—deemphasizing the role of any contextual tasks. When
New Math fell out of favor, the back-to-basics movement brought new emphasis on rote
procedures, again downplaying practical application problems but focusing instead on what we

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now consider “classic” story problems—mixtures, distance-rate-time, coins, and so on (Coxford,
2003). These problems can usually be solved using rote procedures, as “routine applications
without judgment or any higher-level thinking skills” (Verschaffel et al., 2000, p. xiii).
Finally, a reform movement driven by the National Science Foundation (NSF) and
NCTM’s publication of standards documents placed renewed emphasis on real-life applied tasks
with more complex problem-solving scenarios over the last two decades. Two core documents—
the NCTM Curriculum and Evaluation Standards for School Mathematics (1989) and Principals
and Standards for School Mathematics (2000) called for increased use of tasks and practices that
highlight mathematics in real-life situations and the applications of mathematics to these
scenarios. NCTM cites both learning and motivational purposes for contextual tasks (NCTM,
2009). Connections between mathematics and real-world problems made through modeling, for
example, offer several benefits: connecting mathematical ideas and tools, providing opportunities
for mathematical reasoning, providing access to mathematical ideas for diverse learners, and
highlighting the practical need for specific mathematical ideas and offering opportunities to
apply them (NCTM, 2009). Most recently, the Common Core State Standards include
mathematical modeling as a standard of mathematical practice, primarily to provide
opportunities for students to apply mathematics to solve everyday, societal, and workplace
problems (CCSSI, 2010).
Accordingly, curriculum developers have emphasized the role of context in supporting
students’ learning by connecting everyday and school mathematical activity. For instance,
contextual tasks can support the transfer of mathematical knowledge to out-of-school settings
(Meyer et al., 2001; Putnam, Lampert, & Peterson, 1990). Also, problems drawing on familiar
contexts can give students entry into complex tasks by connecting to informal knowledge

12

(Carraher, Carraher, & Schliemann, 1985; Walkington, Sherman, & Petrosino, 2012). A
secondary purpose contextual tasks are hypothesized to serve is to foster student engagement and
interest—or, to promote student motivation to learn mathematics (Chapman, 2006; Meyer et al.,
2001; van den Heuvel-Panhuizen, 2005; Verschaffel et al., 2000).
Writers and developers of curricula and programs based on contextual tasks often draw
on both of these purposes in justifying the emphasis on these tasks (e.g., Civil, 2002; Gutstein,
2003; Lappan & Phillips, 2009; Romberg & Shafer, 2003; Silva et al., 1990). These curricula and
programs have a variety of forms. One form is comprehensive NSF-funded, nationally published
curricula based on the NCTM Standards documents (1989, 2000); namely, the Connected
Mathematics Project (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006; Lappan, Phillips, Fey,
Friel, Grant, & Stewart, 2014) and Mathematics in Context (National Center for Research in
Mathematical Sciences Educational Staff, 1998; Romberg & Shafer, 2003). Another form is
teacher-produced tasks that connect students’ mathematical thinking with community-based
contexts, drawing on students’ multiple funds of knowledge (Aguirre et al., 2012; Civil, 2002,
2006, 2007) and/or social justice issues (e.g., Gutstein, 2003, 2007a, 2007b). Students
themselves can also generate contextual problems, based on tasks provided by teachers or on
their own interests (English, 1997). As an example, I will describe one Standards-based middle
school curriculum and identify the reasons provided by the curriculum developers for this
emphasis.
Connected Mathematics Project—An Example
The Connected Mathematics Project (CMP, Lappan et al., 2006, 2014) originated as an
NSF-funded comprehensive curriculum (grades six to eight) designed to meet recommendations
for teaching and learning school mathematics by NCTM in the Standards documents (1989,

13

2000). CMP has several key features: it is problem-centered; it is organized around big
mathematical ideas to be connected and coherent; it builds conceptual and procedural
knowledge; it is based on research on learning, mathematical concepts, motivation, and policy;
and it provides support for teachers along with students in order to meet increased content and
pedagogical knowledge demands (Lappan, Phillips, & Fey, 2007; Lappan & Phillips, 2009).
According to the authors, the primary reason for designing a mathematics curriculum
based largely on contextual problems is to promote student learning. Students are encouraged to
engage in authentic mathematical activity by “exploring interesting mathematics situations,
reflecting on solution methods, examining why the methods work, comparing methods, and
relating methods to those used in previous situations” (Lappan et al., 2007, p. 73). Problem
contexts also support students in remembering and retrieving the embedded mathematics in new
situations. Individual problems should meet the overall goals of the curriculum—to promote
conceptual and procedural understanding, allow teachers to assess what students know on a
regular basis, encourage student collaboration, and allow multiple strategies and conjectures.
Furthermore, the authors present contextual tasks as having the potential to be interesting
to students and provide familiarity that will give them access to the mathematics (Lappan &
Phillips, 1998). Problem settings should be engaging, and alternative settings should be chosen
by the teachers to increase engagement when appropriate. Original letters written by the CMP
authors (printed in Lappan & Phillips, 2009) indicated that contextual problem scenarios should:
a) promote mathematical thinking, b) generate interest and motivate the study of mathematics,
and c) support collaborative learning and multiple approaches to solve problems. Here, we see
the dual-emphases on contextual tasks to promote student learning and student motivation,
reflecting the purposes the tasks have served over time. Lappan and Phillips do not, however,

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emphasize the role of contextual tasks in helping students transfer or apply mathematical
knowledge to solving everyday tasks.
So far, I have considered the nature and purpose of contextual tasks in mathematics
education. There are significant bodies of literature on the role of these tasks in the curriculum
and student thinking on these types of tasks. For the purposes of this study, however, I want to
consider what we know about how contextual tasks are actually implemented in the classroom,
so I turn next to this topic.
Implementation of Contextual Tasks in Mathematics Classrooms
Few studies exist that specifically address how teachers implement the context of a
mathematics task (c.f. Chapman, 2006; Wernet, 2011). Extant research has shown, however, that
the way tasks are enacted in classrooms can and often does differ from the task as written. The
mathematics task framework, for example, conceptualizes tasks as existing in three phases: as
written in curricular materials, as set up by the teacher, and as enacted in the classroom
(Henningsen & Stein, 1997). The enacted curriculum is defined as an interaction in the
classroom around curricular materials and necessarily involves teachers and students (Remillard,
2005). Set-up and enactment are influenced by the teacher’s knowledge and beliefs, classroom
norms, the nature of the task, and dispositions toward teaching and learning. The framework was
used specifically to investigate changes in the important characteristics of the task and the
cognitive demand, or “the kind of thinking processes entailed in solving the task” (Henningsen &
Stein, 1997, p. 529).
Similarly, I studied contextual tasks as written and as enacted in 8th-grade classrooms and
found that teachers and students used particular moves that explicitly addressed the context
(Wernet, 2011). These moves grouped into five categories: positioning, clarifying, elaborating,

15

referencing, and making meta-level comments on the context. To varying extents, using these
practices changed the centrality and authenticity of the tasks as written. For example, positioning
occurred when students and teachers put themselves and others into the problem situation
through telling personal stories, using personal pronouns, and exhibiting an emotional response
to the context. Positioning served in some cases to raise the authenticity of tasks by making the
problem scenario seem more likely to occur. Elaborating on the context, including inventing
more personal contexts, also raised the authenticity. Referencing, clarifying, and making metalevel remarks about the context made the context more central in problems. These findings
(summarized in Figure 1) support other findings that teachers take particular approaches in
dealing with problem contexts, which either draw attention to or from the context and control the
extent to which students drew on personal knowledge and experiences (Chapman, 2006).

Figure 1. Diagram showing how certain actions by the teacher and students during task
enactment changed the authenticity and centrality of the contexts as written in the curriculum
(Wernet, 2001).
There is some indication that how contextual tasks are implemented may influence
students’ responses to the contexts (Nisbet, Langrall, & Mooney, 2007; Wernet, 2001). Nisbet et

16

al. (2007) found that when middle school students worked on statistics tasks involving contexts
in which one member of a small group was highly interested (e.g., tennis and music), groups
drew more heavily on the context in solving and showed greater levels of engagement. I found
that certain practices (e.g., inventing new contexts) and types of context (e.g., those related to
community issues) generated the most class discussion with extensive student contributions
(Wernet, 2011). These findings, however, were peripheral to the research questions and are
suggestive about the relationship between contextual tasks and engagement, not conclusive. In
the next section, I describe literature on student engagement and motivation, including the
relationship between these constructs, before turning to empirical research more explicitly
linking contextual tasks with engagement and motivation.
Student Motivation to Learn Mathematics and Engage in Mathematics Tasks
Motivation to Learn Mathematics
A large body of literature exists on student motivation in general (see Wentzel &
Wigfield, 2009 for a comprehensive review). Motivation is sensitive to context and content
(Middleton & Spanias, 1999; Turner & Meyer, 2009), so I focus here on literature about
motivation and mathematics. Relatively few studies have explicitly drawn on motivation and
mathematics education constructs (Turner & Meyer, 2009), but a few consistent themes appear
throughout the extant literature across major motivation theories (Middleton & Spanias, 1999).
The following themes and issues are drawn primarily from reviews of literature on motivation to
learn mathematics (Middleton & Spanias, 1999; Turner & Meyer, 2009) and the practitioneroriented book Motivation Matters and Interest Counts: Fostering Engagement in Mathematics
(Middleton & Jansen, 2009) in which the authors use extant literature and extensive personal
experience in mathematics classrooms to propose essential principles related to motivation to

17

learn mathematics. I highlight these themes to frame the “factors of motivation” referenced in
my research questions. Fewer studies have addressed engagement in mathematics classrooms
and the relationship between motivation and engagement (Jansen, 2008), and as previously
stated, motivation and engagement are often used interchangeably in the literature. So although
the following themes emphasize motivation, engagement is implicitly addressed as well.
Motivation to learn is both learned and adaptive. Generally, motivation develops early
in students’ school mathematics experiences, appearing to be relatively stable over time but also
influenced by instruction (Middleton & Spanias, 1999; Middleton & Jansen, 2011). Middleton
and Jansen (2011) argued that motivation to learn mathematics is both learned and adaptive. That
is, levels of motivation are neither innate to an individual nor inherent to mathematics. Students
self-regulate their behavior in the classroom relative to their beliefs and previous experiences
related to mathematics, teaching, and learning.
Learning goals are preferable to performance goals. A second theme is the
importance of intrinsic motivation over extrinsic, and learning (i.e., mastery) goals over
performance goals (Middleton & Spanias, 1999; Middleton & Jansen, 2011). A student with
performance goal orientation shows concern with appearing smart, and success is about
outperforming others or receiving good grades with little effort (Ames & Archer, 1988). A
learning goal orientation stresses the process of learning and gaining skills and knowledge, and
success is based on effort and not ingrained ability (Ames & Archer, 1988). Students need some
expectation they can be successful in order to be motivated in the mathematics classroom
(Middleton & Jansen, 2011; Middleton & Spanias, 1999; Wigfield & Eccles, 2000). The link
between learning and motivation, however, is mediated by the social nature of mathematics—
students want to experience relatedness in the classroom and appear competent in addition to

18

feeling competent (Middleton & Jansen, 2011). It is further mediated by the interpretation of
previous mathematics-related experiences by students, parents, and teachers (Turner & Meyer,
2009).
A growth mindset is preferable to a fixed mindset. Similar to goal orientation, a
students’ mindset toward intelligence influences their engagement and perseverance, and thus
their motivation to learn mathematics. Someone with a growth mindset toward mathematics
believes that ability to learn or be good at mathematics can be developed through effort, while
someone with a fixed mindset believes these qualities are an intrinsic part of a person (Dweck,
2006; Suh, Graham, Ferrarone, Kopeinig, & Bertholet, 2011). Students with a growth mindset
persist when faced with difficult tasks, accept and learn from constructive criticism, and are
inspired by others’ success. Conversely, students with a fixed mindset avoid challenge, give up
easily, disregard constructive criticism, and feel threatened by others’ success (Suh et al., 2011).
Mindsets toward learning shape students’ level of achievement and are related to gender and
racial achievement discrepancies and motivation to pursue careers in mathematics (Dweck,
2008). Importantly, mindsets can be molded. Teachers can promote a growth mindset by directly
teaching students that intelligence is malleable and talent can develop through effort; publically
valuing challenge, effort, and mistakes; praising students’ processes and not just right answers;
and providing meaningful feedback (Dweck, 2008).
Instructional moves can promote motivation to learn. Fourth, several instructional
practices are posited to promote student motivation to learn. A class that promotes a learning
goal orientation, for example, will give students authority in solving problems and contributing
ideas, as well as hold students accountable for disciplinary norms (Engle & Conant, 2002).
Similarly, student motivation is supported by sense-making and personal meaning, which in turn

19

requires that students have opportunities to discuss and collaborate with other students and
multiple routes to appear and feel successful (Turner & Meyer, 2009). Positive teacher-student
relationships are also important (Middleton & Jansen, 2011, Turner & Meyer, 2009), especially
for developing long-term motivation to learn mathematics. Students need both cognitive support,
such as availability of relevant resources (Engle & Conant, 2002), and affective support (e.g.,
enthusiasm, addressing anxiety) from teachers (Turner & Meyer, 2009).
Motivation is situated with tasks as well as cumulative. Finally, on one hand
motivation is situated and related to specific activities (Middleton & Jansen, 2011)—that is,
motivation to learn is related to individual tasks in which students participate. Nyman and
Emanuelsson (2013) posited that students develop and express mathematical interest in tasks and
activities and interest can be researched as such, not just as a psychological state. Accordingly,
Jansen (2006) argued for a shift toward “focusing on dispositions to engage in particular tasks
rather than on learning or doing mathematics generally" (p. 424). Motivation and education
research suggests that motivating aspects of tasks include being appropriately challenging (e.g.,
Copping, 2012; Middleton & Jansen, 2011; Middleton & Spanias, 1999; Turner & Meyer, 2009)
and involving some level of choice (Engle & Conant, 2002; Middleton & Jansen, 2011). On the
other hand, Middleton and Jansen (2011) argue that these localized experiences with individual
tasks are cumulative, and build into long-term motivational patterns.
Linking motivation and engagement. Both within and outside of mathematics
education, research shows that certain motivational factors influence student engagement (e.g.,
Greene, Miller, Crowson, Duke, & Akey, 2004; Shernoff, Csikszentmihalyi, Schneider, &
Shernoff, 2003; Skinner, Furrer, Marchand, & Kindermann, 2008). For example, in a study of
805 children in 4th through 7th grade, Skinner et al. (2008) found through students’ self-reports

20

that perceptions of autonomy was directly related to engagement. Another study of 526 high
school students employing the Experience Sampling Method and based in flow theory found that
student engagement was significantly affected by the balance between the challenge posed by the
task and possession of necessary skills, perceived control over the activity, and relevance of the
task (Shernoff et al., 2003). Likewise, a survey study on high school students’ attitudes toward
mathematics showed that student engagement was predicted by learning goals, a desire to please
the teacher, and perception of future consequences of learning mathematics.
Two studies (Jansen, 2006, 2008) have investigated the relationship between motivation
to learn mathematics and engagement in mathematics classrooms in more depth. Based on
classroom observations and student interviews, Jansen focused on participation in whole-class
discussions, which is a subset of behaviors characterized as cognitive engagement. She found
that students’ motivational beliefs constrained or supported their participation (Jansen, 2006).
Students with constraining beliefs associated high risk with participating in whole-class
discussions and expressed that they learned mathematics best through listening to the teacher and
their peers. These students would participate, however, to help their classmates or to follow class
norms (2006), though they contributed mainly procedural explanations (2008). Students with
supporting beliefs communicated they were less afraid to make mistakes and did not associate
participating with social risk; rather, they participated to learn mathematics. These students
participated mainly to display their competence and to support their own learning (2006), and
offered more conceptual contributions (2008). This research begins to unpack the relationship
between motivation and engagement in mathematics classrooms, and also support the argument
that motivational factors underlie student engagement.

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Empirical Research on Motivating Instructional Practices in Mathematics
Stipek et al. (1998) conducted one of the few existing studies explicitly drawing on both
motivation and mathematics education constructs. They investigated how instruction influenced
student motivation to learn mathematics—specifically, instructional practices recommended by
both motivation literature and mathematics education literature (e.g., NCTM, 1989, 2000). The
instructional practices studied included displaying positive affect and enthusiasm, allowing
students to work autonomously, emphasizing learning and effort, and supporting students in
taking risks. Motivation outcomes studied were focusing on learning in addition to getting
correct solutions (i.e., a learning orientation versus performance orientation), self-confidence,
willingness to take risks, enjoyment, and positive emotions related to task. Three groups of upper
elementary teachers were studied via observations, student and teacher reports, and surveys. The
results suggested that learning orientation and positive affect in the classroom positively affected
student motivation to learn, while differential treatment of students based on achievement
negatively affected motivation. In fact, positive affect had the greatest effect on students’
learning goal orientation, help seeking behaviors, and positive emotions.
Until recently, few other studies explicitly and systematically investigated motivation
outcomes in mathematics classrooms, specifically what instructional practices support
motivation to learn. Middleton and Spanias (1999) wrote,
“[t]he research on motivational variables in mathematics education has been primarily
descriptive and inadequately conceptualized. Often motivation has been thrown “into the
pot” to add a little spice to studies originally focused on other factors—such as
mathematics achievement…In addition, measurement procedures have been primarily
atheoretical and poorly defined.” (p. 83)
In 2011, however, NCTM published a collection of writings on motivation in the mathematics
classroom, a few of which reported empirical results of small-scale studies. These studies

22

confirmed some of the themes previously discussed. For example, one study found that highachieving females reported being motivated to pursue mathematics because of prior success
(e.g., being identified for advanced classes or earning good grades), alignment with personal
goals, the nature and challenge inherent to learning mathematics, and relationships with peers,
teachers, and parents (Soto-Johnson, Craviotto, & Parker, 2011). Another study drew on
expectancy-value theory and found that students reported being motivated to participate in small
group work because of its utility value. It helped to learn the mathematics and develop social
skills and autonomy by encouraging mathematical reasoning and communication (Jansen, 2011).
A third noteworthy study identified four factors that supported students in becoming persistent,
flexible problem solvers with a growth mindset—aspects of motivation and disposition to learn
mathematics: a) opportunities to work on tasks allowing for multiple entry points and learning
styles, b) respectful and clear mathematical communication, c) emphasis on effort as a path to
mastery, and establishing a community that valued challenge (Suh et al., 2011).
Turner and Meyer (2009) argued that motivation research from an educational
psychology perspective has focused on “whys” of motivation with research to support theory.
They encouraged researchers to: a) consider how motivation is influenced in specific learning
situations, b) keep close ties between mathematics—especially the unique and specific
characteristics of mathematics learning—and motivation outcomes, and c) consider the
underlying particulars of how and why math classrooms influence student motivation to learn. In
that light, contextual tasks serve as a specific type of learning situation in mathematics
classrooms that can generate specific instructional moves (Wernet, 2011), thus warranting
investigation into their motivational potential.

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Linking Contextual Mathematics Tasks and Student Motivation
Curriculum developers, mathematics teachers, and mathematics education researchers
have written about contextual tasks as promoting engagement and student motivation to learn
mathematics. Yet, as Middleton and Jansen (2011) argued, problem contexts have the potential
to motivate and demotivate students, pointing toward the complexity I address in this study.
Generally, clear theoretical grounding for claims regarding the relationship between contextual
tasks and motivation or engagement has been lacking, though. Also, studies specifically focused
on contextual problems have investigated more cognitive outcomes such as student performance
on the task and learning outcomes, not the role of such problems in promoting interest or
engagement. Few studies have directly investigated the motivational benefits of contextual
mathematics tasks, linking motivational theory with instructional practices. In the study by
Stipek et al. (1998) described above, for instance, the authors identified the use of multidimensional real-world tasks as a potentially motivating practice in the math education and
motivation literature, but did not include it as a variable in the study.
In this section, I first describe theoretical perspectives that may inform the use of
contextual tasks in mathematics classrooms. I consider theory that supports the motivational
potential of contextual tasks as well as theoretical arguments that problematize the use of
contextual tasks to motivate students to learn mathematics. Then, I present results from empirical
studies relating contextual tasks with motivation and engagement. Two studies systematically
and directly linked contextualization of mathematics with student motivation, and several
qualitative studies supported the use of contextual tasks to motivate students, though these
findings were often peripheral to the core research questions. Finally, I discuss empirical results
that challenge the motivational potential of contextual tasks in mathematics.

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Theoretical Support for Contextual Tasks as Motivators
Three different theories (among probable others) suggest the theoretical potential of
contextual tasks to motivate students to learn mathematics: expectancy-value, interest, and
identity theory. First, expectancy-value theory holds that achievement motivation—choice,
persistence, and performance—can be explained by a person’s expectation for success in an
activity and the value they place on the reward that comes with success (Brophy, 2004; Wigfield
& Eccles, 2000). The value aspect of the theory in particular supports the use of contextual tasks
to motivate students because it addresses the belief in the usefulness and personal relevance of
the content (Wigfield & Eccles, 2000) and “awareness of its role in improving the quality of our
lives” (Brophy, 2004, p. 133).
Interest theory provides a second foundation for considering the relationship between
contextual tasks and motivation. Interest is defined to be “the psychological state of engaging or
the predisposition to reengage with particular classes of objects, events, or ideas over time” and
has both cognitive and emotional aspects (Hidi & Renninger, 2006, p. 112). With respect to
mathematics education, the classes of objects, events, or ideas to which Hidi and Renninger refer
are the mathematics content. Through this theoretical lens, multiple types of contexts could
promote student motivation to learn mathematics. Contexts that are realistic and familiar or
otherwise connect to students’ nonmathematical interests could promote student engagement
through their relevance, but interest theory also provides grounds for using any context that
might surprise students or elicit other emotional responses. Interest theory, for instance, supports
the motivational potential of imaginative, whimsical, or otherwise unrealistic contexts because
they can serve to catch students’ attention and support initial engagement in a task.

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Third, identity theory provides a sound theoretical base for explaining student motivation
through a sociocultural perspective. Sfard and Prusak (2005) define identity to be the “stories
about a person” that are negotiated between the person and others (p. 14). More specifically,
mathematics identity refers to “the dispositions and deeply held beliefs that individuals develop
about their ability to participate and perform effectively in mathematical contexts and to use
mathematics to change the conditions of their lives” (Martin, 2007, p. 150).
Identity theory supports the use of contextual tasks in the mathematics classroom as
effective motivators because of their potential to blur the lines between everyday and school
mathematics knowledge (Nasir, Hand, & Taylor, 2008). Contexts that are of immediate personal
relevance to students—especially community-based contexts like those developed around
students’ funds of knowledge or social justice issues—can validate students’ identities by
connecting mathematics to familiar and personal experiences while increasing their mathematical
content knowledge, and incorporating new mathematical activity into their identities (Nasir,
Hand, & Taylor, 2008). Broadening the mathematical activity that is part of their identity may
lead to greater persistence in mathematics and influence their future choice to participate in it.
These theories speak to motivation to learn, as opposed to motivating students to do well
in mathematics class generally, by emphasizing content. That is, the emphasis is not strictly on
performance, enjoyment, or extrinsic reward. Expectancy-value theory stresses appreciating the
worth of learning specific topics (Brophy, 2008), and interest theory suggests that mathematics
becomes a personal interest when students value the content, pose curiosity questions about
topics, and persist in efforts to learn (Hidi & Renninger, 2006). All three theories suggest that
making the content meaningful and relevant, linking it to other important aspects of students’
lives, will support their motivation to learn.

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Theoretical Objections to Contextual Tasks as Motivators
Some scholars have problematized the use of contextual tasks in the curriculum and have
called for a reconsideration of their role in the curriculum (e.g., Gerofsky, 2004; Verschaffel et
al., 2000). There are two main arguments that challenge the motivational potential of contextual
tasks. The first is philosophical—that contextual tasks cannot ever truly reflect students’ realworld experiences. The notion of “the real-world” itself is problematic because it is difficult to
capture or agree on a definitive meaning (Gerofsky, 2004; Thomas & Gerofsky, 1997). Gerofsky
(2004) argued that rather than reflecting students’ “real lives,” word problems reflect the nature
of other word problems. Thus, students approach contextual tasks with firmly entrenched beliefs
and expectations about how to solve them as rote applications of algorithms (Verschaffel et al.,
2000). This suggests that contextual tasks are unlikely to support meaningful connections
between school and everyday mathematics or communicate the value of mathematics.
Second, the suggestion that contextual tasks can serve to motivate students to learn
mathematics simplifies students’ diverse experiences and interests (Boaler, 1993; Chazan, 2000;
Sullivan et al., 2003). Chazan (2000) wrote, “finding problem contexts that are ‘real’ for a class
full of students on a range of different trajectories seems unrealistic” (p. 55). If contextual tasks
repeatedly fail to relate to students’ experiences and goals, they may in fact alienate or exclude
some students, and result in students disengaging from mathematics (Middleton & Jansen, 2011;
Sullivan et al., 2003). Sullivan et al. (2003) argued that teachers need to be aware of how certain
contexts can exclude marginalized students and work to be explicit about the connection between
the context and mathematical content. This stresses the importance of contexts that are
personally relevant or fit with students’ broader identities, and highlights the difficulty in
developing contexts that will motivate broadly diverse populations of students.

27

Other scholars have argued that too much emphasis on contexts and application could
detract from other aspects of mathematics (Otten, 2011; Sinclaire, 2001; Wu, 1997). They
challenge not so much potential of these tasks to motivate and engage students, but the extent to
which they should be used for this purpose. For instance, although Wu (1997) acknowledged the
necessity for some inclusion of applications in school mathematics he posited that
disproportionate attention to applications threatens students’ opportunity to cultivate “intellectual
appreciation of [the] structure and cohesion” of mathematics (p. 948). Similarly, Otten (2011)
encouraged mathematics teachers not to feel as if they needed a real-life context ready for every
topic. Too much focus on the utility value of mathematics, he wrote, detracts from some other
aspects of the subject, such as mathematical processes, the historical significance of mathematics
as a human achievement, and the intrinsic beauty of the subject.
An extant body of literature focuses on the aesthetics of mathematics, defined by
Sinclaire (2004) to be the potential to combine information and imagination to derive meaning
and pleasure from doing mathematics. Research on mathematicians and the role of aesthetics
(e.g., Wells, 1990) shows that although few precise patterns exist in what they find the most
“beautiful,” mathematicians generally found certain ideas, proofs, and problems to be
aesthetically pleasing, with simplicity, brevity, and surprise recognized as aesthetically pleasing.
Based on the experiences of mathematicians, Sinclaire (2004) argued that in addition to serving
evaluative and generative roles, there is a motivational aesthetic of mathematics related to
interest, values, and emotion. The specific categories of aesthetic motivation Sinclaire proposed
were visual appeal (or simplicity and order), sense of surprise, and a social dimension. Sinclaire
extended this idea to school mathematics, positing that aesthetically rich mathematics learning
environments—those that connect to innate human sensitivities and activities—can be

28

motivating for students (2001), though students with different goal orientations may respond
differently in these environments. She encouraged teachers to elicit and use students’ intuition
and affective responses to mathematical tasks (2004).
The issues raised here accentuate the social, cultural, and mathematical complexities
involved with using contextual tasks to promote interest and positive mathematical identities
among students. Theories supporting the use of contextual tasks to motivate students to learn
mathematics may stop short of explaining the nuances of the relationship between
contextualization and motivation. Next, I turn to empirical evidence that links contextualization
and motivation to learn mathematics.
Empirical Support for Contextual Tasks as Motivators
A review of mathematics education literature yielded only two studies in which
researchers directly and systematically investigated the effects of contextual tasks on student
motivation. First, Cordova and Lepper (1996) investigated how contextualizing and
personalizing mathematics content as well as how provision of choice influenced students’
intrinsic motivation, level of engagement, achievement, perceived competence, and level of
aspiration. In the controlled experimental study, 70 fourth- and fifth-graders completed three 30minute sessions playing a specially designed computer game intended to teach order of
arithmetic operations. The game was contextualized by adding space travel and treasure hunt
scenarios; personalized by adding students’ names, the names of their friends, their favorite toys,
and so on based on preliminary surveys; and choice was added by allowing students to choose
images and characters. Students were split into five groups: 1) a control group, 2) general
contextualized with choice, 3) general contextualized without choice, 4) personalized contextual
with choice, and 5) personalized contextual without choice.

29

Contexts, personalization, and choice significantly (all p < 0.05) influenced each of the
five dependent variables (Cordova & Lepper, 1996). In most cases, students experienced greater
motivation and engaged in more challenging forms of play (suggesting motivation to learn) when
the content was put in imaginative contexts, and these effects were increased when the context
scenario was personalized. Therefore, the study supported the use of contextual mathematics
tasks—particularly with personalized contexts—as a means to promote student engagement with
and enjoyment of specific tasks. There is, however, a question about the influence of novelty in
this study. Over time, the task could become habitual and the effects of contextualization,
personalization, and choice may diminish. Brophy (1986) argued, for example, that focusing on
intrinsic motivation to learn mathematics is problematic because the source of motivation may
lie in features of the task or its implementation (e.g., use of computers, game like nature of the
task) rather than engaging in the content itself. Also, this study came out of the educational
technology literature, and may have limited generalization to contextual mathematics tasks
outside a computer environment.
A study by Ku and Sullivan (2000) in Taiwan did confirm Cordova and Lepper’s findings
related to the personalization of contexts in a traditional classroom environment. Drawing on
interest theory, they performed an experiment with 5th-graders in which students in a treatment
group received instruction with personalized contexts. The authors reported that students in the
treatment group indicated significantly more positive attitudes and higher motivation related to
the lessons and tasks than the control group. One strength of this study was that it was completed
in the students’ classrooms rather than the computer environment used by Cordova and Lepper;
another was the development of a group personalization technique where textbook word
problems were modified according to the most common interests and experiences communicated

30

by students at the class level. Its primary weakness was only basing the measure of student
motivation on a few Likert scale survey questions, not students’ actual engagement in the
classroom through observation or teacher reports. Also, levels of intrinsic motivation (how much
students liked or were interested in the lessons) were captured rather than motivation to learn.
Some qualitative studies have also positively linked contextual tasks in mathematics with
student motivation (Jansen & Bartell, 2011; Mitchell, 1993; Nicol & Crespo, 2005; Weist, 2001),
though most of these studies are only loosely grounded in motivation theory and were not
systematic studies of the effects of contextualization on student motivation. Results from selfreport surveys and interviews with 4th- and 6th-grade students suggested that the students
generally “liked” fantasy and children’s real-world problems (versus adult real-world problems),
and they solved fantasy problems with as much or more success as on real-world problems
(Weist, 2001). Students’ interest and preferences, however, depended not just on the problem
categories, but also on specific problems, suggesting that students tune in to individual contexts.
Through focus groups and open-ended questionnaires of high school mathematics students,
Mitchell (1993) tested his interest model and found that students reported meaningfulness to be
an important element in sustained situational interest. This finding was further supported by a
follow-up quantitative study using interest surveys. The study did not directly address the effect
of contextualization, but did provide evidence that students themselves report increased interest
when content links to their everyday lives, which is one purpose of using contextual tasks.
Other studies include cognitive or behavioral engagement outcomes. Through classroom
observations, Nicol and Crespo (2005) found that mathematics tasks with imaginative contexts
promoted intellectual and emotional engagement and persistence in the tasks for two different
groups of students—preservice elementary teachers and middle school students. To measure

31

engagement, the authors captured instances of students enthusiastically participating in activities,
seeking challenge, and offering unsolicited mathematical ideas. In another study of middle
school students’ beliefs about characteristics of caring mathematics teachers, some students
identified meaningful contextual tasks as interesting, especially when they can relate to them
(Jansen & Bartell, 2011). One student, contrary to Nicol and Crespo’s (2005) and Weist’s (2001)
findings, said that she did not like imaginative contexts, preferring realistic contexts based on
adult experiences. In their study intended to operationalize cognitive engagement, Helme and
Clarke’s (2001) initial findings suggested that novel contextual tasks that were personally
meaningful promoted two eighth-grade students’ cognitive engagement.
Similarly, a few studies outside of mathematics education suggest that relevant and
authentic contexts promote student engagement (Buck, Beeman-Cadwallader, & Trauth-Nare,
2012; Copping, 2012; Mitchell & Carbone, 2001; Rivet & Krajcik, 2008). Based on 100 reports
written by high school teacher-researchers who implemented self-developed tasks intended to
promote quality engagement, Mitchell and Carbone (2011) developed a "typology of task
characteristics" that promote student engagement. They identified eight dimensions of engaging
tasks, including an artificial-authentic dimension. The teacher reports suggested that more
authentic tasks “generate engagement as it is easier for students to perceive either importance for
or relevance to their current or future life” (p. 264). Similarly, studies in science education
indicate relevance is an important factor in student engagement. The results of an investigation
of 7th-grade girls’ experiences in problem-based learning in a science unit suggested that
authentic contexts were one factor of science learning environments that supported girls’
cognitive engagement (Buck et al.; 2011). Rivet and Krajcik (2008) found a strong correlation
between students’ attention to contextual features of science tasks and their achievement through

32

classroom observations and assessments. The authors attribute this relationship to both the
anchors provided by contexts related to students’ prior knowledge and experiences, and the role
of contextualizing science instruction in motivating and engaging students in the learning
process. Based on data from video observations, interviews, journals, and surveys from a study
in which he served as teacher-researcher, Copping (2012) proposed that one aspect of teaching
that supports an effective learning environment in science is a meaningful curriculum that is
aligned with student experiences. The results of these studies support the hypothesis that
contextualized tasks can promote student engagement. However, these conclusions drawn from
research in science or other subject areas—which, by their nature may be easier to connect to
students’ everyday experiences—may not be generalizable to contextual mathematics tasks,
which as discussed elicit distinctive beliefs and emotions in the mathematics classroom.
A few other studies involving context-rich curricula (e.g., CMP, teaching mathematics
for social justice) suggest that contextual tasks promoted student motivation, but evidence from
these studies is largely anecdotal. Lubienski (2000), for example, worked as a teacher-researcher
in a 7th-grade CMP classroom, studying differences in students’ interactions with the curriculum
based on socio-economic status. She reported that realistic contexts were “effective motivators,
prompting students to delve into the problems” (p. 476). This description seems indicative of
cognitive engagement, but it was a peripheral finding in her research.
Gutstein (2003) used both the written MiC curriculum and problems he developed as part
of TFSJ in his classroom. He found that the students thought MiC problems were interesting, but
not particularly relevant, and he reported that students were more interested in the projects he
designed than the MiC contexts. He argued,
Through their practice of reading the world with mathematics, the students began to
change how they felt about mathematics. Although not all loved math, virtually all

33

understood that mathematics was a tool not only to solve both realistic and fanciful,
sometimes enjoyable, problems in books, but it could also be used to dissect society and
understanding equality. (Gutstein, 2003, p. 67)
Through an identity theory lens, students’ recognition of mathematics as something that can
serve to change their community or larger society may support their motivation to learn it. Like
Lubienski, however, this claim was made without significant supporting evidence because it was
not the focus of the research. Gutstein (2007a) also argued that students continued to think about
and discuss the real-world projects after class. He wrote, “Although I am not completely clear
why, and unfortunately did not ask systematically, this project deeply engaged students. I believe
that it was because it tied into their aspirations, experiences, community issues, and sense of
justice” (p. 61). This statement reflects the anecdotal nature of reports on the relationship
between context and student engagement and motivation to learn mathematics.
Empirical Challenges to Contextual Tasks as Motivators
Extant empirical evidence from a sociocultural perspective suggests that contextual tasks
are not necessarily effective in engaging or motivating students to learn mathematics—or more
precisely, that as Boaler (1993) and Chazan (2000) suggested, it is a complicated relationship. As
a teacher-researcher in a high school, low-tracked, task-based algebra course, Chazan (2000)
found that student engagement was highly variable and followed no particular pattern. Students
occasionally disengaged from contextual tasks designed uniquely for the class, but would be
highly engaged in more abstract tasks. “Buy-in” from student leaders in the class seemed to have
far greater influence on student engagement than the contextualization of the content.
Brantlinger (2007) argued that claims are often made about the relationship between
culturally relevant mathematics curricula and motivation to learn for students of color, but these
claims are not grounded in ample empirical evidence. His dissertation work aimed to address this

34

lack of empirical testing. Through practitioner research in which he used reform pedagogy
(including the use of more contextualized tasks), Brantlinger found that student engagement
increased over time in the course. Engagement was measured by the number of elaborate student
contributions, level of resistance, and reliance on teacher help. While some students showed
markedly increased engagement in lessons with social justice tasks, others resisted. His report
provides valuable evidence that tasks drawing on social justice contexts can be engaging for
some students, but not all. It remains unclear why certain students resisted and others showed
higher levels of engagement. Also, it is impossible to separate the effects of reform pedagogy in
general and the specific influence of contextual tasks.
Bevil (2003) conducted a quasi-experiment with 320 students in grades six through eight
to investigate the effects of the use of real-world applications and academic status on student
achievement and perceptions of the classroom environment. Two scales on the survey to measure
student perceptions are of particular relevance—personal involvement in class and satisfaction—
as both serve as measures of student motivation. Treatment consisted of one semester taught
from an applied curriculum developed by Bevil in regular classroom settings. Bevil found that
the emphasis on real-world application tasks only significantly influenced student perceptions of
their personal involvement and satisfaction for high-achieving students, who may have been
bored with the traditional classroom. One weakness of the study was the dependence on
students’ self reports. Also, like the Brantlinger (2007) study, it is difficult or impossible to
differentiate between the effects of contextual tasks and other factors like the higher cognitive
demand of the treatment tasks and use of cooperative groups.

