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EFFECTS OF VIRTUAL MANIPULATIVES WITH OPEN-
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STUDENTS’ KNOWLEDGE OF SLOPE

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EFFECTS OF VIRTUAL MANIPULATIVES WITH OPEN-ENDED VERSUS
STRUCTURED QUESTIONS ON STUDENTS’ KNOWLEDGE OF SLOPE

By

Mustafa F atih Demir

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Educational Psychology and Educational Technology

2009

ABSTRACT

EFFECTS OF VIRTUAL MANIPULATIVES WITH OPEN-ENDED VERSUS
STRUCTURED QUESTIONS ON STUDENTS’ KNOWLEDGE OF SLOPE

By
Mustafa Fatih Demir

Virtual Manipulatives (VMs) are computer-based, dynamic, visual representations
of mathematics concepts, which have become widely used in mathematics instruction in
recent years. Despite their wide use, there is little empirical research about the use of
VMs in mathematics learning and teaching. To ﬁll this gap, this study examined the
effects of using VMs with two different instructional approaches on students’ learning of
slope. Approximately 50 students who were taking a remedial mathematics course at
Michigan State University completed all sessions of the study. They were randomly
assigned into two groups: OVM and SVM groups. After completing a pretest assessing
their initial knowledge of slope, participants were randomly assigned into two groups to
complete four 30- to 45-minute intervention sessions. Students in the OVM group
worked with VMs to respond to open-ended exploratory questions; students in the SVM
group used the same VMs to respond to structured mathematics questions. Then, students
completed a posttest consisting of the same set of questions as the pretest. OVM students
showed considerably higher pre- to posttest gain scores than SVM students on items
requiring conceptual knowledge, whereas SVM students obtained notably higher gains on
items requiring procedural and a combination of conceptual and procedural knowledge.
These results suggest that VMs can be used with various instructional approaches to

improve different types of students’ mathematics knowledge.

Copyright by
MUSTAF A FATIH DEMIR
2009

ACKNOWLEDGEMENTS
I would like to thank Dr. Ralph Putnam who provided valuable guidance and support on
my dissertation. Throughout my doctoral studies, he greatly helped me improving my
research skills and scientiﬁc writing. I want to thank him for his generous time and
commitment. I also want to thank Dr. Patrick Dickson who provided constant support and
encouragement throughout my dissertation and as well as my entire doctoral career.
Additionally, I would like to acknowledge:
0 My other committee members: Dr. Punya Mishra and Dr. Michael Steele for their
support and suggestions throughout my dissertation.
o The other professors whom I have worked with during my doctoral studies: Drs.
Jon Star, Joan Ferrini-Mundy, Sharon Senk, and Sandra Crespo.
o The students who participated in this research project.
Finally, I would like to thank my wife and my parents who provided constant support,

encouragement, and love over the years. Thank you.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vii
LIST OF FIGURES ............................................................................... viii
CHAPTER 1
INTRODUCTION ................................................................................... 1
CHAPTER 2
REVIEW OF THE LITERATURE ................................................................ 4
Media and Learning ........................................................................ 4
VMs and Learning ......................................................................... 5
Theoretical Assumptions ................................................................... 6
Constructivist perspective on learning ........................................... 6
Exempliﬁcations ................................................................... 7
Multiple representations ........................................................... 7
Dual coding theory ................................................................. 9
VMs and Learning Mathematics ........................................................ 10
Learning about Slope ..................................................................... l4
RESEARCH FOCUS ................................................................................ 20
Students’ Knowledge of Slope ............................................................ 21
Research Questions ........................................................................ 23
CHAPTER 3
METHOD .......................................................................................... 24
Participants ................................................................................ 24
Setting ...................................................................................... 25
Materials ................................................................................... 26
Background questionnaire ..................................................... 26
Pretest and posttest ..., .......................................................... 26
Sources of VMs ................................................................. 28
Procedures ................................................................................. 29
Intervention Session 1 (Session 2 — Distance vs. Time Activity) 3l
Intervention Session 2 (Session 3 — Geoboard Activity) ................... 34

Intervention Session 3 (Session 2 — Exploring Graphs Activity) 35
Intervention session 4 (Session 2 — Linear Transformation Activity) 36
Posttest Session .................................................................. 37
Pre- and Posttest Scoring ................................................................ 37

CHAPTER 4

RESULTS .......................................................................................... 4O
Procedural and Conceptual Gains ...................................................... 42
Procedural items .................................................................. 42
Conceptual items .................................................................. 43
Combined conceptual and procedural items ................................. 45
Item by Item Analysis .................................................................... 49
Responses including only statements (Stat.) ................................. 53
Responses with the descriptions of activities on the VMs
(Desc. of acts. on VMs) ....................................................................... 53
Responses consisting of explanations with reasons (Exp) ................. 53
Responses with explanations and descriptions of activities
(Exp. & Desc.) ........................................................................................ 54
Responses with numeric relations or algebraic expressions (Num.) ...... 56
Responses including calculations (Calc.) ..................................... 56
Responses involving qualitative relations (Qual.) ........................... 57
CHAPTER 5
DISCUSSION ...................................................................................... 59
Strengths and Limitations ................................................................ 66
Conclusions ................................................................................. 68
APPENDICES ..................................................................................... 71
Appendix A: Information about Recruitment of Students .................................... 71
Appendix B: Background Questionnaire ........................................................ 73
Appendix C: Pretest (Posttest) .................................................................... 75
Appendix D: Intervention Session Questions ................................................... 85
REFERENCES
References ............................................................................................ 99

vi

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Table 9

Table 10

Table 1 1

Table 12

Table 13

Table 14

LIST OF TABLES

Number of Students Participated to the Sessions ..............................

Categories of Slope Knowledge and Related

Pretest (Posttest) and Session Questions .....................................

Categories of Questions throughout the Pre-and Posttest ................

Design of the Study for the VMs Used with Exploratory

versus Structured Mathematics Questions ..................................

General Rubric for Pre- and Posttest Scoring ..............................

Rubric Used Scoring Students’ Answers on

the Pre (Post) Question 8 (5) .................................................

Interrater Reliability of Overall and Subsections of the

Pre-and Posttest

Means for OVM and SVM Groups’ Overall and Gain Scores ..........