35

Synthesis
In sum, there is empirical evidence to support both sides of the theoretical debate
regarding the relationship between contextual tasks and student engagement and motivation to
learn mathematics. Yet, few studies have directly and systematically investigated the relationship
between contextual tasks in mathematics and motivation and engagement—much of the research
considered the variables peripherally or reported findings anecdotally. Furthermore, most of the
studies reviewed here lacked a strong theoretical foundation and/or did not draw on expectancyvalue, interest, or identity theory, which as discussed above, could provide theoretical grounding
to make sense of how contextual tasks can motivate students.
There is evidence to suggest that contextual tasks can and do promote student
engagement in mathematics, but under certain circumstances that are not quite clear. Middleton
and Jansen (2011) wrote,
We often think that story problems involving contexts from life outside school promote
connection to mathematics and opportunities to make sense of mathematics. Storyproblem contexts can achieve these goals, but only under certain conditions. Sometimes
contexts can interfere with understanding and lead to disengagement! (p. 107)
They suggest that the type of context and instructional support, such as using contexts that are
personally meaningful and culturally relevant can influence student engagement, but these
arguments are based largely on theory, not empirical evidence.
There are several complex variables involved when considering engagement, motivation,
and contextual tasks. First, there is the curriculum in which the contextual tasks exist. Tasks in
NSF-funded, Standards-based, nationally published curricula (e.g., CMP; Mathematics in
Context, National Center for Research in Mathematical Sciences Educational Staff, 1998) and
curricular programs used on a smaller scale intended to increase diverse participation in
mathematics by connecting students’ everyday and school mathematical knowledge (e.g., TFSJ)

36

have sought to create new genres of contextual tasks by: a) putting contextual tasks at the center
of instruction, b) making them more complex so the context is an integral part of solving, c)
making them more personal for students by drawing on community funds of knowledge, and/or
d) explicitly addressing issues of power and equity. By breaking the traditional rules of story
problems, one aim of these tasks is to motivate students to learn mathematics in ways canonical
contextual tasks do not.
A second variable is the type of context—imaginative/whimsical versus realistic,
personalized versus non-personalized, and contexts relevant to students’ immediate lives versus
contexts drawn from adult activities. Additionally, contexts may draw on community-based
knowledge or social justice issues. Third, like any other task, teachers implement contextual
tasks in different ways, and different instructional practices may influence student engagement
with them (Chapman, 2006; Middleton & Jansen, 2011; Nisbet et al., 2007; Wernet, 2011).
Generally, more empirical research is needed that directly investigates the hypothesized
relationship between contextual tasks and student motivation. This study addresses the issues
discussed regarding the nuances of contexts and motivation, grounded in expectancy-value
theory to allow more robust interpretations of results.
Research Questions
Through exploring the relationship between contextual tasks and student motivation, I
hoped to provide a better understanding of what types of contexts and which instructional
practices relate to higher (or lower) levels of engagement and motivation. Toward this end, the
following questions guided the design and implementation of the study:
How does the use of contextual tasks in middle school mathematics classrooms relate to
students’ engagement in the tasks and motivation to learn mathematics?

37

•

What characterizes the contextual tasks as written (e.g., personalized, communitybased, levels of authenticity and centrality) in lessons during which students show
particularly high and low levels of engagement?

•

What characterizes the enactment of contextual tasks, including instructional
practices (e.g., positioning, elaborating, referencing) and student attention to
context, in lessons during which students show particularly high and low levels of
engagement?

•

What factors of motivation (e.g., valuing content, enjoyment of task, alignment with
goals) underlie student engagement relative to the contextual tasks used in class?1

The first two questions address the relationship between contextual mathematics tasks and
student engagement in individual lessons. Building on extant studies suggesting that student
motivation influences engagement in specific learning scenarios (Jansen, 2006, 2008), the third
question was included to explore more deeply students’ experiences in lessons relative to their
expressed levels of motivation to learn and motivation-related beliefs about mathematics.
Theoretical Foundation—Expectancy-Value Theory
As previously discussed, several theories exist that help explain and predict the
relationship between contextual tasks and student engagement and motivation to learn
mathematics. For this study, I chose to use expectancy-value theory because of its specific focus
on students’ beliefs about various aspects of valuing the content—mathematics, in this case—for
1

These research questions differ from those originally proposed. Rather than the first question
being about highly engaging lessons and the second being about low-engagement lessons, I split
the content of the questions so the first is about the tasks as written across the noteworthy lessons
and the second is about the tasks as enacted. I made this change to set up the structure of the
chapters after the study was complete, so it did not change the purpose or nature of the proposed
study. Further, the word “enactment” replaced “implementation” to better reflect the focus on
both the teachers’ instructional practices (e.g., positioning, elaborating, referencing) and student
attention to context.
38

its own sake. Expectancy-value theory holds that achievement motivation—choice, persistence,
and performance—can be explained by a person’s expectation for success in an activity and the
value they place on the reward that comes with success (Brophy, 2004; Wigfield & Eccles,
2000). The theory grew largely out of work in mathematics classrooms (Turner & Meyer, 2009).
Expectancy involves beliefs regarding one’s current competence and future ability in a particular
activity. The value one assigns an activity has several components: a) attainment value, or the
importance of doing well on a task; b) intrinsic value, or level of enjoyment; c) utility value, the
usefulness and correspondence with future plans; and d) cost, the level of effort and sacrifice
required (Wigfield & Eccles, 2000). With respect to mathematics education, a student’s
willingness to work on a particular task is a product of their belief in their ability to successfully
complete the task and how much they value the task. Similarly, a student’s motivation to
participate in and use mathematics throughout their lifetime is a product of their belief in their
ability to learn mathematics and how much they value the subject in general.
Brophy (1986) argued that expectancy-value theory is a promising theory for making
sense of motivation to learn mathematics in particular. Unlike other motivation theories, it
focuses attention on the mathematical content rather than other features of tasks, such as their
social affordances or game-like qualities. In other words, students’ motivation to learn should
arise from the mathematics and the intended learning. As discussed previously, it was a
reasonable theoretical foundation for this study because it supports the hypothesis that contextual
tasks can promote student motivation by highlighting the value of the mathematics. It also
provides a useful framework for analyzing student responses in order to answer the third research
question, regarding factors of motivation underlying their engagement in contextual tasks.

39

CHAPTER 3
METHODS
The purpose of this study was to gain a more nuanced understanding of the conditions
under which contextual tasks do (or do not) engage and motivate students to learn mathematics.
In this chapter, I will describe how I identified the extent to which 7th-grade students were
engaged during lessons observed over the course of one semester, and how I determined the
nature of contextual tasks and their implementation in these lessons. Further, I will describe how
I used surveys and interviews to identify themes in student motivation to learn as it related to
students’ engagement in contextual tasks. The next section includes a general description of the
research strategy used and a rationale of the methodology.
Research Strategy
Fredericks et al. (2004) argued for more qualitative studies on student engagement to
accomplish specific goals: a) to focus on how students experiences the complexities of the school
environment, b) to highlight why students might choose to disengage and how the different types
of engagement are related, and c) to address individual and group differences. A qualitative study
is appropriate to accomplish these goals and answer the research questions because it allowed for
a more in-depth perspective through multiple data sources, including the voices of students
themselves, on the relationship between contextual tasks and student engagement in mathematics
classroom. Drawing on multiple data sources was a central aspect of the research strategy, both
to offer more validity and reliability to the findings through triangulation and to address issues
with limited data sources in extant literature on school engagement (Fredericks & McCloskey,
2012; Middleton & Spanias, 1999, Turner & Meyer, 2009). These issues will be discussed
throughout this chapter to support particular methodological choices.

40

The general approach of the study was to conduct a multiple-case study, where I define
the cases to be middle school mathematics lessons in two 7th-grade classes. Case study is
characterized as “particular, descriptive, inductive, and ultimately heuristic” (Stark & Torrance,
2005, p. 33), making it an appropriate strategy for gaining rich and varied perspectives on the
relationship between the variables of interest. This approach was well-suited for this study
because extant research indicates multiple complex factors, including a variety of instructional
practices, influence student motivation to learn mathematics (e.g., Middleton & Jansen, 2011;
Middleton & Spanias, 1999; Turner & Meyer, 2009). Case study allowed the necessary in-depth,
detailed investigation of student motivation related to solving contextual mathematics tasks.
I had 28 cases of middle school mathematics lessons in two 7th-grade classes that used a
problem-based curriculum. The curriculum included a variety of contextual tasks in terms of
type, centrality, and authenticity of contexts. These constructs will be described further below.
These cases allowed me to study how task characteristics relate to student engagement and
motivation to learn mathematics across classrooms. My aim was to isolate to the extent possible
the influence of contextual tasks and their implementation, and minimize the influence of other
pedagogical or curricular factors. Thus, it was appropriate to study a small number of classrooms
using the same curriculum and look for themes, not comparisons, across classrooms. We also
know that teachers’ practices are fairly consistent in terms of pedagogy and lesson structure (e.g.,
Ho & Kane, 2013), which also supports the ability to consider and compare across lessons the
influence of specific tasks on student engagement and motivation within each classroom.
Another important part of the research design was the opportunity to observe variation in the
type of tasks assigned, the content covered by the tasks (e.g., algebra, geometry, numbers and
operations), and in the implementation of the tasks. I chose to study two classrooms and spend

41

ample time in each in order to develop a richer understanding of the students and environment.
Further, to support the correlation of the results of observations of student engagement with the
teachers’ and students’ perceptions, I needed an adequate number of observations for each
classroom. More than two classrooms would mean less time in classrooms and with students.
The benefit of two classrooms over one was access to more students with more diverse
experiences in and out of the classroom. It also provided an internal means to check the data to
see if themes extend beyond one classroom.
Setting and Participants
The first criterion for selecting participating schools was the use of the Connected
Mathematics Project curriculum throughout the middle grades, because it is problem-based with
ample contextual tasks of various types (Lappan & Phillips, 1998; Lappan et al., 1997). Thus, I
could safely assume that students would have opportunities to engage students in contextual
tasks in a majority of daily lessons, limiting the effect of novelty on student engagement, and
increasing the likelihood of observing the implementation of a diverse range of contextual tasks.
Second, I recruited teachers of 7th grade classes because of the varied content in the 7th-grade
curriculum, including algebra, geometry, and numbers and operations (CCSSI, 2010; Lappan et
al., 2006, 2014). Third, participating teachers needed experience teaching CMP and followed the
curriculum closely in terms of tasks and philosophy, without much supplementing (e.g., with
more procedural materials). Two teachers in two different districts—Mrs. Meyer at Pine River
Middle School and Mrs. Cole at Southpoint Junior High School--fit these criteria. They were
recommended as participants by knowledgeable others, including faculty with experience
working with local teachers and CMP development staff.

42

To select participating classes, I consulted with the teachers to identify classes meeting
certain conditions. I wanted classes in a general mathematics track that covered content required
by all students to increase the likelihood of students having diverse mathematical experiences
and levels of motivation. That is, I wanted to avoid classes in which students were tracked into
advanced (e.g., Algebra 1) or remedial classes. Both teachers had 7th-grade classes in a fourth
hour class, allowing for focus group interviews to take place during lunch immediately after
class. I met with participating classes during the second week of the school year to introduce
myself, inform students about the nature and requirements of the study, and distribute consent
and assent forms. Students and their parents could consent individually to participate in the
surveys, interviews, and be included in observation videos.
Through analyzing students’ initial motivation and belief surveys and consulting with the
teachers, I used purposeful selection to identify six focus students in each class. Selecting six
focus students was consistent with other studies on motivational beliefs and students engagement
in secondary classrooms employing observations and/or interviews (e.g., Jansen, 2006, 2008;
Rivet & Krajcik, 2008). Attending to the same small group of students throughout the study
allowed for more consistent comparison of students’ engagement across classrooms. My goal
was to select students who represented diversity in terms of their motivation to learn
mathematics, prior success in mathematics, and interests; and who would work reasonably well
together for classroom activities and focus group interviews. That is, I wanted groups that were
representative of the class as a whole, avoiding any interpersonal issues or conflicts that would
influence their learning when working together or influence their responses during interviews.
Mrs. Meyer and Ms. Pearson consistently grouped the focus students together for small group
work during the observed lessons.

43

Pine River Middle School2
The first site, Pine River Middle School, is a rural school located in a small town in the
Midwest. It serves students in grades six through eight. A summary of the size and demographics
of Pine River is contained in Table 1. Pine River is an original CMP school; they have used the
curriculum since 1990. At the time of the study, the school was transitioning from the second to
the third edition of CMP (copyright 2014). During the study period from September to January,
the 7th-grade at Pine River worked on the units Shapes and designs: Two dimensional geometry,
Accentuate the negative: Integers and rational numbers, and Stretching and shrinking:
Understanding similarity. Each class period was 60 minutes long.
Mrs. Meyer. Mrs. Meyer was in her 8th year of teaching and had been at Pine River for
four years. She has taught the CMP curriculum all eight years in grades six through eight.
Through classroom observations and many informal conversations with Mrs. Meyer, it was clear
she had a commitment to student inquiry and attending to and building on student reasoning. The
class was organized into “teams” of 4-6 students who sat together at tables, and students were
encouraged to work together and talk with each other about the mathematics in every lesson.
When a student posed a question, she would often respond by asking the student to talk to others
in the group before working with them one-on-one. Mrs. Meyer followed the CMP curriculum
closely and never supplemented with other materials during my observations other than to pilot
one open-ended task for another curriculum project. She did occasionally modify lessons in
terms of how long she spent on each one or specific supports and extensions provided. In a
typical lesson, Mrs. Meyer followed the standard “launch, explore, summarize” pattern of a CMP
lesson. She launched the problem for the day, gave students five minutes to work individually,

2

All school, teacher, and student names are pseudonyms.
44

opened the lesson to working in small groups, then facilitated a summary discussion with the
whole class. The launches often started with students talking about what they had done in the
previous lesson in their small groups, a guiding “focus question” from the CMP teacher
materials, and curriculum-provided launch videos when available.
Students in Pine River participating class. There were 22 7th-grade students in Mrs.
Meyer’s fourth hour mathematics class. Four students were excluded from the study because of
missing forms assent or consent forms. Mrs. Meyer voiced concern early in the school year about
the “struggles” she felt with the class in terms of meeting learning objectives at the same pace as
her other classes (personal communication, October 29, 2013). Overall, the students worked well
together and there were few behavioral disruptions I observed. Mrs. Meyer believed the
difficulties stemmed from the varied academic profile of the class. Of the 22 students, nine were
enrolled in “math lab,” meaning they had an extra math hour with an academic aid. Those
students had Individual Education Plans or 504 plans, having been diagnosed with learning
disabilities, ADHD, and/or other health impairments.
Table 1
School Demographics3
Site
Total
students
Pine River Middle
448
School
Southpoint Junior
High School

951

% Students of
color
4.5

% White
students
95.5

% Eligible for
free/reduced lunch
30.6

33.7

66.3

39.4

3

Demographic information was accessed from www.mischooldata.org and
https://www.michigan.gov/cepi/0,1607,7-113-21423_30451_36965---,00.html on April 16, 2014.
45

Southpoint Junior High School
The second site, Southpoint Junior High School, is a suburban school located outside a
state capitol in the Midwest. It serves students in grades seven and eight. A summary of the size
and demographics of Southpoint is contained in Table 1. Southpoint had used the second edition
of CMP (copyright 2009) for five years, so the school was newer to the curriculum than Pine
River. During the study period, the class at Southpoint worked on material in the units Stretching
and shrinking: Understanding similarity, Comparing and scaling: Ratios, rates, percents, and
proportions, and Accentuate the negative: Integers and rational numbers. Each class period was
55 minutes long.
Ms. Pearson. Ms. Pearson was in her 9th year of teaching at Southpoint. She taught high
school for six years and middle school with the CMP curriculum for three years in grades seven
and eight. Ms. Pearson was committed to standards-based assessment and student selfassessment of major learning goals, which were posted around the room on large posters for each
unit. She was also working on facilitating student work in small groups and developing group
norms appropriate for middle school students. The students sat at desks arranged in rows, which
Ms. Pearson communicated helped them focus during whole-class discussions. On the days I
observed, students typically moved their desks into established groups of three, though they did
not use group work on a daily basis. When in groups, students were occasionally assigned roles
(e.g., facilitator, recorder, resource manager) and were expected to talk only to each other.
Ms. Pearson also followed CMP closely and never supplemented with materials from
other curricula in the observed lessons. She did use individual white boards for students during
unit reviews, and for three of the lessons in the study, she modified the task to be more openended and to include student presentations. In these lessons, Ms. Pearson developed and

46

distributed “task cards” to each group with the modified task and role assignments. For example,
she modified the directions for Stretching and Shrinking problem 2.3 as shown in Appendix A.
In a typical lesson, Ms. Pearson started with going over homework or “learning checks” with the
whole class, launched the problem for the day, asked students to work in small groups, then
facilitated a summary discussion with the whole class.
Students in Southpoint participating class. There were 23 total 7th-grade students in
Ms. Pearson’s fourth hour mathematics class. Three students were excluded from the study
because of missing forms assent or consent forms. In general, the class was diverse in terms of
performance and participation. Ms. Pearson identified a few boys as particularly high-achieving.
They resisted working in small groups during class, preferring to work on their own. She also
communicated that due to a variety of issues, including students who were English Language
Learners and the range of experiences students had with instruction and implementation of CMP
in 6th grade, some students struggled with the language and content demands of the curriculum.
Focus Students
In each class, I selected six focus students for the study based on their expressed
motivation to learn mathematics, their interests outside of school, and teacher recommendations.
Primarily, I sought to select groups that represented the diversity of the class in terms of their
motivation to learn mathematics: their expectations of themselves as mathematics students and
how they value mathematics in and out of school. During my second meeting with students, I
administered a survey on students’ beliefs and values (see Appendix B), which I will describe in
detail in the instruments section below. The survey included questions to capture students’
expectancy for success in 7th-grade mathematics and the extent to which they valued

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mathematics. There were also questions related to mindset, but nearly all students communicated
a growth mindset, so I did not use that data in selecting focus group students.
In selecting focus group students, I first identified participating students who were low in
both expectancy and value, high in both, and students who were low in one and high in the other,
hoping to choose two students in each category per class. “Low” was set as less than an average
value of five for the appropriate survey questions, “high” was set as five or greater. When more
than two students fell into a category, I identified students with a range of interests outside of
school. Finally, teachers reviewed the groups of students being considered. In cases in which I
was choosing between students, the teachers suggested students with different levels of prior
achievement in mathematics and who could work together in class and interviews. In Table 2, I
listed the focus group students who participated in the study with some results from the beliefs
and values survey.
Data Collection and Instruments
Several researchers have emphasized a need for motivation research in mathematics
education that considers students’ self-reports, teacher reports, observations, and/or interviews to
counter the trend in which motivation research focuses on self-report and mathematics education
research focuses on observation (Fredericks & McCloskey, 2012; Middleton & Spanias, 1999;
Turner & Meyer, 2009). This trend is problematic because alone, either approach introduces
potential biases and validity problems (Middleton & Spanias, 1999) and provide limited
information on student engagement and motivation across lessons and tasks. I used four primary
data sources: classroom observations, lesson-specific student surveys, lesson-specific teacher
surveys, and focus group interviews. As discussed in the last section, an initial survey on

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students’ general mathematical beliefs and values served as secondary data to provide baseline
information about student motivation to learn mathematics and select focus group students.
Validity of the instruments and data collection techniques was initially addressed through
two rounds of piloting the instruments and data collection strategies. I conducted the first pilot in
6th- and 7th-grade classrooms at a small private school in the spring of 2013, and the second in a
summer camp program in which students entering 5th and 6th grade engaged in mathematics for
social justice activities every day for two weeks. In each pilot, I considered the extent to which
instruments allowed access to the data I intended and their utility value in answering the research
questions. Based on these experiences, I made some changes to the data collection tools,
including: a) refining fieldnote spreadsheets to improve usability, b) deciding when I rated
cognitive engagement at a 1 (students appeared cognitively disengaged) I would not give a score
for emotional engagement to avoid high engagement scores when students showed positive
affect but were talking about non-mathematical topics, and c) fixing or clarifying language
students found confusing across the surveys, adding sample responses to the open-ended
questions. Generally, the results suggested the lesson-specific surveys were valid. In follow-up
interviews, students and teachers talked about the aspects of engagement in the way I intended to
capture cognitive and emotional engagement (e.g. how hard they tried, how “fun” the lesson was,
the level of cognitive demand of the task). Their responses were also internally consistent; they
gave similar and supporting answers regarding their self-assessment of their engagement
throughout the subsequent interviews. In the following sections, I describe each data source and
more results from checks on validity and reliability as appropriate.

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Table 2
Study Focus Students
Student
Name

Site

Felix

Pine River

Average
Expectancy
of Success
Response
2.8

Average
Value
Response

Interests

Favorite Lesson

Other Information
Provided by Teacher

3

Skeet shooting, making
things out of wood,
fishing, knife collecting

Didn’t have a
favorite—struggled

Was diagnosed with
ADHD, has a 504 plan
and attends math lab

Drayton

Pine River

4.4

3.83

Lunch, hunting,
fishing, eating

Division. It’s easy, I
like it

New student at Pine
River—previously
homeschooled

Sophia

Pine River

2.6

5.67

Music, drawing, riding
bikes, swimming

Fractions. Easy, fun

Attends math lab

Lilly

Pine River

6

4.67

Reading, writing, cross
country, playing
clarinet

Graphing, because I
liked drawing graphs

McKenna

Pine River

6.4

5.5

Reading, writing, ice
cream, playing oboe,
saxophone, and piano

Multiplication the
traditional way. It was
easier and the teacher
explained it well

Jeff

Pine River

6.8

7

Baseball, basketball,
football, hanging out
with friends, fishing,
watching TV

Long division. It was
challenging, and I like
to be challenged in
things I’m good at

Lena

Southpoint

1.8

2.67

Reading, writing,
playing piano and
trumpet, running,
talking, organizing

Percent lessons,
actually kind of fun

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Table 2 (cont’d)
Emily

Southpoint

3.6

3.17

Cheer, dance, spending
time with friends and
family

My favorite math
lesson from last year
was learning how to
add taxes onto the total.
Because it’s something
that you use often.

Jacob

Southpoint

4.4

5.8

Go camping, shoot
guns, archery, air soft
battles, and sharpening
my Bowie knife

Measuring angles and
lengths

Adelyn

Southpoint

4.8

4.5

Reading, writing,
drawing, listening to
music, going exploring
in the woods…pulling
pranks, joking around

Learning about
fractions. Learning
fractions always came
easier to me and I loved
them since 2nd grade ☺

Brianne

Southpoint

3.8

5.33

Reading, roller skating,
extra work

All the lessons I
understand

Kim

Southpoint

5

5.83

Softball and winter
guard

Plotting points. It was
fun

Elijah

Southpoint

6.4

6.33

Running, soccer,
basketball, video
games, Science
Olympiad, math
competitions

My favorite math
lesson last year was
when we had to find
missing dimensions
with shapes when our
teacher only give me a
percent of how big it is
compared to the
original.

[Elijah] is very task
oriented and more
concerned about getting
his work and
homework done than
working in a group,
figuring things out
together, and being
further challenged,
although he would
accept challenging
questions.
Note: Brianne was originally selected as a focus student, but left Southpoint in September (after the second observation) and was replaced by Jacob.

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Classroom Observations
I collected all data from September 2013 through January 2014. I observed the two
participating classes weekly from the third week of school through the end of the first semester.
With a break for state testing and schedule changes due to weather-related school cancellations, I
conducted a total of 14 observations at each site. Observing provided important data in the study,
because it provided a systematic way for me to capture student engagement for comparison
across lessons. Fredericks and McCloskey (2012) wrote,
The prime advantage of using observation techniques to study engagement is that they
can provide detailed and descriptive accounts of…factors occurring with higher or lower
engagement levels. These descriptions enhance our understanding of unfolding processes
within contexts. Observational methods also can be used to verify information about
engagement collected from survey and interview techniques. (pp. 767-768)
Each observation consisted of one class period, which typically included work on one CMP
problem. To the extent possible, I observed on consistent days (Tuesdays at one site and
Thursdays at the other) to minimize the possible effect of patterns of engagement through the
week (e.g., students might be more engaged early in the week than as they approach the
weekend). I did, however, observe three consecutive days at both sites late in the semester to see
how student engagement and ideas progressed over an Investigation (these are included in the
total 14 lessons per site). Observing weekly also allowed me to see a range of content in the 7thgrade curriculum, including algebra, geometry, and number, and to observe lessons with and
without contextual tasks. My role in the observations was as observer-as-participant (Creswell,
2009), because demands on my attention while observing allowed minimal participation in
lesson activities, though when students occasionally asked me questions about tasks I would

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answer if possible. In the following sections, I describe the specific data sources collected during
observations, including written tasks, fieldnotes, video, and evidence of student engagement.
Tasks. For each observed lesson, I collected materials related to the task or tasks
implemented, whether or not they were part of the written curriculum. These provided data to
characterize the nature of the contextual tasks, as I will describe in the data analysis section. This
allowed me to address the first research question.
Fieldnotes and video. Answering the first two research questions necessitated
appropriate evidence of student engagement and how the teacher set up and implemented tasks.
During each lesson, I took detailed fieldnotes, capturing verbal statements made by the teachers
and students to the extent possible and nonverbal behaviors indicative of student engagement
(such as tone of voice, gestures, taking out materials, and so on). I recorded as much of the
classroom activity as possible including the classroom structure (i.e., whole group discussion,
small group work, individual work) and times when the class was not working on mathematics
(e.g., announcements, dealing with discipline issues, moving desks, cleaning notebooks).
I also took video and audio recordings of each lesson, using table microphones in the
center of the focus student groups to capture audio and a video camera fixed on the focus
students. The video camera was positioned such that non-participating students were not
included in the videos, but it was near enough to the teacher to capture whole-class discussions.
Throughout the data collection period, I used the audio and video to expand on my fieldnotes and
generate transcripts of lessons.
Engagement protocol. During the observations, I used a protocol that outlined specific
student behaviors indicative of cognitive and emotional engagement (Fredericks et al., 2004) to
identify events for future analysis and to assign holistic engagement ratings. In real time, I

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recorded student discourse as much as possible, focusing on capturing the behaviors in the
protocol (see Appendix C for the list of behaviors and ratings used during data collection). The
protocol was adapted from a list of behaviors generated from research on cognitive engagement
in mathematics classrooms (Brantlinger, 2007; Helme & Clarke, 2001; Nicol & Crespo, 2005). I
started with a framework developed by Helme and Clarke (2001) around specific episode types
(i.e., whole group, small groups, individuals working in parallel), then consolidated the list when
there was overlap to make it easier to reference during observations. For example, I used student
contributes a mathematical idea to represent “verbalizing thinking,” “exchanging ideas,” and
“contributing ideas” from Helme and Clarke’s (2011) list of engagement indicators. I also
modified the list based on other research on engagement in mathematics classrooms. For
instance, I added the behaviors “student perseveres in the face of challenge” and “student seeks
challenge” based on Nicol and Crespo’s (2005) description of student engagement in
mathematical tasks.
In my reading, I did not encounter literature providing similar protocols for capturing
emotional engagement in mathematics classrooms, possibly because it is difficult to capture
students’ emotions through their behavior and it can require a high level of inference (Fredericks
& McCloskey, 2012). Based on definitions and descriptions of emotional engagement
(Fredericks et al., 2004; Nicol & Crespo, 2005), I added behaviors to the protocol where students
explicitly make a statement about their feelings toward the mathematical activity, such as
enjoyment and interest. I also included behaviors that indicated excitement and enthusiasm such
as encouraging one’s peers, starting in on a task right away, and using animated gestures and
tone of voice.

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To capture an overall assessment of the students’ engagement in real time, I used two
holistic scales, one for cognitive and one for emotional engagement. Modeled after the
techniques used by Stipek et al. (1998), I assigned a rating for each type of engagement once
every five minutes (approximately) during the lesson, or sooner if students shifted from one
major activity to another (e.g., when the class transitioned from lecture to small group work).
Whereas Stipek et al. (1998) used several 3-point scales, I incorporated general behaviors into
two 4-point scales based, again, on literature on engagement in mathematics classrooms
(Brantlinger, 2007; Buck et al., 2012; Fredericks et al., 2004; Nicol & Crespo, 2005). I rated
cognitive engagement from 1 (disengagement) to 4 (elaborate engagement). The term and
concept elaborate engagement are drawn from Brantlinger (2007), who defined it as students
contributing ideas measuring more than two lines of written text in the transcript. In real time, to
code student engagement as a 4, I watched for students providing extended (multiple-sentence)
mathematical explanations or ideas. I also rated emotional engagement from 1 (active resistance)
to 4 (enthusiasm, excitement, eagerness). See Appendix C for a brief description of each holistic
code. I assigned cognitive and emotional engagement ratings based on the general state of the
focus students as a group, or the behaviors that most of the focus students or the class as a whole
were exhibiting.
I attended to the validity and reliability of the process of taking fieldnotes and
transcribing both through the pilot studies and through a second observer who attended class and
took fieldnotes during two lessons, one at each site. Our discussions and comparisons of notes
and reflections following each lesson led to clarification of engaged behaviors and more
thorough descriptions of evidence of engagement. In particular, the second coder noted far more
instances where students were disengaged while I focused on evidence of engagement, though

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our holistic impressions of student engagement were fully aligned. This highlighted the
inferential nature of capturing student engagement during classroom observations and the
importance of multiple data sources in this study. I also reviewed her notes of non-verbal student
engagement to check against my transcripts. She noted the same behaviors I identified in the
transcripts (e.g., students raising hands, excited tone of voice indicated by exclamation points,
gestures that helped communicate mathematical thinking, students trading papers), although she
also noted behaviors that I ultimately did not count toward evidence of engagement, such as
taking out books and materials when asked. That is, I may have “undercounted” non-verbal
evidence of student engagement because I excluded these examples of what I interpreted to be
indicators of behavioral engagement, or being “on task” (Fredericks et al., 2004).
Lesson-Specific Surveys by Teacher and Students
At the end of each observed class period, I administered brief (i.e., two to three minutes)
lesson-specific surveys to the teacher and all participating students (see Appendices D and E for
sample surveys). These surveys included two Likert-scale questions that asked the participant to
rate students’ cognitive and emotional engagement; the student version also asked students to
relate the mathematics lesson to a food, sporting event, or movie. The format of the Likert-scale
questions mirrored the real-time holistic ratings from the observations. Together with the
observation data, the responses to these questions primarily provided information needed to
identify the relative levels of student engagement as required to answer the first two research
questions. In particular, the self-reports provided by the survey data helped to counter researcher
bias introduced by the observation data.
The metaphor questions on the student survey asked students to complete the
statement(s), “If today’s math lesson were a (food, sporting event, movie), it would be _____,

56

because ____”, These questions served a different purpose, as they provided data about the
motivational factors underlying student engagement by soliciting students’ reactions to the
lesson at a more complex level. They were adapted from a survey instrument used to “reliably
assess students’ mathematical dispositions and map the emotional terrain of each classroom
regarding students’ assent to being instructed” (Cai & Merlino, 2011, p. 147). The authors based
the survey on the notion that metaphors provide a way to express one’s thoughts and feelings
regarding complex ideas (Cai & Merlino, 2011). I modified the question to be specific to
individual lessons rather than about students’ mathematical dispositions in general. By asking
why students feel the way they do about the lesson, I was able access some of the reasons behind
student engagement (or lack of engagement) across the class.
Focus Group Interviews
Finally, to gain a fuller understanding of students’ experiences from their perspective, I
conducted five monthly semi-structured focus group interviews with the six focus students at
each site (see Appendix F for the interview questions and an overview of the lessons preceding
each interview). Though individual interviews may have allowed each student to say more about
a lesson, it was not possible to interview every student after a lesson due to time constraints—I
conducted interviews during lunch immediately after class to support students in accurately
reflecting on their experiences. Group interviews allowed me to hear from every focus student
for specific lessons and thus compare responses across students. The purpose of these interviews
was to explore students’ self-perceived level of engagement and the reasons underlying their
feelings about tasks in greater depth than allowed by the surveys. Specifically, I explored the
links between their choice to engage in the task (or not), their underlying motivation to learn
mathematics, and the nature of tasks as evidence to answer the third research subquestion.

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During the interviews, I ensured that every student had multiple opportunities to answer
each questions and tried to maintain an environment in which students felt comfortable sharing.
Each interview took approximately 20 minutes. I took notes during the interviews and audio
recorded the group using a table microphone. Each evening following an interview, I used the
audio to complete the notes and fully transcribe the interview.
Beliefs and Values Survey
During my second visit to participating classes, I administered a beliefs and values survey
(Appendix B) as a secondary data source to gain a baseline measure of students’ attitudes toward
and motivation to learn mathematics, and to select focus students. Students completed the survey
in approximately fifteen minutes. It included mainly Likert-scale questions focused on student
beliefs related to expectancy-value theory—namely, how confident students were in their
mathematical ability, their beliefs about utility value, their enjoyment of mathematics, and
mindset related to learning mathematics. These questions were adapted from two existing
instruments: Items #1-11 were adapted from Wigfield & Eccles (2000; available at
http://www.rcgd.isr.umich.edu/msalt). Information on validity and reliability of the items related
to expectancy and value can be found in a report by Eccles, Wigfield, Harold, & Blumenfeld
(1993). Items #12-15 were items from Dweck (2007) to assess students’ mindsets, which I
adapted to make mathematics-specific and to clarify language as needed. I added items 16 and 17
to capture some of students’ interests in and out of the classroom, which helped in selecting more
diverse groups of focus students. In Table 3, I summarize the data source and how they
supported each research question.

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Table 3
Map of Research Questions to Data Sources
Research
Contributing Data Sources
Question
Question 1 Task as written by teacher or
appearing in curriculum

Question 2

Description
Having the tasks allowed me to characterize the
nature of the contextual task in terms of type,
centrality, and authenticity of the context.

Classroom observation –
fieldnotes

Scripted fieldnotes provided a record of student
behaviors to code for student engagement as well
as records of teaching moves. Thus, the notes
contributed to the identification of the level of
student engagement in each lesson and the
characterization of the implementation of the
tasks.

Classroom observation – video

The video data allowed me to fill in missing data
in the fieldnotes taken in real time. These
transcriptions were used to code teaching moves
and student contributions to characterize the
enactment of the tasks.

Classroom observation –
coding protocol

In conjunction with the fieldnotes, the protocol
was used to code student behaviors and provided
the means to record a holistic assessment of
student engagement. This data was used to
identify the lessons in which students were most
and least engaged.

Student survey

The student survey provided a second measure of
student engagement from their own perspective.
It provided a validity check on my researcher
evaluation of engagement.

Teacher survey

The teacher survey provided another measure of
student engagement from a third perspective. It
provided a validity check on my researcher
evaluation of engagement.

Same as question 1

As with question 1, the observation fieldnotes,
video, coding protocol, and lesson-specific
surveys contributed to identifying the lessons in
which students were most and least engaged and
to the characterization of the teacher and student
actions that contributed to the enactment of
contextual tasks.

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Table 3 (cont’d)
Question 3 Student surveys

The open-ended question on the survey (i.e. the
question that asks why students chose the
metaphor they did) provided data on why
students engaged the way they did and their
responses to specific tasks for every participating
student. These responses also served as a basis
for prompts in the student interviews.

Interviews

The interviews provided opportunities to ask
students questions about their engagement (both
from their own and my perspectives) and to
access their perspective on the lesson and tasks
in more detail than allowed by the survey.

Initial attitude/beliefs survey

A secondary data source, students’ responses to
the initial survey were used to characterized
students’ motivation to learn mathematics and
checked against students’ responses to various
tasks in interviews and lesson-specific surveys.

Note on Validity of Measures
Triangulating the data through the use of teacher and student surveys, researcher
observations, and interviews helped ensure concurrent validity for the measures of student
engagement. Table 3 shows how I used at least three data sources in answering each question. As
I describe in the analysis section, I correlated the results of these measures, and based the
selection of high- and low-engagement lessons on results from all four measures. Finally, in
addition to informal conversations on student engagement and task implementation after each
lesson throughout the semester, I conducted member checks with Ms. Pearson and Mrs. Meyer at
the end of the data collection period in which they reviewed and provided feedback on my initial
results and interpretations of the findings.
Data Analysis
Three phases of data analysis were necessary to answer the first two research questions.
In the first phase, I ranked the observed lessons at each school according to the level of the

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class’s engagement in order to identify the lessons in which students were most and least
engaged. In the second phase, I characterized the contexts in the tasks as written and as
implemented in the identified lessons, and in the third I identified patterns and commonalities
between the contextual tasks and instruction in the lessons with highest and with lowest
engagement. Finally, to answer the third research question, I analyzed students’ survey and
interview responses using expectancy-value theory for theoretical coding. In the following
sections, I describe each of these phases of analysis in greater detail.
Ranking Lessons by Engagement
In the first phase of data analysis, I ranked the observed lessons by students’ engagement
level in order to identify the high- and low-engagement lessons for each class. To do so, I took
into account four data sources for each lesson: students’ numerical survey responses, teachers’
survey responses, the holistic engagement ratings, and counts of evidence of engagement. I chose
to generate aggregate scores for cognitive and emotional engagement rather than address each
individually. Research on student engagement has used widely varied definitions of engagement
incorporating elements of behavioral, cognitive, and emotional engagement (e.g., Arnold, 2010;
Brown, 2009; Buck et al., 2012), but this research typically generated single measures of student
engagement. For the purposes of this study, both cognitive and emotional engagement played
important roles. I considered the level of cognitive effort students put into a task to be valuable
alongside student affect relative to the task, particularly in supporting motivation to learn
mathematics. That is, students might work on and think hard about a task (exhibit cognitive
engagement), but feel discouraged negative toward what they are doing. They could also be
happy and energetic in the mathematics classroom (exhibit emotional engagement), say because

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the teacher told a joke unrelated to the content. Neither type of engagement alone supports
motivation to learn mathematics as well as the two working in tandem.
Students’ and teachers’ lesson-specific survey numerical responses. To generate a
single value representing students’ self-reported engagement for each lesson, I added students’
numerical responses representing their self-reported cognitive and emotional engagement, then
averaged these aggregate scores across each lesson for the six focus students in each class.
Similarly, I added the teachers’ ratings representing cognitive and emotional engagement for
each observed lesson. This yielded single values ranging from 2 to 8 for teachers’ and students’
reports on student engagement for each of 27 lessons. I eliminated Observation 14 at Southpoint
Junior High from the study because the students did not take the survey at the end of the lesson;
students were preparing for their final semester assessments that day, which meant they did not
work on specific mathematics tasks and did not have time for the survey.
Holistic engagement ratings. I averaged my real-time cognitive and emotional
engagement ratings over each lesson to get two ratings between 1 and 4. Then, I added them to
yield an aggregate holistic engagement score for each lesson between 2 and 8, parallel to the
student and teacher results. I considered weighting the holistic scores by time—that is,
determining how many minutes of each lesson focus students were rated at a 1, 2, 3, or 4—to
generate a holistic engagement score. I chose to use averages, however, because I aimed to look
across each lesson (and not, say, focus on moments of peak engagement) to identify high- and
low-engagement lessons. These moments of particularly high or low engagement were accounted
for in later analyses of lesson enactment.
Counts of evidence of engagement. Counts of evidence of cognitive and emotional
engagement provided another observation-based measure of student engagement. These counts

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required coding each lesson transcript for evidence of engagement using the engagement
protocol discussed previously (see Appendix C for a list of engaged behaviors). As I coded two
initial lessons, one at each site, I expanded the list of engaged behaviors in Appendix C into a
lesson engagement codebook (see Appendix G).
When coding a lesson, I considered each contribution and noted non-verbal behavior for
the focus students. I used the list of engaged behaviors generated from prior literature on student
engagement in the mathematics classroom and iteratively developed and used the codebook to
identify evidence of student engagement. Then, I noted the actor(s) in the event; the type of
cognitive and/or emotional engagement (e.g., student contributes a mathematical idea, student
shows visible excitement through tone of voice); and whether student engagement was related to
mathematical content, the context of a task, or both. After coding the lesson, I counted the total
number of distinct events showing evidence of student engagement across the focus students.
Finally, I recorded the number of distinct events showing evidence of each student’s engagement
for more in-depth analysis of individual students’ engagement. This contributed to findings
related to student motivation to learn mathematics as the underlying factors of engagement.
After coding all 28 lessons, I calculated the number of engagement counts per hour. I
found how many minutes students were working on mathematics content by eliminating the time
spend on non-mathematical activity identified during the observations, and scaled the counts of
engagement evidence accordingly. This provided more consistent observation data across lessons
and sites, since the two classes were different lengths.
Notes on coding. The initial coding raised issues regarding situations when student
engagement was ambiguous. One was when students participated in interaction segments with
other students or the teacher, defined as “student- or teacher-initiated series of turns in an

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interaction around a single topic” (Jansen, 2008, p. 77). Within interaction segments, it was
sometimes unclear when students showed new evidence of engagement. Interaction segments
often included a short series of statements and questions in which the student repeatedly
exhibited the same type of engagement, such as completing a peer or teacher utterance or
answering a direct question from the teacher. In these cases, I chose to count the interaction
segment as a single piece of evidence of cognitive engagement. So, for example, if in an
interaction segment the teacher asked a student a question, the student responded with a brief
answer, the teacher asked a follow up question, and the student again responded with a brief
answer, I identified one count of student engagement. As an example, consider the following
interaction segment from Lesson 10 at Southpoint Junior High School.
Ms. Pearson: So if I go five miles in 20 minutes, how many minutes are in an hour?
Emily: Sixty.
Ms. Pearson: So if I go 5 miles in 20 minutes, how many 20 minutes are in 60 minutes?
Emily: Three.
Ms. Pearson: Three, so if there are three “20 minutes” in 60 minutes, that ratio would be
equivalent to what I would have to do to the 5.
Emily: To get 20?
In this interaction segment, Emily provides brief answers to Ms. Pearson’s direct questions
around a single topic. Using the guideline I described, I counted this as one count of cognitive
engagement. If students had a pattern of extended responses or showed different types of
engagement in an interaction segment, multiple counts were possible.
When coding for student engagement within an interaction segment was unclear,
particularly when students exhibited multiple types of engagement in an interaction segment, I
considered the following questions: Is this student engaged in this moment? How do I know? Did

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the student just prepare a single answer to a teacher’s question or a single question to ask? Or, do
they persist in engaging? Do they ask a relevant follow-up question or continue explaining an
idea further, worthy of another count of engagement evidence? This helped to clarify whether
one or more examples of student engagement existed in the segment.
A second scenario where student engagement was ambiguous was when students made a
comment related to the context but not necessarily the mathematical content of the task. In some
of these cases, I coded students’ contributions as self-monitoring or reflective self-questioning if
it seemed they were checking or making sense of their solution. For instance, in Lesson 14 at
Pine River Middle School, Drayton found his solution and exclaimed, “a 25-foot shadow? […]
That’s like, bigger than this room!” In this and other cases where a student contributed to a
discussion about the context and it was closely linked to the mathematics in the task, I counted
evidence of cognitive engagement. Other comments on the context were peripheral or seemingly
random and did not count as engagement at all. In other situations, students’ comments indicated
they were excited about the task (at least the storyline of the task), indicating emotional
engagement. For example, in Lesson 8 at Southpoint, Kim said to her small group, “Wow, that’s
a lot of calories! Look how much calories that is.” Kim’s statement did not necessarily show
evidence of cognitive engagement in the mathematical content, but did show evidence of
excitement or surprise (i.e., emotional engagement) related to the context of the problem.
A third scenario requiring clarification was when students raised their hands in a lesson.
Unless a student was called on, it was sometimes unclear why the student was raising their hand
or what they were thinking in the moment (e.g., maybe they were going to ask to use the
restroom or sharpen their pencil.) Thus, I only coded these instances as evidence of engagement
when students raised their hand in response to the teacher posing a question, making it clearer

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they intended to share a mathematics idea, complete a teacher utterance, and so on. Without
knowing the nature of students’ potential responses, I counted hand-raising as emotional
engagement because the gesture indicated some level of confidence or enthusiasm in
participation. When multiple focus students raised their hand, I noted each as showing emotional
engagement, but only gave the event one count of evidence of engagement. I clarified and made
coding decisions on these issues and several others through the initial coding, and included them
as notes in the lesson engagement codebook.
Reliability coding. A second coder double-coded two lessons for a check on reliability. I
selected the lessons because my research memos and holistic engagement scores suggested they
had high and complex levels of student cognitive engagement, providing opportunities for
discussions about multiple coding issues. The second coder was a colleague at another university
with experience in secondary mathematics classroom observation and video coding. First, she
coded the transcript for Pine River Lesson 9 using the lesson engagement codebook. I compared
our two sets of codes for how we identified the actors, type of engagement, and counts for each
interaction segment. Then, we considered each other’s codes without discussing them and
resolved several issues where there were initially disagreements by referring back to the
guidelines in the codebook. Finally, we discussed the remaining events and resolved our coding
to full agreement. We iteratively revised the codebook through this process, focusing on the
points identified in the last section.
Next, we coded the transcript for Southpoint Lesson 9 using the revised lesson
engagement codebook. We reached full agreement by discussing individual events. An example
of a point we needed to clarify was regarding evidence of negative affect—for example, a
student saying, “I don’t like this!” or “I don’t care!” Since the focus of the study was on

66

engagement that supports motivation to learn, we collaboratively decided to only include
indications of positive emotion as emotional engagement. Using the final, revised codebook, I
went back to the first lessons I had coded to revise prior codes that changed as a result of
checking inter-rater reliability.
Comparing engagement results by data source. After analyzing the data for student
engagement, results from four different measures were available to support identification of the
most and least engaging lessons—teacher and student average ratings, average holistic ratings,
and counts of evidence of cognitive and emotional engagement. This was the first step in
answering the first two research questions on the relationships between student engagement and
specific characteristics of mathematics tasks related to context. To consider the relationship
between the four measures, I first found Pearson Correlation Coefficients for each pair of data
over the lessons at each site. The correlation matrices for each class are included in Tables 4 and
5. I found that most correlations between ratings of engagement were moderately (0.4-0.7) or
highly (> 0.7) correlated. Generally, the correlations were higher between focus students, the
teacher, and myself at Pine River than Southpoint. The counts of evidence of engagement,
however, had weak or no relationship between the focus student responses. This was an
unanticipated result, because the engagement counts were based on specific observable behaviors
across lessons and underwent inter-rater reliability coding, and thus were in some ways the most
rigorous and objective measure.
Table 4
Correlation Matrix for Pine River Engagement Results
Measure

Student average response
Teacher response
Researcher holistic average
Researcher counts

Focus student
average
response
1
0.523
0.756
0.255

Teacher response

Researcher holistic
score average

Researcher
counts

0.523
1
0.707
0.499

0.756
0.707
1
0.568

0.255
0.499
0.568
1

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Table 5
Correlation Matrix for Southpoint Results
Measure

Student average response
Teacher response
Researcher holistic average
Researcher counts

Focus student
average
response
1
0.480
0.422
-0.005

Teacher response

Researcher holistic
score average

Researcher
counts

0.480
1
0.402
0.451

0.422
0.402
1
0.435

-0.005
0.451
0.435
1

For a better perspective on these results allowing the identification of most and least
engaging lessons, I generated dot plots for each of the four measures, and pairwise scatterplots
for each pair of bivariate data (see Appendices H and I). I considered the variation in each
measure and looked for outliers that indicated individual lessons that stood out for some reason.
The dot plots show visually that the student and holistic ratings were within a small range (2.5
points for average student ratings and 2 points for average holistic ratings), whereas the counts of
evidence of engagement showed greater variation with several high outliers (roughly, above 80
for Southpoint and above 100 for Pine River). This suggests that perhaps the variation in student
responses was too low for a strong association with the engagement counts.
Another possible explanation is that the different measures were sensitive to different
aspects of lesson and classroom structure. That is, students may have more (or fewer)
opportunities to participate in the kinds of engaged behaviors captured in the transcripts in
certain tasks or lessons, which would be captured in the counts but may not be something
students attended to when reflecting on their own level of engagement. For example, at Pine
River the engagement counts were moderately correlated with the number of minutes the
students spent in small groups (r = 0.545). That is, student engagement was higher when they
spent more time working as a group of six students than in a whole-class setting. In small groups,
individual students generally have more opportunities to share mathematical ideas, explain their
reasoning, and pose questions that in a whole-class setting. Students, or at least some students,
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may be more likely to take up those opportunities in smaller groups. The counts did not
distinguish between small-group and whole-group participation.
This is consistent with existing research on students’ motivation beliefs about classroom
participation, which suggests some students, particularly those who associate whole-class
participation with high risk, prefer to discuss mathematics in small groups (Jansen, 2006, 2008).
Students communicated similar beliefs during focus group interviews in the current study.
Following Pine River Lesson 5, Lilly and Sophia both stated they felt they were particularly
engaged, especially during the small group work. When asked about their favorite part of the
lesson, Lilly responded, “the working parts [in small groups]. I don’t like discussing stuff with a
bunch of people. But when you don’t understanding something in the small group, you can just
ask them.” Later in the interview, Sophia stated, “I hate being wrong. I feel embarrassed in front
of the whole class.” Although these findings do not necessarily mean students in this study were
more engaged overall than in small groups than in other classroom structures, the prevalence of
particular behaviors captured by the observation protocol seems to be associated with students’
work structure.
Other factors outside the amount of small group work time may also have influenced the
alignment of student responses and engagement counts. The correlation between time in small
groups and engagement counts was lower at Southpoint (r = 0.241), possibly because the small
group structure was different (recall 2-3 students worked together during small group work at
Southpoint, versus the entire focus group at Pine River). In both classes, however, some specific
lessons in which the counts disagreed with average student ratings of their own engagement had
atypical characteristics. For instance, Lesson 9 at Southpoint resulted in the greatest number of
counts of student engagement per hour (144.71) across the 14 lessons, but students did not report

69

high engagement in the lesson. In the lesson, Ms. Pearson introduced the chip model for making
sense of integer operations. There was little small group work in the lesson, and it was
interspersed with whole-class work as students worked through problems briefly in groups of
three then presented the results to the class. It was the only lesson observed with this format, and
although there were many opportunities for student participation (captured in the counts of
engagement), there was also “downtime” when a majority of students were not involved in the
whole-class discussion (which students might have attended to in their response).
Lesson 6 at Pine River had opposite results—the students reported a high level of
engagement, though the counts of engagement were the lowest for any lesson (65.08). Again, it
was an unusual lesson in that students were organized into small groups but were primarily
working individually in parallel on a graphing assignment. Students indicated in their surveys
that they put effort into and enjoyed the task, but as outside observers, Mrs. Meyer and I did not
capture as much evidence of engagement.
These examples and analyses suggest that the counts of evidence of student engagement
may be more sensitive to activity structure and expectations for student work on specific tasks
than my holistic ratings and teacher and student ratings for a lesson. In sum, the counts may be a
better predictor of the other measures of student engagement across lessons more similar in task
type and lesson structure than I observed in this study. Taken together, the generally moderate
association between the students’, teachers, and observers’ measures of student engagement
across the 14 lessons in each class indicate that we may attend to different aspects of students’
behavior (and in the case of students, their own thoughts and experiences) when assessing levels
of engagement. This highlights the likelihood that observer-centered ratings of student
engagement are a limited measure and the importance of drawing on multiple data sources.

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Identifying lessons with high and low engagement. Due to the lack of strong
correlation between the different measures of student engagement and the importance of
accounting for multiple perspectives on student engagement in the study, I chose not to use any
single measure to rank lessons in terms of student engagement. Rather, to classify the top and
bottom lessons in terms of student engagement, I first identified the five highest and lowest-rated
lessons for each of the four measures4. Reflecting my commitment to and valuing of students’
self-perceptions of their engagement, I next considered the high- and low-engagement lessons as
reported by students of these 10 lessons and identified any lessons where at least two other
measures (between the counts, holistic rating averages, and teacher ratings) “agreed” in terms of
the highest and lowest rated lessons. I identified any lessons that met the criterion of being one of
students’ top or bottom five lessons and being in this category for two other data sources—I did
not further prioritize data sources. This strategy resulted in three high- and three low-engagement
lessons in each class. Three was a coincidental and not a predetermined number, but provided
consistency for the following analyses.
Thus, the results presented in subsequent chapters are based on the students’, teachers,
and my own perceptions of student engagement, privileging students’ reports. Identifying most
and least engaging lessons in this way afforded a level of confidence moving forward with
characterizing tasks and identifying patterns that multiple sources agreed the noteworthy lessons
were or were not engaging for students. This approach was also consistent with literature
emphasizing the use of multiple data sources when capturing and measuring student engagement
in classrooms. I continue to explore in greater depth the lessons in which the students, teacher,
and I disagreed on students’ relative level of engagement in Chapter 6.
4

With the exception of Ms. Pearson’s ratings for the Southpoint class for whom I identified the
top four lessons, because after the top four lessons the next three were “tied” at a rating of 6.
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Characterizing Lessons
Characterizing contexts as written. In the second phase of data analysis, I characterized
tasks as written in each of the lessons identified as high or low engagement to address the first
research question. For each task, I considered the content it addressed—the mathematical topic,
the learning objectives, the placement of the task in the unit, and whether the task was a
contextual task. For contextual tasks, I looked at three aspects of the context. First, I identified
the type of context, considering several binaries: personalized versus nonpersonalized,
community-based versus general, future (adult)-oriented versus student-oriented,
imaginative/whimsical versus realistic, and addressing social justice issues or not. Different
types of contexts are described with examples in Table 6. It was possible for tasks to be
identified as more than one type. For example, many social justice tasks are community-based,
and some tasks, like the Wumps task discussed later, have elements that are realistic and others
that are imaginative.
Table 6
Different Types of Context
Type
Personalized

Description
Contains some element—
name of student, local
landmark, school name—
specific to an individual
student or students in the class

Example
Joe [a student in class] is
buying gum at Quick Stop
[local convenience store]…

Future (adult)-oriented

Incorporates an idea or
activity relevant to adult life,
especially career-oriented
tasks or personal finance tasks

Martha and Bill are
considering different mortgage
options…

Student-oriented

Incorporates an idea or
activity relevant to the life of a
teen, such as pop culture or
applying to colleges

Alex borrowed her mom’s car
and needs to fill the tank
before she brings it home. She
only has $7.50…

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Table 6 (cont’d)
Imaginative

Describes an “impossible”
situation with some level of
whimsy

Calculating areas of land on
Mars given a fixed perimeter
(CMP, Lappan et al., 2006,
2014)

Community-based

Addresses some situation
relevant to the local
community

The number of family-owned
farms in [local town] has
decreased in the last ten years.
The table below shows the
number of farms operating
each year…

Social justice

Tasks emerging from
students’ lived experiences
and interests that help
“students develop
sociopolitical consciousness, a
sense of agency, and positive
social and cultural identities”
(Gutstein, 2003, p. 40)

Using mathematics to
investigate displacement of
people of color through
gentrification

The second aspect of contexts I considered was the centrality of the context in the task.
Different tasks require different levels of attention to the context and to the mathematics. In
some tasks, the context is a cover story and the problem can be mathematized easily; in others,
the context plays an integral role and must be considered throughout the solution process in order
for students to be successful on the task. To capture this in the tasks as written, I determined if
the context was: (a) peripheral—the context was unnecessary in making sense of and solving the
task; (b) helpful but not necessary in making sense of the task; or (c) necessary in making sense
of the task (Wernet, 2011).
Third, I considered the level of authenticity of each contextual task defined by Palm
(2006) as “the concordance between mathematical school tasks and situations in the real world
beyond the mathematics classroom” (p. 43). His framework was intended to describe aspects of
real-life situations that are important to consider when simulating these situations. The following

73

is a partial list of aspects of authentic tasks that were relevant to this study and a brief description
of each:
•

Event: The scenario described in the task has occurred or is likely to occur.

•

Question: The question posed in the task is likely to be posed during the real-life
event.

•

Information/data: Values provided are realistic, the amount of information provided
reflects what would actually be available

•

Solution strategies: The availability and plausibility of strategies for solving a
contextual task reflects strategies available and plausible in the situation being
simulated

•

Solution requirements: There is a close alignment between the mathematics required
for completing a school task and the mathematics required to do the task outside of
school, as well as the level of accuracy required in the two situations and the types of
assumptions allowed

•

Purpose: The purpose of solving the task and the purpose of solving the associated
real-world problem are clear to students.

In determining authenticity, I first “scored” the tasks on a three-point scale for each of the
six aspects—a score of 2 indicated full alignment between task and real-life scenario, a 1 meant
partial alignment, and 0 meant no alignment. Then, I took into account the scores for each of
these aspects of the authenticity framework together, and make an overall judgment of the
authenticity of the task (low, partial, full). If the task scored mostly 0s with some 1s, it had low
authenticity as written. If a majority of the scores were 1s, it had partial authenticity, and if it
scores mostly 2s with some 1s, it had full authenticity as written.

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Characterizing contexts as enacted. To characterize the tasks as enacted, I first coded
teacher statements and questions related to the context using the framework I developed
previously (Wernet, 2011). I coded for specific types of events in five general categories:
positioning self and others in the problem scenario, clarifying aspects of context, elaborating on
the context as written, referencing context while making sense of task or content, and making
meta-level comments on contexts (e.g., comments regarding why curriculum writers might have
used a certain context, explicit direction on how the problem context should influence the
solving strategy). I counted these teaching moves and tracked each move by type. That is, for
each lesson, I recorded how many times the teacher made elaborating, referencing, positioning,
clarifying, and meta-level statements (or posed questions asking students for these types of
responses) either in the whole-group setting or with the small group of focus students. Attending
to context in one of these ways required the teacher to do more than read the task as written in
the curriculum. Again, when counting these moves I used interaction segments to delineate
excerpts in which the teachers attended to context. For example, in the interaction segment
between Ms. Pearson and Emily presented earlier, although Ms. Pearson used contextual
language several times, I counted one instance of referencing the context. Had she also clarified
some aspect of the context, I would have counted two instances (one referencing, one clarifying).
Finally, I identified each instance of student engagement that was related to task contexts.
These events were any questions, statements, or exclamations about or using the language of the
problem scenario when working on a contextual task. Using open coding, I generated categories
for ways students used and talked about task contexts. Together, teachers’ instructional moves
and students’ attention to context contributed to descriptions of how tasks were enacted.

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Identifying patterns and commonalities in tasks. Once I characterized contexts and
determined the lessons in which students are most and least engaged, I looked across tasks
enacted in these lessons to identify patterns and commonalities. Specifically, I considered: a) the
types of contexts that were addressed in the high- and low-engagement lessons; b) the centrality
of the contexts; c) the authenticity of the contexts; and d) the instructional moves used to address
contexts, and the frequency with which these moves occurred. That is, I described the nature of
contextual tasks and their enactment when students exhibited high and low levels of engagement.
I considered a pattern or commonality to be noteworthy when a characteristic of the tasks
or their implementation (e.g., tasks contained realistic contexts, or the teacher referenced the
context more than four times) appeared in about four of the lessons within a group and did not
occur in more than two lessons in the “opposite” group. For example, when teachers attended to
contexts more than 20 times in five high-engagement lessons and only one low-engagement
lessons, I considered it a relevant finding. I completed this analysis for each of the two
classrooms, then looked for patterns across the two classrooms.
Coding Interview and Survey Data
The goal of the third phase of analysis was to provide a richer understanding of why
students exhibited higher or lower levels of engagement in specific tasks. It allowed me to build
on and speak to broader motivational theory. Student engagement and motivation are complex,
and the reasons students communicated for choosing whether to engage in a task and to what
extent (e.g., feeling tired, being distracted by a personal situation) helped unpack and make sense
of their levels of engagement in specific tasks and lessons. I sought to identify motivational
factors that serve as the foundation for student engagement relative to their work on contextual

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tasks and looked for evidence of how students’ experiences with contextual tasks might
influence their motivation-related beliefs about mathematics.
To answer the third research question on motivation factors underlying students’
engagement in specific lessons, I considered students’ responses to the open-ended question on
the lesson-specific surveys across all the lessons in each class, students’ contributions in focus
group interviews, and any relevant comments by students in the lesson itself. I identified any
statement in which students linked their general response to a lesson or their engagement in a
lesson to aspects of a task. I used expectancy value theory to code these statements along the
central constructs in the theory: ability beliefs, expectancy for success in the task, attainment
value (importance), intrinsic value (enjoyment), utility value (usefulness), and cost (Wigfield &
Eccles, 2000). For example, to code for students’ beliefs about the utility value of a lesson, I
attended specifically to students’ responses to the interview question, “How beneficial (or useful)
do you think the stuff you learned today will be for you, either in this class, future classes, or life
outside of school? Why?” When students made statements about their feelings of confidence or
competence, or about a task being easy, hard, confusing, and so on, I coded the statements as
being related to their expectancy for success or ability beliefs. Comments about a task being fun
or boring indicated their level of enjoyment in the lesson.
After considering these statements for individual students, I identified themes in how
students responded to contexts relative to students with similar expressed motivation to learn
mathematics. That is, I grouped the focus students into low expectancy/low valuing, high/high,
and mixed/neutral and looked for multiple students repeating similar attitudes toward lessons.
Commonalities among students with similar beliefs about mathematics and themselves as

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learners of mathematics supported claims about students’ motivation to learn underlying their
engagement in contextual tasks.
Limitations and Delimitations of Research Design
As with any study, this investigation required several decisions that bounded and
influenced the direction of the study. One such set of choices was around identifying high- and
low-engagement lessons. I chose to privilege student perceptions of their own engagement level,
while accounting for teacher reports and multiple aspects of my own researcher perceptions. This
choice was influenced by calls for research on student engagement and motivation in secondary
classrooms that draws on multiple data sources (Fredericks & McCloskey, 2012; Middleton &
Spanias, 1999; Turner & Meyer, 2009) and prior research on student engagement and motivation
to learn mathematics (Jansen, 2006, 2008; Middleton & Jansen, 2011). Using different or fewer
data sources, or emphasizing a different data source, may have yielded different results in terms
of the lessons in which students were most or least engaged.
Another choice bounding the study was to use CMP tasks and classrooms. This
essentially eliminated the possibility of observing certain types of contextual problems, such as
those designed by teachers to draw on students’ funds of knowledge (Aguirre et al., 2012),
problems posed by students themselves around their own interests (English, 1997; Lavy &
Shriki, 2010; Mason, 2000), or TMSJ tasks (Brantlinger, 2007; Gutstein, 2003, 2007a, 2007b).
Yet, the affordances of CMP made the study feasible by providing frequent and regular lessons
with a variety of contextual tasks as part of the core curriculum, which would likely not have
been possible in a traditional or typical classroom.
Finally, although I coded specific behaviors indicative of cognitive and emotional
engagement to help identify evidence of the two types of engagement, I did not differentiate

78

between them when identifying noteworthy lessons in terms of student engagement. For
example, asking a clarification question, offering a brief response to a teacher’s direct prompt,
and contributing a mathematical idea to a discussion were all examples of cognitive engagement
captured in this study (Helme & Clarke, 2001). Grouping these behaviors together as cognitive
engagement rather than differentiating between them may have influenced the outcome of the
study, as there may be a difference in the cognitive demand of these examples of engagement.
Attending to and analyzing specific types of evidence of engagement may offer a more detailed
picture of the relationship between contextual tasks and student engagement in future research,
but ultimately was not necessary in identifying most- and least-engaging lessons and was outside
the scope of the study.

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CHAPTER 4
STUDENT ENGAGEMENT AND CONTEXTS AS WRITTEN
The purpose of this study was to explore the relationship between contextual tasks and
student motivation and engagement in learning mathematics. To gain a better understanding of
the nature of contexts and instructional practices that relate to higher (or lower) levels of
engagement, I considered the characteristics of contexts and instructional practices during task
implementation across 14 lessons in each of two classrooms. The goal of this analysis was to
gain a more nuanced perspective on how the type of context in a task as well as its authenticity
and centrality relate to student engagement and students’ experiences with these problems.
The first research question asked, “What characterizes the contextual tasks as written in
lessons during which students show particularly high and low levels of engagement?” The
variables of interest in answering this question are the task characteristics and student
engagement. Task characteristics include whether or not the context was personalized, adult- or
student-oriented, imaginative or realistic, whether or not the context was community-based, and
whether the context addressed social justice issues. I also considered the centrality of the
context—how necessary it was to attend to the context in making sense of a problem—and the
authenticity of the context, or how well the story aligns with a real-life situation. I attended to
both cognitive and emotional engagement exhibited by students, that is, evidence of student
investment in learning mathematics as well as positive affect as captured in student and teacher
surveys and my observations.
In this chapter, I present findings on the lessons for which students reported and/or
exhibited the highest and lowest levels of engagement and describe the tasks used in those
lessons. I found that students were more likely to show high levels of engagement in contextual

80

tasks than noncontextual tasks. Their engagement in contextual tasks was also related, however,
to the placement and learning goals of the task and the function of the context in problem
solving. In high-engagement lessons, the tasks tended toward open-ended modeling tasks with
contexts central in solving the problem.
These results suggest that contextualizing mathematics has potential to engage students
both cognitively and emotionally, especially in open-ended modeling tasks when students can
use the context to support problem solving in a variety of ways. Based on these findings, I argue
that contextual tasks engage students by eliciting genuine interest in the context itself, providing
entry into and support in solving the problem, and anchoring the instruction to provide students a
shared experience on which to develop their understanding of the mathematical concepts.
I unpack these findings by first providing general information about the 28 observed
lessons, then identifying the three high- and three low-engagement lessons in Pine River and
Southpoint. To provide the reader a clear picture of the tasks given to students, I describe the
task as written or as modified by the teacher for each of these noteworthy lessons, focusing on
characterizing the contextual scenario(s) in the tasks. Finally, I consider the trends across the
lessons in which students exhibited high and low engagement in order to explicate what aspects
of contextual tasks might support or hinder students’ engagement in mathematics tasks. In the
next chapter, I will consider in greater depth the enactment of these tasks by teachers and
students.
Description of Observed Lessons and Tasks
Lesson Overview
Table K1 in Appendix K summarizes the content, placement, learning goals, and key
aspects of the tasks and their implementation in the 28 lessons observed at Pine River and

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Southpoint. Most of the lessons were “typical” in that they followed the Launch-ExploreSummary structure of CMP tasks, and students focused on one core problem in small groups.
Any lessons that did not follow this common structure (e.g., the class completed a task from the
prior day before starting a new task or the teacher significantly modified a task as written) are
described in the last column of Table K1, “Other Notes on Lesson Activities.” The lessons
covered a range of mathematical content, including numbers and operations, proportional
reasoning, geometry and algebra. They also covered a range of learning goals, from providing an
introduction to the ideas in a unit to reviewing concepts before an assessment.
A majority of the lessons observed at both schools involved contextual tasks to some
extent; that is, tasks with some realistic or imaginary scenario described using nonmathematical
language or images. Five of the lessons did not include any contextual problems, 11 included
references to contexts (e.g., money or changes in temperature) that were not part of the main
problem presented to students, and in 12 lessons, the main problem was a contextual task. The
lessons with core tasks that were non-contextual primarily came from the units on integer
operations (Accentuate the Negative) and characteristics of polygons (Shapes and Designs).
Identification of High- and Low-Engagement Lessons
Of the 28 observed lessons, I identified those at each site in which the focus students
reported and/or exhibited the highest and lowest levels of engagement, using the processes
discussed in the methods chapter. Table 7 presents the student and teacher reports, holistic
ratings, and counts of evidence of engagement for each of the lessons. The high- and lowengagement lessons for each class are also indicated in the table.

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Table 7
Lesson Engagement Results
Pine River Middle School
Lesson Average StudentTeacherAverage Holistic
Weighted Counts
Identified as
Reported
Reported
Researcher
of Evidence of
High- or LowEngagement
Engagement
Rating
Student
Engagement
Rating
Rating
Engagement
1
6
4
5.06
72.00
Low
2
5.67
6
5
81.00
Low
3
6.8
5
5.3
81.00
4
7.5
6
5.51
83.00
5
6.6
5
6
115.00
6
7.4
5
5.3
65.08
7
7.33
8
6.3
89.00
8
7.5
8
6
80.40
High
9
7.6
8
6.36
133.00
High
10
7.5
8
6.09
147.00
High
11
5.2
6
4.8
76.00
Low
12
5.6
6
5.37
127.12
13
7.3
7
5.73
141.00
14
6.33
4.5
5.4
87.00
Southpoint Junior High School
Lesson Average StudentTeacherAverage Holistic
Weighted Counts
Identified as
Reported
Reported
Researcher
of Evidence of
High- or LowEngagement
Engagement
Rating
Student
Engagement
Rating
Rating
Engagement
Lesson
1
5.2
4
5.73
54.44
2
6
7
5.82
61.09
High
3
5.5
6
5.3
82.91
4
5.2
7
6.6
70.38
High
5
4.41
5.5
6.1
108.98
6
5.17
3
5.72
51.76
7
5.42
8
5.7
123.75
High
8
4.59
6
5.08
40.00
Low
9
4.84
7
6.2
144.71
10
3.84
5
5
90.55
Low
11
4.8
6
4.9
43.33
12
4
2
4.95
63.60
Low
13
4.07
5
5.5
63.27
Note: Recall that counts of evidence observed in each lesson were weighted to be a measure of counts per hour of
mathematical activity, because class time lengths were different and time spent on nonmathematical activity was
excluded.

As presented in the methods, a relatively low association existed between the student
reports and the counts of evidence of student engagement. Recall that in identifying the high- and
low-engagement lessons, I started with student reports, and then found agreement between
students’, teachers’, and my measures. Thus, the results presented in the remainder of this

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chapter are based on the students’, teachers, and my own perceptions of student engagement, but
privilege student reports.
Contexts in Highly Engaging Lessons
In the remainder of this chapter, I focus attention on the relationship between various
characteristics of contextual tasks, including centrality and authenticity, and student engagement.
To begin to address the first research question, I will describe the main tasks as written for each
of the six high-engagement lessons identified in the two classrooms. Interestingly, all six of the
tasks in these lessons were contextual tasks, though as you will read, the types of contexts and
their centrality and authenticity varied. The lessons are organized into two subsections: the first
set consists of tasks that posed a contextual question as the primary question for students to
answer, and the second includes other contextual tasks. After describing each lesson, I
summarize the trends and patterns related to the contextual elements of tasks across the six
lessons.
Lessons with Open-Ended Contextual Questions as the Core Activity
Analysis of the tasks in high-engagement lessons revealed that most of the tasks were
fairly open-ended contextual tasks. In each case, the teacher focused on one part of the task or
phrased their own question for students to answer based primarily on the storyline on the task.
On the surface, some questions did not even seem to be particularly mathematical. Students
encountered the tasks early in their respective units. I organized these lessons and tasks together
to highlight these important characteristics.
Pine River Lesson 8: Solving the mystery of the teacher in disguise. Lesson 8 at Pine
River took place mid-semester. The class began a new unit, Stretching and Shrinking, and spent
the class period working on the opening material for Problem 1.1 (see Figure 2). It is a

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contextual task in which the Mystery Club at a middle school is determining the height of a
teacher from a photograph so they can figure out the teacher’s identity. It is important to note
that Mrs. Meyer enacted this task differently than as written in the CMP curriculum, because she
used the opening question (the first bullet point in Figure 2) as the core task for the lesson. The
actual problem in 1.1 used the same general context, but asked students to enlarge an image from
a flyer to use on a poster and then compare angle measures and segment lengths. The task as
given to students was more open-ended and allowed students to use the storyline in whatever
way they wanted.
As written, the Mystery Club context is non-personalized, general (not communitybased), is student-oriented, and is realistic (versus imaginative)5. The context is necessary in
making sense of and solving the task, since the questions students are to answer—how tall is the
teacher, and how do you use the picture to estimate the height—are drawn from the Mystery
Club storyline. That is, students’ solutions to the task answer real-world questions and thus
require an interpretation of the contextual features of the task. As for authenticity, the task has
full authenticity as written because although they may be unlikely, the event and question in the
problem scenario are plausible; and the information provided, strategies available to students,
and the required accuracy of the solution are aligned with they real-life scenario. Furthermore,
both the mathematical purpose of the task (exploring and using mathematical similarity) and the
real-life purpose (identifying the mystery teacher) are clear.

5

None of the tasks observed clearly addressed social justice issues, as this is not a focus of the
CMP curriculum. Thus, although I coded for that feature of contexts and believe it is an
important aspect to consider when relating student engagement with types of contexts, I will not
include it in my descriptions of the contexts as written.
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•

What do you think Daphne has in mind? Use the picture and the information about
the height of the magazine to estimate the teacher’s height. Explain your reasoning.

•

The teacher advisor to the Mystery Club says that the picture is similar to the actual
scene. What do you suppose the advisor means by similar? Is it different from
saying that two students in your class are similar?

Figure 2. Problem statement and image from Stretching and Shrinking 1.1
Pine River Lesson 9: Meeting the Wumps and impostors. Lesson 9 at Pine River was
the first of two days in which students worked on Problem 2.1 in Stretching and Shrinking (see
Figure 3). The problems in the investigation have two layered contexts. The first involves a pair
of fictional students named Zach and Marta who are designing and programming a computer

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game. Specifically, this aspect of the context involves how to animate and change sizes of
figures using coordinates. The second context involves video game characters called Wumps and
determining which characters are Wumps and which are impostors.
As written, the contexts in the task are non-personalized and general (not communitybased). There are both adult- (designing computer games) and student- (determining who is in
the Wumps family) oriented aspects to the context. Similarly, there are both realistic (designing
computer games) and whimsical/imaginative (Wumps family and impostors) aspects of the
context. The computer design context is peripheral to solving task, because once students start
calculating coordinates and graphing the figures, they do not need to reference or use that
storyline. The Wumps versus impostors context is helpful but not necessary in this part of the
task. Students could, in theory, ignore the story and focus on coordinates, lengths, and similarity.
As for authenticity, the computer programming context has partial authenticity as written.
Although the scenario is plausible and programmers or game designers do use coordinate points,
this kind of work (by hand) would be unnecessary if you were actually programming. You would
know ahead of time which figures were similar. The Wumps context is imaginative and thus has
low authenticity as written.
In this first day of implementing the task, Mrs. Meyer centered students’ work
around a contextual question—who is similar to Mug Wump is this belongs in the Wump family,
and who is an impostor. Students spent most of the lesson graphing the characters. So the lesson
started with an extensive exploration of the computer programming context, but it shifted to
focus on the Wumps context.

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Figure 3. Problem statement and image from Stretching and Shrinking 2.1.
Pine River Lesson 10: Mathematizing the Wumps and impostors. This lesson was the
second day students spent on the Wumps task. 2.1 As students get into the task as written,
though, the actual questions (see Figure 4, problems A-C) do not refer to the computer game
design context, but focus on the Wumps versus impostors context. So for this lesson, the context
is non-personalized, general (not community-based), student-oriented, and
whimsical/imaginative. The Wumps versus impostors context is necessary for making sense of
this part of the task because whereas the day before students were primarily graphing, in this
lesson students focused on the question, “who is similar to Mug Wump is this belongs in the

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Wump family, and who is an impostor?” The Wumps context is imaginative and thus considered
to have low authenticity as written.
The unit of analysis in this study was individual lessons rather than time spent on
individual CMP tasks. Thus, it is not particularly surprising that both Lessons 9 and 10 were
high-engagement lessons, as they worked on the same task. It is noteworthy, however, that
students maintained high engagement for a second day on the task, because it was also likely the
novelty would have worn off and students would start to disengage.

Figure 4. The problem as written in Stretching and Shrinking 2.1. Mrs. Meyer focused on one
core question: Who are the Wumps, and who are impostors?

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Southpoint Lesson 4: Making sense of cola advertising. In this lesson, students worked
on the first part of Problem 1.1 in Comparing and Scaling. Like the Mystery Teacher problem,
this was the first lesson in a new unit and provided an opportunity for students to explore
comparison statements in an open-ended task. The lesson opened with a brief introduction to the
new goals of the unit before Ms. Pearson launched the task. The context in the problem involved
a soda company using the results of a taste test comparing their product with another brand in a
new advertisement. Students are given four comparison statements (Figure 5) and directed to
make sense of the statements in the first part of the problem. For example, the text asks them to
describe each statement in their own words. Ms. Pearson focused on the question, “Which of the
above statements do you think would be best in an advertisement for Bolda Cola? Why?” (CMP
2nd edition, Comparing and Scaling 1.1).

1. Ina t ast e t est, people whopreferred Bolda Cola
out numbered those whopreferred Cola-Nola by
a rat ioof 17,139 t o11,426.
2. Ina t ast e t est, 5,713 more people preferred
Bolda Cola.
3. Ina t ast e t est, 60%of the people preferred
Bolda Cola.
4. Ina t ast e t est, people whopreferred Bolda
Cola out numbered t hose whopreferred
Cola-Nola by a ratioof 3 t o2.