Means for OVM and SVM Groups’ Overall and

Gain Scores on the Subset of Procedural Knowledge Questions . ..

Means for OVM and SVM Groups’ Overall and Gain Scores on the

Subset of Conceptual Knowledge Questions ..............................

Means for [OVM and SVM Groups’ Overall and Gain Scores on the

Subset of Combined Con. and Proc. Knowledge Questions .............
Mean Gain Scores Item by Item ............................................................

Categories of Students’ Responses in Session 1 and 4 (by %). . . . . . . .

Percentage of Students’ Responses by Category in Session 1 and 4

(by %) ............................................................................

vii

.. 25

..27

..28

..31

37

38

39

40

42

44

46
.. 50
. 55

..57

LIST OF FIGURES

Figure l A screenshot of the VM with exploratory open-ended questions .. 32
Figure 2 A screenshot of the VM with structured mathematics questions 33
Figure 3 A screenshot of the VM that students used in Intervention Session 2 34
Figure 4 A screenshot of the VM that students used in Intervention Session 3 35
Figure 5 A screenshot of the VM that students used in intervention Session 4 ..... 36
Figure 6 Comparison of OVM and SVM groups’ overall gain scores ............... 41
Figure 7 Comparison of OVM and SVM groups’ gain scores on procedural
knowledge questions .............................................................. 43
Figure 8 Comparison of OVM, and SVM groups’ gain scores on conceptual
knowledge questions .............................................................. 45
Figure 9 Comparison of OVM and SVM groups’ gain scores on the combined
conceptual and procedural knowledge questions ............................ 47
Figure 10 Conceptual question 3 ............................................................ 51

Figure 11 Procedural question 2 ............................................................ 52

viii

CHAPTER 1
INTRODUCTION

In the last two decades, computer and the Internet technology have shown
breathtaking advances, and people have already experienced the effects of these
technologies in various areas of their life. Education is one of the areas where the impact
of technology can be clearly observed. Because various applications of computer
technology have already been widely used in many areas of everyday life (e.g.,
communication, business, medicine), it is impossible to neglect the developments of
computer technology in the area of education. Today, students are more often using
computer technology and the Internet to improve their learning of school subjects than
students in the past. School mathematics is one of the areas that today’s students use
computer technology to enhance their learning.

Many researchers and leading mathematics education organizations (e. g., NCTM,
2000; Steen et al., 2006; Wenglinsky, 1998) have argued that using technology can
advance students’ mathematics learning. For example, Wenglinsky (1998) examined the
impact of using simulation and higher-order drinking technologies on students’
performance in mathematics. He analyzed a US. national sample of 6,227 fourth and
7,146 eighth-grade students’ mathematics scores on the National Assessment of
Educational Progress (N AEP) and found that students who worked with the technologies
improved their mathematics achievement. The National Council of Teachers of
Mathematics (N CTM) is one of the professional mathematics education organizations in
the United States. NCTM’s Principles and Standards for School Mathematics (2000)

declared that use of technology is one of the six major principles of a high-quality

mathematics instruction. The technology principle states that “Technology is essential in
teaching and learning mathematics; it inﬂuences the mathematics that is taught and
enhances students’ learning.” (NCTM, 2000, p. 11). With the aid of technology, students
effectively focus on decision making, reﬂection, and problems solving; thus students can
develop deep understanding of mathematics concepts by using computer technology
(NCTM, 2000). Many types of computer technologies have been used to support learning
and teaching of mathematics. For example, there are a variety of educational mathematics
software such as Geometer’s Sketchpad (J ackiw, 2001), Cognitive Tutor Algebra
Soﬁware (Koedinger et al., 1997), SimCalc Mathworlds (Roschelle & Kaput, 1996);
computer simulations such as Java applets; and Virtual Manipulatives (VMs). Among
these technologies, VMs are a recent player in the mathematics education scene.

Recent advances in computer and Internet technology have made it possible for
students to manipulate objects on the computer screen as easily as they can manipulate
physical manipulatives. These VMs are dynamic visual representations of physical
manipulatives, have additional features, and provide various opportunities to improve
learning and teaching. VMs provide interactive learning environments where students can
engage with various cognitive activities (e.g., generalizing, planning, testing) to learn
mathematical ideas. Since VMs present interactive learning activities, they go beyond the
other media that only present or deliver information. Therefore, Clark’s (1983) famous
controversial claim that media only delivers the information without inﬂuencing learning
does not hold for using VMs in mathematics instruction. However, Clark’s suggestion
about exploring the impact of different instructional methods used with media on learning

rather than only studying the impact of one speciﬁc media on learning is crucial. With

such knowledge, mathematics teachers can better choose the most appropriate
instructional methods to use with VMs after looking at the results of the research studies
exploring the effects of different instructional approaches used with VMs on students’
mathematics learning. However, there is a need for research that will study the impact of
various instructional approaches with VMs on students’ learning of mathematics
concepts. Therefore, this study examined the effects of VMs used with two different
instructional approaches on students’ knowledge of slope, an important mathematical
concept that is often problematic for students.

With the help of current computer technology, a wide variety of VMs have
become available and practical for both students and teachers. This availability increased
the need for research studies analyzing the affordances and constraints of VMs in
learning and teaching mathematics. However, few studies have examined the impact of
using VMs on students’ understanding of mathematical concepts. Many researchers (e. g.,
Reimer & Moyer, 2005; Steen et al., 2006; Sub & Moyer, 2007) have pointed out the lack
of research on using VMs in mathematics learning.

This study aimed to ﬁll the gap in research literature and inform current practices
with VMs in mathematics instruction by exploring the impact of VMs used with two
different instructional approaches on students’ learning of slope. Slope is one of the
concepts that students often have difﬁculties in their courses. Because slope is a widely-
used concept in various areas of school mathematics, students’ knowledge of slope
inﬂuences their learning of other mathematics subjects (Thompson, 1994). Thus, several
researchers examined students’ difﬁculties in learning slope and suggested various ways

for developing instructional activities to improve students’ learning.

CHAPTER 2
REVIEW OF THE LITERATURE

In this chapter, ﬁrst, the relations between media and learning will be discussed
through presenting a brief summary of different views about media and its effects on
learning. Second, theoretical assumptions of various learning perspectives will be
examined to identify possible effects of using VMs on students’ mathematics learning.
Then individual research studies analyzing the effects of using VMs on learning and
teaching mathematics will be presented. Finally, research about students’ learning of
slope will be reviewed.
Media and Learning

Many researchers (e.g., Kaput, 1992; Kulik et al., 1983) have examined the
impact of various types of media on students’ learning of particular subject areas. Studies
analyzed the relationships between media and learning unveiled conﬂicting views among
researchers about the effects of media on learning. Researchers have been discussing the
effects of media on learning for a long time. Clark (1983) stimulated the discussion about
media and its effects on learning by asserting that choice of media has no inﬂuence on
learning. After reviewing media comparison studies that scrutinized the effects of media
on learning, Clark concluded that media do not inﬂuence learning under any condition.
He claimed that ﬁnding larger positive effect sizes for one media than another can be
attributed to differences in instructional method, novelty of the media, and editorial bias.

Clark (1983) viewed media based instructional materials as “mere vehicles that
deliver instruction but do not inﬂuence student achievement any more than the truck that

delivers our groceries causes changes in our nutrition” (p. 445). He suggested that

researchers should avoid conducting additional studies that explore the relationship
between media and learning. In response to Clark claims about media and learning,
Kozrna (1994) emphasized that as technologies develop and their use is systematically
examined, the effects of media on learning will eventually be identiﬁed and utilized.

As Clark (1983) reviewed the media comparison studies that were conducted
prior to 1983, his view of media is quite limited in the light of various types of current
media. Clark considered media as presentations of information, or the tools that deliver
information. Many types of current media, however, do more than presenting and
delivering information. Since computer and the Internet technologies developed
dramatically in the last two decades, there are various technologies that have been used in
instruction, and have advanced capabilities (e.g., interactivity) and obviously they do
more than only presenting and delivering information. One of these technologies that
have been recently used in educational settings is VMs.

VMs and Learning

VMs can be considered as visual representations of dynamic objects that support
students’ understanding of mathematics concepts. They are a different type of computer
technology that has potential to meet the shortcomings of physical manipulatives in
learning and teaching (Dorward, 2002; Sarama et a1. 1996). VMs provide interactive
learning environments where students can engage with various cognitive activities (e.g.,
generalizing, planning, testing) to understand mathematics concepts. Because VMs
enable students to interact with mathematics concepts, they differ from the media that
simply present or deliver information. Therefore, Clark’s claim about the media and

learning is not valid for using virtual manipulatives in learning and teaching of

mathematics. However, his suggestion about exploring the effects of different
instructional methods within various media on learning rather than only studying the
impact of one speciﬁc media on learning is important. Thus, research studies need to
examine the effects of various instructional methods with VMs on students’ learning.
Theoretical Assumptions

According to assumptions of various learning perspectives, VMs have potential to
support learning and teaching mathematics in the following ways.

Constructivist perspective on learning. VMs provide interactive learning
environments where students can instantly observe the effects of their actions, monitor
their own learning, and form and test hypotheses in order to construct or modify their
knowledge of mathematical concepts. The constructivist perspective (Anderson et al.,
2000; Brown et al., 1989; Greeno et al., 1996) on learning assumes that learners construct
their own knowledge by interacting with their physical and social environment.
Therefore, students’ interactions (e. g., receiving immediate and speciﬁc feedback) with
VMs may enable them to understand mathematical concepts.

Researchers (Clements & McMillen, 1996; Kaput, 1995; Reimer & Moyer 2005;
Sub et al., 2005; Sub & Moyer, 2007) have emphasized that VMs can support students’
understanding of mathematical concepts by providing interactive learning environments.
For example, Kaput (1995) used the term cybernetic manipulatives to refer the
manipulatives presented on the computers. He emphasized that cybernetic manipulatives
can enable students to make connections between their actions in different notation
systems. While using physical materials, students exhaust all their cognitive resources to

perform actions in one or the other notation systems, and therefore they cannot track

relations between their actions. However, cybernetic manipulatives can provide
immediate feedback to students’ actions, and this enables students to identify
relationships between their actions in different notation systems.

Exempliﬁcations. Another perspective on learning that might be used to support
the contributions of VMs on students’ understanding of mathematical ideas is Nesher’s
(1989) Learning System (LS) model. Nesher developed a LS model based on two
components: a knowledge component and an exempliﬁcation component. Knowledge
component refers to the unit of knowledge to be instructed, based on expert knowledge.
The exempliﬁcation component includes an illustrative domain that corresponds to the
knowledge component and is purposely chosen to serve as an exempliﬁcation. In this LS
model, learners make connections between the objects, relations, and operations in
exempliﬁcation and knowledge components to understand abstract concepts.

VMs may help students exemplify mathematical concepts by providing concrete
representations of formal mathematical concepts. The exempliﬁcation components
include illustrations that are familiar to learners’ experiences, thus learners can match the
objects, operations, and relations in exempliﬁcation components with those in knowledge
components in order to understand abstract concepts. VMs can be thought as an example
of Nesher’s LS model. For example, some VMs use balance scales to support students’
learning of solving linear equations. While working with these VMs, learners can use
balance scale as an exempliﬁcation component to comprehend mathematical properties of
solving linear equations.

Multiple representations. The multiple representations view of learning can also

be employed to support the educational beneﬁts of using VMs in learning and teaching

mathematics. The multiple representations perspective of learning assumes that people
can improve their understanding of mathematical ideas through working with multiple
representations of the same concepts. Researchers (Goldenberg, 1995; Heid & Edwards,
2001; Kaput, 1998) have also pointed out that using multiple representations (e. g.,
graphical, symbolic) and making connections among these representations is one of the
important factors in modeling and understanding mathematical concepts.

Students can move from less abstract representations of mathematics concepts to
more abstract ones while working with VMs. This enables them to begin with familiar
concepts then continue with less familiar ones in their learning processes. Kaput (1995)
suggested that computer software should enable students to extend their knowledge
moving from familiar, concrete contexts to less familiar, abstract contexts. He pointed
out that students can make this movement by using the links (provided by the computer
software) between more familiar representations and less familiar ones. Kaput maintained
that in software-based learning environments students should engage with a set of
representations, beginning at the concrete level and romping upward in abstractness to
more abstract representations. VMs can help students initially work with less abstract
representations of mathematics concepts and then engage with more abstract ones.