Figure 5. Comparison statements from taste test results for Comparing and Scaling 1.1
As written in the CMP text, the Bolda Cola and Cola Nola taste test context is nonpersonalized, general (not community-based), and realistic. It is both adult- and studentoriented—it is more adult-oriented if analyzing the information from the perspective of an
advertiser, but the soda context might be more aimed at students. The storyline does not clearly

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address social justice issues, though it could if students focus on being discerning consumers of
advertising or the health risks involved in drinking soda heavily. The context is necessary in
making sense of and solving the task, because every question required students to analyze
contextual statements and solve problems related to the two companies and the results of the
taste test. The context has partial authenticity as written because although conducting taste tests
and using results in advertising is common, the specific questions asked of students and available
information are not realistic. That is, copywriters would not need to determine some of the
specific solutions asked of students.
Other Highly Engaging Contextual Tasks
The tasks in the other two high-engagement lessons did not center around a core
contextual question, and both came later in their respective units. They were contextual tasks,
however. In both lessons, Ms. Pearson adapted the tasks as written to be group challenges with
specified roles.
Southpoint Lesson 2: Exploring similarity in the Wumps’ noses and mouths. This
lesson took place early in the school year, and as described in the Methods, Ms. Pearson
modified Stretching and Shrinking 2.3 in CMP (see Appendix A for the written task provided to
students) to use in the lesson. Students were organized into small groups of two to three students
to complete the task and created a poster to present their work. The modifications to the task
were intended to make it more open-ended and emphasize group roles, but maintained the
contextual aspect of the task as written in the curriculum.
The task in this lesson continued with the Wumps context. Here, the fictional students
Zach and Marta are analyzing pictures of characters’ noses and mouths to determine which are
those of Wumps and which are those of impostors. The context is non-personalized, general (not

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community-based), student-oriented, and whimsical/imaginative. Both the Wumps and computer
game design contexts are peripheral in making sense of the task. Zach, Marta, and the Wumps
are basically a cover story for analyzing shapes for similarity are not necessary to understand for
solving a task. The Wumps context is imaginative, and in this task, the context of designing a
computer game and not having the information asked for in the task is highly unlikely. Thus, it
has low authenticity as written.
Southpoint Lesson 7: Analyzing Sascha’s bike trip. As in Southpoint Lesson 2, Ms.
Pearson developed a modified version of Comparing and Scaling Problem 3.2 for use in this
lesson (see Appendix J for the written task provided to students). The context involves a man
named Sascha who is going on a bike trip. There are three legs to the trip. Students were told
how long each leg is and how many minutes it took to finish. Then, they were asked to find and
compare unit rates and think about how fast they would need to ride to race Sascha.
It is important to note that the problem as written for students represented an instructional
choice that was part of the implementation of the lesson, and also influenced how the task was
enacted with students. For example, Ms. Pearson changed the word “Stops” to “Legs” and then
explained what a leg was in the launch. She also added questions about which legs had the most
uphill and downhill portions, which required students to make sense of the biking context in a
new way. Finally, in part E about tying in a race, Ms. Pearson wrote an explanation of what a tie
would look like mathematically in the problem text, so in that way she clarified an aspect of the
context and added to what the textbook had actually asked.
As written in the modified version of the task, the bike trip context was non-personalized,
general (not community-based), age-neutral, and realistic. In this task, the context is necessary in
making sense of and solving the task, because each part of the problem requires analyzing given

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information about Sascha's bike trip, making sense of the rates, and tying solutions to units. The
problem has full authenticity as written because the event—taking a bike trip and recording
distances and times for each leg is common and is likely to occur. Moreover, finding average
rates and analyzing information on when one rode uphill and downhill is appropriate. The
information provided in the problem is reasonable based on Internet searches on typical speeds,
and the solution requirements and strategies available to students are appropriate, assuming
Sascha did not have access to certain technologies. The mathematical and contextual purposes of
the problem are somewhat unclear, though the aspect about racing Sascha (part E) helps clarify
one purpose of solving the task.
Trends and Themes in Contexts as Written in Highly Engaging Lessons
The first research question for this study asked what written features characterize
contextual tasks in high- and low-engagement lessons. In the previous sections, I described and
provided descriptive statistics on the contexts of tasks in each of the six lessons in which students
exhibited and reported the highest levels of engagement. Key information and features are
summarized in Table 8. In this section, I identify patterns and themes in the six lessons, to be
compared and contrasted with low-engagement lessons in subsequent sections.
Table 8
Summary of Features of Tasks in High-Engagement Lessons
Lesson
Contextual
Centrality
Authenticity
Task
Pine River 8
Yes
Necessary
Full
Pine River 9
Yes
Peripheral (game design),
Partial (game
helpful not necessary
design), low
(Wumps)
(Wumps)
Pine River 10
Yes
Necessary
Low
Southpoint 2
Yes
Peripheral
Low
Southpoint 4
Yes
Necessary
Partial
Southpoint 7
Yes
Necessary
Full

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The first notable shared feature of the six lessons is that they all had contextual tasks as
the primary learning activity. The contexts of the problems were of the same types—they were
location-general (not community-based), non-personalized, and did not address social justice
issues. The five tasks included both student- and adult-oriented tasks, but each of the tasks was at
least relatable to most middle school students in that students likely have some experience with
the various problem scenarios. Contexts such as the Mystery Teacher and the Wumps are stories
appropriate for middle school students, whereas contexts such as Sascha’s bike trip and
comparing Bolda Cola and Cola Nola are more age-neutral. All of the tasks had some realistic
aspect in the context.
The authenticity and centrality of the tasks varied. All tasks except Southpoint Lesson 2
(an extension of the Wumps storyline), however, had partial or full authenticity as written. The
Wumps context had low authenticity because it was an imaginative, whimsical context. Further,
almost all of the lessons had contextual features that were at least helpful in solving the task, and
it was necessary to understand the contextual storyline in four of the six lessons.
Finally, four of the six lessons (three of the five tasks, since Problem 2.1 extended over
Pine River Lessons 9 and 10) posed open-ended contextual questions as the core question
students were expected to answer. In Pine River Lesson 8 students were asked to find the height
of the Mystery Teacher, and in Lessons 9 and 10 they were asked which characters in the game
were Wumps and which were impostors. In Southpoint Lesson 4, students were asked which of
the four statements was most convincing and should be used in an ad for Bolda Cola. Although
students needed to mathematize the problems to reach reasonable solutions, the questions
themselves were drawn from the problem scenario. Moreover, in the other two tasks—Sascha’s

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bike trip and the Wumps’ mouths and noses—students were asked several different questions as
part of the task, and most of these sub-questions drew on the context in some way.
Contexts in the Least Engaging Lessons
In contrast to the high-engagement lessons, only three lessons in which students reported
and exhibited low engagement included contextual tasks. Lessons 1 and 2 at Pine River and
Lesson 12 at Southpoint had noncontextual tasks, meaning there were no non-mathematical
storylines in any part of the tasks as written or implemented. The two lessons at Pine River came
from the unit Shapes and Designs; students were analyzing angle properties of polygons. The
Southpoint lesson came from Problem 3.3 in Accentuate the Negative on dividing integers. Ms.
Pearson produced an assignment for this lesson that actually removed the contextual part from
the written text that returned to a previous story about a Number Relay. For each of the three
low-engagement lessons that included contextual tasks, I will discuss the contexts of the task as
written, including the type of context and the centrality and authenticity of the context. Unlike
the tasks in high-engagement lessons, these three tasks are distinct in terms of their format and
purpose of their contexts. Thus, I discuss them individually before summarizing the trends and
patterns related to the contextual elements of tasks across the low-engagement lessons.
Pine River Lesson 11: Counting and Pricing Apples, Bananas, and Eggs
In Lesson 11, Mrs. Meyer piloted an alpha version of a Formative Assessment Lesson6
called Real Life Equations. The lesson followed a typical lesson structure with an opening launch
that reflected back on earlier activities, time for students to explore in small groups (of two or
three rather than the usual four to six students), and a whole-class summary discussion. The task
6

Formative Assessment Lessons and more information about their development are available at
http://map.mathshell.org, a project led by Mathematics Assessment Resource Service University
of Nottingham and UC Berkeley.

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identified the meanings of different variables (see Figure 6), and students were asked to cut out
cards with statements and match them with appropriate expressions or equations to make a
poster.

Figure 6. Variables defined in task given to students.
One context embedded in the task involved two fictional students—a boy and a girl
whose cartoon images appear in the slides—who disagree about the equations relating two
variables. The other contexts in this lesson arose from the everyday referents of the variables.
Students interpreted equations and expressions involving the number and price of eggs, apples,
and bananas. The task did not have a storyline, other than the single statement, “suppose you are
buying apples and bananas in a shop,” which was in the suggested script for teachers and not
students’ version of the problem. This food context is non-personalized, general (not communitybased), adult-oriented because grocery shopping is often an adult activity, and realistic (versus
imaginative).
The context is necessary in making sense of the task, because although the objects could
have been switched out for anything, the entire task required students to link the meaning of
variables with costs and numbers of the objects. Thus, students needed to go back to the context
throughout. The context had low authenticity as written, because although the “event”—grocery
shopping—is part of everyday life, the actual questions posed, strategies students are expected to

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use, values given, and mathematical requirements are unrealistic. Shoppers would probably not
use expressions or equations in this scenario, making the purpose unclear.
Southpoint Lesson 8: What do Puffins, Airplanes, Enchiladas, and Student Council
Elections Have in Common?
In this lesson, students worked on the final task in the unit Comparing and Scaling. The
purpose of the task was to synthesize their knowledge about proportions and apply that
knowledge to new situations. Thus, this task was different in nature than the other high- and lowengagement lessons. Although students worked on a single problem, each part of the problem
used a different context. So the class worked on and/or discussed four different contexts in the
lesson: estimating puffin populations (from a homework problem the students went over in
class), determining how many miles a jet would take to descend 5280 feet, estimating the
number of calories Jack consumed when he ate enchiladas over the course of a year, and
comparing a middle school population with the numbers of students on student council by grade.
I analyzed each of the contexts as written separately:
•

Puffins: The context in the puffins problem is non-personalized, adult-oriented, realistic,
and general. It is necessary in making sense of task. It has full authenticity as written
because this kind of tag-and-release technique occurs and is appropriate for estimating
animal populations.

•

Jet: The context in the jet problem is non-personalized, adult-oriented, realistic, and
general. The context is peripheral because it is easily mathematized without needing to
make sense of the jet scenario. It has partial authenticity as written because although
landing a jet is a reasonable event and the values given are appropriate, one would never
use pencil and paper and proportions to determine the landing distance, and it is left
unclear why the information is needed.

97

•

Jack and the enchiladas: This context is non-personalized, is both adult- and studentoriented, and realistic. The storyline is helpful but not necessary, as it is easily
mathematized but students need to go back to the context in some parts of the question. It
has partial authenticity as written because counting calories is a common event and the
calorie counts provided are realistic, but the question asked, though plausible, is unlikely.

•

Student council: The context of determining the number of student representatives for
student council is non-personalized, both adult- and student-oriented, realistic. It is
necessary in making sense of task, because each part requires students to make sense of
quantitative relationships related to grades and the number of students. It has full
authenticity as written; although there are many possible strategies that students do not
need to explore and the purpose of solving the real-life problem not be clear to students,
it is an authentic situation and one that is likely to occur.

So within this task, the multiple contexts introduced had fairly diverse characteristics, ranging
from peripheral to necessary in solving the different parts.
Southpoint Lesson 10: Money Models
Like the last lesson, this lesson was atypical compared with the other observed lessons at
Southpoint and Pine River, including both the high- and other low-engagement lessons. The core
task of the lesson was a teacher-produced, non-contextual assignment made up of problems
drawn from the homework problem set from Accentuate the Negative Investigation 1. The lesson
started, however, with an extensive (approximately 38 minutes) discussion reviewing students’
work on Problem 1.4, and a few parts involved money contexts. These were stories in which the
actors mow lawns, walk dogs, and borrow money from their siblings, or more generally, spend,

98

earn, and owe money. Also, Emily, a focus student, spontaneously referenced an abstract “point”
context in which points were gained and lost.
The money contexts are non-personalized, general (not community-based), studentoriented, and realistic. They are peripheral cover stories in which the scenario does not matter,
just the numbers and whether the actor is "owing" or "earning," which translates to adding or
subtracting positive and negative numbers. The stories have partial authenticity as written,
because although the events are plausible and likely to occur, and the questions and info are
reasonable, the chips and number sentences would not be used in these scenarios and thus there
is weak alignment between the purpose of the task and the purpose of figuring out how much
money one has.
Trends and Themes in Contexts as Written and Enacted in Least Engaging Lessons
To further address the first research question, in this section I considered the contextual
features of tasks as written in low-engagement lessons. Only three of the six lessons in which
students exhibited and reported the lowest levels of engagement involved contextual problems in
any part. The key information and features of these three tasks are summarized in Table 9. One
striking difference between this group of low-engagement lessons and the high-engagement
lessons is that whereas the most engaging lessons had contextual tasks as the main mathematical
activity, half of the low-engagement lessons did not include any contextual aspects to the
mathematics. Three of the five lessons with non-contextual tasks observed across the study
period were in this group.

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Table 9
Summary of Features of Tasks in Low-Engagement Lessons
Lesson
Contextual Task
Centrality
Pine River 1
No
Pine River 2
No
Pine River 11
Yes
Necessary
Southpoint 8
Yes
Puffins
Necessary
Jet
Peripheral
Enchiladas
Helpful
Student Council
Necessary

Authenticity
Low
Full
Partial
Partial
Full

Southpoint 10

Core task not contextual; but
reviewed contextual
homework problems

Peripheral

Partial

Southpoint 12

No

-

-

In some ways, the contexts in the two groups of lessons were similar as written. The
contexts of the problems in low-engagement lessons were generally of the same types as those in
the high-engagement lessons—they were location-general (not community-based), nonpersonalized, and did not address social justice issues. The three problems tasks included both
student- and adult-oriented tasks, but each of the tasks was at least relatable to most middle
school students. Grocery shopping, earning and spending money, and the various contexts in
Southpoint Lesson 8 were situations generally familiar to seventh-grade students.
Also, like with the high-engagement lessons, the authenticity and centrality of the tasks
varied. There were tasks in which the context was peripheral, helpful-not-necessary, and
necessary in making sense of the mathematical concepts. Likewise, there were tasks with low,
partial, and full authenticity. Pine River 11, however, was the only lesson observed that had a
“realistic” context with low authenticity—the other low authenticity tasks had the imaginative
Wumps context.

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Though these characteristics of the contextual tasks were similar across the lessons, the
three contextual tasks in the low-engagement lessons were different in nature than the highengagement lessons. All of the high-engagement lessons but Southpoint 2 had core tasks (i.e.,
students’ main mathematical activity) with a single, clear non-mathematical storyline that was
central in solving the task. This was not true of the tasks in the low-engagement lessons. Lesson
11 at Pine River had a core contextual task—the entire task involved interpreting variables in
expression and equations in terms of their real-life quantities. But the context had low
authenticity and did not have any kind of story to clarify the purpose of solving the mathematical
or everyday problems. The core task in Southpoint Lesson 8 consisted of three different
contextual problems, all of which had more qualities of traditional application problems (i.e.,
story problems) than the more open-ended tasks in the high-engagement lessons. Finally, the
contextual tasks in Southpoint Lesson 10 were not part of the core problem in the lesson, but
were from the discussion of the prior day’s homework. The class moved on to work on a
problem set developed by Ms. Pearson, none of which were contextual as written.
Discussion: Engagement Potential of Contextual Tasks as Written
The first goal of this study was to identify the lessons in which students exhibited high
and low engagement and characterize the context, if any, in these lessons’ main mathematical
tasks. I found that, generally, students showed the most engagement in lessons with contextual
tasks, and those contexts were both relatable for students and a central part of problem solving.
The low-engagement lessons either did not involve contextual elements, or the problem
scenarios lacked the complexity of those in the more engaging lessons. Before considering these
tasks as enacted in the next chapter, I unpack these findings on the contexts as written and begin

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to frame an argument about the role contexts may play in student engagement in mathematics
tasks.
Meaningful Contexts Can Engage Students
The first aspect of tasks I considered in characterizing the contexts as written was the
type of context. Though students showed different levels of engagement in contextual tasks, little
variation existed in the types of contexts across the observed lessons. This was an expected
limitation in choosing to use CMP classrooms in the study, because as a nationally-published
curriculum, it does not include personalized or community-based tasks. An important finding,
however, is that contexts can be engaging for students. In fact, the data suggests students are
more likely to engage cognitively and emotionally in contextual tasks than noncontextual tasks.
This finding agrees with literature about meaningful contextual tasks being motivating for
students to engage in mathematics (Cordova & Lepper, 1996; Helme & Clarke, 2001; Jansen &
Bartell, 2011; Mitchell, 1993; Nicol & Crespo, 2005; Weist, 2001). It also supports the
theoretical arguments given by curriculum designers and researchers that contextual tasks are
included in curricula in part to promote student engagement and spark their interest in tasks (e.g.,
Civil, 2002; Gutstein, 2003; Lappan & Phillips, 2009; Nisbet et al., 2007; Romberg & Shafer,
2003; Silva et al., 1990.)
The findings also agree with literature that showed imaginative contexts with low
authenticity can be engaging for students, (Nicol & Crespo, 2005; Weist, 2001), as three of the
six most engaging lessons involved the Wumps versus impostors context. Yet, all but one lesson
with a contextual task (Southpoint 2, Wumps extension) involved some realistic element,
whether based on adult or student experiences. Even the Wumps context is prefaced with a realworld context about programming computer games that links the Wumps with the mathematical

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content. The important thing to note is that these contexts—imaginative or realistic, aimed at
adults or students—were relatable for students. That is, they were realistic in the Realistic
Mathematics Education (RME) sense of the word.
In Dutch, the verb ‘zich realiseren’ means ‘to imagine’. In other words, the term
‘realistic’ refers more to the intention that students should be offered problem situations
that they can imagine…than that it refers to the ‘realness’ or authenticity of problems.
However, the latter does not mean that the connection to real life is not important. It only
implies that the contexts are not necessarily restricted to real-world situations. The
fantasy world of fairy tales and even the formal world of mathematics can be very
suitable contexts for problems, as long as they are ‘real’ in the students’ minds. (Van den
Heuvel-Panhuizen, 2003)
This description of realistic as “imaginable” emphasizes the need for problem contexts to be
personally meaningful for students (Helme & Clarke, 2001; Jansen & Bartell, 2011; Mitchell,
1993; Silva et al., 1990). Moreover, because the tasks in both the high- and low-engagement
lessons ranged from low to high authenticity, the results suggest that authenticity as
conceptualized in the analytical framework used in this study may not be as important as the
extent to which the context is relatable or meaningful for students.
Importance of Task Purpose and Centrality of the Context
Whereas the type of contexts in tasks and their authenticity did not seem to directly
influence student engagement, the centrality of the context did appear to be linked to higher
levels of engagement. All of the tasks in high-engagement lessons had contexts that were at least
helpful in solving the problem except one (Southpoint 2, Wumps extension). Also, as I present in
the next chapter, Mrs. Meyer and the Pine River students attended significantly to context when
it was necessary for solving the problem even in the low-engagement lesson with the apples and
bananas task. This makes sense—regardless of what the context is, if it is necessary to make
sense of the context to solve the task, it will likely come up in discussions throughout lesson. It

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suggests, though, that the more central the context is in solving the problem, the greater the
potential to support student engagement.
This is likely because the problem scenarios can provide entry into and support students’
work on the task. As argued by others, (Carraher et al., 1985; Walkington et al, 2012), providing
everyday contexts can encourage informal strategies to make sense of the problems, with less
opportunity to immediately mathematize the problem. Problems that marry meaningful contexts
with contexts necessary to solve the task promote students’ appropriate use of the context in
solving and may encourages the use of the context to check reasonableness of solutions and
strategies. So these two purposes of contextualizing math—supporting learning and
engagement—are interrelated, because having access to the task mediates student engagement.
Building on this idea, the relationship between student engagement and task contexts as
written seems to be more about the purpose and learning goals of a task and the function of the
context in problem solving than the type of context. In the most engaging lessons, task prompts
were open-ended and exploratory in nature. They often occurred at or near the beginning of the
unit or an Investigation and did not prescribe the mathematical concepts needed to solve. The
purpose of these tasks was to introduce new ideas or to give students opportunities to explore,
make conjectures, and try to develop strategies for solving a novel problem. In most of these
tasks, the context played a central role in making sense of and solving the problem. In these
tasks, just the description of the context and accompanying images took up over a page of the
textbook! Students could not answer the questions asked without attending to context. In four
high-engagement lessons (Wumps, Bolda Cola, Mystery Teacher) the primary questions posed
were contextual questions: How tall is the teacher and how do you use the picture to figure it

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out? Who are the Wumps, and who are the impostors? Which statement would you use in the
Bolda Cola ad?
In the spectrum of contextual tasks, these were closer to modeling problems than
application problems (Ness, Blum, & Galbraith, 2007). Modeling and application are sometimes
used interchangeably for any connection between mathematics and the “real world.” But as Ness
et al. (2007) describe,
The term "modelling", on the one hand, tends to focus on the direction
"reality mathematics" and, on the other hand and more generally, emphasises the
processes involved. Simply put, with modelling we are standing outside mathematics
looking in: "Where can I find some mathematics to help me with this problem?" In
contrast, the term "application", on the one hand, tends to focus on the opposite direction
"mathematics reality" and, more generally, emphasises the objects involved—in
particular those parts of the real world which are (made) accessible to a mathematical
treatment and to which corresponding mathematical models already exist. Again simply
put, with applications we are standing inside mathematics looking out: "Where can I use
this particular piece of mathematical knowledge?" (pp. 10-11).
In most of the high-engagement lessons, the idea was to introduce and support student
exploration of mathematical ideas that were new to them, not to apply established mathematical
principles to real-world situations. The open-endedness of the tasks gave students opportunities
to use the contexts in a variety of ways, as multiple strategies were possible. These results are
consistent with motivation literature arguing that opportunities to work on tasks with appropriate
levels of challenge and that allow multiple entry points and learning styles promote student
motivation and engagement (e.g., Copping, 2012; Engle & Conant, 2002; Middleton & Jansen,
2011; Middleton & Spanias, 1999; Stipek et al., 1998; Suh et al., 2011; Turner & Meyer, 2009).
The modeling tasks in the study may promote in students a sense of authority over problem
solving more than traditional application problems, and studies have shown this sense of
authority is related to student engagement and motivation to learn (Engle & Conant, 2002;
Shernoff, 2003; Skinner et al., 2003).

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In contrast, task contexts in the low-engagement lessons served different functions. On
one hand, they seem to be exceptions to the findings. For example, Southpoint Lesson 8 had
contextual tasks in its core activity, and those contexts were fairly central to solving the
problems. On the other hand, the function of the context was different than in the highengagement lessons. The placement of this lesson was at the end of the unit, and thus students
were expected to apply what they had already learned to new situations. Though the contexts
were still new to them (thus, novelty was not the issue), by that point in the unit students had
established procedures for solving such tasks. This was an important point in their learning of the
concepts, but the contexts were now something that needed to be mathematized quickly and not
necessarily explored. This made the questions more similar to traditional story problems with
one primary solution path, which many have argued do not promote student engagement (Boaler,
1993; Chazan, 2000; Gerofsky, 2004; Sullivan et al., 2003; Thomas & Gerofsky, 1997;
Verschaffel et al., 2000).
Likewise, students needed to use the apples and bananas context in Pine River Lesson 11
to complete the task. In this case, the difference may have been the type of context. The grocery
shopping storyline was more of a pseudo-context (Olive et al., 2010) or microcontext (CTGV,
1992)—that is, it is a “real-life” and relatable context, but students need to (or can) ignore any
complexities in the problem scenario. This creates some risk that students will draw
inappropriately on the context, for example, using aspects of the context too literally and letting
that stand in the way of mathematizing the problem (Olive et al., 2010; Lubienski, 2000). The
openness of the tasks coupled with the centrality of the contexts in the high-engagement lessons
allowed students more flexibility in how they used the context to make sense of the problem,
helping them avoid these pitfalls. Finally, the highly engaging task in which context was

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peripheral (Southpoint 2) was still based on the Wumps context, which had been more central in
previous tasks, meaning it was more embedded in the mathematics they were studying in the
lesson (i.e., similarity and scale factor).
Summary and Preliminary Argument
In this chapter, I addressed my first research question regarding the relationship between
contextual tasks as written and student engagement. In sum, the lessons in which students
reported and/or exhibited the highest levels of engagement all included contextual tasks,
indicating contextual tasks promote student engagement. Though the level of authenticity and the
types of context did not have a strong relationship with the level at which students engaged with
tasks, the level of centrality of the context and its function relative to the purpose of the task
appeared to be important. The results suggest that contextualizing mathematics can have
potential to engage students both cognitively and emotionally, especially in open-ended
modeling tasks when students can use the context to support problem solving in a variety of
ways.
It is worth considering how contexts supported student engagement in some mathematics
tasks. Based on the results presented here, together with those in the next chapter, I argue that
contextualizing mathematics plays a role in student engagement in mathematics lessons by: a)
catching student interest, b) providing entry into and support in working on mathematical tasks,
and c) providing opportunities to anchor the instruction in shared experiences. Here, I simply
introduce this argument with a promise to discuss it in more depth after presenting results on the
enactment of the high- and low-engagement lessons.
First, the fact that students showed the highest levels of engagement in tasks with
meaningful contexts suggests that one reason contexts can promote student engagement in

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mathematics is that students enjoy or like the contexts themselves. As suggested in the literature,
the stories or scenarios can generate students’ interest by eliciting an emotional response or
relating to something familiar for students (Hidi & Renninger, 2006). Second, all highengagement lessons had contexts that were at least helpful in solving the task (except Southpoint
Lesson 2 with the extension of the Wumps context), suggesting that the more central the context
is in solving the problem, the greater the potential to support student engagement. As others have
posited (Carraher et al., 1985; Walkington et al., 2012), contexts can provide entry into and
support students’ work on mathematics tasks by encouraging informal strategies to make sense
of problems, with less likelihood they will immediately mathematize the scenario. The results of
this study indicate that this access to working on the task mediates the relationship between
meaningful contexts and student engagement. Third, students’ level of engagement seemed to be
more closely related to the purpose of the task and the role of the context than the type of
context. In the more engaging lessons, tasks were more open-ended and exploratory in nature
and the context was a central part of the question students were expected to answer. The tasks
were more about modeling than applying established mathematical ideas (Ness, Blum, &
Galbraith, 2007). In this way, the problem scenarios served to anchor the instruction, allowing
students to build understanding of new mathematical concepts on shared prior experiences and
knowledge (CTGV, 1992a, 1992b).
So far, I have focused on descriptions of tasks and their contexts as they appear in
curriculum materials to highlight the differences in the two sets of high- and low-engagement
lessons. In the next chapter, I continue to develop the argument presented here—that contexts
can promote student engagement catching student interest, providing entry into and support in
working on mathematical tasks, and providing opportunities to anchor the instruction—by

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considering the relationship between student engagement and contexts as enacted. Using
excerpts from the observed lessons with context-related contributions from teachers and students,
I consider in greater depth how attention to problem contexts supported student engagement in
these lessons.

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CHAPTER 5
STUDENT ENGAGEMENT AND CONTEXTS AS ENACTED
The purpose of this study was to explore the relationship between contextual tasks and
student motivation and engagement in learning mathematics. In the last chapter, I presented data
on the contexts as written in the lessons where students exhibited or reported high and low levels
of engagement. Those results suggested that contextualizing mathematics has potential to engage
students both cognitively and emotionally, especially in open-ended modeling tasks when
students can use the context to support problem solving in a variety of ways. In the preliminary
discussion at the end of that chapter, I introduced the argument that contextual tasks engage
students by eliciting genuine interest in the context itself, providing entry into and support in
solving the problem, and anchoring the instruction to provide students a shared experience on
which to build their understanding of the mathematical concepts.
To further explore and substantiate this argument, in this chapter I present results focused
on the enactment of contextual features of tasks in high- and low-engagement lessons, including
both teachers’ and students’ attention to contexts. Specifically, I considered when and how
teachers focused students’ attention on the contexts and what student engagement related to
context looked like in high- and low-engagement lessons. Identifying students’ context-related
engagement provided another perspective on the roles contexts tasks play in student engagement
in mathematics.
I found quantitative and qualitative differences in the way students and teachers attended
to contextual features of tasks between the high- and low-engagement lessons. The ways students
engaged in contextual tasks appeared to be related to the extent to which their teachers attended
to the context during lessons. Students drew on the context more in the high engagement lessons,

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and were more likely to connect the context to the main mathematical ideas in the lesson.
Teachers also paid more attention to contexts across the high-engagement lessons, and in more
ways (such as elaborating, clarifying, and positioning) than in the low-engagement lessons.
In the following sections, I first describe the frequency and nature of students’ contextrelated engagement, with illustrative examples of student engagement related to the context in
high- and low-engagement lessons. Then, I summarize the teachers’ attention to the context
across the nine lessons with descriptive statistics on the frequency and types of context-related
discussion. Finally, I consider the relationship between teachers’ and students’ attention to task
contexts, and present in greater detail two lessons with high student engagement and two lessons
with low engagement with representative or noteworthy examples of statements and questions
related to the context. The chapter closes with a full discussion of the results on the tasks,
instruction, and student engagement related to problem contexts.
Student Engagement Related to Context
When coding evidence of student engagement, I identified each event that was related to
task contexts. These events were any questions, statements, or exclamations about or using the
language of the problem scenario when working on a contextual task. Through this analysis, I
found that the extent to which students attended to the context varied across lessons and across
the two classes. There were 266 total instances of students’ context-related engagement across
the high- and low-engagement lessons. A summary of these counts, including whether the
references to contexts were evidence of cognitive or emotional engagement, are provided in
Table 10. I will present these descriptive statistics then consider student participation more
qualitatively, providing examples of context-related engagement and comparing the nature of
this engagement between high- and low-engagement lessons.

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Table 10
Summary of Context-Related Engagement by Lesson Phase
Lesson

Contextrelated
engagement

Percent
of total

Cognitive
engagement

Emotional
engagement

Launch

Explore

Summary

High-engagement lessons
PR 8
26
38.8
24
2
8
13
5
PR 9
62
46.6
30
32
5
55
2
PR 10
75
51.0
32
43
6
61
8
S2
3
5.4
3
0
0
1
2
S4
21
34.4
17
4
12
5
4
S7
34
34.3
32
2
0
29
4
Low-engagement lessons with contextual tasks
PR 11
29
38.2
23
7
8
17
4
S8
7
23.3
3
4
0
2
3
S 10
6
7.23
5
2
0
2
0
Note: The context-related evidence of engagement column includes references to prior contexts
when reviewing homework or other classroom activities that are not one of the three main phases
of the lesson.
What Does Context-Related Engagement Look Like?
Each instance of student engagement related to contexts involved students’ verbal
participation (including laughter) directly related to the nonmathematical part of a task. Recall
that a student raising their hand to offer a contribution to a class discussion in response to a
teacher prompt counted as evidence of emotional engagement regardless of whether they were
called on, as it showed they were invested in the task enough to voluntarily participate. But these
instances were not counted as context-related because there was no way to be sure whether a
potential contribution was linked to the context or not. So each example is a student statement or
question in which they explicitly used language drawn from the context or story in the task.
Using open coding, I identified three ways students used contextual language in the nine
high- and low-engagement lessons with contextual tasks:
•

Context only: Commenting on some aspect of the context or story itself without any
mathematical reference;
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•

Making sense of and solving tasks: Using language of the context as written or a
spontaneously-generated context while selecting a strategy or making sense of the task
requirements, while solving or making sense of the mathematical content, or while
checking solutions; and

•

Connecting to learning goals: Connecting the context to the core mathematical concept
or learning goals of the Investigation or lesson.

Each example of students’ context-related engagement in the nine lessons fell into one of these
three categories. In the following sections, I provide descriptions of each type of context-related
engagement, giving specific examples of focus students’ statements and questions to illustrate
engagement in the categories. I identify speakers by name for a clearer view of interactions in
these examples and to show engagement across the focus students—it was not limited to a few
students. Table 11 below breaks down counts of student engagement by type.
Commenting on context without connecting to mathematics. Each of the 73 examples
of student engagement in this first category was purely context-related. Students talked about
some aspect of the story itself without relating it to the mathematical content in that particular
moment in the discussion. For example, Jeff, a focus student at Pine River, commented on the
name of one of the fictional students solving the teacher mystery during the exploration phase of
Lesson 8. He said, “I think it’s kind of ironic that the sleuth’s name is Daphne. You know, like
from Scooby-Doo?” Several instances of this type of context-related engagement took place at
Pine River during the Wumps lessons. For instance, in Lesson 9, Lilly and McKenna had a
conversation about how the Wumps did not have arms, and why they should. In Lesson 10, Lilly
asked her group, “Why are we leaving out Bug and Glug? They’re my favorites!” Students
commented on contexts in the low-engagement lessons as well, such as when Kim read the

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problem in which Jack eats an enchilada every day and remarked, “Wow, that would get boring
after awhile…Ah, wow, that's a lot of calories! Look how much calories that is.” These
examples, like most others in this category, were examples of emotional engagement because
students showed they were excited about or otherwise involved in the problem, but their
contribution was not mathematical so they were not necessarily engaged in mathematical
thinking at that moment. Some instances where student engagement was purely context related
were coded as cognitive engagement, however (9 out of the 73). For example, while discussing
the puffin problem during Southpoint Lesson 8, Adelyn asked, “Is that so if one of them died
they would know if it was natural causes or a poacher or something?” Although Adelyn was
focusing only on the storyline and not solving the problem, she was asking about the purpose of
these types of mathematical tasks, and thus it was counted as cognitive engagement.
Using the context to make sense of and solve a problem. Students’ references to the
context in this second category, unlike the first, related to the mathematics in the task. A closer
analysis revealed that students’ references to the context could serve different purposes while
working on a task. First, in some lessons (10 times across the nine high- and low-engagement
lessons) students used the context to select a solution strategy or make sense of task
requirements. For example, when launching the Mystery Teacher task, Mrs. Meyer asked the
students to talk in their groups about how a photograph would help them solve the mystery.
Drayton and McKenna offered several ideas, including identifying the teacher by their wrinkles;
by looking at whether the teacher was particularly tall, short, or wide; or by finding the brand of
shoes the teacher is wearing then asking around at shoe stores. Most of these suggestions were
not helpful for solving the problem eventually posed about the height of the teacher, but the
problem scenario helped the students begin to think and talk about how to answer the questions

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in the task. When beginning the “Real-Life Equations” task in Pine River 11, the class attempted
together to interpret an equation. They debated whether the fictional boy or girl’s interpretation
was correct, and McKenna asked, “Wouldn’t the boy be right too, because one box equals six
eggs?” Here, McKenna aired a misconception, but showed that she was using her prior
knowledge about how eggs are packaged to make sense of the equation, which was necessary for
solving the upcoming apples and bananas task.
Second, in a few cases students used the context to check and interpret their numerical
solutions. In the summary discussion for Pine River Lesson 8, for example, Lilly said she had an
answer for how tall the teacher was but did not think it was right. She explained her strategy, and
when Mrs. Meyer asked how she knew it was inaccurate, Lilly responded, “Cause I somehow got
that she was 7.5 feet tall…that’s really tall.” Going back to the context to make sense of her
solution helped Lilly determine her answer was probably incorrect because it was unrealistic.
Third, the students’ referenced contexts while they were solving and communicating
about the mathematics. These events included use of appropriate units or other language from the
context during problem solving or talking about the mathematical work. It is important to note
that students used the context this way most frequently within this category (130 times over the
nine lessons), especially in tasks with contexts necessary for solving the task, because it was
nearly impossible not to reference the context when talking about those tasks. For example, the
following discussion took place while students at Southpoint worked on the Sascha’s bike trip
task:
Elijah: So the fastest, which one did he go the fastest?

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Lena: Leg 1? Leg 1—he went the fastest in Leg 1? Wait, no. All right, so in Leg 1, for
every hour he only goes 15 miles. In Leg 2, Leg 3 [points to her paper]…how do you
know if he’s going uphill or downhill?
Elijah: [Leaning over the desk to point to Emily’s paper] All right, so in one hour he only
travels 15 minutes.
Emily: Oh, so Leg 1 would be the slowest.
Elijah: Yeah, and do you go faster or slower when you’re going uphill?
Throughout this excerpt, the students use the language of the context (faster, slowest, legs, hours,
miles, uphill, downhill) to answer the questions posed and to communicate about the problem
and the mathematics. Further, they use prior experience with the scenario—riding bikes up and
downhill—to determine which legs of the trip were the slowest and fastest.
Students also made up their own contexts to support their thinking about tasks. For
instance, Emily spontaneously created an abstract “points” context to explain her thinking about
an integer operation problem in Southpoint Lesson 10. She said, “I just put down -3 minus 5—
okay, I just think about, like—if you have -3 points, and you have to subtract positive 5—I don't
know, I just did it in my head!” It seems she had done the problem but had trouble explaining her
thinking, so she tried (though with little success) to generate a context that would help her to
think about and justify her work.
Connecting the context with the lesson’s core mathematical concepts. The third way
students referenced the context was when making sense of or talking about the main
mathematical learning goals (see Table K1 for a description of the learning goals for all observed
lessons) in the problem or the investigation. Often, this type of engagement took place during the
summary phase in the lesson (12 times of the 48 total instances of connecting to core

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mathematical concepts). Students’ contributions related the context to the mathematics, but more
to generalized concepts than solving a particular task as in the previous category. For example,
near the end of the Mystery Teacher lesson, Felix stated, “That the scene in the picture is exactly
the same as how it would be in real life, just smaller,” evidence that he was starting to think
about what it means for two figures to be mathematically similar. In the first Wumps lesson, as
McKenna tried to identify Wumps and impostors, she shared the following idea: “The y goes up
and down. So for Glug, the y-axis—he’s tall, and the y-axis is three times, so he's going to be
taller, and Lug, he's going to be wider because it's 3x.” She was starting to make sense of how to
tell if two figures are similar by thinking about their coordinate rules. In the cola ad lesson,
Elijah explained to the class how he interpreted a comparison statement to determine the
relationship between the numbers of people who preferred each cola:
Well with the results it's kind of big, it's like 15 percent…you have to compare it with
something, that's kind of what we're doing. Otherwise it's just a number, if it's 5,713 to 1
that's a huge amount, but comparing it to 11,000 it's okay, not huge. (Emphasis added)
In each of these examples, students were linking the context to the main ideas of the lesson.
There were fewer of these examples across the nine lessons high- and low-engagement
lessons, likely because it takes longer for those concepts to develop than within a single lesson.
There were, however, references in these lessons to past contexts that suggest those contexts
were memorable for students and helped them make connections between ideas. For example,
the focus group students at Pine River had a short conversation during the Wumps task that they
related to a lesson in Shapes and Designs from several weeks earlier in which they had
determined the minimum number of facts you could tell someone to draw a unique triangle. Mrs.
Meyer had used a context in which the students were trying to text someone directions to draw a
triangle and needed to use as few pieces of information as possible.

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Jeff: There is such a thing as texting about triangles, but there is no such thing as a Mug
Wump video game!
Lilly: Why would you text about triangles?
Jeff: Well you might be texting about triangles from your tree stand but there's no such
thing as a Mug Wump video game!
While finishing the task later in the lesson, Jeff exclaimed, “Oh my gosh, I think we should
change texting about triangles to mailing triangles and texting about Mug Wumps. Because Mug
Wump's so much better!” It seems in this case that the texting storyline supported Jeff in linking
the triangle lesson to the texting context of the related task and helped him remember the lesson
and its main mathematical ideas (how to give instructions for generating unique triangles).
Comparing Context-Related Engagement in High- and Low-Engagement Lessons
Students’ attention to context varied both in quantity and nature between the high- and
low-engagement lessons with contextual tasks. As seen in Tables 10 and 11, there was more
evidence of student engagement related to problem contexts in the six high-engagement lessons
than the three low-engagement lessons with contextual tasks—on average, 35.1% of the total
counts of student engagement in high-engagement lessons were related to problem contexts,
compared to 22.9% in low-engagement lessons. This indicates that students were not just more
engaged in those six lessons, but also that the contexts played a more central role in the way
students make sense of tasks and mathematical ideas. No observable differences existed between
the two groups of lessons in terms of when (in which phase of the lesson—launch, explore, or
summary) students exhibited this engagement. Across the lessons, most of the context-related
engagement occurred during the explore phase, which makes sense because that is when students
had the most opportunities to share ideas or ask questions.