As VMs provide various representations of mathematical concepts, ﬁrst, they may
present representations which are concrete to students, and then gradually increase the
level of abstractness of representations in order to support students’ mathematics
learning. For example, some virtual manipulatives offer various forms (e.g., verbal,
tabular, graphical, and algebraic) forms of linear functions. These VMs can be designed

in a way that enables students to work with less abstract representations in the beginning

and then work with more abstract forms of linear functions. For example, while working
with VMs, students can initially engage with verbal forms of linear functions then they
can study tabular and graphical representations.

Dual coding theory. One of the perspectives that can be used to support the
effectiveness of using VMs on learning and teaching is Dual Coding theory. The theory
assumes that individuals can enhance their understanding of concepts when they receive
both verbal and visual forms of the same information (Clark & Paivio, 1991). VMs may
improve students’ learning of mathematical concepts by providing visual and verbal
forms of the same concepts in a coordinated way. They can present verbal information
(either as narrative or written format) while students manipulate objects on the computer
screen. Thus, students may recognize and focus on mathematical ideas and concepts
behind their actions with VMs by having verbal information coordinated with their
manipulations.

Having verbal forms of information coordinated with individuals’ actions on VMs
also supports teachers’ use of VMs in their courses. For example, Crawford and Brown
(2003) designed a survey to identify teachers’ evaluations of VMs and their views about
integration of VMs into classroom settings. In the study, after reviewing a number of
VMs on the web, many classroom teachers emphasized that because the VMs (that they
examined) provided neither clear instructions about their use nor feedback while students
working with the VMs, they rated these manipulatives as unsatisfactory for their
instructional practices. Therefore, as dual coding theory suggested, it is important to
provide verbal information coordinated with students’ actions while they work with VMs

in order to improve their learning.

Researchers (e.g., Suh & Moyer, 2007) have already used some of these learning
perspectives (multiple representations, constructivism) to explore the effects of VMs on
students’ understanding of mathematical concepts. However, dual coding theory and
exempliﬁcations approaches to learning have not been used in the research on VMs in
mathematics education. It is important to use these learning perspectives or their various
combinations (e. g., dual coding theory and multiple representations) to increase the
quality of research studies that examine the impact of VMs on mathematics learning.
VMs and Learning Mathematics

In this section, individual research studies exploring the effects of using VMs on
learning and teaching mathematics will be presented. Although there are few studies on
VMs and mathematics learning, this review aims to present research evidence regarding
the affordances and constraints of VMs on mathematics learning and teaching.

One of the few studies examining the features of VMs that support students’
learning of mathematics concepts was conducted by Suh et al. (2005), who designed a
qualitative study that included interviews with students, observation notes, and
videotapes of classroom sessions to identify the characteristics of the VMs that supported
students’ learning of equivalence and fraction addition. Forty-six ﬁﬁh-grade students
were identiﬁed as low-, average-, and high-level achievers based on standardized test
results used at their school. These three groups of students independently worked with
VMs after the same elementary teacher introduced a unit on fractions and presented
mathematical tasks. The researchers found that a virtual fraction applet enabled students
to make conjectures, connections between symbolic and iconic forms of fractions, and

discover various properties of fractions. Furthermore, the study indicated that playing a

10

fraction track game supported students’ communications of mathematical ideas and their
ability to apply previously learned ideas into a different mathematical context. The
ﬁndings of the study implied that as students learn mathematics concepts, they may have
beneﬁts of some aspects of VMs tutorials such as having combined visual and symbolic
information in a linked format, and being able to test their hypotheses in a secure
environment.

Steen et al. (2006) examined the impact of using VMs on students’ academic
achievement, attitudes, and interactions during a geometry unit. In the study, 31 Grade 1
students in an urban elementary school were randomly assigned to the treatment or
control group for a geometry unit. Students were given a pretest that included two tests,
one is Grade 1 and the other is Grade 2 level (provided by the publishers of the
textbooks) to assess each student’s prior knowledge of geometry concepts such as
patterns, symmetry, and shape identiﬁcation. Students in the control group used their
regular textbooks, and worked with physical manipulatives for practice. Treatment group
students interacted with VMs, but used the same textbooks. The corresponding forms of
Grade 1 and Grade 2 tests were given as a posttest to both groups. All students completed
the pre- and posttest on the paper-pencil format.

Steen et al. (2006) found that the use of VMs as an instructional tool was helpful
for the treatment group. The researchers conducted a t-test to analyze the changes in both
control and treatment group students’ scores over the pre-and posttest. On the Grade 2
test, students in the treatment group had a mean change of 7.25 (SD = 4.74) and control
group students had a mean change of 3.33 (SD = 3.20). Steen et al. reported that the t-test

indicated a signiﬁcant difference between groups on the Grade 2 test (p < 0.05).

11

Similarly, t-test showed a signiﬁcant difference between control and treatment groups on
the Grade 1 test (p < 0.05). In addition to the pretest and posttest data, the researchers
examined the treatment teacher’s daily journal to identify the teacher’s observations
about students’ attitudes and interactions while working with VMs. The analysis of
teacher’s daily journal indicated that VMs enabled students to make changes on their
learning, do more number of practices than they do in traditional classroom settings, and
. have beneﬁts of the same high quality lessons and activities.

Reimer and Moyer (2005) also examined the effects of using VMs on students’
mathematics learning. They analyzed the impact of using VMs on 19 third-grade
students’ conceptual and procedural knowledge of fractions in a classroom setting. These
students worked with several VMs in a computer laboratory during a two-week unit on
ﬁactions. Reimer and Moyer designed pre- and posttest to identify students’ conceptual
and procedural knowledge of fractions at the beginning and end of the study. They also
conducted interviews to identify the changes in students’ knowledge of fractions, and
used student attitudes survey. The analysis of the pre-and posttest data indicated that
students considerably improved their conceptual knowledge of fractions. Speciﬁcally,
researchers found that students displayed higher performance on the posttest (M = 11.0,
SD = 3.61) than they did on the pretest (M = 9.58, SD = 4.53). Furthermore, the analysis
of student interviews and attitude surveys also revealed that VMs improved students’
learning of fractions by providing immediate and speciﬁc feedback. Reimer and Meyer
found that VMs helped students enhance their mathematics learning. However, since the
researchers used neither control nor comparison group in their study, it is difﬁcult to

identify various effects of using VMs on students’ conceptual knowledge of fractions.

12

Suh and Moyer (2007) studied students’ learning while working with physical
manipulatives and VMs in order to identify the features of these manipulatives that
enhanced students’ learning. In the study, 36 third-grade students worked with physical
and virtual balance scales in their regular mathematics course sessions. They were
assigned into two groups; one group worked with the Virtual Balance Scale applet from
the National Library of Virtual Manipulatives (N LVM) to solve linear equations. The
other group worked with a physical manipulative called Hands-On Equations®
(Borenson, 1997) to respond questions. Suh and Moyer used observational ﬁeld notes and
interviews to identify the effects of using physical manipulatives and VMs on students’
solving linear equations. They found that the virtual balance applet improved students’
mathematical thinking by providing explicit links of symbolic and visual models, step-
by-step support in algorithmic processes, immediate feedback and self-checking system.
However, the physical balance manipulative provided opportunities for invented
strategies, mental mathematics, and tactile features to support students’ learning. The
study also indicated that although physical manipulatives and VMs have distinctive
aspects to support students’ learning, and both were helpful in improving students’
knowledge and algebraic reasoning.

Studies of VMs in mathematics learning and teaching have reported that VMs
supported students’ learning by providing immediate feedback, multiple representations
of mathematics concepts and interactive learning environment. However, these studies
examined the effects of VMs while students studying mathematical ideas according to
traditional instructional methods (e. g., using standard textbook). They had no attention to

the impact of VMs with different instructional approaches on students’ learning of

13

mathematical concepts. Therefore, there is a need for research that will examine the
effects of using VMs with different instructional approaches on students’ mathematics
learning.

Learning about Slope

Many researchers have emphasized the importance of the students’ understanding
of slope concept in mathematics learning (e.g., Leinhardt et al., 1990; Thompson, 1994;
Lobato et al., 2003). Slope can be thought of as a powerful concept that can help students
make connections between a line and its algebraic equation (Leinhardt et al., 1990).
Students’ knowledge of slope can affect their understanding of advanced mathematics
concepts (Thompson, 1994). After analyzing college seniors’ and graduate students’
understanding of the fundamental theorem of calculus, Thompson found that students’
impoverished knowledge about slope and rate of change caused difficulties in their
learning of the theorem. Students’ familiarity with point-slope formula for a line may
enable them to improve their comprehension of the fundamental theorem of calculus
(Macula, 1995). Furthermore, Macula mathematically showed how the point-slope
formula of a line leads to the fundamental theorem of calculus.

Although several researchers have mentioned the importance of deep
understanding of slope concept for students’ mathematics learning, few studies have
speciﬁcally focused on students’ understanding of slope concept. Rather, researchers
have often examined the concept of slope as a part of students’ understanding of linear
ﬁrnctions. As Lobato and Siebert (2002) pointed out, since most researchers and reform
documents have focused on the ability to make connections among multiple

representations of [linear] functions, students’ knowledge of particular concepts such as

14

slope has not been adequately examined to develop a type of instruction to support
students’ learning of slope. Few researchers have examined particular aspects of students’
knowledge of slope. For example, Lobato et al. (2003) underlined two major parts of
conceptual understanding of slope in their study. These are considering slope as a rate of
the change in one quantity as related to the change in another quantity, and viewing slope
as forming ratios for measures of particular attributes such as constant speed or the
steepness of a ramp.

Research exploring students’ knowledge of slope has focused on various aspects
of slope knowledge, such as identifying the value of slope and interpreting its meaning
from various representations (e. g., algebraic, geometric, physical, functional), making
connections among these representations of slope (Leinhardt et al., 1990; Knuth, 2000),
and considering slope as a measure to identify steepness or rate of change in functional
situations where two quantities co-vary (Stump, 2001; Lobato et al., 2003). For example,
Stump (2001) examined high school students’ understanding of slope as a measure in two
different contexts: physical contexts where slope can be used to measure the steepness,
and functional contexts in which slope can be considered as a measure of rate of change.
In another study, Stump (1997) identiﬁed six different representations of slope concept
(geometric, algebraic, physical, trigonometric, functional, and ratio) and studied
secondary mathematics teachers’ knowledge of slope by analyzing the types of
representations that teachers mostly used in their responses to the survey questions about
slope.

Research has also revealed students’ various difficulties in learning slope. The

most common difﬁculties include (a) slope versus height confusion (McDermott et al.,

15

1987), (b) making connections from an algebraic formula of slope to its graphical
(Leinhardt et al., 1990, Knuth 2000) and functional (Lobato et al. 2003) representation,
(0) identifying the distinction between the slope of a line and the slope of a function while
ﬁnding SIOpe value on a non-homogenous coordinate system that has different scales on
each axis (Zaslavsky et al., 2002; Rasslan & Vinner, 1995), and (d) interpreting slope as a
measure of rate of change (Stump, 2001). In addition to these common difficulties,
researchers have also underscored students’ other difﬁculties in learning slope such as
considering m in y = b + mx as a difference rather than a ratio (Lobato et al., 2003);
identifying the relations between the concept of slope and the angle that a line forms with
the x-axis (Rasslan & Vinner, 1995, Stump 1997).

Slope-height confusion is one problem commonly reported for students having
difﬁculties in identifying slope values while working on graphs. In their review of studies
on the learning and teaching of functions and graphs, Leinhardt et al. (1990) underlined
slope-height confusion as one of the three major categories of students’ difficulties
regarding analysis of graphs. McDermott et al. (1987) reported the slope-height
confusion of students who were trying to compare the velocity of two moving objects by
analyzing their position versus time graph. In that study, researchers presented students
(who were enrolled at introductory physics course at college) position versus time graphs
of two moving objects, and then asked them to compare the velocity of two objects at a
particular time. Most students provided incorrect answers to the question, and their
responses implied that instead of looking at the difference between slopes of two lines on
the position versus time graph of two moving objects, students focused on the heights of

the lines in order to compare the velocity of the objects at a particular time.