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Table 11 also shows, however, that the nature of students’ attention to context was
somewhat different across the two sets of lessons. Most notably, in the high-engagement lessons
(with the exception of Southpoint Lesson 2, which only had a peripheral context in one part of
the larger task), a significant portion of students’ context-related engagement connected to the
main learning goals of the lesson. In the three low-engagement lessons, there were no examples
of student engagement that connected to learning goals, though students did talk about the
context itself and used it to solve the tasks as needed. That is, although in low-engagement
lessons the focus students did engage in tasks to an extent, there was no evidence they were
engaging with the central ideas in tandem with contexts.
Table 11
Summary of Students’ Context-Related Engagement by Types
Lesson

Total Evidence
of Engagement

Context-related
evidence of
engagement

Category 1:
Context only

Category 2:
Connects to
mathematics in
task

Category 3:
Connects to
learning goals

High-engagement lessons
Pine River 8
67
26
3
14
9
Pine River 9
133
62
26
29
7
Pine River 10
147
75
29
28
18
Southpoint 2
56
3
0
3
0
Southpoint 4
61
21
6
13
2
Southpoint 7
99
34 (33)
0
22 (21)
12
Low-engagement lessons with contextual tasks
Pine River 11
76
29
3
26
0
Southpoint 8
30
7 (5)
5 (4)
2 (1)
0
Southpoint 10
83
6 (2)
1 (0)
5 (2)
0
Note: Numbers in parentheses represent the counts of evidence of student engagement in the core
task of the lesson.
Also, students paid relatively high attention to the context in Pine River Lesson 11, the
only low-engagement lesson with a single context necessary to solve the core problem of the
lesson. Several examples of their context-related engagement, however, were situations in which
students challenged the authenticity of the task—a task that was titled “Real-Life Equations.” For

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example, when students started working on the task, Lilly asked, “These are real-life equations,
but who would ever want to know this?” Her comment indicated her skepticism about the “reallife” aspect of the task, a type of negative attention to the problem scenario that did not occur in
the high-engagement lessons, even those without full authenticity (e.g., Wumps, cola
advertising).
In sum, students attended more to the contexts in high-engagement lessons than they did
in low-engagement lessons with contextual tasks, and were more likely to connect the problem
scenarios to the central mathematical concepts in the high-engagement lessons. It is difficult to
interpret students’ attention to the problem contexts, however, without considering how teachers
did or did not talk about the contexts in these lessons. Thus, I turn next to teachers’ attention to
contexts during enactment of contextual tasks in the high- and low-engagement lessons.
Themes in Teachers’ Enactment of Tasks in High- and Low-Engagement Lessons
Because teachers are “active designer[s] of curriculum” (Remillard, 2005, p. 214), I
anticipated that Mrs. Meyer and Ms. Pearson would modify tasks in the CMP texts at least in
part. As expected, they rarely implemented tasks exactly as written, and that included the
contextual features of tasks. The two teachers talked about and focused on the problem scenarios
in different ways across the observed lessons. In this section, I present descriptive statistics on
the teachers’ implementation of contextual tasks and the relationship between implementation
and students’ context-related engagement. I also preview some qualitative differences before
presenting specific cases of lesson enactment in the next section.
Attention to Context by Type and Lesson Phase
After identifying interaction segments in which teachers attended to context in the nine
contextual high- and low-engagement lessons, I coded each as positioning, elaborating,

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clarifying, referencing, or making a meta-level comment about the problem scenario. I counted
these events for each lesson, noting the lesson phase in which the events occurred. The results
are summarized in Table 12. Also, recall that descriptions of noteworthy aspects of the teachers’
implementation of tasks across the 28 lessons in the study can also be found in
Table K1.
Table 12
Summary of Teachers’ Attention to Context
Lesson

Total Counts of
Teacher
Attention to
Context

High-engagement lessons
Pine River 8
25
Launch
7
Explore
1
Summary
17
Pine River 9
31
Launch
23
Explore
3
Summary
5
Pine River 10
Launch
Explore
Summary

38

Southpoint 2
Launch
Explore
Summary

3

Southpoint 4
Launch
Explore
Summary

34

Southpoint 7
Launch
Explore
Summary

23 (17)

Reference

Clarify

Elaborate

Position

MetaLevel

13

6

1

3

1

0
1
13
16

4
8
3
5

32
3
8
27

1
0
2
5

4
0
0
2

0
0
4

0
0
1
2

5
0
0

2
0
0

0

0

0
0
2

0

0

0

0

2

10

5

0

0
0
3
17
10
1
6
18 (12)

2
3
12

4

4

3

0
0
1

4
0
0

3
8
21

0
0
3
27
1
7

6
0
0

2
0
0
4

0
3
9

9
0
1
0

5
0
0
1

2
0
2

Low-engagement lessons with contextual tasks
Pine River 11
23
17
3

121

0
0
0
1

2

0

1

Table 12 (cont’d)
Launch
Explore
Summary
Southpoint 8
Launch
Explore
Summary

16
1
6
18 (9)

13
1
3
(7)

0
0
9

3
0
0
(2)

7

0
0
2
(0)

0
0
1
(0)

(0)

2

Southpoint 10
6 (0)
2 (0)
0
2 (0)
1 (0)
1 (0)
Launch
0
Explore
0
Summary
0
Note: Numbers in parentheses represent the counts of teacher attention to context in the core task
of the lesson, including launch, explore, and summary phases. The total number includes
references to prior contexts while reviewing homework. All of Ms. Pearson’s references to
contexts in Lesson 10 took place when reviewing homework.
Mrs. Meyer and Ms. Pearson paid significant attention to task contexts in their respective
high-engagement lessons. The exception was Southpoint Lesson 2, but at that point in the unit
the Wumps context had already been introduced and explored in previous lessons and likely did
not need further explanation. On average, the teachers drew on contextual elements in 22
different interaction segments in the six lessons. These events spanned the lessons; that is, they
took place in all three lesson phases—launch, explore, and summary. The low numbers in the
explore phase make sense, as I only observed interactions between the teacher and the focus
group and the teacher’s monitoring during explorations was spread out between groups. Further,
in the high-engagement lessons, teachers used at least three types of questions and statements
related to the context. All but Southpoint Lesson 2 included referencing and clarifying, and at
least one of elaborating and positioning.
Because there were only three low-engagement lessons with contextual tasks (and only
two with contextual tasks that were the main lesson activity), it is difficult to identify consistent
patterns in their implementation. However, the average number of teaching moves that drew

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attention to the context in these three lessons was 16, lower than in the high-engagement lessons.
Mrs. Meyer had 23 teaching moves that referenced the context in Lesson 11, because although it
was inauthentic and without an actual storyline, the apples and bananas context was central and
necessary in completing the task. Still, this number was lower than her level of attention to
context in any of the high-engagement lessons. Moreover, in these lessons, there was less
diversity in the way Ms. Pearson and Mrs. Meyer attended to the context than in the highengagement lessons. In Southpoint Lesson 8, for instance, Ms. Pearson only drew on the context
in the summary phase, and in Lesson 10, she only talked about contexts when reviewing the
homework from the night before.
Relationship Between Teachers’ and Students’ Enactment
The numbers in Tables 10, 11, and 12 suggest association between the extent to which
the teachers and students attended to contextual elements of tasks. There is strong correlation
between the total counts of teacher attention to contexts and students’ context-related
engagement across the main tasks in the nine lessons (r = 0.819). The lesson phase in which
teachers attend to context may have some relationship with student engagement as well, with the
launch and exploration being more closely associated with students’ context-related engagement
(r = 0.746 and 0.779, respectively) than the summary (r = 0.595).
Qualitative Differences in Teachers’ Enactment of Contextual Tasks
The teachers’ enactment of contextual features of the tasks in high- and low-engagement
lessons also differed in other more qualitative ways not captured in the counts in Table 12. More
specifically, they tended to pay attention to contexts in high-engagement lessons in ways that
highlighted contexts as being relevant, important, or interesting. I noted three interesting features

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in teachers’ enactment of tasks with high student engagement, which I introduce here then
elaborate on in the descriptions of lesson cases that follow.
The first distinction was the teachers’ positioning students’ experiences relative to
contexts. For example, Ms. Pearson orchestrated an extensive discussion about ads students had
seen that compared two products using numerical information in the launch of Lesson 4 (with the
cola ad context). Several students had the opportunity to share personal stories, and Ms. Pearson
ended with a warning to be smart consumers. This discussion served to position students relative
to the advertising scenario and elaborate on the context as written, and was unlike any wholeclass discussion in the low-engagement lessons.
The second theme in lesson implementation of high-engagement lessons was centralizing
the role of contexts by opening up the problem, meaning teachers made the problem more openended, allowing for more solution strategies, than written in the text. In Pine River Lesson 8,
Mrs. Meyer implemented the Mystery Teacher task differently than as written in the CMP
curriculum. As I mentioned in Chapter 4, she used the opening question as the core task for the
lesson, which the authors refer to as a box question, an open-ended question that can be used in
lieu of the written problem. The actual problem in 1.1 used the same Mystery Club storyline, but
asked students to enlarge an image from a flyer to use on a poster and then compare angle
measures and segment lengths. So from the planning phase, Mrs. Meyer’s choice about enacting
the task made the storyline about the Mystery Club and teacher more central to the lesson. As
written, a teacher could have passed over the highly contextual opening questions altogether or
just asked the questions hypothetically rather than using them as the primary task in the lesson as
Mrs. Meyer did. Neither teacher increased the centrality of the context in problem solving to this
extent in a low-engagement lesson.

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Another difference was in teachers’ emotional response to tasks, with Mrs. Meyer and
Ms. Pearson showing more positive affect in high-engagement lessons. For instance, whereas
Mrs. Meyer exhibited a lot of positive emotion related to the context in the Wumps lesson, she
became frustrated (showed negative affect) in Lesson 11 when students brought up how
unrealistic the context was. I will explore these themes further in the next section by describing
the enactment of four lessons in detail.
Cases of Enactment of Contextual Tasks
The purpose of this section is to provide a clear picture of the differences in how teachers
and student enacted contextual tasks across lessons. Through four diverse lesson cases—one
from each class in each of the low- and high-engagement sets—I illustrate how teachers attended
to task contexts, providing examples of different types (e.g., referencing, elaborating,
positioning), and how this attention was related to student contributions in the enactment of these
tasks. I chose the first Wumps lesson in Pine River because the students showed the highest
levels of cognitive and emotional engagement. I chose the Southpoint lesson with Sascha’s bike
trip because it was a fairly typical task relative to the fourteen observed lessons at the site, and
the only high-engagement lesson that took place in the middle of a unit. Because Southpoint
Lesson 10 offered little to discuss in terms of enactment of contexts, I chose to describe the other
two low-engagement lessons with contextual tasks.
Pine River Lesson 9: “Let Me Tell You, You Will Not Forget the Wumps”
The first noteworthy characteristic of the enactment of this Wumps lesson was an
extensive build-up of the two contexts, with Mrs. Meyer using all five types of attention to the
context. She opened the lesson by showing the launch video provided by the publisher that
accompanies the problem. The video provided images of a computer game design program in

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which coordinates are used to design and animate a figure. The voiceover explained the context
in more detail. Using the video was an instructional choice that elaborated on and clarified the
computer programing context in a different way than setting it up verbally, since it provided a
shared experience with a visual representation.
Mrs. Meyer highlighted both contexts in the problem launch, but focused on the
computer game design storyline. In addition to using the video, she facilitated a discussion
positioning students relative to the context. She started the launch with the following:
Once you have the focus question on your paper, I want you to think about what your
favorite game is on your computer, your tablet, app on your phone or your parent's
phone. Or a video game, like you have an Xbox or whatever. So you're just thinking
about that. I want to know what some of your favorite games are.
Several students, including Drayton and Felix who generally showed low engagement, raised
their hands to share their favorite games. After reading the problem from the text, Mrs. Meyer
added,
Have you ever thought about that while you're playing those games? Like thought about
who was the programmer behind the scenes who put it all together? Can you imagine all
the things they have to enter into the computer to get those graphics to move the way you
want them to move?
This series of questions positioned students relative to and elaborated on the programming
context by emphasizing the human aspect of the game. Mrs. Meyer used these discussions to
focus students’ attention on the role of coordinates, stretching, and shrinking in computer game
programming.
Before students started exploring the task, Mrs. Meyer turned to the Wumps context and
linked it to the Zach and Marta storyline. She stated,
[Slyly] We have some impostors here, too. We have some people who want to be in the
Wump family, but they aren't similar. They are the impostors. So what you're going to be
doing today is working with several rules that Marta has put together for Zach and
playing with all these rules and coordinate points. And you are going to be finding out

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who is actually in the Wump family, or who is similar, and who are the impostors, who
are not similar.
Here, Mrs. Meyer highlighted the purpose of the task, elaborated on the context (“some people
who want to be in the Wump family”), and also made meta-level statements on the context by
giving explicit expectations for how to use the Wumps context to solve the problem. From that
point in the lesson, the Wumps context (not the programming context) was central in solving the
task. Mrs. Meyer referenced it throughout, asking students during the explore phase who were
Wumps or impostors and how they knew, and drawing on the context in the brief summarizing
discussion. Rather than stripping away the storyline, Mrs. Meyer continued to use the language
of characters, Wumps, and impostors when discussing the rules and shapes.
The second noteworthy feature of this lesson was the extent of students’ context-related
engagement. All of the focus students talked about and used the Wumps context during the work
on the task. There were 26 instances when the students talked about the context itself, separate
from the mathematics. While they plotted points and graphed the characters, they made frequent
statements and explanations about the figures, such as, “Oh, he looks cool” (Jeff), “He looks
funny!” (Lilly), and “He looks suspicious!” (Drayton). There were also 29 instances when focus
students used the language of the context to support their work on the task. For instance, Lilly
commented, “It [the impostor] has to be Glug, or nobody. Everybody else fits in.” McKenna
noted that one character’s eyes were in line with his nose, which allowed her to recognize a
mistake in her calculation of the coordinates. Throughout the lesson, the Wumps context was a
central part of students’ discussion and work.
The third striking aspect of Mrs. Meyer and her students’ enactment of the Wumps task
was strong positive affect related to the context. Mrs. Meyer showed significant positive

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emotional engagement in the task herself, specifically related to the Wumps storyline. When
introducing the task, she started with,
I have to tell you guys that this is one of my favorite problems. I really actually like the
whole book of Stretching and Shrinking to be honest. But this problem is just so much
fun! What you will notice is that Zach and Marta's computer program involves a family
called the Wumps…and let me tell you, you will not forget the Wumps.
This seemed to set the tone for the rest of the class period. As students started working on the
problem, McKenna said she had seen someone wearing a Wumps shirt, and Mrs. Meyer laughed
with students about the fact that she owns a Wumps t-shirt and planned to wear it the next day.
When a student asked if the figures would be in 3D, Mrs. Meyer responded, “you just wait,
they’re going to be pretty fancy.” These statements made it clear that she enjoyed this task, was
excited to implement it with the class, and believed the students would also enjoy and remember
the context.
The focus students also showed enthusiasm and excitement about the Wumps context. As
the examples above show, their talk was consistently positive as they talked about the characters
and tried to determine who was a Wump and who was an impostor. There were 32 examples of
context-related emotional engagement in the lesson, and all six students showed some level of
emotional engagement relative to the context. Their enthusiasm continued into the summary
discussion. McKenna was excited to discover (or come up with language for) the relationship
between Zug and Mug Wump, exclaiming, Zug looks like, just a, a blown-up Mug. They’re
similar!”
In sum, there were several interesting characteristics in the way Mrs. Meyer and her
students enacted the Wumps task. Mrs. Meyer spent several minutes of the task launch focused
on the computer programming and Wumps contexts. Then, students talked extensively about the
Wumps context throughout the lesson as they made sense of the task and the mathematics

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involved. They appeared genuinely excited to work on the task, which mirrored the excitement
Mrs. Meyer exhibited from the beginning of the launch through the end of the lesson.
Southpoint Lesson 7: Using the Bike Context to Make Sense of Unit Rates
Of all the high-engagement lessons, the bike trip task had the most realistic and adultoriented context. Unlike Mrs. Meyer’s implementation of the Wumps task, in this lesson, most of
Ms. Pearson’s attention to the context occurred in the summary phase. She primarily referenced
the context, and there were no instances of elaborating on or making meta-level comments
regarding the context. It is possible that Ms. Pearson thought the biking context was familiar
enough for students that it did not need further elaboration, or much attention in general.
Ms. Pearson’s discussion of the bike trip context in the launch served to clarify key
aspects of the context. For instance, she explained the meaning of the legs of the trip when she
said,
So when this says "Leg 1," leg refers to the part of his trip. So on the first part of his trip
he rode 5 miles in 20 minutes, then he stopped and wrote that down. Then he went an
additional eight miles and rode for another 24 minutes, stopped wrote that down. Then he
got on his bike again rode 20 more miles in 40 minutes stopped and wrote that down.
After students started working on the task, Ms. Pearson mainly referenced the context using key
language, such as appropriate units. For example, in the summarizing discussion Ms. Pearson
stated, “In part B, Leg 1 should have been 15 miles per hour and there were a variety of ways
you can get it,” and later asked, “Given that information then, when was he going the fastest?
Which Leg?” These references focused attention back on the storyline rather than just numerical
solutions.
Likewise, students frequently used contextual language to solve and communicate about
the mathematics of the problem during the exploration and summary. In particular, throughout
their work on the problem students used the appropriate terms and units drawn from the task—

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miles, hours, biking, and so on. Again, because the context was so entwined in the questions
posed (i.e., the context was central to problem solving) it makes sense that the class would attend
more to the context. In this task, all parts but the final graphing question required the use of units
of some kind. In fact, the focus students never talked about Sascha’s bike trip apart from the
mathematical ideas, and it was the lesson with the highest number of instances of students
connecting the context to the major idea of the lesson (exploring unit rates). For example, Emily
and Elijah engaged in the following discussion early in the exploration:
Emily: Hmm…don't you do like 20 divided by 5? 4. So that's four minutes per mile?
That's fast.
Elijah: I don't know…what'd you get for the first one for miles per hour?
Emily: Four.
Elijah: Four? Miles per hour? That would mean she would be biking fifteen minutes per
mile.
Later, in the summary, Ms. Pearson asked, “When was he going uphill?” Adelyn responded,
“Leg 1…because it’s the slowest and usually when you go up hill you go slower.” These
examples of the focus students’ engagement relative to the context are representative of the ways
they talked about and used the context throughout the lesson. They generally did not focus on the
biking storyline itself, but rather used it to support their mathematical work.
Pine River Lesson 11: Inauthenticity of Apples and Bananas
The first noteworthy characteristic of this lesson was the relatively high attention to the
context by the class, particularly in close relationship to the mathematics. Like the enactment of
the bike trip task at Southpoint, in the enactment of the apples and bananas task at Pine River,
Mrs. Meyer and the focus students drew frequently on the context throughout the lesson by using

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contextual language while solving the task. This makes sense, as the context (the number and
prices of apples and bananas, not the grocery shopping storyline) was central in correctly
matching expressions and equations with different scenarios. Most of Mrs. Meyer’s attention to
the context (17 instances) involved referencing the eggs, apples, and bananas in discussions
about the equations and expressions. For example, after a student suggested during the launch
that there were six full boxes of eggs, Mrs. Meyer said, “Six [eggs] per box. So you’re telling me
this number, so let’s try that. If we have three boxes of eggs, how many eggs do we have total?”
Or, in the summary, she asked, “Okay, so twice as many apples as bananas. So here, you’re
adding them together?” These statements and questions referencing the food items while
interpreting equations and expressions are representative of the kind of context references Mrs.
Meyer used during the lesson.
The students also used these types of references throughout the lesson, particularly during
their work in small groups. Though they reported low engagement, the level of their engagement
related to the context nearly matched that in the high-engagement lessons. Nearly all of these
instances were connected to the mathematics in the task as they interpreted various statements.
Consider the following discussion between Jeff and Lilly as they worked through the problem (it
is worth noting that Drayton, who was also in their group of three, did not have any relevant
contribution during the work time):
Jeff: Let me see this equation. Total cost in dollars—x is two times y, yup. There are half
as many apples as bananas, is that statement correct? [He pauses while thinking,
murmuring.] No, half as many—
Lilly and Jeff: Yes!

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Lilly: These two go here. Do these two go here, do you think?
Jeff: Drayton, figure out where these two go.
Drayton: No, that's Lilly's job.
Jeff: You need to help, Drayton!
Lilly: He's gluing stuff down. It goes right here, it goes right here!
Jeff: … Now, y = 2x and x = 2y. Half—
Lilly: This would be...they're the same numbers. Here.
Jeff: They are the same number of apples for this one.
Lilly: Is this the same cost, then?
Jeff: Yeah, the same cost. Same number of apples.
Lilly: Ba-na-na!!
Jeff : And bananas. [Pause, the two are working quietly.] I'm pretty sure we're done now!
Lilly: I'm not done writing. There's one more box down here.
Jeff: That's the cost of bananas and apples are the same. Apples and bananas are the
same cost.
Lilly: [Writing] Ba-na-nas!!
Here, Jeff and Lilly make several references to apples, bananas, and cost, all contextual features
of the task. Twice, Lilly says “banana” in a singsong voice as she writes. Most of the pair’s
discussion as they worked drew on the contextual language in this way.
The second noteworthy feature of the enactment of the task was that the context seemed
to create a distraction for the students in the class and drew negative attention that I had not
observed in the high-engagement lessons. Most of the teacher and students’ attention to the
context that was not simply referencing it during problem solving addressed the inauthenticity of

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the problem scenario. Recall, for example, that early in the lesson Lilly commented, “These are
real-life equations, but who would ever want to know this?” When a student skeptically asked
about a girl buying individual eggs, Mrs. Meyer clarified that yes, “she needed two extra, and she
could buy those individual.”
Later, as Mrs. Meyer was working through a problem with the whole class, they came to
an accurate but unrealistic solution.
Mrs. Meyer: Let's say the apple costs 5 dollars.
Student 1: That's an expensive apple!
Mrs. Meyer: I know it. It must be a honey crisp, those are really expensive.
Student 2: Or it's made of gold. [Students laugh.]
Mrs. Meyer: Yeah!
Drayton: What about a Macintosh one?
Mrs. Meyer: So if the apple is 5 dollars, how much is a banana?
Lilly (and others): Ten!
Student 3: Ten dollars?!
Jeff: That's a lot of money for a banana! [Students laugh.]
Mrs. Meyer: It must be delicious.
Jeff: If it's not delicious I'm suing the company. [Students started to laugh across the
class.]
Mrs. Meyer: Guys? Obviously, we're just trying to work with some friendly numbers
here.
When a student commented on the five-dollar apple, Mrs. Meyer elaborated on the context by
suggesting it was a honey crisp apple. At the end, she made a meta-level comment when she

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acknowledged to students that the situation was unrealistic and inauthentic, but emphasized the
point was the mathematics. This conversation drew attention away from the mathematical idea
Mrs. Meyer was trying to address. She had to pull students back into the task and explain the
unrealistic values in the problem. In sum, the class did use the context throughout this lesson, but
it was primarily when referencing units while solving and discussing the task. When they talked
about the context itself, it was primarily to challenge the authenticity and purpose of the
problem.
Southpoint Lesson 8: Lots of Contexts, Little Engagement
Recall that the core task in this lesson had several different contexts, including an
airplane in flight, a man named Jack who ate enchiladas every day for a year, and student council
elections in a middle school. Regardless of the range of problem scenarios in this lowengagement lesson, Ms. Pearson and the focus students attended minimally to the contexts
overall. Ms. Pearson only talked about these contexts nine times during the lesson. Interestingly,
all of them took place during the summary phase. She did not clarify or highlight the different
problem scenarios in the launch or as she worked with the focus group during the exploration. Of
the nine events in the summary, two were clarifying and seven were referencing. The
clarification was for the student council context, as some students had a difficult time
understanding why there should not be equal numbers of students from each grade. Ms. Pearson
explained, “When it says fairly, that doesn't necessarily mean equal. You have more seventh
graders in this case so you should have more seventh graders on the 35 person committee.” She
continued, saying, “They want the student council to represent the student body. So they want
more seventh graders because that makes it fair. Fair doesn't mean even though and I think you're
thinking even…. I should have the same number of 6th, 7th, and 8th graders.” These statements

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are noteworthy because they are the only instances across the nine lessons in which a teacher
addressed student confusion about the problem scenario in the summary, after students should
have finished the task. In her references to the context, Ms. Pearson posed questions such as, “So
how many miles?, ”How many calories for 240 enchiladas?,” and “In [question] B5 when it says
how many calories did he eat per day, what type of number are you finding?” These references
drew on the relevant units and other contextual language.
All but one instance of the focus students’ context-related engagement during the main
task involved the part of the problem with Jack eating enchiladas at work. These were cases in
which the students commented on the context itself during the exploration and summary without
necessarily connecting it to the relevant mathematical ideas. Kim and Emily both commented
that Jack consumed a lot of calories, and Jacob announced, “You would be fat!” Kim also
commented about eating the same thing every day, saying, “wow, that would get boring after
awhile.” So although this particular problem scenario (eating enchiladas) seemed to catch their
interest, it was not in relationship to the ideas of ratio and proportion they were using, and the
contexts in the task did little to support their overall engagement in the mathematics.
Summary, and a Note on Comparisons Across Classrooms
This study was not intended to compare classrooms, but I would like to highlight some
differences in the enactment of contextual tasks across the two sites. First, there were lower
levels of engagement overall in Southpoint. I am not particularly surprised by the lower count of
evidence of student engagement at Southpoint, because the small groups only included two to
three students to observe during small group work versus all six focus students contributing to
discussions in the small group at Pine River. Further, Ms. Pearson used more whole-group
discussion (when individual students have fewer opportunities to contribute) than Mrs. Meyer.

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However, the focus students at Southpoint also attended to the context less than the
students at Pine River—for instance, their average percent of context-related engagement in the
high-engagement lessons was 24.7%, compared with 45.5% at Pine River. Also, they did not
exhibit the high levels of emotional engagement as in Pine River (see Table 10), and were less
likely to connect the context with core mathematical learning goals (see Table 11). Finally, Ms.
Pearson attended to the context less overall than Ms. Meyer.
This suggests that teachers, or possibly the mathematical culture in a school, may play an
important role in setting the tone when it comes to attending to the context in classroom
discussions and while working on tasks. The idea that teachers might influence the extent and
nature of students’ context-related engagement is supported by the results presented in this
chapter, as I found a strong association between how much the teachers and students talked about
the contexts and used them when solving problems. Also, in the individual lessons described
above, the nature of student engagement around the contexts often reflected the teacher’s
attention to the contexts. Specifically, students seemed to follow the teacher’s lead in terms of
how close they stayed to the context when exploring mathematical ideas and their emotional
response to the contexts themselves. I explore these relationships further in the discussion.
Discussion: Multiple Roles of Contexts in Student Engagement
The results in this chapter addressed the second research question regarding
characteristics of the enactment of contextual tasks in lessons during which students showed
particularly high and low levels of engagement. I found that levels of student engagement in
contextual tasks were related to the extent to which their teachers attended to the context during
lessons. Students drew on the context more in the high engagement lessons, and were more
likely to connect the context to the main mathematical ideas in the lesson. Teachers also paid

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more attention to contexts across the high-engagement lessons, and in more ways (elaborating,
clarifying, and so on) than in the low-engagement lessons.
The results in Chapter 4 suggested that the contextual features of tasks as written have
potential to engage students both cognitively and emotionally. I now add that this potential is
heightened when the teacher maintains a focus on the context throughout the lesson in varied
ways. In that chapter I also introduced an argument to unpack how contexts can support student
engagement: Contextual tasks can engage students by eliciting genuine interest in the context
itself, providing entry into and support in solving the problem, and anchoring the instruction to
provide students a shared experience on which to build their understanding of the mathematical
concepts. Now, I explore this argument further and consider how teachers’ instructional practices
can enhance these three roles of contexts in promoting student engagement.
Contexts as Likeable and Interesting
The level of attention to context students exhibited in the high-engagement lessons
suggests that one reason contexts might promote student engagement in tasks is that students find
genuine enjoyment in the contexts themselves. As suggested in interest theory literature (e.g.,
Hidi & Renninger, 2006; Mitchell, 1993), the stories or scenarios can generate students’ interest
by providing an element of surprise, personal relevance or familiarity with the situation, or
identification with characters. The results on students’ context-related engagement support this
argument, since students frequently talked about the contexts apart from the mathematics when
making sense of and working on tasks. They even took note of the contexts in some of the lowengagement lessons, though as in Pine River Lesson 11, in some cases it was to draw attention to
particularly unrealistic aspects of the context (and away from the mathematical discussion).
Examples of evidence of emotional engagement related to problem contexts are particularly

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salient, since early phases of interest development are characterized by both cognitive focus and
positive feelings toward an activity (Hidi & Renninger, 2006).
The results also highlight the influence of teacher affect related to context on students’
engagement in tasks. Students’ interest in the contexts seemed to be supported in part by the
teachers’ enthusiasm related to context. This finding agrees with extant research on motivating
instructional moves (Stipek et al., 1998; Turner & Meyer, 2009) that found a relationship
between the classroom teacher’s positive affect and enthusiasm and increased student
engagement and motivation to learn mathematics. Likewise, evidence from this study indicates
that positive affect is related to increased student engagement, including within individual
lessons relative to specific contextual scenarios. In two of the lessons with the highest levels of
engagement (the Wumps lessons), Mrs. Meyer showed particularly positive feelings. She
commented, for example, that the Wumps lessons are her favorite lessons, that the Wumps
would be unforgettable for students, and that she was excited to wear her Wumps t-shirt. In so
doing, she exhibited clear and consistent enthusiasm for the lessons, and specifically for the
context itself.
Contexts Provide Entry Into and Support in Problem Solving
Across the nine focus lessons, the most frequent form of students’ attention to context
was connecting the problem scenario to the mathematics in the task as they made sense of and
solved the task. As discussed in the last chapter, the extent to which students engaged with the
context related to how central the context was in problem solving in both the high- or lowengagement. For example, in the apples and bananas task, the context was central and a large
portion of student engagement was related to the context even though their overall engagement

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was low. Or in the high-engagement Wumps lesson at Southpoint, the context was peripheral to
problem solving and students’ attention to the context was low.
Thus, the second way contexts might promote student engagement is by providing entry
into and support in solving mathematical problems, especially when the context was central in
solving the task. By entry, I mean students have a way to start the problem even if they are
unfamiliar with or unsure about the mathematics required. The storyline in the tasks gave
students something to talk about as they started thinking about and working on the task. For
example, Drayton and Felix, two students who reported low motivation to learn, actively
participated in the launch discussion in the first Wumps lesson because they took a personal
interest in the context (computer games). Or in the Mystery Teacher lesson, students suggested
ideas for identifying the teacher that were mostly unrelated to the mathematics in the task but
gave students initial ideas about what to look for in solutions. Consistent with other literature on
students’ performance on contextual problems (Carraher et al., 1985; Koedinger & Nathan,
2004; Walkington et al., 2012), the storylines of the tasks also supported students’ problem
solving. This role the context played in student engagement took different forms, including
assisting with the selection of solving strategies, providing language for students to use as they
discussed their work with others, and offering a way for students to check the reasonableness of
their solutions.
As with their affective response to problem contexts, the ways in which teachers enacted
the contextual features of tasks was also related to student engagement. Across the nine highand low-engagement lessons with contextual elements, a strong association existed between the
extent of teacher attention to context and the percent of student engagement related to context.
This finding suggests that if and how teachers emphasize the contextual elements of a problem

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relates to students’ engagement and attention to context. It seems teachers helped establish norms
for emotional and cognitive engagement when it came to the contexts, modeling the extent to
which and how students should attend to the problem scenario when solving and discussing
contextual tasks.
Literature on mathematics and motivation argues teachers’ instructional practices can
influence student engagement and motivation, and that these are malleable student characteristics
(Middleton & Spanias, 1999; Middleton & Jansen, 2011; Stipek et al., 1998; Turner & Meyer,
2009). The results of this study on the relationship between teachers’ and students’ enactment of
contextual tasks agree with this argument. Further, the results add to the literature by showing
that teaching moves not only influence student motivation and beliefs about mathematics
generally, but that attention to context can influence the extent of student engagement in specific
tasks and the way students draw on the context in problem solving and mathematical discussions.
Contexts as Anchors for Instruction
Finally, I argue that contexts anchored instruction and student learning for some highengagement lessons, and that teachers’ implementation of contextual tasks can augment this role.
The Cognition and Technology Group (CTGV) at Vanderbilt University define anchored
instruction as the use of lessons with meaningful—realistic and interesting—problem-solving
contexts, in which students can use prior knowledge and experiences to understand the problem
scenario and build on that knowledge as they gain relevant understanding of new concepts
(CTGV, 1992a,b). Anchors can be visual or textual.
The fact that students connected problem scenarios to the core mathematical ideas in
high-engagement lessons provides evidence that the contexts helped anchor student
understanding of concepts. Studies have shown that anchored instruction can lead to

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improvement in students’ performance on solving contextual problems, use of problem-solving
strategies, ability to transfer knowledge to new situations, and attitudes toward mathematics
(Bottge, Rueda, Serlin, Hung, & Kwon, 2007; CTGV, 1992b; Rivet & Krajcik, 2008). For
instance, one study found anchored instruction in mathematics promoted student achievement for
middle school students, including students with learning disabilities (Bottge et al., 2007). The
authors posited that anchors allowed students to “see” problems, and provided multiple elements
(text, multimedia, visual representations) that interacted to support students—particularly those
with learning disabilities—in solving problems.
The results also suggest that teachers’ enactment can enhance the role of contexts in
student engagement, allowing them to be anchors for students’ understanding of specific content.
When Mrs. Meyers and Ms. Pearson attended to contexts in a variety of ways across a lesson,
particularly in the launch and summary, they made an explicit effort to create a shared
experience for students to draw on as they made sense of the mathematics in the task. Often, they
called attention to students’ prior everyday experiences. The discussions they led about video
games (Pine River Lesson 9) and advertisements (Southpoint Lesson 7) are the best examples of
the way they drew on students’ prior knowledge and experiences through the contexts to help
students make sense of tasks and their respective purposes. Further, some questions posed by the
teachers explicitly encouraged students to connect contexts with the main mathematical learning
goals in a lesson. For example, by asking Lilly, “why is [Bug] so skinny?,” Mrs. Meyer
encouraged Lilly to consider the character’s coordinates and the role they played in his size and
shape, which related to the lesson’s core concept of mathematical similarity.
These findings on the influence of teachers’ instructional practices involving contexts on
student engagement agree with extant literature. Results of a large-scale study in CMP

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classrooms designed to describe the relationship between task set-up and student opportunities to
learn mathematics in summarizing whole-class discussion indicated that certain aspects of a task
launch might relate to higher quality student engagement in the summary phase (Jackson,
Garrison, Wilson, Gibbons, & Shahan, 2013). Two of the aspects of setup the researchers studied
were discussion of key contextual features of the task scenario and development of common
"taken-as-shared" language to describe contextual features, mathematical ideas and relationships,
and other vocabulary. These aspects of focus were hypothesized to help students understand
problem scenarios and align with the framework used in this study to code implementation of
contexts (Wernet, 2011).
Jackson et al. (2013) posited, “students were more likely to make connections to one
another's ideas and to provide conceptual evidence for their reasoning in the whole-class
discussion when taken-as-shared understanding of the contextual features of the problem-solving
scenario was established in the setup" (p. 677). Because making connections to each other’s
ideas and justifying their reasoning using mathematical concepts are fine-grained examples of
student engagement, their findings suggest that attending to the context leads to more highquality engagement in the summary. My study adds to this by relating both the teacher’s and
students’ attention to context with cognitive and emotional engagement across entire lessons.
Whereas Jackson et al.’s (2013) study did not consider the explore phase of the lesson, my
results suggest that higher-quality student engagement in the summary may be the result of
higher-quality engagement in the task itself—that is, how students worked on the task during the
explore phase may mediate the relationship between attention to contextual features in the setup
and student engagement with mathematical concepts in the summarizing whole-class discussion.
Further, the results presented in this chapter suggest there may be more to the relationship

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between attending to context and student engagement than just making sure everyone
understands and/or is familiar with the problem scenario to promote equity of learning
experience and opportunities. Discussing and clarifying the contexts may promote emotional
engagement as well, and centralize the context to anchor students’ work on tasks.
Considering Alternate Explanations and Other Factors Influencing Engagement
There were certainly other forces at play influencing student engagement in the two
classrooms, and other possible reasons why students engaged more in some tasks than others.
For example, there is an ebb and flow in student engagement over a week and a school year,
though patterns in engagement played out differently across the two classrooms. There is also the
question of novelty of certain tasks and their implementation, a factor known to promote student
engagement (Mitchell & Carbone, 2011). Novelty alone cannot explain student engagement in
the lessons observed in this study, however, as certain novel tasks such as the apples and bananas
task at Pine River did not promote high levels of student engagement, and all of the highengagement lessons were quite typical of CMP lessons.
The content students were learning likely played a role. All of the high-engagement
lessons at Pine River came from the unit Stretching and Shrinking, while none of the highengagement lessons in either class came from Accentuate the Negative. The role content plays in
student engagement is likely related to the open-endedness of tasks, another factor that seemed to
influence student engagement. The high-engagement lessons tended toward those with tasks that
allowed for multiple solution strategies and different possible solutions. Tasks in Accentuate the
Negative, however, generally had a single correct answer with fewer reasonable strategies. Tasks
without a clear-cut solution path allowed students more autonomy and choice, which have been

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shown to support student engagement and motivation (Engle & Conant, 2002; Mitchell &
Carbone, 2011; Skinner et al., 2008; Turner & Meyer, 2009).
Finally, I acknowledge that one limitation of the study is the descriptive nature of the
findings—specifically, I described the characteristics of contexts as written and as enacted by
teachers and students in high- and low-engagement lessons. Though the evidence suggests
relationships between these factors, I make no claims about causality or directionality. For
instance, it is possible that a teacher can emphasize a task’s context, or a context can be central to
problem solving and meaningful for students, but still not promote student engagement. It would
be necessary to analyze more lessons to further explore these types of claims.
Conclusion
It would be reasonable to ask, Do we want students to be talking about problem contexts
in mathematics class? What value is there in spending class time just talking about the problem
scenario? How much does it add to or take away from a focus on the mathematics? These
questions are part of the myriad of decisions teachers make in planning and enacting a lesson.
True, in a few cases students’ discussion about aspects of problem contexts temporarily led them
off track mathematically. Yet, the relationship between high student engagement, contextual
tasks, and teachers’ attention to contexts indicates that talking about contexts can be worthwhile
in promoting student effort and enjoyment. The fact that students were cognitively engaged and
reported greater effort in the high-engagement lessons with contextual tasks suggests that the
contexts, and time spent specifically addressing them, supported students in solving problems
and making connections to mathematical ideas. Over time, these positive experiences with
mathematics may lead to greater motivation to learn. In the next chapter, I turn to how these
motivational factors relate to student engagement relative to problem contexts.