16

Another area where students often struggle while studying the slope concept is
making connections between the algebraic formula of slope and its graphical and

functional representations. When students are asked to ﬁnd the slope of a line, they often

Y2 ”yr

 

apply the algebraic formula of m = and do the computation (Crawford & Scott,

x 2 l
2000). Finding the numeric value of slepe from the formula, however, does not show that
students have mentally constructed slope as the ratio of the change in one variable to the

change in the other related variable, or slope as the ratio of the length of perpendicular

line segment (y2 —y1) to the length of horizontal line segment (x2 —x1) on the Cartesian

plane (Lobato et al., 2003, Schoenfeld et al., 1993). For example, Schoenfeld et al.
examined one student’s understanding of linear functions in a computer setting where the
student was playing a game and working with a tutor. They found that the student’s initial
knowledge of slope included no connection between slope formula and its graphical
representation on the Cartesian plane. In his study with high-school students who were
enrolled in ﬁrst-year algebra through calculus, Knuth (2000) found that the majority of
the students were successful in ﬁnding slope values by using algebraic formula, but their
responses to the questions about slope showed no connection between graphical and
algebraic representations of slope.

Research focused on identifying slope of a line on a non-homogenous coordinate
system (e.g., Rasslan & Vinner, 1995; Zaslavsky et al., 2002) indicated that students
often experience difﬁculties in ﬁnding slope value. Zaslavsky et al. asked their subjects
with various levels of experiences in learning and teaching mathematics (from high-

school students to mathematics educators and mathematicians) to ﬁnd the slope of a

17

function whose graph was presented in two different coordinate systems: homogenous
versus non-homogenous systems. Both students and inservice mathematics teachers
experienced confusion while trying to ﬁnd the slope of a linear function on a non-
homogenous system, and they could not identify the distinction between the rate of
change of the ﬁrnction and the slope of its line. In particular, most participants in the
study could not recognize that the rate of the change of a function (or the slope of a
function) is the slope of the line representing the function in a homogenous system;
however, there is no relation between the rate of change (or slope) of the function and the
slope of its line on a non-homogenous system.

Leinhardt et al. (1990), Crawford and Scott (2000), and Stump (2001) emphasized
that students often have difﬁculties in considering slope as a measure for the rate of
change where two quantities covary. Only 3 out of 22 high school students in Stump’s
(2001) study used the slope of a line to identify the rate of change in functional contexts
where two quantities changed relatively. Similarly, after examining preservice and
inservice secondary mathematics teachers’ mathematical understanding of slope in
various representations, Stump (1997) found that most teachers did not consider slope as
a rate of change while analyzing the changes in two quantities. As Crawford and Scott
(2000) pointed out, reform based projects and documents (e. g., NCTM, 2000) placed
considerable emphasis on studying variability and change, thus it has become essential
for students understanding slope as a rate of change.

After identifying students’ difﬁculties in understanding slope concept, researchers
have recommended various instructional strategies to help students overcome their

struggles and improve their learning of slope concept. Their most common suggestions

18

for teaching slope can be summarized in the following way. First, introduce slope
concept as a rate of change by using real life examples, such as distance versus time
relation for traveling cars in order to help students develop some understanding of slope
(Crawford & Scott, 2000; Leinhardt et a1, 1990). Second, revisit students’ knowledge of
Cartesian coordinate system and help students refresh and improve their knowledge about
the Cartesian system in order to enable them to understand slope as a ratio by making
connections between the algebraic formula of slope and its graphical representation
(Leinhardt et a1, 1990; Schoenfeld et al., 1993; Stump, 2001). Finally, help students
realize the distinction between the slope of a linear function and the slope of a line by
allowing them to work on both homogenous and non-homogenous coordinate systems

while studying slope concept (Rasslan & Vinner, 1995; Zaslavsky et al., 2002).

19

Research Focus

VMs have the potential to meet the shortcomings of physical manipulatives in
learning and teaching (Dorward, 2002; Sarama et al. 1996) by providing multiple linked
representations (e.g., algebraic, graphic, and tabular) of linear functions to help students
identify slope value in various forms of linear functions. Furthermore, VMs may enable
students to recognize the relations between making changes on the slope value in one
representation of linear functions and the effects of these changes on the other forms of
linear functions. With some VMs, for example, once students make changes on the slope
value in a particular algebraic form of a linear function, they can instantly observe the
effects of their changes on the graph of the linear function.

An analysis of several VMs on the web revealed that various instructional
approaches can be used with VMs to improve students’ learning of mathematics. Among
these approaches, two instructional approaches were chosen for this study because both
of them were commonly used in mathematics teaching and they also underlined different
aspects of mathematics learning. In one of these two instructional approaches, one group
of participants (OVM) used VMs to answer open-ended exploratory questions asking
them to observe the activities on the VMs, make reﬂection on their interactions with the
VMs, and identify the relations among various mathematical ideas based on their
observations. In the other instructional approach, the other group of participants (SVM)
worked with the same set of VMs that OVM group used to answer structured
mathematics questions requiring them to mostly focus on numeric expressions,

quantitative relations, calculations and mathematical procedures.

20

This study examined the effects of using VMs with these two different
instructional approaches on students’ knowledge of slope. In the following sections, the
knowledge of slope that served as a focus for the study will be described and the research
questions that guided the inquiry will be presented.

Students’ Knowledge of Slope

Review of research literature about students’ learning of slope indicated that there
are major areas of slope knowledge that need particular attention to improve students’
understanding of slope. This study focused on the following aspects of students’
knowledge of slope.

Knowledge of making connections between slope concept and real life settings:

0 Considering slope as a rate of change while exploring graphs that compare
two quantities from real life contexts (e.g., distance-time vs. velocity-time
graphs), and being able to identify slope values from these graphs.

0 Forming a graph from real life data, then ﬁnding slope value on the graph, and
identifying its contextual meaning (e.g., slope as the rate of increase in the
perimeter of circle).