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CHAPTER 6
THE RELATIONSHIP BETWEEN MOTIVATION TO LEARN MATHEMATICS AND
STUDENTS’ RESPONSES TO CONTEXTUAL TASKS
In this chapter, I turn from characterizing contexts as written and enacted in high- and
low-engagement lessons and focus on students’ experiences with contextual tasks more
generally, based on what they reported in class observations, surveys, and interviews about
particular lessons and problems. In doing so, I discuss how student motivation to learn as
assessed through their initial surveys related to their response to contexts. This addresses the
third research question: How does student motivation to learn mathematics (e.g., valuing content,
enjoyment of task, alignment with goals) relate to student engagement relative to the contextual
tasks used in class? This question was driven by the fact that, anecdotally, students avoid
traditional story problems, which often come at the end of each homework problem set and are
considered to be the hardest problems for students to do (Nathan & Koedinger, 2000a, 200b).
Yet, contexts are also believed to promote student engagement by representing everyday
scenarios where mathematics is needed or useful (e.g., Mitchell & Carbone, 2011). Further, this
question addresses a call to research “how individual student differences…moderate the
relationship between task characteristics and engagement” (Fredericks et al., 2004, p. 79).
Motivation and engagement—and the relationship between them—are complex
constructs. Thus, I pursued operationalized research questions. To explore how students’
underlying motivation to learn influences how they respond to contextual contexts, I considered
the question, How do students who reported different degrees of motivation to learn mathematics
engage with and talk about problem contexts?, To consider the other direction of the relationship
regarding how engaging with contextual tasks might influence students’ motivation-related
beliefs, I asked, Do problem contexts support students beliefs about their success in the task and

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the value of the content? (See Figure 7 for a visual representation of these operationalized
questions). To answer these questions, I analyzed students’ individual responses to questions on
lesson surveys (Appendix D) and interviews (Appendix F) as well as any comments about tasks
that students made in class. I coded statements related to their motivation to learn the
mathematics using expectancy-value theory, described in Chapter 3 and summarized in the
following section. Here, engagement is not based on what the teacher or I observed but on
students’ reactions to and reflections on lessons and tasks.
I found that the aspects of tasks students attended to (including contexts) when reflecting
on the value of mathematical content and their experiences in lessons was related to their
underlying motivation to learn mathematics. Trends across groups of students, however, indicate
that task contexts play little role in promoting students’ valuing of mathematics or beliefs that
they can be successful on a task. Based on these findings and those in previous chapters, I argue
that contexts in the mathematics tasks studied have potential to engage students in particular
lessons or to serve as a cognitive support for students’ problem solving, but do not necessarily
have the same potential to motivate students to learn. Further, students may respond to contexts
differently than intended by curriculum developers and teachers who implement the tasks.
The first part of this chapter presents two major themes in the findings across the whole
group of focus students. To illustrate these themes, the second part focuses on three students who
represent diverse beliefs in terms of motivation to learn mathematics and what they
communicated about their classroom experiences with contextual and noncontextual tasks. I end
with a discussion of the findings, highlighting the potential influence of contextual tasks on
student motivation and engagement in CMP classrooms.

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How do different groups of students describe their experience
working on contextual tasks?

Students’
Motivation
to Learn
Mathematics

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Do contextual tasks support students’ expectation of success?
Do contextual tasks support students’ valuing of the content?
Figure 7. Operationalized questions guiding analysis for research question 3. I conceptualized
the relationship between a student’s motivation to learn mathematics and their responses to
contextual tasks to be bidirectional, with questions to guide each aspect of the relationship.
Theoretical Background
Recall that motivation to learn is the “tendenc[y] to find academic activities meaningful
and worthwhile and to try to get the intended learning benefits from them” (Brophy, 2004, p. 16).
Motivation more generally is a construct used to explain the reasons underlying one’s behavior
(Brophy, 2004; Middleton & Jansen, 2011) and is largely non-observable and relatively enduring
(Skinner et al., 2008). It is different than engagement, which is observable and can change
quickly with changes in, for example, classroom activities. Thus, in this study I focused on
engagement as students’ reported and observed levels of effort and enjoyment in mathematics
lessons. My investigation was guided more broadly by expectancy-value theory, which I describe
next along with how I used it to analyze data to address the third research question.

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Expectancy-Value Theory
Like other motivation theories, expectancy-value theory explains a person’s achievement
motivation—or their choice to engage in an activity, their persistence, and their performance—as
being a product of their level of expectations of success and the value they place on the reward
for success (Brophy, 2004; Wigfield & Eccles, 2000). In this study, the activity of focus is
learning through mathematics tasks in 7th-grade classrooms. Thus, I considered students’
motivation to learn mathematics—their expectation of success solving tasks and how they valued
mathematical knowledge.
The two main constructs in the theory are expectancy, or students’ beliefs about their
ability to successfully complete mathematics tasks, and value, or how much they value the
knowledge gained through these tasks. These are subjective values, because different people
assign different value to the same activity (Wigfield, Tonks, & Klauda, 2009). There are four
different aspects of value. The first is attainment value, or the importance of doing well on a task.
Attainment value relates to one’s identity, as “tasks are important when individuals view them as
central to their own sense of themselves, or allow them to express or confirm important aspects
of self” (Wigfield et al., 2009, p. 58). The second is intrinsic value, or students’ level of
enjoyment when working on mathematics tasks. The third is utility value, or students’ perception
of a task’s usefulness and its correspondence with future plans. Finally, cost refers to the level of
effort and sacrifice required to be successful.
Taken together, these constructs allowed me to characterize how students reacted to
different tasks. Students regularly communicated these motivation-related beliefs through their
comments in class and responses to survey and interview prompts. Expressing confidence in
their ability to complete a task, enjoyment while working on the task, or belief in the importance

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of the embedded mathematics indicated a positive response to the task. Expressing confusion,
boredom, a sense that the mathematics was useless, or that the cost of engaging in a task was too
high indicated a negative response. Looking across tasks and identifying patterns in how
different students responded to contextual and noncontextual tasks allowed me address the
research question by relating student motivation to learn and their experiences with task contexts.
Analytical Framework
To explore the relationship between students’ motivation to learn mathematics and how
they described their experiences working on mathematics tasks (particularly contextual tasks), I
considered their responses to the open-ended questions on the lesson-specific surveys across all
the lessons in each class, students’ contributions in focus group interviews (Table F1 in
Appendix F describes the lessons preceding interviews), and comments made while working on
mathematics in class. These comments included complete phrases in which students explicitly
linked their perceived engagement in a lesson to aspects of the tasks they had worked on in the
lesson. Table 13 provides a summary of how I identified students’ statements using expectancyvalue theory. For the final stages of analysis, I considered patterns in lessons to which each
student responded especially positively or negatively, identifying these lessons based on
students’ lesson-specific self-reports of engagement as well as my observations of their
engagement. Thus, from this point when I refer to how students “responded to,” “reacted to,” or
“reflected on” tasks and lessons, I am drawing on students’ explicit comments about tasks and
their engagement in them.

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Table 13
Analytic Framework for Students’ Motivation-Related Statements
Expectancy-Value Theory
Nature of Student Comments and
Construct
Responses
Ability beliefs (focused on current A task being easy, hard, confusing, simple,
task)
straightforward, challenging
Expectation of success (focused
on upcoming tasks, immediate or
long-term)

Feelings of competence or confidence, being
challenged, struggling, comments on what
kind of math student they are

Intrinsic value (enjoyment)

A task being fun, exciting, boring, a lesson
going by quickly

Attainment value

Necessity of learning the content in a task,
the importance of doing well for a future
math class or to get a good grade

Utility value

The content being useful (or not useful)
outside of school, a task’s relevance, how
closely a task relates to their interests

Cost

Being tired, losing social status, fear of being
wrong, rather be doing something else, a task
is too much work, stressful
As discussed in Methods, students’ motivation to learn mathematics was determined from

the initial motivation beliefs surveys administered at the beginning of the study. Felix, Lena, and
Emily were in the low motivation group; Drayton, Jacob, Adelyn, Lilly, and Sophia were in the
neutral/mixed motivation group because they gave neutral responses to the initial survey
questions on motivation beliefs, or had high expectations for success but did not value
mathematics or vice versa; and Elijah, Jeff, Kim, and McKenna were in the high motivation
group. Again, these groups were determined from early student reports and not what I observed
during the study period. The survey results, however, were supported by other evidence as
students talked about their perceived mathematics ability, how much they enjoyed mathematics,
and so on. Some of the unprompted statements I heard during observations and interviews
included:
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•

Felix telling his peers during class that he goes to math lab, which he described as being
“for people who aren’t very good at math.”

•

Drayton commenting in an interview, “I feel like I’m not that good of a mathematician.”

•

Adelyn commenting, “I usually like math class, though. I used to be good at math class,
now I'm just…I didn’t even try, because it was hard.”

•

Sophia saying in an interview, “I hate being wrong. I feel embarrassed in front of the
whole class.” Later, she said she doesn’t like math, except for some days when she feels
smart.

•

Elijah remarking to Jacob, “Math is my friend!”

•

McKenna commenting, “I kind of like math now, this year it's easier.”

The beliefs students communicated about the usefulness and benefits of learning content in
specific lessons also reflected their initial survey results, as presented in the forthcoming results.
The results in this chapter are organized into two parts. In Part 1, I discuss trends across
focus group students with diverse motivation to learn. In Part 2, I discuss illustrative cases of
three students—Felix, Elijah, and Sophia—who represent different levels of motivation to learn
mathematics. Next, I describe the central themes in the data on how students who expressed
diverse motivation to learn mathematics responded to different types of tasks.
Results Part 1: Trends in Focus Group Students’ Responses to Tasks
The focus students regularly and openly expressed beliefs about their expectations for
success on tasks and the extent to which they valued the content of lessons—both spontaneously
while working in class and when prompted in surveys and interviews. My analysis focused on
how these responses related to the contexts of tasks and students’ underlying motivation to learn.
The findings suggest students’ experiences with certain tasks and their contexts may be more

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strongly related to their underlying motivation to learn (whether they are low motivation, high
motivation, or somewhere between) than to the nature of the contexts. I identified two main
themes in the data. The first related to how students discussed the value of mathematical content
in the lesson interviews, and the second related more broadly to what different students attended
to when reflecting on tasks.
Contextual Tasks and Beliefs About the Value of Mathematical Content
The first trend in the data addressed the question, “Do contextual tasks support
students’ valuing of mathematics content?” as well as how students’ motivation to learn related
to what they attended to in contextual tasks. Most of the relevant data came from students’
responses to the interview question, “How beneficial (or useful) do you think the stuff you
learned today will be for you, either in this class, future classes, or life outside of school? Why?”
This question elicited students’ beliefs about the attainment and utility value of mathematical
concepts explored in tasks (refer to Table 13 to review these constructs).
Overall, the focus students consistently described potential applications for the content
they learned in lessons. Their responses and beliefs about the value of the material, however,
differed between motivation groups. Low motivation students were less likely to express a belief
that the mathematics they studied was useful and tied possible applications to problem contexts,
whereas high motivation students considered broader mathematical goals. Students’ responses to
money-related contexts were noteworthy, as these contexts seemed to particularly support
students’ beliefs that rational number operations are valuable concepts in their everyday lives.
Generating ideas for future use of mathematical concepts and skills. Across all 28
lessons I observed, neither Ms. Pearson nor Mrs. Meyer explicitly emphasized the usefulness of
mathematics content students were learning in the lesson (e.g., saying something like, “you need

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to know these concepts for/if you…”). Yet, the focus groups expressed beliefs that the material
they learned was valuable in each interview. That is, though not all students may have agreed,
whenever I asked, more than one student in each group stated that what they learned would be
beneficial or useful. For example, they responded with statements such as, “This is useful in
other math classes” (Lilly, Lesson 7) and “Everybody's going to have to know this” (Elijah,
Lesson 10). Comments of this nature communicated a belief in the general attainment or utility
value of the content they were learning. In these examples, the content was similarity and integer
operations, respectively.
Students also identified specific applications for the content or situations in which the
mathematical ideas would be useful. When asked if similar figures would be beneficial or useful
to him, for example, Jeff responded, “Packaging. If you get a job in packaging you need to know
area and perimeter of the product so you can package it” (Jeff, Lesson 10). In two different
interviews, Jacob suggested that the mathematics in their lesson would be helpful to physicists
(Lesson 7) and accountants (Lesson 13).
Students’ ability to generate these types of examples and the likelihood that they would
find the mathematics beneficial did not, however, seem related to whether the lesson contained
contextual tasks. As shown in Table 14, students expressed beliefs in the attainment and utility
value of the content they learned for both contextual and noncontextual tasks. This implies that
engaging in contextual tasks did not necessarily support students in identifying how the
mathematics could be useful in future careers or everyday situations, and working on problems
without a storyline did not necessarily hinder them from seeing the value of the content or
generating applications.

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For instance, almost all focus students at Southpoint responded positively regarding the
value of learning about scale factor following a lesson in which the Wumps context was highly
peripheral to the task (Lesson 2) and discussed its application in carpentry (Kim). Likewise, the
focus students at Pine River communicated they believed that multiplying positive and negative
integers would be useful and beneficial following a lesson in which no contextual tasks were
used (Lesson 7), and identified applications such as banking (Jeff) and business (McKenna).
Following Lesson 14, which had a highly contextual task, most of the students at Pine River said
they did not think that using similarity to estimate unknown heights would be beneficial or useful
in everyday life. Thus, the students did not necessarily perceive more or better applications for
mathematics through contextual tasks than through noncontextual tasks.
Table 14
Belief in the Value of Mathematics Expressed in Focus Group Interviews
Lesson

Description of Context

Applications generated by
students

None

Number of students
expressing belief that
content was
useful/beneficial,
neutral, or not
useful/beneficial
3,0,1

Pine River 2
Pine River 5

Money contexts

2,2,1

Drafting

Pine River 7

None

3,0,1

Working at McDonalds (dealing
with money), banking, business

Pine River 10

Identifying Wumps and impostors

3,0,1

Working at McDonalds, designing
apps, packaging

Pine River 14

Using shadows to find heights

2,3,1

Architecture, sports, telling time

Southpoint 2

Analyzing Wumps’ noses and
mouths

2,1,3

Construction, carpentry

Southpoint 5

Determine which cookie mix is the
most “chocolatey”

3,1,0

Sales, cooking

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Carpeting, packaging

Table 14 (cont’d)
Southpoint 7

Sascha’s bike trip

3,1,0

Physics, pilot, music producer,
general scaling problems in
everyday life, biking

Southpoint 10

Money contexts

5,0,1

Student loans, taxes, general
personal finance

Southpoint 13

Temperature context

4,1,0

Taxes, jobs, accounting, parenting
(paying for kids’ expenses)

Note: A neutral response was one where students thought the mathematics would be useful but
only for certain professions, would be useful in school but not in real life, and so on.
Motivation to learn and value-related responses to contexts. Though students’ valuerelated responses to tasks did not seem related to whether the tasks were contextual, they did
relate to students’ motivation to learn mathematics. Students’ beliefs regarding the value of the
mathematics content explored during the lessons and the applications students generated differed
across motivation groups.
Low motivation group. Because they expressed low valuing of mathematics in the initial
beliefs survey, it was not particularly surprising the students in this group were least likely to see
the material they were learning as useful when asked in interviews. Drayton (neutral on
expectation of success, low on valuing mathematics) never expressed a belief in the value of
specific topics. He openly challenged the relevance of some contexts, asking, for instance, “when
will you need number lines outside of school?” (Lesson 5) and “what about if the sun’s not out
when you need the height of an object?” (Lesson 14). Similarly, Emily said that the usefulness of
mathematics “depends on what you want to do. Like if you want to be a hairstylist you wouldn’t
use it” (Lesson 2) and remarked about integer operations, “I think we’d all forget about this
when we’re adults” (Lesson 10).
When students in this low motivation group did talk about the value of the mathematics
they were learning outside of school, it was either very general (e.g., “I think it could be useful,”

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Lena, Lesson 10) or closely tied to context when it was a contextual task. For instance, following
Lesson 14 in which students found heights of tall objects using similar triangles, Felix said the
topics was, “not really that useful, except in architecture. Tearing down a building, putting it up,
measuring how high it is before you tear it down.” Similarly, after the lesson on using ratios to
determine which cookies were the most “chocolatey,” Lena said, “I guess if you’re cooking and
stuff it would help” (Lesson 10). This phenomenon is noteworthy because it indicates that, for
some students, contexts may actually narrow their perception of the value of mathematics and
promote the belief that it is only useful in the situations presented in their books.
Neutral and mixed motivation group. Most students in the neutral/mixed motivation
group expressed that what they learned was somewhat valuable and recognized the importance
of what they were learning, but generally only for future mathematics courses or for other
people. That is, they discussed the attainment value and utility value of the content, but for
situations they did not consider personally relevant. For example, Jacob explained that different
topics they learned would be helpful if one is in certain professions such as being a physicist or
accountant, but these were careers he is not interested in. He told me, “I’m not going to be an
accountant” (Lesson 13). Lilly gave similar responses. In talking about integer operations, for
instance, she said, “Yeah, my dad does math a lot. He works at Meijer warehouse…It will be
useful in school, but for what I want to be [an author] I don't think it's going to be very useful. I
want to write fantasy books, not math books”.
Often, students in this group took a literal view of contexts. Following the lesson on
finding heights of tall objects using similar triangles formed by shadows, Lilly commented that
the concept is, “not very useful…it might be useful tomorrow and next week, but not outside of
school. What if it's in the middle of winter?” (Lesson 14). Adelyn’s responses were mixed. Early

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in the school year, she challenged the practical value of a concept, saying, “I don't think this is
ever going to come up when I stranger walks up to you and says, ‘find the scale factor’!" (Lesson
2). For the lesson on rates, however, she described, “It depends on what you want to be, like it
would be good if you want to be a pilot or a music producer. I would say very beneficial!”
(Lesson 7). Generally, this group of students saw more value in their lessons, but not necessarily
for themselves personally.
High motivation group. The high motivation students consistently communicated the
usefulness of the mathematical content they learned, and in more specific and meaningful ways
than students in the other groups. They often identified specific careers where the skills would be
used, or certain kinds of everyday problems one could solve using the ideas. The examples these
students generated were not tied to applications immediately related to problem contexts; rather,
they related more to the core concepts and objectives of the lesson. Elijah, for example,
discussed how comparing ratios in the chocolatey cookies “would be really helpful if you were
like a sales rep, because you would need to know stuff about unit rates, and you'd have to be able
to make things appealing to the audience. So they'd buy your product, and use all the
information.” At different points in the interviews, Jeff talked about how the topics they were
learning were relevant to packaging, drafting, and using the sun for telling time—none of which
were contexts used in the problems they encountered. For example, after the lesson on similar
figures he said, “If you get a job in packaging you need to know area and perimeter of the
product so you can package it” (Lesson 10). McKenna generally thought that the mathematical
concepts were useful to support learning future topics in school, but also expressed its relevance
to her own future. She explained, for example, “I would use [multiplying and dividing integers]
at my job, because I want to own a business…I don't know, what jobs don't use numbers?” In

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sum, the students in the high motivation group attended to broader mathematical goals of tasks
when describing the value of a task and did not consider only applications connected to the
immediate contexts they explored in the text.
Money-related contexts and rational number operations. The money-related contexts
from the unit Accentuate the Negative were a special case in that students were able to generate
ideas for future use of rational number operations regardless of level of motivation to learn.
These contexts supported students across groups in valuing the content in the unit. Most students
repeatedly communicated beliefs about the attainment and utility value of rational number
operations both because certain professionals use the skills and concepts and because they will
need them for personal finance.
Examples of students’ responses when asked about the benefits and usefulness of the
material, all from three different Accentuate the Negative lessons, include:
“I guess it could be useful for adding college debt and student loans and stuff” (Jacob).
“Yeah [it is beneficial], like taxes and stuff…If you were using this to do taxes then you
would not forget. You would have to relearn it” (Adelyn).
“Probably, yeah, yeah, very [useful]. Taxes…When you're a mother or father, and you
have to pay for your kids' stuff” (Kim).
“Banking…banking! You have negative and positive numbers and you have to multiply
them” (Jeff).
“I said [this lesson was like eating] carrots, because they don't taste the best but I know
they'll be good for me” (Emily).
Though money contexts were used in multiple lessons in the study, these specific examples
students offered—determining debt, paying taxes, and banking—were not explicitly referenced

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in the lessons I observed. This indicates students bring to bear some personal experience as they
interpret task contexts.
Student Responses to Tasks—Expectation of Success and Intrinsic Value
The results in the last section indicated students can think and talk about applications for
mathematics encountered through contextual and noncontextual tasks when asked about the
value of the content they learned. The second trend in the data, though, was that the students
across motivation groups focused almost entirely on the level of ease or challenge they faced
(expectation of success) or enjoyment (intrinsic value) in their responses to lessons. Students in
different motivation groups did attend to different aspects of experiences with tasks when talking
about their individual high- and low-engagement lessons. Again, students’ positive or negative
responses to lessons seemed unrelated to whether or not the lesson included a contextual task,
though several students referenced their favorite lessons by context.
Patterns in student responses to lessons by motivation group. For each focus student,
I identified “top” and “bottom” lessons based on their reports of their level of effort and
enjoyment in the lesson surveys7. The important point in the second theme is that in the more
open prompts for reflection on lessons, students almost never talked about the utility or
attainment value of mathematics. Rather, they focused on how easy or difficult and fun or boring
the task was. How students responded to different lessons, however, was related to students’
level of motivation to learn mathematics. More specifically, students in different motivation
groups attended to varied aspects of tasks when describing their class experiences in surveys and
interviews.

7

Note that this process was different than in Chapters 4 and 5, where the analysis was for focus
students as a group and included more data sources that considered groups of students rather than
students’ individual responses.
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Low motivation group. Students with low motivation to learn mathematics most often
responded to prompts about their reactions to lessons based on how easy they felt the lesson was.
How much they liked or disliked lessons was closely connected to how confident they felt in the
material or how difficult they found it.
Students’ individual low-engagement lessons varied amongst the low motivation group,
but they were generally lessons where the students expressed trouble understanding the content.
For example, Felix described one of his low-engagement lessons (finding heights using shadows)
as “difficult” and likened his experience in the task to “Rugby, because I barely even know what
it is” (Lesson 14). Lena responded negatively in her survey to the Bolda Cola lesson. During
work time, she commented, “I guess I don’t get that,” and when the teacher offered a challenge
activity, she said, “I don’t want a challenge!” (Lesson 4). Emily’s lowest lesson was the
chocolatey cookies task. While working on the problem, she commented, “I don't know,
because…I don't know! I'm better at scale factors than this thing!” (Lesson 15). Later, in the
interview, she said, “It was really hard to understand…I’m usually good at math but I did not
understand what we were doing today.”
The low motivation students’ individual high-engagement lessons seemed to be lessons
in which they felt most confident. One of Lena and Emily’s most highly-rated lessons was
Sascha’s bike task. Lena struggled at first but began to understand the major concepts. In the
summary discussion, Lena shared an idea and Ms. Pearson responded, “So Lena is absolutely
correct.” In the interview, Lena said, “I found [the task] interesting because I learned a lot…I
was putting a lot of work into it and trying really hard” (Lesson 7). Similarly, in the interview
Emily commented, “I would say it was challenging, but a good kind of challenging, because I

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could understand it.” Though she expressed confusion throughout the small group work time, she
eventually grew more comfortable, saying, “this is easy!”
The students in this group also mentioned task structure and/or expectations—which
reflect instructional decisions—when describing the lessons they particularly liked and disliked.
Lena enjoyed Lesson 2 because she “liked working in groups and making posters.” She even
commented at one point in the lesson that making posters makes her “feel smart.” Lena and
Emily both disliked the chocolately cookies lesson, during which Ms. Pearson asked students to
move into groups to argue for the most chocolately cookie. They agreed that they “didn’t like
standing up over there [in the corner of the room] and sharing.” Emily also stated multiple times
that she did not like using the chip model because she found it confusing. Taken together, these
examples point to low motivation students’ focus on how successful they felt at understanding
the material when reflecting on a lesson.
Neutral and mixed motivation group. Like the low motivation group, the students who
reported neutral or mixed motivation to learn mathematics sometimes described lessons in terms
of their level of difficulty or ease. They focused more, however, on how “fun” or “boring” the
lessons were—that is, they attended to their intrinsic value—and for this group, the lessons they
described as fun were likely to involve contextual tasks. Also, they often expressed that they
liked lessons even when they struggled or found them difficult and disliked lessons when they
were too easy or boring. So, these students did not show the same strong correspondence
between liking lessons and ease in understanding.
Drayton, for instance, described the lesson in which students estimated the distance
across a river as “fun,” even though he struggled with it. During the lesson, he asked, “Hey
team? I need help. I’m totally lost” (Lesson 12). Similarly, he described the Wumps lessons as

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awesome and fun, saying he loved graphing the Wumps. Yet, again, he expressed having trouble
with the task during class. He told his group, “Good thing I got these, because all my other things
were wrong.” Later, when the teacher asked him how graphing was going, Drayton responded,
“Not so good; I don’t think I’m going to get that [what another student graphed]” (Lesson 10).
Lilly also loved the Wumps lessons, describing them as fun repeatedly on surveys, in class, and
in the interviews. At one point, she announced, “The Wumps are amazing!” (Lesson 10). Jacob’s
high-rated lessons included the Bolda Cola and chocolatey cookies tasks. He wrote they were
exciting, fun, and not too boring. Like Drayton with the river and Wumps tasks, Jacob struggled
somewhat with the cookies lesson but still liked it. At one point he remarked, “this is confusing!”
But after the teacher explained it, “the pieces kind of fit together” (Lesson 5).
These students responded negatively to lessons typically because they found them boring.
Adelyn described her low-engagement lessons (5 and 8) as boring and anticlimactic. She also
said it was hard to focus, she was easily distracted, the material was hard to grasp, and she
“didn’t get it” (Lesson 5), indicating her belief in the high cost of engagement in the lesson.
Jacob’s negative responses were all to lessons from Accentuate the Negative. He likened one
lesson to a kids’ cartoon, for example, because it was “boring and for young kids…it was stupid
and easy” (Lesson 13).
Finally, some of the neutral/mixed motivation students based their lesson responses on
how much they participated in the lesson or on the nature of the lesson activity—whether it was
a game, writing on whiteboards, they particularly like graphing, and so on. Adelyn, for example,
liked the whiteboards, describing them as “cool” (Lesson 13). Lilly rated Lesson 5 (using the
chip model) highly. She described it as interesting, then highlighted her own participation. “I

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thought I did good, entered the discussion more than usual. I actually don’t talk very much.”
Lesson 13 was similar—she liked the lesson better than others “because she felt useful.”
In sum, students in this group did not necessarily associate easy with “good” and hard
with “bad.” Rather, they expressed that lessons that were too easy were boring, and talked about
lessons being fun even when they struggled to understand. Like the low motivation group, these
students sometimes attended to the nature of the lesson activities. Contexts played a bigger role
in this group’s engagement, though, such as Drayton and Lilly with Wumps tasks and Jacob with
Bolda Cola and the chocolatey cookies.
High motivation group. The students highly motivated to learn mathematics attended to
some of the same things in their responses to lessons as the neutral/mixed motivation group. For
example, they disliked lessons they found boring. Kim described her low-rated lessons as “not
exciting and confusing” (Lesson 1) and “boring” (Lesson 6). They talked positively about
lessons they found fun even when they were difficult. McKenna particularly liked Lessons 12
and 13, which she described as “easy to take in, and fun” though she expressed confusion
throughout the lesson.
For this group, however, contextual tasks did not play much if any role in their positive or
negative responses to lessons. They liked lessons with noncontextual tasks or tasks with
peripheral contexts, and did not necessarily like the lessons with highly contextualized problems.
This may be related to the fact that, as described earlier, specific contexts did not hinder or
support these students in identifying other meaningful applications for the mathematics. They
gave very consistent responses to lessons, with Elijah rating most lessons low (no fours for
engagement) and Jeff and McKenna describing almost every lesson positively. Most notably,
though, the high motivation students explicitly communicated that they liked feeling challenged,

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figuring out solutions to challenging tasks, and being active learners with opportunities to work
on mathematics.
Elijah, for instance, often said lessons were boring because there was not “enough to do”
(e.g., Lesson 7). The lessons he liked, he described as “fast” and “lots of events jam-packed
together.” Similarly, Jeff related a lesson on negative numbers to golfing because it was “slow
but boring. If you’re playing, it’s actually pretty fun.” McKenna’s favorite part of the Wumps
lessons was “figuring out…how to find the x and y—figuring out the rule.” She rated her effort
as a 4, saying, “I tried to pay attention and I liked this math class, I wanted to learn more about
it” (Lesson 10). Her favorite parts of Lesson 14 were “figuring out the problems, figuring out the
heights” and “discussing it so I could make sure I got the right answers.”
So, students in the high motivation to learn group based their responses to lessons on
how active they were in the lesson in terms of having opportunities to engage in mathematical
thinking and solve problems. This was in contrast to the other groups, whose high-engagement
lessons were easy (low motivation students) or fun (mixed/neutral). Although none of the
students attended to the utility or attainment value of the mathematics content when talking about
their general reactions to tasks, the students with neutral or mixed motivation to learn did seem
to be drawn to certain contexts, particularly the Wumps and Bolda Cola contexts. Interestingly,
all the students at Pine River responded positively to the Wumps tasks, but note that different
motivation groups liked these tasks for different reasons.
Contextual tasks not related to individual high- and low-engagement lessons. As seen
in the previous section, another trend in student responses was that across the focus students,
their high- and low-engagement lessons included a mix of contextual tasks and noncontextual
tasks. That is, some of their high engagement lessons were contextual and some were not; some

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low engagement lessons were contextual, some were not. So students’ positive or negative
reactions to tasks did not seem to be closely related to whether or not the task was contextual nor
their underlying motivation to learn mathematics.
For example, Adelyn (who was neutral in both expectancy of success and valuing
mathematics) reported two high-engagement lessons. The first was the group poster activity with
a peripheral Wumps context, and the second was a whiteboard review activity with no contextual
tasks. Her low-engagement lessons both included contextual tasks--“chocolatey cookies” in
Lesson 5 and Lesson 8 with multiple contexts (e.g., plane in flight, Jack’s enchiladas), which was
a low-engagement lesson overall for the focus students. Similarly, Jeff (high motivation to learn)
reported lower-than-typical engagement in a noncontextual lesson from Shapes and Designs
(Lesson 2) and another with some contextual elements from Accentuate the Negative. Two of his
particularly high-engagement lessons were when students used similar triangles to: a) find the
distance across a river (Lesson 12) and b) to find missing measurements in polygons, a
noncontextual task (Lesson 13).
Moreover, most students responded positively to noncontextual tasks. In Pine River, for
instance, the five focus students present described Lesson 3 (from Shapes and Designs) as
“good,” “enjoyable and fun,” “[like eating] steak, because it was super hard to chew or
understand but in the end it was like Jell-O, easy and understandable,” and how “at the end we
had to finish and I wanted to keep going.” The Southpoint students described the lesson that had
no contextual elements as “easy,” “alright,” and “[like eating] oranges—it’s tasty and kept me
occupied.” Together, these examples illustrate a lack of pattern between students’ positive or
negative responses and the contextualization of the content. Also, students never talked about the

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utility or attainment value (or lack thereof) in these open descriptions of their experiences with
the tasks, and contexts were not necessary for students to find a task easy or fun.
Several students did, however, identify tasks or content by their contexts when asked
about their favorite lesson from the class so far at the end of the semester. Four of the 12 focus
students—including students from all three motivation groups—said the Wumps tasks were their
favorite by name. They responded with statements like, “I liked the Mug Wump lessons because
they were fun” (Lilly), and “Mug Wump because it was entertaining to draw and guess” (Jeff).
The Wumps context seemed to be memorable and meaningful to students, and these results show
the potential for contexts to anchor students’ classroom experiences. This trend extended to the
entire set of study participants, as approximately half of the total students identified a contextual
task as their favorite, mostly Wumps or using shadows to estimate heights. To summarize,
although results presented in earlier chapters showed that students generally showed high
engagement in contextual tasks, it is clear that contextualization is only one factor in student
engagement and reactions to tasks.
Results Part 2: Illustrative Student Cases
The purpose of this section is to introduce in greater depth three focus students and
provide a fuller picture of their experiences in their 7th-grade mathematics class, including their
reactions to and engagement in tasks. I chose one student from each motivation group to
illustrate and highlight the themes described in Part 1 of the results: Felix from the low
motivation group; Sophia, who expressed low expectation of success but high valuing of
mathematics; and Elijah from the high motivation group. Felix and Elijah are good
representatives because they articulated fairly typical responses for their motivation group and
each represents a different class. I chose Sophia because she was the one student who had a

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negative view of herself as a learner of mathematics but highly valued the subject. Drawing on
context-related interview and survey responses as well as their contributions in class, I will
describe important characteristics of each student, their interests, and noteworthy observations.
Felix—I feel smart on weekends, because I'm not being told I'm wrong!
Felix expressed both low expectation of being successful in mathematics and low valuing
of mathematics. In his opening survey, he wrote that he did not have a favorite lesson in 6th grade
mathematics and said he “struggled.” His interests outside of school included skeet shooting,
making things out of wood, fishing, and knife collecting. Felix had a 504 plan and was labeled
by the school as a special needs student. He attended the school’s “math lab” every day to work
with an academic aid who also attended his class to work with various students.
Felix was generally quiet in class and got along well with other students in the group. He
rarely made it very far through problems, however, or contributed to discussions unless directly
asked. On one occasion he was off-task most of the lesson and eventually Mrs. Meyer moved
him to a table to work alone.
There were few instances in which Felix attended to the context while solving or
discussion problems, though he did use contexts in various ways. When discussing the Mystery
Teacher task, for example, he connected the context to the central mathematical idea of the
lesson (similarity) during the whole class summary, saying, “The scene in the picture is exactly
the same as how it would be in real life, just smaller.” He talked about contexts themselves in the
first Wumps lesson by sharing about his favorite video game and commenting on how the
characters looked like “double-sided porcupines or hedgehogs.” Later, he used his graph of Glug
to check his work. “I feel bad for Glug. His eyes are underneath his nose…Oh wait, three times

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five, fifteen!” These examples of his contributions show how Felix did make use of the context
when solving tasks though his engagement was generally low.
As with the other students in the low motivation group, Felix did not think the
mathematics they learned would be very beneficial. He said, for example, that he would not use
what they were learning about polygons “too much outside of school because I’m going into
finance” (Lesson 2). His desire to go into finance did not seem to translate into believing that
other mathematics content would be valuable, however, since he did not say that any of the other
lessons would be beneficial, even the lessons on integer operations that most other students said
would help with personal finances. The only time Felix expressed a belief that the content would
be valuable was after the lesson on estimating heights of tall objects. He described it as “not
really that useful, except in architecture. Tearing down a building, putting it up, measuring how
high it is before you tear it down.” (Lesson 14). As described in Part 1 of the results, this one
application extended directly from the context as written in the task.
The lessons to which Felix responded most negatively were clearly linked to when he had
trouble understanding. He used the following descriptions for the lessons he reported (and I
observed) his lowest engagement: “It was really confusing” (Lesson 1), “it was extremely
confusing” (Lesson 11), “I found it difficult” and it was like “rugby because I barely even know
what it is” (Lesson 14). Note that the last lesson was one where Felix did recognize utility value.
Students played a game in another of Felix’s least favorite lessons, and students liked it overall.
But Felix called it “cheesy,” and announced during the lesson, “I never win!” (Lesson 5).
One of the lessons Felix responded most positively to was the Mystery Teacher task,
which was one of the overall high-engagement lessons. Though he did not contribute much in the
lesson, recall that it is the lesson where he shared an important idea during the summary that

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linked the context to core idea of lesson. This likely increased his sense of success in the lesson
and related to his report of high engagement. Felix’s other high-engagement lesson did not
involve a contextual task, but did include use of the polystrips manipulative to construct
polygons.
Felix’s response to the Wumps lessons, which were generally highly engaging for
students, was interesting. Immediately after the lessons, he described them as “fun,” though he
did not rate them particularly high nor did he exhibit high engagement by my observation. He
rated his effort and enjoyment as three out of four, saying, “I was engaging in it a lot until I got
distracted giving them mustaches” and “it wasn’t…exactly the most fun I’ve ever had in math
class.” Lilly pushed back on that last comment with “But it was the Mug Wumps! It was the
Wumps!” and Felix responded, “I know, but in my old school we graphed bigger things, like
buildings and machines and stuff. One time we graphed an engine and it was pretty cool.” I am
unsure what kind of task he might be referring to, but the message was clear—Felix enjoyed the
lessons well enough, but they did little to support his valuing of mathematics or the specific
topic. In a later interview, some students spoke about how they disliked mathematics and again,
Lilly asked, “What about Mug Wump?” Felix replied, “I feel smart on weekends. I feel smart on
weekends, because I’m not being told I’m wrong!” Yet, at the end of the study period, Felix
identified his favorite lesson to be “Wump—it was easier than the rest of the year.” This
response indicates that he did recall that he had relative success in the Wumps lesson, and also
that the context itself was memorable for him.
Felix is a case of a student who struggled to learn mathematics and did not see much
value in learning it. Though some evidence exists suggesting that task contexts supported Felix’s
work in the classroom, they had little positive or negative relationship to how he responded to a

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lesson or how much he valued the content studied. Rather, he focused on the extent to which he
could make sense of the mathematics (i.e., how confusing or easy it was) and how successful he
felt in the lesson.
Elijah—Math is My Friend!
Elijah expressed both high expectations of being successful in mathematics and high
valuing of the subject. His interests included running, soccer, basketball, video games, Science
Olympiad, and competing in mathematics competitions. He seemed to love mathematics and did
not mind being challenged. On his initial survey, he wrote, “my favorite math lesson last year
was when we had to find missing dimensions with shapes when our teacher only give me a
percent of how big it is compared to the original.” Ms. Pearson wrote in email communication,
“Elijah and I had a discussion today—he’s very task oriented and more concerned about getting
his work and homework done than working in a group, figuring things out together, and being
further challenged, although he would accept challenging questions.” He was eventually moved
to an 8th-grade course after the end of the study period.
In my observation, I saw that Elijah was a strong student and picked up on ideas quickly.
As Ms. Pearson said, he did not like working with classmates and was sometimes even
condescending or resisted explaining problems and concepts to them. He completed the assigned
homework even though it did not count toward his grade and often worked on upcoming
assignments during group discussions. Elijah held high social and academic status in the class,
though other students expressed that they preferred not to work with him because he made them
feel deficient.
During lessons, Elijah used the context to solve and make sense of problems whenever it
was necessary or helpful. The fact that he talked about and used the contexts is not surprising, as

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he consistently exhibited high engagement in each lesson. In the cola advertising task, for
instance, he frequently referenced the two companies in his explanations. A representative
example is when he said, “I chose [statement] 2, because it makes it sound like a ton more people
liked Bolda Cola…when I see that number I think of a really big number, and I think they must, I
think they didn't have that many people. So 5,000 looks like a lot.” Another example is from the
bike task. Elijah explained to his group, “What you want to do is finish at the exact same time as
Sascha does…so what you're writing down is the miles per hour how fast you should be going.”
Although he referenced contexts while working on tasks like in these examples, Elijah almost
never talked about the problem contexts apart from the mathematics. The only exception was at
the beginning of the cola advertising lesson, when he contributed ideas about familiar
commercials that compared two products. This was consistent with his focus on the mathematics
and desire to complete tasks.
Elijah communicated in every interview that the content they had studied was beneficial
and useful, and he could generate specific and relevant applications for the material. He
described scale factors as “Very useful. For almost everything, even if you're just building a
table, if you need to scale it down.” For integer operations, he said, “Everybody's going to have
to know this. Everybody needs to deal with financial stuff.” He described the “sales rep”
example as an application for unit rates. In other interviews, he stated that what they learned was
“very helpful for everyday life” and “very useful, because you're constantly getting statistics that
get punched into your mind.” So Elijah, like the other students in the high motivation group,
generally viewed what they learned in class to be useful and important in one’s everyday life
regardless of career path.