0 Identifying slope as the steepness of physical objects (e. g., ramps) while
solving problems given in real life settings.

Knowledge of making connections between algebraic and graphical forms of

linear functions, and identifying slope value on different coordinate systems:

0 Forming lines that have the same or different slope values on the Cartesian
coordinate system and identifying how making changes on an algebraic form

of a linear function affect its graph on the Cartesian plane, and vice versa.

21

0 Identifying the relations between different numeric values of slope and the
steepness of a line.

Knowledge of identifying slope value from tabular and various algebraic forms

(e.g., slope-intercept form) of linear functions:

0 Finding the slope of a linear function from its tabular representation, and
identifying slope value from various algebraic representations of linear
functions (e.g., slope intercept form).

0 Forming an algebraic equation by using the slope value and a pair of the
points that the equation satisﬁes.

The study also examined students’ knowledge of slope through three types of
questions: The questions mainly concerning use of conceptual knowledge of slope
constitute the ﬁrst type of questions. The second type of the questions mostly requires use
of procedural knowledge of slope. The other type of questions primarily entails use of
combination of conceptual and procedural knowledge of slope. Hiebert and Lefevre’s
(1986) deﬁnitions of procedural and conceptual knowledge were extended to describe
conceptual, procedural, and combination of conceptual and procedural knowledge of
slope. Hiebert and Lefevre deﬁne conceptual knowledge as the knowledge that has rich
relationships, and consider procedural knowledge as rules or procedures to solve
mathematics problems. Based on these deﬁnitions of conceptual and procedural
knowledge in mathematics, conceptual, procedural, and combination of conceptual and
procedural knowledge of slope can be described in the following way:

0 Conceptual knowledge of slope includes identifying mathematical relations

among various representations (e. g., algebraic, graphic) of slope concept,

22

deﬁning slope concept based on its different representations, and interpreting
the meaning of slope values in real life settings.

0 Procedural knowledge of slope consists of ﬁnding slope values in algebraic,
tabular, and graphic representations of linear functions by using different
formulas or procedures.

0 Combined conceptual and procedural knowledge of slope contains various
amounts of procedural and conceptual knowledge of slope. For example,
ﬁnding the value of slope in a graphical representation, and identifying the
relationships between algebraic and graphical representations of slope.

Research Questions

The purpose of this study is to examine the effects of using VMs with two
different instructional approaches on students’ knowledge of slope through the following
overall question:

What are the effects of VMs used with open-ended exploratory questions versus

structured mathematics questions on students’ knowledge of slope?

This overall question gives rise to two more speciﬁc research questions:

1. What are the effects of using VMs with open-ended exploratory questions
versus structured mathematics questions on the pre- to posttest gains?

2. What are the effects of using VMs with open-ended exploratory questions
versus structured mathematics questions on the pre- to posttest gains on
the procedural, conceptual, and combined conceptual and procedural

questions?

23

CHAPTER 3
METHOD

College students taking a remedial mathematics course completed a pretest that
assesses their knowledge of slope, randomly assigned into two groups, individually
worked with a set of VMs to answer open-ended exploratory versus structured
mathematics questions on their laptops throughout four 30-to-45 minute intervention
sessions, and then completed a posttest having the same set of questions with the pretest.
Participants

Sixty-ﬁve students taking Intermediate Algebra (MTH 1825) at Michigan State
University in the Fall 2008 Semester participated in the study, with 48 of the students
completing all of the experimental tasks to be included in the analysis (See Table 1).
MTH 1825 is a three-credit remedial mathematics course. According to the university’s
regulations, students who score 7 or below on the Mathematics Placement Service (MPS)
exam, or have a subscore below 12 on the Elementary Algebra section of the ACT exam
(a national college admission examination) are required to take MTH 1825. The course
covers basic mathematics topics such as properties of real numbers, factoring, roots,

radicals, ﬁrst and second degree equations, and linear inequalities.

24

Table 1

Number of Students Participated to the Sessions

 

 

 

 

 

 

 

 

Sessions Number of
Participants
OVM SVM
Session 1 32 33
Session 2 24 25
Session 3 24 25
Session 4 24 24
Session 5 24 24
Session 6 24 24

 

 

 

 

 

Students volunteered to participate in the study through their MTH 1825
instructors’ announcement about the study, ﬂyers around their classrooms, and a recruiter
who was hired by the researcher. Students’ participation to the study was encouraged in
two ways. First, the researcher offered each student $40 to participate in all six sessions
of the study. Students received $5 for their participation to each session of the study, and
they received $10 as bonus in addition to $30 after participating in all six sessions.
Second, the researcher informed students that through their participation in the study,
they could learn mathematics concepts pertaining to the future topics of their current
MTH 1825 course. The details about the recruitment of students can be found in
Appendix A.

Setting

The study was conducted in the rooms that have wireless connection. All

participants responded to sets of questions by using the VMs and typing their answers on

the computer.

25

Materials

Each participant in the study completed a background questionnaire, pretest,
posttest, and used the VMs to answer the questions on the computer.

Background questionnaire. The background questionnaire (see Appendix B)
consisted of open-ended questions that aim to identify students’ experiences in learning
mathematics, and using computer and the lntemet technology in (or outside of) their
courses.

Pretest and posttest. All participants answered to the questions about slope on a
pretest and posttest (see Appendix C). Both tests have the same set of questions,
presented in different order. The pre- and posttest measured students’ knowledge of slope
by including questions in three major areas (see Table 2):

0 Making connections between slope concept and real life experiences.

0 Identifying relations between algebraic and graphical forms of linear functions,
and ﬁnding slope of the functions on different coordinate systems.

0 Finding slope value from tabular and various algebraic forms (e.g., slope-intercept

form) of linear functions.