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Also like others in the highly motivated group, Elijah rated his own engagement
consistently across the lessons in the study. Interestingly, though, he reported relatively low
engagement, never rating his effort or enjoyment above a three. In over half the lessons he rated
his effort as a one and his enjoyment as a two or three, which I believe was his way of saying the
course was too easy for him. He did rate two lessons somewhat high; both involved contextual
tasks with peripheral contexts (Lessons 2 and 6, which mainly involved going over homework).
Elijah connected his positive response to these lessons with how active he could be in the lesson,
describing these lessons as “fast” and “lots of events jam-packed together.” He also said, “It's
more fun when you actually do the problems and not just write them down.” Elijah rated one
lesson (Lesson 10) particularly low. He saw value in learning about negative number operations
but said it was “boring because we didn’t really do anything….we didn’t get to do anything for
40 minutes. I didn’t feel involved.” This was often his complaint about lessons.
Elijah is a case of a student highly motivated to learn mathematics, who believed he
would be successful in class and that what they were learning was valuable. Whenever asked, he
communicated that the mathematics topic they had studied would be beneficial and useful in
everyday life, regardless of one’s career path. Also, he was quick to come up with meaningful
applications for lessons. Like Felix, however, these beliefs did not seem to be connected to
whether or not a task had contextual elements. Instead, he emphasized the lessons’ cognitive
demand, which was related more to the activities and structure of lessons than task contexts.
Sophia—What if I Become a Famous App-Maker?
Sophia was the only student who expressed low expectancy of success in mathematics
but high valuing of the subject. In fact, her view of her ability to learn mathematics was one of
the lowest of all the students across the two classes, while her valuing was one of the highest.

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She listed her interests as music, drawing, riding bikes, and swimming. “Fractions” was her
favorite lesson from the previous year because it was “easy and fun.” Sophia also attended the
math lab, and occasionally left class during observed lessons with some other girls to work with
an academic aid.
I noticed in the observations and interviews that Sophia struggled to understand the
mathematics but worked hard toward understanding. She was generally highly engaged, but
often in the form of asking questions of or requesting help from Mrs. Meyer and her group
mates. Sophia seemed excited when she understood how to complete problems, and appeared
pleased when the teacher called on her to share an answer or an idea.
Along those lines, Sophia’s self-reported high engagement and the lessons she responded
positively to were those in which she experienced some level of success and enjoyment. Sophia
described the lesson with the integer product game as being easy and fun, responding, “I was
close to winning!” and “When you have a game, the time just flies.” She rated her effort and
enjoyment high but struggled to understand the problem in Lesson 13, asking her peers, “How
did you get the answer? I just don't know how to do it!” Later, she talked positively about the
lesson and said it was easy. Sophia’s reaction to a task on subtracting integers was particularly
noteworthy. She described it as easy, fun, and interesting; then turned to a reflection on her
participation. “I was actually participating, like I was talking more.” Her favorite part was
“discussing it out loud to everybody.” Sophia showed productive engagement in the lesson and
received significant teacher encouragement. Mrs. Meyer asked her to share her solutions in the
whole-class summary and publically praised her, saying, “You’re sharing some good ideas.”
Later, another student referenced Sophia’s strategy.

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On the other end, Sophia described the lesson on estimating heights using similar
triangles as “slow and boring.” During the lesson, she said she was confused and did not get the
concepts, but in the interview said, “I really didn’t think a lot; I was just bored.” These
reflections on the lessons Sophia found most and least engaging show that she attended most
closely to her level of participation and how fun she found the task, which was characteristic of
the neutral/mixed motivation group. Further, struggling with the content did not deter her from
enjoying and putting effort into a lesson.
It is important to note, though, that unlike other students in that group, Sophia’s response
to tasks had little to do with their contexts. Her three highest-engagement lessons primarily
included noncontextual tasks on integer operations and using scale factor. Her lowest rated
lesson—finding heights of objects—had a context central to solving the task, but it did not seem
to promote Sophia’s engagement. Thus, it seems task contexts did not support Sophia’s
engagement like they did for Lilly, Drayton, and Jacob.
This lack of relationship between contexts and engagement may be because of Sophia’s
beliefs in the general importance and value of learning mathematics. More similar to the students
in the high motivation group, Sophia expressed in each interview that she believed the content
was useful and beneficial outside of school and offered applications for the content. For instance,
after a lesson from Shapes and Designs on interior and exterior angles, Sophia argued that the
lesson was important for “If you, when you’re older, get a job in carpeting and you have to
measure angles and stuff” (Lesson 2). Or, after playing the integer product game, said it would
be beneficial in college and that “You still have to use numbers at any job, even if you have to
work at McDonalds or something! Yeah, you use negative and positive numbers because some
people are short the money. They would have to pay you the next time” (Lesson 7). Sometimes,

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though, the applications Sophia suggested were quite vague or somewhat irrelevant. Similar
figures, she said, would be useful if you worked at McDonalds, “Like if you order some meat or
something.” She then asked, “What if I become a famous app-maker, and I made the Wumps
game?” to which Lilly responded, “That would be awesome!” Finally, she suggested that
estimating heights of objects would be helpful in “sports.” In these examples, it was unclear how
the mathematical topic would be useful in the situations Sophia identified, but the examples are
consistent with the high value she placed on mathematical ideas.
In sum, Sophia is a case of a student who learns mathematics with difficulty, but who
believes it is worth the effort—that mathematics is an important and useful subject. She
consistently communicated this high valuing of mathematics, particularly in the interviews when
asked about the usefulness of particular topics. Her responses, however, were not closely related
to whether or not lessons included contextual tasks, and the extent to which she reflected
positively on lessons reflected her level of participation and the intrinsic value she placed on the
activities over her perceived utility value of the mathematics content.
Discussion
An analysis of students’ responses to tasks based on surveys, interviews, and class
contributions across the study period indicated that task contexts had a fairly weak relationship
with the value focus students found in the mathematics content and how positively they
responded to lessons. Yet, there were noteworthy patterns in what students attended to in their
responses, and certain contexts stood out as particularly meaningful for some students. Based on
these findings, I argue that contexts can play different roles in students’ response to mathematics
tasks. A particular context may engage students by offering intrinsic value to lessons, or it may
support their understanding and problem solving. Although it is highly possible that certain

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contexts can promote students’ motivation to learn by supporting their valuing of the subject,
little evidence from this study exists to suggest this is the case for CMP tasks. These roles are not
mutually exclusive, and contexts may not have the effect anticipated by curriculum authors or
teachers.
Contexts, Value, and Application
Hidden applications. An exploration of students’ beliefs about the attainment and
utility value of the mathematics they learned revealed that focus students as a group consistently
described potential applications (some more reasonable than others) for the content they learned
in lessons. Responses about value and applications for mathematics had little if any relationship
to whether students had engaged in contextual tasks, however, and differed between motivation
groups. Money-related contexts seemed especially effective in supporting students’ beliefs that
rational number operations are valuable in their everyday lives. So although this study showed
that contexts can be engaging for students during in-class work, these students’ work with
contextual tasks did not appear to translate to greater recognition of the value of the content.
Students’ varied responses regarding the usefulness and potential application for what
they learned—and the fact that contextualizing the material does not seem particularly helpful—
might not be of concern. As Otten (2011) and others have argued (e.g., Wu, 1997), mathematics
educators should not feel obligated to have an application ready for every topic. Content in the
curriculum can serve multiple purposes, such as developing skills or ways of thinking to support
learning of future content. The mathematics we teach does not necessarily have or need a direct
application. No other subject has quite the same demand to answer the question, “when will I
need to know this?” Expecting contextual problems to bear the responsibility to answer that
question, or for students to glean much meaning from the stories, may be unrealistic.

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Yet, the material students encountered in 7th-grade mathematics had applications far
beyond what was represented in the classroom problems. For instance, most people use rational
number operations in their careers and everyday life in a broad range of contexts, and these skills
are supported by working with positive and negative numbers in school mathematics. Further,
these concepts support quantitative and algebraic reasoning necessary for success with future
mathematics topics and courses, interpreting statistics, and so on. Similarity and scale factor are
used in numerous situations, including engineering, construction, architecture, design, art, and
travel. These many applications remained largely hidden from the students in the study,
indicating that students do not transfer task contexts to a belief in the value of the content. It may
be that the students simply do not have the life experiences on which to hang these applications.
Boaler suggested this idea as well (1993):
[C]ontexts may be useful in relation to learning transfer even though contexts as they are
generally used are not useful, and…factors which determine whether the context is useful
or not are numerous and complex and have little to do with a description or depiction of
real world events which students will eventually encounter. (p. 13)
So although the mathematics students are learning is useful and used to solve problems outside
of school, contextualizing school mathematics problems do little to expose students to these
applications.
Differences related to motivation group. Moreover, a relationship exists between how
students with different levels of motivation to learn mathematics respond to problem contexts,
with highly motivated students in this study being better at generating meaningful and relevant
applications for specific content than students with low motivation to learn. It is possible,
therefore, that students are motivated in part because they can see the “mathematical horizon”
(Ball, 1993, p. 376), look beyond a given situation, attend to more general learning goals, and
transfer the use of mathematics introduced through a specific context beyond that context. This

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points to the potential importance of students’ future goals, personal experiences, and beliefs
about themselves in relation to schooling in the relationship between student motivation to learn
and engagement with contextual tasks.
Based on this study, it is impossible to clarify the direction of influence in this
relationship. That is, are motivated students like Elijah and Jeff more disposed to think about and
come up with applications for what they learn? Or does this inclination promote valuing the
subject and thus lead to increased motivation? And the same for low motivation students—if
students in a problem-based curriculum interpret problem contexts literally, as Felix and Lena
did, does it hinder them in seeing personal relevance in the content? Or does low valuing of
mathematics lead to a generalized perception that school mathematics is not useful? These are
open questions, though other researchers have noted how students relate to and make use of
contexts differently (Christiensen, 1997), particularly along lines of socio-economic status
(Cooper & Dunne, 1998; Lubienski, 2000). There might be interplay, then, between students’
motivation to learn, socio-economic status, and attention to contexts while solving problems and
reflecting on ideas in mathematics.
Thus, if we want to build on the potential of contextual tasks to promote valuing of
mathematics, we may need to be more explicit with students about contexts—what they mean,
the role they play in problem solving, and how the context may point to realistic applications for
the mathematical ideas (Jackson et al., 2012, 2013; Middleton & Jansen, 2011; Sullivan, et al.,
2003). It is also possible that other types of tasks such as those with community-based or social
justice contexts—which I did not observe in this study—have greater potential to support
students’ beliefs in the value of mathematics. I am not suggesting, of course, that all contextual
tasks will be engaging and/or motivating for all students, but that mathematics educators and

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curriculum designers need to be thoughtful about intent when it comes to writing and enacting
contexts.
Money contexts as motivators. The finding regarding students’ responses to money
contexts was somewhat surprising. Those contexts supported students’ valuing content in
Accentuate the Negative lessons (rational number operations), but the intent of these contexts
seemed to be to support students’ problem solving rather than catching students’ interests or
motivating them to learn. Consider an example typical of the contextual tasks in that unit:
“Jeremy had $7 on Saturday. He earned $5 walking his aunt’s dog and spent $10 going to a
movie at the theater. Find how much money Jeremy has now, and write a number sentence for
this situation.” From my perspective, there is nothing particularly interesting or relevant about
the storylines in these problems (I imagine most people do not normally write number sentences
when they encounter these problems in their everyday lives), and none of the lessons in this unit
were found to be highly engaging for students. Representing authentic applications was not the
intention of the authors, either. In the teacher’s guide for the unit, the authors state that weather,
money, and game score contexts are used to give students opportunities to build on informal
integer operation strategies (Lappan et al., 2013).
Yet, the connection students made between rational number operations and future careers
and everyday matters related to money seemed to promote beliefs in the value of the content.
One possible explanation is that the students were attracted to the accessible but seemingly adult
financial scenarios, which would be consistent with Jansen and Bartell’s (2011) finding that
some middle school students preferred adult contexts such as mortgage rates and rates on car
loans. Whatever the reason, it highlights the point that task contexts may have an effect on
students quite different than intended by teachers or curriculum authors.

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Contexts, Expectations of Success, and Intrinsic Value
When reflecting on lessons and the factors that influenced student engagement, students
generally focused on the level of ease and fun they had in the lesson and not on attainment or
utility value. Again, contexts did not seem to play a role in students’ reactions to lessons, though
these results were somewhat different based on their underlying motivation to learn. On one
hand, this trend may be unexpected in light of the findings presented in previous chapters.
Specifically, all high-engagement lessons included contextual tasks as the primary activity and
half of the low engagement lessons had only noncontextual tasks. Of course, these earlier
findings were for engagement across each group of focus students rather than for individual
student engagement.
On the other hand, students’ focus on feelings of success in lessons was not surprising
given extant literature. Brophy (2008) wrote that research has generally focused heavily on
expectations of success over how students value what they learn. Motivation theories such as
self-attribution theory (e.g., Weiner, 1985), self-efficacy theory (Pajares, 1996), and goal
orientation theory (e.g., Ames & Archer, 1988) also emphasize students’ beliefs about
themselves as learners of mathematics over beliefs about attainment or utility value. It might be
that existing theory and literature reflects what students tend to communicate about, or what is
most apparent to them, which seems to be how successful and able they feel in mathematics. The
trends in students’ responses are also consistent with a theme in literature on motivation to learn
that students need some expectation they can be successful in order to be motivated in the
mathematics classroom (Middleton & Jansen, 2011; Middleton & Spanias, 1999).
Together, the findings suggest that not all contexts have the same potential to engage
students, and that different students are attracted to different types or aspects of tasks and

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activities. This backs the assertion in the literature that assuming contextual tasks are motivating
for students simplifies students’ diverse experiences, goals, beliefs, and interests (e.g., Boaler,
1993, Chazan, 2000; Middleton & Jansen, 2011; Sullivan et al., 2003). We should not expect all
students to respond the same way to contexts. Though this study did not consider students’
racial, ethnic, and cultural experiences and thus cannot speak directly to arguments that contexts
could link content to these personally relevant experiences, I saw that even within this small
group of students who were diverse in terms of their motivation to learn mathematics attended to
different aspects of tasks and lessons. Although contexts may play a role in helping students link
school mathematics and the “real world,” of primary importance is developing personal meaning
of mathematical ideas through exploring their own solution strategies and recognition of “the
individual nature of students’ learning” (Boaler, 1993, p. 16). For low motivation students, this
personal meaning might involve a stronger understanding of the material, while for high
motivation students, this might reflect more active engagement and solving challenging tasks. Of
course, several other factors may influence how students with different levels of motivation to
learn respond to mathematics lessons, including motivation-related beliefs about the risks and
benefits associated with classroom participation (Jansen, 2006).
Not all students responded positively to all contextual tasks, but for these twelve students,
contextual tasks did not turn them off, either. They were not the “hated word problems” (Thomas
in Thomas & Gerofsky, 1997, p. 21) Students took contextual tasks in stride, accepting them as a
regular part of their experience in the mathematics classroom. Contextual tasks were a regular
part of their experience, as all of the focus students but Drayton were in their second year of
CMP. Some contexts were particularly enjoyable for most students (e.g., Wumps, Sascha’s bike
trip, and Bolda Cola advertising), but for the most part contexts did not seem to play a major role

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in how students viewed tasks. This is likely because of students’ frequent and consistent
encounters with contextual tasks as the main lesson activity in the CMP curriculum, versus
application problems that come at the end of assignments after “naked” number tasks. The fact
that students referenced their favorite lessons by context also indicates that these storylines
helped topics to “stick” in students’ minds. This suggests, again, that contexts helped anchor
students’ understanding, and reflects the anecdotal claim by the curriculum authors that CMP
students refer to concepts and strategies by the problems in which they encountered them.
(Lappan, Phillips, & Fey, 2007)
Conclusion
Students’ experiences with individual mathematics tasks are complex and varied.
Evidence from this study suggests that contexts influence how students solve problems and
engage in tasks. But there seems to be a weaker connection between contexts and what value
students see in mathematics or how successful they felt they were in learning. These findings
may be informative to teachers as they consider how to implement contextual tasks and to
curriculum designers as they consider the purposes and goals of contextual storylines in
mathematics tasks. In the next and final chapter I return to the research questions and propose
implications of the results discussed here on contextual tasks and student motivation to learn,
along with earlier results on the relationship between contextual tasks and lesson engagement.

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CHAPTER 7
CONCLUSIONS AND IMPLICATIONS
With this study, I set out to explore the relationship between contextual tasks and student
engagement and motivation to learn mathematics. Specifically, I intended to determine
characteristics of contextual tasks that were particularly engaging and those associated with low
student engagement in middle school classrooms in which contextual tasks were used on a
frequent and regular basis. To close the presentation of results and the discussion on the
motivational potential of contextual tasks, I return to the research questions and summarize the
main arguments and takeaway points from this investigation.
Potential of Contextual Tasks to Motivate and Engage
Returning to the Research Questions
To determine features of contextual tasks that were particularly engaging or disengaging,
the first research question was: What characterizes the contextual tasks as written in lessons
during which students show particularly high and low levels of engagement? As discussed in
Chapter 4, I found that some of the characteristics I had focused on, such as level of authenticity
and personalization, seemed to have little bearing on student engagement in tasks. Contextual
tasks in both the high and low-engagement lessons shared many of the same characteristics—
they were not personalized or community-based, some were adult-oriented and some were
student-oriented, and they represented a range in terms of authenticity and centrality. What
seemed more important in engaging students was the role contexts played in problem solving.
The high-engagement lessons had contextual tasks that were more open-ended modeling tasks in
which the context was at the root of the problem, where students were trying to answer questions
within the problem scenario. Determining the identity of the Mystery Teacher, identifying which

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characters were Wumps and which were impostors, analyzing advertisements for Bolda Cola—
these contexts seemed to be most engaging for students, allowing for flexible thinking about the
mathematics and the use of the context in tandem. Most of these tasks came early in units, when
students were just beginning to explore the mathematical ideas and building their understanding
on informal, everyday reasoning.
The second research question asked: What characterizes the enactment of contextual
tasks, including instructional practices and student attention to context, in lessons during which
students show particularly high and low levels of engagement? In Chapter 5, I discussed how in
high-engagement lessons, students attended more to context (i.e., a higher percentage of their
engagement was related to contexts), and they were more likely to link the core mathematical
ideas to the problem scenario. Student attention to context was related to teacher attention to
context—teachers talked about the contexts more, and in more varied ways, in higher
engagement lessons. This suggests students are sensitive to teachers’ cues as to how important or
interesting a context is.
Finally, I asked, What factors of motivation underlie student engagement relative to the
contextual tasks used in class? I explained in Chapter 6 how students talked about the value of
the mathematical content they learned, but what they attended to in tasks focused far more on
how easy (hard) or fun (boring) they found the material. These factors had little relationship to
whether or not lessons involved contextual tasks. Interestingly, though, students’ underlying
level of motivation did seem to have bearing on how students perceived problem contexts, and
how they perceived contexts relative to the value of the mathematics content. The highly
motivated students could come up with more relevant and varied examples of applications for the
content, and these applications were not as closely linked to contexts as they were for less

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motivated students. It was the neutral/mixed motivation group of students, however, who seemed
the most interested in the contexts themselves and talked about them when reflecting on tasks.
So What’s the Story on ‘Story Problems’?
Mathematics education literature presents different beliefs and perspectives regarding the
motivational potential of contextual tasks. Contextual tasks are often described as being
interesting or motivating to students (e.g., Chapman, 2006; Meyer, Dekker, & Querelle, 2001;
Middleton & Jansen, 2011; Mitchell & Carbone, 2011; van den Heuvel-Panhuizen, 2005), but
researchers and teachers have challenged that notion (e.g., Boaler, 1993; Chazan, 2000; Sullivan
et al., 2003; Verschaffel et al., 2000). The purpose of this study was to explore a more nuanced
understanding of the relationship between contextual tasks and student engagement and
motivation. What, then, can we conclude?
It is important to interpret the findings while keeping in mind the nature of the tasks in
the study. I considered lessons with tasks from the CMP curriculum—some were imaginative
and whimsical, others were realistic; some were based on experiences a young teen might
encounter, others were based on more adult experiences. Most of the tasks bore little
resemblance to classic story problems, but as part of a widely used, nationally published
curriculum, they also were neither personalized or based on community-specific scenarios. Thus,
although I observed a variety of contextual tasks, they were still a small subset of the tasks used
in school mathematics.
To summarize the whole of the findings and core arguments, contextual tasks of the types
observed in this study can be cognitively and emotionally engaging for students. This study also
offers some clarity about when and how contextual tasks might be most likely to engage students
in mathematics. Open-ended modeling tasks that offer multiple ways to draw on contexts while

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solving with contexts that are meaningful to students may have the most potential for promoting
engagement. The results for high- and low-engagement lessons in this study suggests that how
these tasks are enacted matters—students respond best when teachers attend to and highlight the
contextual storylines in a variety of ways through out the lessons. Students’ attention to
contextual tasks indicated that contexts can promote student engagement by supporting students’
entry into and work on the problems, by catching students’ interest, and by anchoring students’
understanding of the core ideas to something familiar to them. Based on the findings in Chapters
4 and 5, Figure 8 offers some questions one might consider when designing or using contextual
tasks if the context is intended to engage students.

If a context is intended to engage
students:
Is the context familiar or relatable
for students?
Are there multiple, flexible ways for
students to use the context in
problem solving?
Is the context an application of the
content or the foundation of
mathematical modeling?
How might a teacher draw attention
to the context throughout the lesson
(e.g., position students relative to the
context, use a visual aid)?

Figure 8. Questions to consider when designing or using contextual tasks to engage students
based on results from high- and low-engagement lessons.
Contextual tasks of the type observed in this study, however, might not have the same
potential to motivate students to learn or influence how they value mathematics. What students

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communicated about their responses to tasks indicated that contexts did little to promote positive
beliefs about the value of the content they learned for the tasks in the lessons I observed.
Influencing these beliefs may depend more on contexts that are personally relevant, community
based, or address social justice issues. In the next section, I address this limitation and suggest
ideas for further study.
The results suggest, as others have argued (Boaler, 1993; Walkington et al., 2012), that
the opportunities students have to make meaning and sense of the mathematics are of great
importance. The contextual features of tasks can serve to engage and motivate students to the
extent they support students in developing understanding that is personally meaningful. We
should not assume, then, that all contextual tasks can be engaging and/or motivating for students.
Nor should we assume that noncontextual tasks are intrinsically less engaging or motivating.
The findings do emphasize a need to be thoughtful about intent when it comes to creating
and using contextual tasks by clarifying the role of the context for ourselves (and, when
appropriate, for students) when planning and enacting a task (Sullivan et al., 2003; Jackson et al.,
2013). For instance, if the purpose of a context is to support student learning (e.g., temperature
change as a context for adding negative numbers), it should not necessarily be expected to
promote student engagement. Or, like some of the contexts in the study, a context could have
potential to engage students but might not promote positive motivation-related beliefs. The
purpose of a context can guide task development and implementation. Accordingly, Figure 9
outlines three potential purposes contexts serve in school mathematics as suggested in extant
literature (e.g., Carraher et al., 1985; CCSSI, 2010; NCTM, 1989, 2000, 2009; van den HeuvelPanhuizen, 2005, Verschaffel et al., 2000; Walkington et al., 2012). It builds on Figure 8 by
offering questions to consider when using contextual tasks to motivate students and support

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students’ problem solving. These considerations do not follow directly from the results of this
study, as I found little relationship between contextual tasks and students’ motivation-related
beliefs, and investigating students’ problem solving and learning was outside the scope of this
study. Rather, the questions are based on implications of theoretical and empirical literature (e.g.,
CCSSI, 2010; NCTM, 1989, 2000, 2009; Jackson et al., 2012, 2013; Wigfield & Eccles, 2000).

What purpose is the context intended to serve?
Engage students
Is the context familiar
or relatable for
students?
Are there multiple,
flexible ways for
students to use the
context in problem
solving?
Is the context an
application of the
content or the
foundation of
mathematical
modeling?
How might a teacher
draw attention to the
context throughout the
lesson (e.g., position
students relative to the
context, use a visual
aid)?

Motivate students to
learn mathematics

Support student
problem solving

Does the context help
students understand the
importance of learning
the mathematics to
solve relevant
problems?

Will all students
understand the
contextual language?

Does the context help
students see the utility
value of the
mathematics they are
learning?
Does the context help
students see the
attainment value of the
mathematics they are
learning?

To what extent will
prior experience help
students understand the
contextual features of
the scenario?
How might a teacher
support students in
using the context
appropriately during
problem solving?

Figure 9. Purposes of contextual tasks and questions to consider when using them to serve these
purposes. The first column follows from the findings of the study. The second and third columns
are italicized to show these are my suggested questions for consideration based on extant
literature.

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The purposes of contextual tasks included in Figure 9 are not mutually exclusive. We
know, for example, that students’ learning experiences are cumulative and build into long-term
motivation patterns (see Middleton & Jansen, 2011, for a discussion of this topic), so frequent
experiences where students are engaged in mathematics because of an interesting context might
lead to increased motivated to learn. Or, if a context supports students’ problem solving, they are
more likely to engage in the lesson (e.g., Rivet & Krajcik, 2008).
Limitations and Directions for Future Research
Of course, one needs to consider the results in light of limitations of the study,
particularly the choice to use CMP classrooms. CMP afforded confidence that I could observe
many contextual tasks enacted over several different mathematics concepts. Yet, the types of
contexts encountered in the observed lessons represented just a part of the full spectrum of
contextual tasks used in school mathematics. On one hand, many CMP tasks are somewhat of a
“new breed” of contextual tasks that reflect few of the characteristics of traditional story
problems. On the other hand, I expected teachers to use more personalization or to refine the
written tasks into more community-based contexts based on extant research (Wernet, 2011), and
these practices rarely occurred. This may help explain why I found that different types of
contexts did relate to different levels of student engagement. Student motivation might have been
better promoted through different types of contexts, such as personalized or community-based
contexts, by supporting beliefs in the attainment and utility value of the mathematics.
Furthermore, the findings have limited generalizability beyond CMP classrooms. Most of
the students in this study were in their second year using CMP and were used to working on
contextual tasks as a frequent core activity in the mathematics classroom. Students in classrooms
with more traditional curricula, who do not engage in contextual tasks as the main lesson activity

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on a regular basis, may respond quite differently to these types of tasks. Jackson et al. (2013)
made a similar argument about their findings regarding the relationship between task launches
and summary discussions—that the ways teachers launch tasks (including their attention to
contexts and contextual vocabulary) is probably more important in cognitively challenging tasks
aimed at developing conceptual understanding. Thus, future research should include classrooms
using diverse curricula, including teaching for social justice lessons and more traditional
mathematics curricula. Also, participants should represent more diverse age, racial, ethnic and
economic demographics than those in this study. The results and questions raised in this smallscale study should be explored with larger number of students or over a greater number of
lessons to lessen effects of the numerous factors related to student engagement and motivation.
Part of this broader research might include an investigation of the intersections between
student motivation and engagement in contextual tasks and student characteristics such as future
goals, general beliefs about schooling, identity, gender, race, ethnicity and socioeconomic status.
This would allow us to answer open questions about how contexts that are immediately
personally relevant for students might help to connect mathematics content to racial, ethnic, and
cultural experiences (Nasir et al., 2008; Tate, 1996) or how contextual tasks that do not relate to
students’ experiences and goals may alienate and exclude students (Middleton & Jansen, 2011;
Sullivan et al., 2003). How can we interpret, for example, the fact that Sophia twice generated
applications for mathematics related to working at McDonalds, while Elijah and Jeff (both high
motivation students) offered examples such as “sales rep,” packaging engineering, banking, and
drafting? Similarly, though it was not the focus of this study, instances in which teachers brought
attention to students’ work seemed to promote positive responses to lessons. Analyzing the data
with a different theoretical lens such as self-efficacy theory or identity theory would likely yield

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different perspectives on students’ classroom experiences with specific mathematics tasks.
Potential studies could more closely investigate engagement with contextual tasks and student
motivation mediated by other student characteristics as well as interactions with teachers.
Curriculum analysis is another direction for future research. As discussed below,
curriculum designers should clearly communicate the intended purpose(s) of problem contexts
and support teachers in making instructional decisions around contexts. My experience teaching
CMP indicates that the curriculum provides some support in implementing contexts, but analysis
of the teacher materials was outside scope of this study. Comparing mathematics textbooks in
terms of the guidance provided teachers for using contextual tasks would provide insight on the
role of teachers’ guides in whether and how teachers attend to contexts. This research could
ultimately inform mathematics teacher education.
Finally, this study focused on cognitive and emotional engagement, but certain engaged
behaviors are likely more productive or of higher quality than others, and this study did not
capture those nuances. For example, a student posing a relevant question because she does not
understand how to start a problem may be qualitatively different than contributing a meaningful
mathematical idea, and I counted both as evidence of cognitive engagement. Future studies
might differentiate between specific engaged behaviors for a more in-depth understanding of the
relationship between working on contextual tasks and student engagement.
I propose three potential studies as the next steps to follow up on this research. The first
is really a set of studies to check the generalizability of these results in a broader range of
settings. Multiple-case studies analogous to this one (comparing student engagement across
lessons) could be conducted in elementary and high school mathematics classrooms in which
contextual problems are a significant part of the curriculum to see if results on student

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engagement and motivation extend to other age groups. Similarly, a multiple-case study in
classrooms using more traditional mathematics curricula would offer insight into students’
experiences with a range of contextual tasks. Second, an action research study would be
appropriate to address the original intended question of this study regarding differences in
context types such as personalized/non-personalized and student/adult-focused. In my own
classroom, for instance, I could modify task contexts according to my students’ interests or
community needs without introducing other novel aspects of tasks and thus more directly
investigate the relationship between certain characteristics of contexts and student engagement.
Lastly, I suggest a longitudinal study of a small number of students that addresses the
relationship between aspects of students’ identities, engagement and motivation factors, and
students’ experiences with contextual tasks.
Implications for Research and Practice
Research
This study offers a set of research tools for measuring student engagement in individual
mathematics lessons that were reasonably valid, reliable, and efficient. The student and teacher
surveys and observation tools developed have potential for future research on how tasks or other
features of lessons relate to student engagement, an area of research with many questions to
investigate. I encourage others to refine and revise these tools to serve specific purposes of
prospective studies, including a larger scale for the lesson-specific surveys to allow more
variation in student and teacher responses.
Further, the result of identifying high- and low-engagement lessons can inform
considerations of data sources when studying student engagement. Specifically, there was a
fairly low association between the student reports and the counts of evidence of student

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engagement across the 14 lessons at each school. The literature on researching student
engagement calls for the use of multiple data sources, including student reports and observations,
to provide a detailed and multifaceted interpretation of engagement (Fredericks et al., 2004;
Fredericks & McCloskey, 2012). My findings, however, highlight the importance of: a)
identifying which vantage point(s) and measures will be weighted most heavily in case of
disagreement, and b) carefully selecting lessons when using observations. If observations are
used to make comparisons across lessons, then researchers should strive to select lessons that are
similar in terms of structure, timing, and activities. Counts of engaged behaviors seemed more
sensitive than other measures to differences in these factors, including, for example, the amount
of time spent in small groups.
My experience conducting the research also highlighted the inherent difficulty of gauging
engagement via observation. For example, I had noted Felix’s particularly high engagement in a
Wumps lesson while observing because he appeared determined to finish graphing, working into
the whole-class summary. In the interview, however, I learned that he had been drawing
mustaches on his characters and not following the discussion. Again, this should emphasize the
value of multiple data sources to make the best possible interpretations of student behavior and
responses to tasks. Moreover, data collection technologies such as video cameras in eyeglasses
or hat brims (Chval, Pinnow, & Thomas, 2014) or Livescribe Smartpens™ might be appropriate
for capturing student engagement in classroom observations.
Curriculum Design
The implications of the study results for curriculum design are similar to those for
instruction, but begin with determining whether or not a context is appropriate for a problem
given the learning objectives of a task, lesson, and unit. If a task is to have a context, it should

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serve a clear purpose (or purposes). If that purpose is to promote student interest or engagement,
one needs to consider the nature of the context and how students will be expected to use it. The
fact that students were most engaged in lessons with open-ended tasks that offered opportunities
for students to draw on the context in multiple ways while problem solving suggests that giving
these types of tasks a regular role in mathematics curricula may promote student engagement.
Many modeling tasks fit these characteristics. Mathematical modeling is a Standard for
Mathematical Practice in the Common Core State Standards, making it a recommended practice
throughout the curriculum and grades. I encourage curriculum writers to consider the potential of
modeling in promoting not only mathematical proficiency but also student engagement. The least
engaging lessons in this study had tasks that were noncontextual, had contexts peripheral to
problem solving, or were application problems that closely resembled more traditional story
problems. Such contexts might be completely reasonable in achieving the mathematical goals of
a task, but the data challenge the notion that such contexts would go far toward promoting
students’ cognitive and emotional engagement.
The potential of contextual tasks to engage students is boosted when the contexts are
meaningful to students. Results suggest that problem storylines do not need to be realistic, but
should at least be accessible to students. Every effort should be made to ensure that contexts are
imaginable (Van den Heuvel-Panhuizen, 2003) for diverse groups of students in the appropriate
age range. Also, although it was not a focus of the study, the most engaging tasks had strong
visual elements in the form of photographs, cartoons, graphs, or introductory launch videos. So
these elements seemed to emphasize and clarify contexts in ways that supported student
engagement.

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Teachers’ attention to context can also enhance the engagement potential of contexts; in
the next section, I discuss specific strategies we can employ to make contexts more meaningful
for students. Thus, like teachers with their students, curriculum designers may need to make the
purposes of contexts explicit for teachers to guide their decisions about implementation.
Teachers’ decisions regarding context would be easier to make and likely be more effective with
clear communication from the authors about the intent of context as written.
Instruction
The implications for mathematics teachers and the teacher educators who support them
follow directly from the findings and central arguments. Engagement and motivation to learn are
issues directly related to our practice. As I have been reminded of on my return to the middle
school classroom after finishing this project, we can lead students to the “perfect” task, but we
cannot make them engage with it. This study offers support for one promising way to engage
students in thinking about mathematics and even enjoy the process. Incorporating open-ended
contextual tasks with a meaningful storyline can promote student cognitive and emotional
engagement, even when implemented on a regular basis (thus decreasing the effect of novelty).
Further, contexts do not have to be personalized or community-based to promote widespread
engagement. Imaginative contexts have potential to engage, but any context needs to be
accessible to and meaningful for students. Because no widely published curriculum can match all
students’ experiences and interests, some of the responsibility to make contexts accessible and
meaningful falls on teachers.
When preparing to teach a contextual task, the first thing to consider is the intended role
of the context relative to the mathematical goals of the problem as in Figure 9, and determine the
extent to which it helpful or necessary for students to attend to the context. Once a teacher has

195

decided the role the context should play, they can plan how to launch and use the context
throughout the lesson based on knowledge of their students. The purpose of the context may
need to be made explicit to students, through statements such as, “I know this situation is not
very realistic, but it will help you think about how to write an equation,” or “This problem shows
how important exponential functions are in understanding the spread of disease.” If the goal of
the context is to promote students’ motivation to learn, teachers should highlight what the
context shows about the usefulness and application of the content—a practice I rarely observed
in this study, and the results showed that contexts did not necessarily help students (especially
those who already had low motivation to learn) note meaningful applications of the mathematics.
Being explicit about the nature of the problem context can also help toward resolving
equity issues presented by tasks (Sullivan et al., 2003) by supporting students’ access to the
mathematics in the problem. Teachers can identify what terms or aspects of the context may need
to be discussed and what prior knowledge students bring to bear (Jackson et al., 2012, 2013).
These decisions need to be made as part of a complex web of instructional decisions. Jackson et
al. (2013) put it well:
The scenarios could lend themselves to extended talk about a number of contextual
features, which many or may not be critical to solving the task...Clearly, time is of the
essence in classroom instruction; therefore, teachers need to make judgments regarding
what to focus on in the setup. These judgments must be made against a clear set of
mathematical goals for instruction and knowledge of what is likely to be unfamiliar
(contextually, mathematically, and linguistically) to students. (p. 656)
These decisions about how to draw on the context extend past the launch phase. If the purpose is
to support student engagement or motivation to learn, the results indicate implementation of
contexts is important across the lesson. Drawing students’ attention to the context in a variety of
ways in class discussions and while supporting student work on the task can influence the way
students use and attend to the context, and promote their engagement in tasks overall.

196

Final Thoughts
Contextual tasks, in a variety of forms, have probably been a part of mathematics
education since its inception. They are unlikely to go anywhere, as leaders in mathematics
education have advocated for increased emphasis on mathematics problems drawn from realworld scenarios and tasks that promote mathematical modeling and quantitative reasoning (e.g.,
NCTM, 2000; CCSSI, 2010). Researchers, teachers, and curriculum designers need to continue
to think about how to support students’ work in contextual tasks if that work is to lead to robust
understanding. In particular, we should consider the issues of how to promote students’
productive engagement in thinking about these problems, and how students’ experiences with the
tasks might motivate them to learn mathematics.
If nothing else, the results presented here provide an existence proof that students can
engage meaningfully in contextual tasks in a way that supports positive affect. Yet, we should
always be cognizant of the purpose of developing and using contexts for school mathematics,
recognizing that not all contextual tasks have the same potential for promoting all students’
engagement or motivation. Refining our intentions and decisions involving contextual
mathematics tasks might ultimately lead to more students choosing to learn and use mathematics,
as they “learn it with understanding and see its beauty and the possibility of applying it to
matters that interest them, including games as well as more practical matters” (Willoughby,
2010, p. 83). For me, that is always the goal.

197

APPENDICES

198

APPENDIX A
SAMPLE LESSON MODIFIED BY MS. PEARSON

2.3 – Scale Factors

Name ______________________ H r__

Stretching and Shrinking
Facilitator: Make sure everyone knows the instructions and that everyone does all parts of the
problem. It will help to read all the directions first before beginning
Recorder/Reporter: Your group needs to organize a poster with all your results. Your poster needs
to have everyone’s work, be well organized, and use math tools, arrows, and/or colors to show your
thinking.
Resource Manager: Manage materials needed. Make sure all questions are group questions! Don’t
let your group stay stuck – call me over!
Team Captain: Make sure everyone understands the solution to the problem. No talking outside
your group!
Your group will work together to explore properties of similar figures and their scale factors. On the
resource paper, there are many rectangles and triangles. Use them to answer the following questions.
Put your answers on your notepaper. You will create a group poster after.
The rectangles represent the mouths of some Wump family members and some imposters. The noses are
represented by the triangles.
1. Decide which pairs of rectangles are similar and find the scale factor for each pair.
2. Decide which pairs of triangles are similar and find the scale factor for each pair.
3. Pick one pair of similar rectangles and one pair of similar triangles. Use the scale factors you found
from #1 and #2 to predict the relationship between the perimeters for each pair of similar shapes.
Explain. (You do not need to find the actual perimeter, just talk about the relationship between the
perimeters of the two similar shapes.)
4. After studying the noses and mouths in the diagram, Marta and Zack agree that rectangles J and L
are similar. However, Marta says the scale factor is 2, while Zack says it is 0.5. Who is correct?
How would you describe the scale factor so there is no confusion?
5. On graph paper, draw a rectangle that is similar to rectangle J, but is larger than any rectangle shown
in the diagram. What is the scale factor from rectangle J to your rectangle?
6. Draw a triangle that is not similar to any triangle shown in the diagram. Why is it not similar?
7. Draw a rectangle that is not similar to any rectangle shown in the diagram. Why is it not similar?
8. Explain how to find the scale factor from one figure to a similar figure.
When EVERYONE is finished, organize your work into a large poster. You can glue the shapes
directly onto the poster. Your poster should clearly communicate your solutions and justifications!
Make connections between ideas, the shapes, and your justifications with color, arrows, or other tools.

199

2.3 – Scale Factors

Name ______________________ H r__

Stretching and Shrinking

Figure A1. Accompanying resource page distributed by Ms. Pearson.

200

APPENDIX B
BELIEFS AND VALUES SURVEY - FALL
Please answer all of the following questions (17 total). Remember, your responses will not be
shared with other students and will be anonymous for research.
For questions 1-11, circle the ONE number that best describes your beliefs about math.
1) How good are you in math?
1
Not at all
good

2

3

4

5

6

7
Very good

2) If you were to list all the students in your class from the worst to the best in math,
where would you put yourself?
1
One of the
worst

2

3

4

5

6

7
One of the
best

3) Some kids are better in one subject than in another. For example, you might be
better in math than in reading. Compared to most of your other school subjects,
how good are you in math?
1
A lot worse
in math
than in
other
subjects

2

3

4

5

6

7
A lot better
in math
than in
other
subjects

5

6

7
Very well

4) How well do you expect to do in math this year?
1
Not at all
well

2

3

4

201

5) How good would you be at learning something new in math?
1
Not at all
good

2

3

4

5

6

7
Very good

6) Some things that you learn in school help you do things better outside of class, that
is, they are useful. For example, learning about plants in science might help you
grow a garden. In general, how useful is what you learn in math?
1
Not at all
useful

2

3

4

5

6

7
Very
useful

7) Compared to most of your other activities, how useful is what you learn in math?
1
Not at all
useful

2

3

4

5

6

7
Very
useful

5

6

7
Very
important

8) For you personally, being good in math is:
1
Not at all
important

2

3

4

9) Compared to most of your other activities, how important is it to be good at math?
1
Not at all
important

2

3

4

5

6

7
Very
important

10) In general, I find working on math assignments:
1
Very
boring

2

3

4

202

5

6

7
Very interesting
– fun

11) How much do you like doing math?
1
Not at all

2

3

4

5

6

7
Very much

For questions 12 to 15, think about the statement and circle the ONE number that best represents
your level of agreement.
12) No matter how much math ability you have now, you can always change it quite a
bit.
1
Strongly
disagree

2
Disagree

3
Mostly
disagree

4
Mostly agree

5
Agree

6
Strongly
agree

13) You can learn new things, but you can’t really change your basic math ability.
1
Strongly
disagree

2
Disagree

3
Mostly
disagree

4
Mostly agree

5
Agree

6
Strongly
agree

14) You have a certain amount of math ability, and you can’t really do much to change
it.
1
Strongly
disagree

2
Disagree

3
Mostly
disagree

4
Mostly agree

5
Agree

6
Strongly
agree

15) No matter who you are, you can change your math ability a lot.
1
Strongly
disagree

2
Disagree

3
Mostly
disagree

4
Mostly agree

203

5
Agree

6
Strongly
agree

For 16 and 17, answer as completely as you can.
16) What are some things you’re interested in, or that you like to do? These could be
hobbies, favorite pastimes, or other things you like to do in or out of school.

17) Describe your favorite math lesson from last year. What did you like about it?
Why?

204

APPENDIX C
OBSERVATION PROTOCOL FOR STUDENT ENGAGEMENT
Adapted from Helme and Clarke (2001) with reference to Nicol and Crespo (2005), Stipek et al.
(1998), Brantlinger (2007), Fredericks et al. (2004)
Behaviors to capture and code in fieldnotes
Student behaviors characteristic of cognitive engagement, with description or examples*:
•

Student contributes mathematical idea (e.g., posing conjectures, offering a solution
strategy, making a generalization)

•

Student(s) self-monitor while working on a task

•

Student concentrates on work (resists distractions and interruptions)

•

Student gestures while working or talking (externalizing their thought process, Helme &
Clarke, 2001)

•

Student poses question seeking information and/or feedback (e.g., how did you get
negative ten?; is it okay if we make a table? Non-example would be, “how do you do
this?” when student has put little effort into the task.)

•

Student completes peer or teacher utterance (e.g., teacher says, “we know it’s linear
because the rate if change is…” and student fills in “constant.”)

•

Student gives directions or information about the task to a peer (e.g., we’re supposed to
go through number five; I think we should graph first then find the equation)

•

Student justifies an argument or explains procedure or reasoning (e.g., responds to
teacher “why” or “how did you get that” question)

•

Student shows reflective self-questioning (e.g., how did I get that?; I got 60, wait, what
should the units be?)

•

Student perseveres in the face of challenge (e.g., says, “I don’t get this” or, “this is too
hard” but continues working on the task)

•

Student seeks challenge (e.g., continues working on task after a solution is reached,
trying something beyond what is asked for, generating an additional representation, etc.)

*Also indicate if these behaviors are related to context rather than mathematical content

205

Student behaviors characteristic of emotional engagement, with descriptions or examples:
•

Student offers encouragement to a peer (e.g., we can do this!)

•

Student(s) begin working on a task with confidence and/or enthusiasm (e.g., starting
work right away, not waiting for teacher to visit table)

•

Student makes a positive emotional statement about a task or the lesson (e.g., this is fun;
this is interesting; wow, class really went fast today; I like this problem)

•

Student(s) show visible excitement when working on or talking about the content through
gesturing or tone of voice (e.g., animatedly trying to talk over one another during a
discussion)

Real-time assessment of student engagement
For most students in the class, the best description of their cognitive engagement is:
1 – disengaged (sleeping, talking about topics not related to the mathematics, being disruptive,
not working on task or visibly not paying attention to teacher)
2 – somewhat engaged (appear to be working on the task or paying attention to the discussion
with some distraction, but not actively taking part in discussion)
3 – engaged (actively participating in class discussion, talking with peers about the mathematics
task, generally “on task”)
4 – elaborately engaged (students are sharing ideas, justifying arguments, or explaining their
reasoning with two or more consecutive sentences)
For most students in the class, the best description of their emotional engagement is:
1 – active resistance (negative emotion)
2 – boredom
3 – neutral or interested
4 – eagerness, enthusiasm, excitement

206

APPENDIX D
LESSON-SPECIFIC STUDENT SURVEY
Adapted from Cai & Merlino (2011)

I am interested in learning what you thought and how you felt about the lesson today.
Please take a minute to think about the following questions and write how you truly feel.
Your response is confidential, and there is no right or wrong answer!

1) Today’s math lesson felt like watching or participating in a sporting event such as
________________________________ because ________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
Example: In today’s math lesson, I felt like I do when I’m watching a sporting event like
baseball, because I like baseball but there were many slow and boring parts.

2) Rate the level of effort and thinking you put into today’s lesson, where 1 means you
weren’t really paying attention or trying in class, and a 4 means you thought hard about
what you were learning and tried your best to do the work.
1
Low

2

3

4
High

3) Rate how much you enjoyed today’s lesson, where 1 means you hated the lesson, and
a 4 means you thought the lesson was interesting and exciting.
1
Low

2

3

207

4
High

APPENDIX E
LESSON-SPECIFIC TEACHER SURVEY
Rate the level of effort the students seemed to put into today’s lesson relative to the “average”
lesson, where 1 means you think they weren’t really paying attention or trying in class, and a 4
means they were thinking hard about what you were teaching and they were trying their best to
do the work.
1
Low

2

3

4
High

Rate how much the students seemed to enjoy today’s lesson relative to the “average” lesson,
where 1 means they seemed to hate the lesson, and a 4 means they seemed interested and
excited.
1
Low

2

3

208

4
High

APPENDIX F
FOCUS GROUP INTERVIEW PROTOCOL
We’re going to talk a while about your lesson today. I understand it might seem strange to think
and talk generally about today’s class. Just try to remember what it was like to learn math today,
and be honest! Anything you say here is confidential.
1) If a friend who is in the same class later in the day (or a parent, etc.) asked you what math
class was like today, what would you tell him or her?
[Follow up on this question with the following, as appropriate: What would you tell them
you learned? Was class pretty normal today, or particularly boring, interesting, etc.?
Why?]
2) What (food, sporting event, movie) did you compare the lesson to?
[Follow up on this question with the following, as appropriate: What was it about the task
[name the specific problem(s)] today that made you feel that way?]
3) What was your favorite part of class today? Least favorite? Why?
4) How beneficial (or useful) do you think the stuff you learned today will be for you, either
in this class, future classes, or life outside of school? Why?
[Remind students about specific content from the lesson, as needed.]
5) What did you rate your effort for the day?
[Follow up on this question with the following, as appropriate: Do you mind telling us
why you answered that way? Was there something about the problem(s) [name the
specific problem(s)] today that affected how hard you worked?]
6) What did you rate your enjoyment for the day?
[Follow up on this question with the following, as appropriate: Do you mind telling us
why you answered that way? Was there something about the problem(s) [name the
specific problem(s)] today that affected how much you enjoyed class?]

209

Table F1
Summary of Lessons Preceding Interviews
Lesson

Lesson Content

Description of Context

Pine River 2

Interior and exterior
angles

None

Pine River 5

Subtracting positive and
negative integers

Money contexts (spending
and earning) were used in
some problems

Felix absent

Pine River 7

Multiplication and
division of integers

None

All present

Pine River 10

Making similar figures

“Wumps” characters-identifying which figures
are similar to Mug Wump
and are in the Wump
family, and which are
impostors

All present

Pine River 14

Applying properties of
similar shapes

Determining heights of tall
objects (in this case, a
clock tower) using
shadows

All present

Southpoint 2

Scale factors

Continuing the Wumps
context, analyzing the
characters’ noses and
mouths

Jacob did not participate

Southpoint 5

Comparing ratios

Julia and Mariah are
making chocolate cookies
for their fellow campers.
They consider four
different mixes; students
need to determine which is
the most “chocolatey”

All present

Southpoint 7

Finding rates

Sascha is going on a bike
trip with three legs.
Students are told how long
each leg is and how many
minutes it took to finish.
They find and compare
unit rates and think about
how fast they would need
to ride to race Sascha.

All present

210

Focus group
attendance
All present

Table F1 (cont’d)
Southpoint 10

Using the chip model for
integer addition and
subtraction

Southpoint 13

Operations on integers

A few tasks in the
homework reviewed use
money contexts (spending,
earning, owing money);
student spontaneously
references having a certain
number of abstract
“points”
One problem used
temperature context;
spontaneously used money
as support in operating
with decimals

211

All present

All present

APPENDIX G
LESSON ENGAGEMENT CODEBOOK
Table G1
Lesson Engagement Codebook for Evidence of Student Engagement from Lesson Transcripts
Student behaviors characteristic of cognitive
Examples from data
engagement, with description or examples
Student contributes mathematical idea (e.g., posing
McKenna raises hand and offers, “I got 105 degrees,” contradicting
conjectures, offering a solution strategy, making a
another student’s answer. T: So what did you get for the sum? McKenna:
generalization)
I got 525. (Site A, Observation 1)
Student(s) self-monitor while working on a task (e.g., Jacob: I said all the angle measures are the same, but—T: Which Hat are
“wait, that’s not right”; “I think that makes sense”; two
you talking about? Jacob: All but Hat 4. Most of them are the same.
or more students exchange papers to check own and/or
Actually, no, never mind, I messed up. (Site B, Observation 1)
others’ work)
S4: Yeah, but there’s got to be three red chips though. Emily: This is
like—I’ve got to think about it.
This should be explicitly noted in the comments or transcript
Student concentrates on work (resists distractions
and interruptions)
This is should be explicitly noted in the comments or transcript.
Student gestures while working or talking
(externalizing their thought process, Helme & Clarke,
2001)
Adelyn: I don’t understand how you pair [the hats] […] So what do you
Student poses question seeking information and/or
feedback (e.g., how did you get negative ten?; is it okay say then, that they have the same angles. (Site B, Observation 1)
if we make a table? Non-example would be, “how do
NON-Examples: McKenna: I have a question on C. I don’t get it [….]
you do this?” or “has anyone done A4 yet?” when
Did anyone get A4 yet? I didn’t get it. (Site A, Observation 1)
student has put little effort into the task.)
Asking (in whole class) to go over a homework problem, or general
question about an assessment (e.g., can we use our notes on the exam?)
Sophia to McKenna: What are you working on right now? (Site A,
Observation 14)

212

Table G1 (cont’d)
Student completes peer or teacher utterance or
answers direct question from teacher (doesn’t
require explanation or elaboration) (e.g., teacher
says, “we know it’s linear because the rate if change
is…” and student fills in “constant.”)
Student gives directions or information about the
task to a peer (e.g., we’re supposed to go through
number five; I think we should graph first then find the
equation)
Student justifies an argument or explains procedure
or reasoning (e.g., responds to teacher “why” or “how
did you get that” question)

T: What do you notice about which polygon has the smallest angle?
McKenna: The triangle. (Site A, Observation 1)

Elijah: You’re trying to find if they’re similar, they have the same
general shape. […] So you’re seeing if going from Mug’s hat to Hat 1, if
they have the same angles. Compare the angles and side lengths.
Compare each hat to Mug’s hat. (Site B, Observation 1)
NON-Examples: Jeff: I’m on A4 right now. (Site A, Observation 1)
T: Well the same, right? Why would they be the same? McKenna:
Because the sun is shining on both of them at the same time. And so the
shadows are going to be the same, since it’s a similar shape. (Site A,
Observation 14)
Drayton: A 25-foot shadow? […] That’s, like, bigger than this room!
(Site A, Observation 14)
Lilly: Ugh, this is so confusing…the last one should not be 1440! I’m so
confused. [Continues to work with McKenna on the table.]

Student shows reflective self-questioning (e.g., how
did I get that?; I got 60, wait, what should the units be?)
Student perseveres in the face of challenge (e.g., says,
“I don’t get this” or, “this is too hard” but continues
working on the task)
Student seeks challenge (e.g., continues working on
task after a solution is reached, trying something beyond
what is asked for, generating an additional
representation, etc.)
Student behaviors characteristic of emotional
engagement, with descriptions or examples
Student offers encouragement to a peer (e.g., we can Jeff: See? [McKenna] did do it right! (Site A, Observation 9)
do this!)
This should be explicitly noted in the comments or transcript.
Student(s) begin working on a task with confidence
and/or enthusiasm (e.g., starting work right away, not
waiting for teacher to visit table)

213

Table G1 (cont’d)
Student makes a positive emotional statement about
a task or the lesson (e.g., this is fun; this is interesting;
wow, class really went fast today; I like this problem)
Student(s) show visible excitement when working on
or talking about the content through gesturing or
tone of voice (e.g., animatedly trying to talk over one
another during a discussion, raising or waving hand,
showing surprise or agitation)

Drayton: This is easy! Felix: I’m done, I’m done! (Site A, Observation
14)
This should be explicitly noted in the comments or transcript.
Ex: A student makes an excited statement about watching a launch video
or cranes neck to see what’s on the board.
Emily: I didn't know you can do that! I would have thought you have to
start with one color! (Site B, Observation 9)

Notes
When there is an exchange between two people: Often, when two (or more) students interact or a teacher and student act, there are a
short series of statements and questions in the event. I counted each of these “events” as single pieces of evidence of cognitive
engagement. So, for example, a teacher asks a student a question, the student responds, the teacher asks a follow up question, the
student responds—this counts as ONE example of the student explaining their reasoning or responding to the teacher’s question.
When this is unclear: If this is unclear, think about—is this student engaged in this moment? How do I know? Did the student
just prepare a single answer to a teacher’s question, or a single question to ask someone? Or, do they persist in engaging—do
they ask a relevant follow-up question or continue explaining an idea further, worthy of another count of engagement
evidence?
When there is an exchange between two people involving different types of engagement: If an event such as the one just
described starts with the student asking a question for information and/or feedback, the teacher responds and asks a question, the
student responds—that could count as two separate pieces of evidence of engagement—asking the question, then responding
appropriately to the teacher’s question.
When a student verbalizes thinking for no one in particular: When students are working individually “in parallel” and verbalizing
thinking, this can count as explaining their procedure, self-monitoring, etc.—even if no one appears to be listening (consistent with
Helme & Clarke, 2001).

214

When it’s noted that a student raises their hand: If a student raises their hand to offer response to teacher’s question, count as
emotional engagement regardless of whether or not they are called on.
If multiple students raise hands to respond: If in one instance, multiple focus group students raise, hands, note names and
count as one piece of evidence. If different students raise hands in response to different questions, count as different evidence
of student engagement.
If multiple students respond in unison: Multiple FG students respond to the teacher or to each other in unison—this counts as one
piece of evidence, but note both students’ names because it can count for each student.
If it’s not clear whether engagement is related to content or context: Only mark content or context if this is clear—for example,
when students are persisting or concentrating/resisting distraction, it is not possible to clearly identify if they are focusing on content
or context.
When students comment on something context-related: When students make comments on something context-related, it could be
an example of self-monitoring or reflective self-questioning (see Drayton’s example of this above) if it seems students are checking or
making sense of their solution. If a student contributes to a discussion (e.g., completes a teacher utterance) on the context and it is
closely linked to the mathematics in the task, also counts as CE. But if it’s just a random comment, it might not count as engagement
at all, or if it is an indication they’re really “into” the task, it’s likely emotional engagement.
When a student says something task-related (might only be context related) that’s indicated in the fieldnotes with an
explanation point: Mark as evidence of emotional engagement when it falls into the category of “positive emotional engagement” (as
most research describes it—enthusiasm, excitement, confidence). Multiple students may make these “enthusiastic statements” within
an interaction segment or conversation. Count each as a new count of emotional engagement.
When students exchange papers to check one’s own or others’ work: Mark as one count of evidence of cognitive engagement
(self-monitoring). Though multiple students are involved, it is usually hard to discern if one student (and who) initiated.
When a student checks in with peers about status on a task WITH evidence they have been working on task: It can be
ambiguous because the student’s intent may be unclear. But can assume they want to know if they are on track either in terms of time
or mathematically, so count as self-monitoring.

215

APPENDIX H
ENGAGEMENT ANALYSIS CHARTS FOR PINE RIVER:
MRS. MEYER’S CLASS

Figure H1. Dotplots of student and teacher engagement ratings over lessons at Pine River.

216

Average Engagement Counts Weighted per hour vs Average
Focus Student Response
160.00
140.00
120.00
100.00
80.00
60.00
40.00
20.00
0.00
0

1

2

3

4

5

6

7

8

Figure H2. Scatterplot showing average weighted engagement counts per hour versus the
average focus student engagement rating at Pine River

217

APPENDIX I
ENGAGEMENT ANALYSIS CHARTS FOR SOUTHPOINT:
MS. PEARSON’S CLASS

Figure I1: Dotplots of student and teacher engagement ratings over lessons at Southpoint.

218

Average Engagement Counts Weighted per hour vs.
Average Focus Student Response
160.00
140.00
120.00
100.00
80.00
60.00
40.00
20.00
0.00
0

1

2

3

4

5

6

7

Figure I2: Scatterplot showing average weighted engagement counts per hour versus the average
focus student engagement rating at Southpoint.

219

APPENDIX J
COMPARING AND SCALING 3.2 MODIFIED BY MS. PEARSON

3.2 – Finding Rates

Name ______________________ H r__

Comparing and Scaling – Goal 8
Facilitator: Make sure everyone knows the instructions and that everyone does all parts of the
problem. It will help to read all the directions first before beginning
Recorder/Reporter: Your group needs to organize a poster with all your results. Your poster needs
to have everyone’s work, be well organized, and use math tools, arrows, and/or colors to show your
thinking.
Resource Manager: Manage materials needed. Make sure all questions are group questions! Don’t
let your group stay stuck – call me over!
Team Captain: Make sure everyone understands the solution to the problem. No talking outside
your group!
Your group will work together to explore finding different unit rates. Answer the questions on your
notepaper. You will need a copy of your own answers. At the end, you will create a poster.

Sascha cycled on a route with different kinds of conditions. Sometimes he went uphill; sometimes he went
mostly downhill. Sometimes he was on flat ground. He stopped three times to record his time and distance
for the section he just biked.
•
•
•

Leg 1: 5 miles in 20 minutes
Leg 2: 8 miles in 24 minutes
Leg 3: 15 miles in 40 minutes

A. What is the total time and distance Sascha biked?
B. Find Sascha’s rate in miles per hour for each part of the route.
C. Explain mathematically on which leg of the trip do you think Sascha was cycling:
a. The fastest.
b. The slowest.
c. Uphill.
d. Downhill.
e. On flat ground.
D. You can maintain a steady rate of 13 miles per hour on a bike. How much time will it take you to
travel the same total distance Sascha traveled?
E. Suppose you’re going to race Sascha. What steady rate would you have to maintain on the bike to
tie him? (Remember, “tie” means finish the same distance in the same amount of time. You will
probably have to go a different rate than that in part D.)
F. On a sheet of graph paper, create a graph that shows Sascha’s time and total distance he biked. Time
should go on the x-axis. Distance should go on the y-axis.
When EVERYONE is finished, organize your work for just parts A, B, C and F (if you get to it) into a large
poster. Your poster should clearly communicate your solutions and justifications! Make connections between
ideas, the shapes, and your justifications with color, arrows, or other tools. You may tape one person’s graph
on the page.

220

APPENDIX K
TABLE K1: SUMMARY OF OBSERVED LESSONS
Table K1
Summary of Observed Lessons
Pine River Middle School
Lesson
Date
Main Task

Lesson
Content
Angle sums of
regular
polygons

1

9/17/13

Shapes and
Designs
Investigation 2,
Problem 1

2

9/26/13

Shapes and
Designs
Investigation 2,
Problem 4

Interior and
exterior angles

3

10/3/13

Shapes and
Designs
Investigation 3,
Problems 2 and
3

Drawing
triangles and
quadrilaterals
with certain
combinations
of angles and
side lengths

Learning Goal(s)

Placement in Unit

Determining how to
find the size of each
angle and the sum of all
angles in a regular
polygon with n sides

In Investigation 1,
students explored general
properties of polygons,
how to measure and
estimate angles, and using
information to draw
uniquely-determined
figures
Students continue to
analyze properties of
polygons before learning
to design polygons with
certain properties in
Investigation 3
These problems are near
the end of the unit;
students are synthesizing
ideas about polygons and
their side lengths and
angle measures

Identify exterior angles
and relate their
measures and sum of
their measures to the
measures of interior
angles
Determining the
smallest number of side
lengths and angle
measures that will tell
how to draw an exact
copy of any given
triangle; determining
what combinations of
side lengths can make a
quadrilateral and how
many different shapes
are possible with these
combinations

221

Description of
Context
None

Other Notes on
Lesson Activities

None

Problem 3.2 asks
students to
determine the
shortest test
message possible
to tell a friend how
to draw a specific
triangle; 3.3 is not
contextual task

Mrs. Meyer started
with a discussion
summarizing
Problem 3.2,
leading to a
mathematical
argument between
some students about
the Angle-AngleSide rule. Then
transitioned to 3.3,
using polystrips to
investigate
quadrilaterals

Table K1 (cont’d)
4

10/22/13

Accentuate the
Negative
Investigation 1,
Problems 3 and
4

Using number
lines and the
chip model

Modeling number
sentences with number
lines and the chip model,
and using these tools to
generate number
sentences

5

10/29/13

Accentuate the
Negative
Investigation 2,
Problem 2

Subtracting
positive and
negative
integers

Using the chip model
and number lines to
develop an algorithm for
subtraction of integers

6

11/05/13

Graphing
activity from
Accentuate the
Negative in
CMP 2nd
edition,
Investigation 2,
Problem 5

Coordinate
graphing in
four quadrants

Identify coordinates of
points and plot points in
all four quadrants of the
coordinate plane

222

These lessons come
early in the unit, after
students have explored
integers generally and
have been introduced
to the number line as a
way to compare
integers. The number
line and chip model
will be used
throughout the unit to
develop understanding
of integer operations
Students have begun
to develop an
algorithm for addition
of integers, and will go
on to relate the
addition and
subtraction algorithms
and explore
multiplication and
division of integers
This was a
supplemental lesson to
provide some students
practice with positive
and negative integers
and graphing in all
four quadrants
between exploring
addition and
subtraction of integers
and multiplication and
division of integers

Temperature and
money contexts
(earning and
spending, being in
the “red” or
“black” were used
in some problems
involving integer
operations

Mrs. Meyer
orchestrated a
summarizing
discussion of 1.3,
then students worked
on 1.4, followed by a
whole-class
discussion of 1.4 part
(a) only. Students
were introduced to
the chip model for
operating with
integers.

Money contexts
(spending and
earning) were used
in some problems

For challenge
activity at end of
lesson, some
students graphed
picture or figure to
exchange with
another student

Some students
finished a quiz
(including Sophia
and McKenna). Other
students worked on a
graphing activity—
organized in groups,
but worked
individually in
parallel.
Challenge/extension
graphing activity
available for students
who finished before
the end of the hour.

Table K1 (cont’d)
7

11/12/13

Accentuate the
Negative
Investigation 3,
Problem 4

Multiplication
and division of
integers

Exploring patterns in the
products and quotients
of integers while playing
the “Integer Product
Game” in pairs

8

11/21/13

Stretching and
Shrinking
Investigation 1,
Problem 1

Mathematical
similarity

Exploring what it means
for two figures to be
mathematically similar

9

12/04/13

Stretching and
Shrinking
Investigation 2,
Problem 1

Making similar
figures

Determining if two
shapes are similar by
looking at the rule for
producing specific
coordinates for the
image

223

Students have
generated algorithms
multiplication and
division of integers
and are relating these
operations before
synthesizing across all
four operations in
Investigation 4
This is the
introductory lesson in
the unit, providing an
open-ended
opportunity for
students to explore the
major mathematical
ideas in the
investigation and unit

None

Students have
explored what it
means for figures to be
similar, and have
generated informal
rules for similarity.
This is the first lesson
in the investigation in
which students define
scale factor and
consider similarity in
graphed images

Two contexts: a)
Fictional students
Zach and Marta
designing a
computer game,
and b) the
“Wumps”—
identifying which
figures are similar
to Mug Wump and
are in the Wump
family, and which
are impostors

There’s a contest in
a middle school,
and the Mystery
Club is working
together to figure
out who the
mystery teacher is
from a photograph.

Students did not
actually get into the
problem as written in
this lesson. Mrs.
Meyer asked them to
investigate the more
open-ended
introductory question
in the text—finding
the teacher’s height
using any strategy.
Mrs. Meyer had
differentiated plans in
place. After awhile,
struggling students
receive a completed
table so they can
begin graphing.

Table K1 (cont’d)
10

12/05/13

Stretching and
Shrinking
Investigation 2,
Problem 1

Making similar
figures

Determining if two
shapes are similar by
looking at the rule for
producing specific
coordinates for the
image

11

12/19/13

Mathematics
Assessment
Project pilot
Formative
Assessment
Lesson Reallife Equations8

Interpreting
real-life
expressions
and equations

12

1/09/14

Stretching and
Shrinking
Investigation 3,
Problem 4

Finding lengths
using similar
triangles

Connecting algebraic
expressions and
equations to real-life
situations, addressing
misconceptions related
to the meaning of
variables in expressions
and equations
Using similar triangles
to find distances that are
difficult to measure
directly

8

Students have
explored what it
means for figures to be
similar, and have
generated informal
rules for similarity.
This is the first lesson
in the investigation in
which students define
scale factor and
consider similarity in
graphed images
This lesson was not
part of their 7th-grade
curriculum sequence.
It built on 6th-grade
content in Variables
and Patterns

Students should have a
firm understanding of
similarity and scale
factor, and are now
applying these
concepts to solve
contextual tasks
involving missing
measurements in
shapes

Same context as
lesson 9, but
focused on
identifying which
figures are similar
to Mug Wump and
are in the Wump
family, and which
are impostors

Mrs. Meyer had
differentiated plans in
place. Students who
finish receive more
figures to graph.

Students
interpreted
equations involving
the number and
price of eggs,
apples, and
bananas.
Determining the
distance across a
river to build a
boardwalk across

Formative Assessment Lessons and more information about their development are available at http://map.mathshell.org, a project led
by Mathematics Assessment Resource Service University of Nottingham and UC Berkeley.
224

Table K1 (cont’d)
13

1/15/14

Stretching and
Shrinking
Investigation 4,
Problem 3

Applying
properties of
similar shapes

Using properties of
similarity and given
information about shapes
to find unknown side
lengths, perimeters, and
areas

14

1/16/14

Stretching and
Shrinking
Investigation 5,
Problem 1
(CMP 2nd
edition)

Applying
properties of
similar shapes

Using properties of
similarity and given
information about
objects to find heights
that are difficult to
measure directly

Southpoint
Lesson
Date
1

9/19/13

Main Task
Stretching and
Shrinking
Investigation 2,
Problem 2

Lesson
Content
Changing a
figure’s size
and location
in the
coordinate
plane

Students should have a
firm understanding of
similarity, scale factor,
and ratio, and are now
applying these
concepts to solve
contextual tasks
involving missing
measurements in
shapes
This lesson comes
near the end of the
unit. Students should
have a firm
understanding of
similarity, scale factor,
and ratio, and are now
applying these
concepts to solve
contextual tasks
involving missing
measurements in
shapes

Learning Goal(s)

Placement in Unit

Determining
coordinate rules that
produce similar
figures and nonsimilar
figures; using
coordinate rules to
predict side lengths in
pairs of similar figures

Students have
explored what it means
for figures to be
similar, and have
generated informal
rules for similarity. In
this lesson, students
continue to consider
similarity in graphed
images and coordinate
rules

225

None

Lesson began with
going over Problem
4.2 and going over a
Learning Check

Determining
heights of tall
objects (in this
case, a clock
tower) using
shadows

Description of
Context
Continuing the
Wumps context,
analyzing
characters’ hats

Other Notes on Lesson
Activities
Continued the lesson from
the day before.

Table K1 (cont’d)
2

9/24/13

Stretching and
Shrinking
Investigation 2,
Problem 3

Scale factors

Generalizing strategies
for deciding if two
figures are similar,
finding and using scale
factors

3

10/1/13

Stretching and
Shrinking
Investigation 4,
Problem 1

Ratios within
similar
parallelogram
s

Exploring what
information is
provided by ratios of
adjacent side lengths
in rectangles and
parallelograms;
relating this
information with
similarity

4

10/25/13

Comparing
and Scaling
Investigation 1,
Problem 1

Analyzing
comparison
statements

Exploring what
different comparisons
of quantities tell about
the relationship
between the quantities

226

After exploring
similarity in shapes
and graphed images,
students have a
definition for
similarity and scale
factor and generalize
these ideas to other
shapes.
Students should have a
firm understanding of
similarity and scale
factor, and are now
exploring the role
ratios play in
generating similar
figures and solving
problems involving
similar shapes
This is the
introductory lesson in
the unit, providing an
open-ended
opportunity for
students to explore the
major mathematical
ideas in the
investigation and unit

Continuing the
Wumps context,
analyzing the
characters’ noses
and mouths

Ms. Pearson modified the
task somewhat, making a
task card with small group
roles and a resource page
consistent with Complex
Instruction practices. Also,
had students develop
posters to present their
work.

Investigation
begins with
photo image
being “dragged”
to change the
size and shape;
Problem 1 is not
contextual

A soda company
is using results
from a taste test
comparing their
product with
another in their
add. A
copywriter
suggested four
comparison
statements to use
in the ad

Lesson started with
students doing a short selfpre-assessment on the new
learning objective and
exchanging books, since
this is a new unit.

Table K1 (cont’d)
5

10/31/13

Comparing
and Scaling
Investigation 2,
Problem 1

Comparing
ratios

Determining and
analyzing different
strategies to compare
quantities for their
accuracy and
efficiency

Students used rations,
fractions, percents, and
differences to compare
quantities in the first
Investigation; now
they begin to analyze
and choose appropriate
strategies for solving
different problems

6

11/7/13

Class goes over
Comparing
and Scaling
Investigation 2
homework
problems, then
completes a
“ticket out the
door”
assessment and
begins
answering the
Mathematical
Reflection
questions for
the
Investigation

Comparing
ratios,
percents, and
fractions

Review and assess
student understanding
of content in the
Investigation

Students have
developed strategies
for selecting and using
appropriate solution
methods for
comparing quantities
and are reviewing this
content before
investigating rates and
proportions

227

Julia and Mariah
are making
chocolate
cookies for their
fellow campers.
They consider
four different
mixes; students
need to
determine which
is the most
“chocolatey”
One homework
task involved
determining the
most “appley”
juice. In another,
Duane and
Miriam make
pottery bowls;
students need to
determine who is
faster. A third
problem involves
a party; students
need to
determine at
which table
someone can get
the most pizza.

Ms. Pearson modified the
task, making a task card
with small group roles
consistent with Complex
Instruction practices. Also,
had students develop
posters to present their
work and changed the
context from determining
the most “orangey” juice
to the most “chocolatey”
cookie.
Ms. Pearson had been
gone for a few days
unexpectedly, so this
lesson was mainly about
reviewing problems
students worked on in her
absence and assessing
their understanding of the
Investigation.

Table K1 (cont’d)
7

11/14/13

Comparing
and Scaling
Investigation 3,
Problem 2

Finding rates

Calculating and
making sense of rates
and the relationship
between distance, rate,
and time

Students have
developed strategies
for using rations to
compare quantities are
now solving problems
involving rate before
working with
proportions

8

11/26/13

Comparing
and Scaling
Investigation 4,
Problem 3

Solving
proportions

Developing general
strategies for solving
problems involving
proportions

This is the final lesson
in the unit; students
are solidifying their
understanding of
proportions to solve
different contextual
problems

228

Sascha is going
on a bike trip
with three legs.
Students are told
how long each
leg is and how
many minutes it
took to finish.
They find and
compare unit
rates and think
about how fast
they would need
to ride to race
Sascha.
Multiple parts of
problems had
different
contexts:
estimating puffin
populations, a
traveling jet
plane, Jack
eating enchiladas
at work and
estimating his
calories
consumed, and a
middle school
population by
grade compared
with the numbers
of students on
student council.

Ms. Pearson modified the
task somewhat, making a
task card with small group
roles consistent with
Complex Instruction
practices and adding parts
to the problem. Also, had
students develop posters to
present their work.

Lesson started with going
over Investigation 4
homework and Ms.
Pearson introducing the
essay prompts for an
assignment due the
following week. Then
students were allowed to
work individually on 4.3
or could choose to work
with a partner.

Table K1 (cont’d)
9

12/10/13

Accentuate the
Negative
Investigation 1,
Problem 4

Using the chip
model

Using the chip model
to represent number
sentences with
addition and
subtraction of integers

10

12/12/13

Using the chip
model

Using the chip model
to represent number
sentences with
addition and
subtraction of integers

11

12/16/13

Accentuate the
Negative
Investigation 1,
Problem 4;
work on
teacherproduced small
group
assignment
containing
homework
problems from
CMP
Accentuate the
Negative
Investigation 2,
Problem 2

Subtracting
positive and
negative
integers

Using the chip model
and number lines to
develop an algorithm
for subtraction of
integers

229

This lesson comes
early in the unit, after
students have explored
integers generally and
have been introduced
to the number line as a
way to compare
integers. The chip
model will be used
throughout the unit to
develop understanding
of integer operations
Students have been
working with the chip
model for two lessons;
this was a formative
assessment
opportunity before
using the chip model
to develop algorithms
for integer addition
and subtraction

Temperature and
money contexts
(earning and
spending, being
in the “red” or
“black” were
used in some
problems
involving integer
operations

Students mainly worked as
a whole class as they are
introduced to the chip
model, but also spend
several brief sessions in
small groups to solve
problems.

A few tasks in
the homework
reviewed use
money contexts
(spending,
earning, owing
money); student
spontaneously
references having
a certain number
of abstract
“points”

The class reviewed 1.4
homework together, then
students worked on
teacher-produced small
group assignment
containing homework
problems from
Investigation 1

Students have begun to
develop an algorithm
for addition of
integers, and will go
on to relate the
addition and
subtraction algorithms
and explore
multiplication and
division of integers

Spontaneous use
of quarters to
help add
multiples of
0.25; forgiving
debts and
temperature
change to make
sense of
subtracting
negatives; money
contexts
(spending,
earning, debt)

Table K1 (cont’d)
12

1/10/14

Accentuate the
Negative
Investigation 3,
Problem 3

Division of
integers

Determining an
algorithm for dividing
integers and relating
integer multiplication
and division

This lesson focuses on
the last operation—
division—before
synthesizing these
ideas and moving
toward properties of
operations

None

13

1/13/14

Operations on
integers

Review properties of
negative and positive
numbers and practice
adding, subtracting,
multiplying, and
dividing

Students have
algorithms for all the
integer operations and
are reviewing to get
reading for a Learning
Check and the
semester exam

14

01/14/14

Review activity
(Accentuate the
Negative) with
students
responding to
prompts
individually on
white boards
Go over review
sheets, do a
“learning
check” (i.e.,
quiz), begin
working on
exam review

Scale factor,
ratio, rate,
operations on
integers

Review and assess
student understanding
of three units of
content

One task used
temperature
context;
spontaneously
used money as
support in
operating with
decimals
One task used
money contexts
(spending,
earning, owing
money

230

Teacher generated an
“investigation sheet” that
mirrored the CMP
problem with some
modification—took out
the one contextual part of
the task (about a Number
Relay) and added
scaffolding for students to
determine an algorithm for
division

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231

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