26

Table 2

Categories of Slope Knowledge and Related Pretest (Posttest) and Session Questions

 

 

 

 

 

 

forming an algebraic equation by using the
slope value and a pair of the points that the
equation satisﬁes

 

 

Categories of Examples Related Related
Slope Pretest Intervention
Knowledge (Posttest) Sessions
Questions

Connections considering slope as a rate of change while Pre (Post) Session 1
between slope exploring graphs that compare two Question 5 (7),
concept and quantities from real life contexts and 8 (5), and 9 (3)
real life identifying slope values from these graphs
experiences

building a graph from a real life data, then

ﬁnding slope value on the graph, and

identify its contextual meaning

identifying slope as the steepness of

physical objects (e.g., ramps) while solving

problems given in real life settings
Identifying constructing lines that have the same or Pre (Post) Session 2
relations different slope values Question 1(1), and 3
between 4 (6), 7 (9), and
algebraic and how making changes on an algebraic form 10 (10)
graphical of linear function affects its graph
forms of linear identifying the relations between the
functions numeric value of slope and the steepness of

a line

‘ ﬁnding slope of a function when its graphs

is drawn on a Cartesian plane or a non-

homogenous coordinate system
Finding slape ﬁnding the slope of a linear function from Pre (Post) Session 3
value from its tabular representation Question 2 (4), and 4
tabular and 3 (2), 10 (10)
various identifying slope value from various
algebraic algebraic representations of linear
forms of linear functions (e.g., slope-intercept form)
functions

 

27

 

As shown in Table 3, pre- and posttest also consisted of three types of questions
assessing students’ conceptual, procedural, and combined conceptual and procedural
knowledge of slope.

Table 3

Categories of Questions throughout the Pre-and Posttest

 

 

 

 

 

 

 

 

 

 

Pretest Posttest Type of knowledge needed to answer the question
Q1 Q1 Conceptual

Q2 Q4 Procedural

Q3 Q2 Procedural

Q4 Q6 Combined conceptual and procedural

Q5 Q7 Combined conceptual and procedural

Q7 Q9 Procedural

Q8 Q5 Conceptual

Q9 Q3 Conceptual

Q10 Q10 Combined conceptual and procedural

 

 

 

 

 

Sources of VMs. In the study, students used the VMs from the National Library of
Virtual Manipulatives (N LVM) (http://nlvm.usu.edu), SeeingMath
(http://seeingmath.concord.org), and National Council of Teachers of Mathematics
(NCTM) Illuminations (http://illuminations.nctm.org) web sites. All of the VMs were
incorporated into a standard computer interface, which presented each VM along with

exploratory open—ended or structured mathematics questions.

28

Procedures

Each student participated in six sessions, one session to complete the background
questionnaire and pretest, four sessions for the intervention, and one session to complete
the posttest (see Table 4). In the ﬁrst session, participants completed the background
questionnaire and the pretest on the paper-pencil format, with 45 minutes to complete the
pretest. Then, students were randomly assigned into two groups: VMs with open-ended
exploratory questions (OVM) group and VMs with structured mathematics questions
(SVM) group.

Throughout the intervention sessions, OVM students used VMs to answer open-
ended exploratory questions asking them to observe the activities on the VMs, think
about their interactions with VMs, and identify the relations among different
mathematical ideas based on their observations. While working with VMs, OVM
students used their own initial values based on their choices to answer open-ended
exploratory questions that rarely required using numeric expressions, algebraic relations,
calculations, and mathematical procedures. In contrast to OVM group, SVM students
worked with VMs to answer the structured mathematics questions that mostly entailed
using formal mathematics language such as numeric values, expressions, calculations and
mathematical procedures during the intervention sessions. Structured mathematics
questions mostly have one correct answer and required students completing particular
steps or executing speciﬁc formulas. Moreover, these questions posed particular initial
values that SVM students had to use while working with VMs. Therefore, SVM students
had considerably less freedom than OVM group in their activities with VMs to answer

the intervention questions.

29

Students participated in four 30-to-45 minute intervention sessions, they
completed ﬁrst two sessions with 10-15 minutes break between the sessions in one day,
and at least one day at most one week later they completed the other two sessions with
10-15 minutes break between the sessions. In the intervention sessions, students worked
with VMs individually in a computer laboratory, with OVM students working on open-
ended exploratory questions, and SVM students working on structured mathematics
questions.

Students in the SVM group were presented with structured mathematics questions
on the right side of the computer screen, next to the VM. After working with VMs to
answer the intervention session questions, SVM students typed their answers on the
spaces provided on the computer. Students in the OVM group worked with the same
VMs, but used them to answer open-ended exploratory questions. They, too, typed their
responses on the computer. Throughout the intervention sessions, students’ written

answers were automatically saved by the computer.

30

Table 4

Design of the Study for the VMs Used with Exploratory versus Structured Mathematics

 

 

 

 

 

 

 

 

Questions

Sessions VMs with VMs with Order of

Exploratory Q. Structured Q. Sessions

Pretest OVM SVM Session 1

8 Distance vs. Time Activity OVM SVM Session 2

:3; E Geoboard Activity OVM SVM Session 3
E g Graphs Activity OVM SVM Session 4
.5. Linear Trans. Activity OVM SVM Session 5
Posttest OVM SVM Session 6

 

 

 

 

 

 

Intervention Session 1 (Session 2 - Distance vs. Time Activity). This session aimed
to help students understand the meaning of slope in a real-life setting by providing a VM
that provides the simulation of two runners’ actions along a track and presents position-
time graphs of these two runners. Students in the OVM group worked with the VM (see
Figure l), guided by open-ended exploratory questions (see Appendix D) about the rate
of change and its relation to slope. Students in the SVM group used the same VM (see
Figure 2) to answer structured mathematics questions (see Appendix D) focusing on the

relations between slope and rate of change.

31

 

 

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Intervention Session 2 (Session 3 - Geoboard Activity). In this session, students
were expected to improve their ability to identify the slope of a linear function when they
had two different points that satisfy the function. Students used the VM (see Figure 3)
that enabled them to draw lines between different points on a Geoboard to respond the
open-ended exploratory or structured mathematics questions (see Appendix D).

Figure 3. A screenshot of the VM that students used in Intervention Session 2.

Contents 1. Slope Given Two Points ( 1) '3 Prev M

 

 

 

 

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34

Intervention Session 3 (Session 4 - Graphs Activity). This session aimed to help
students develop their ability to ﬁnd slope value from algebraic and graphical
representations of linear functions and identify various meanings of slope value through
these representations. Students worked with the VM (see Figure 4) that provides both
algebraic and graphical representations of linear functions to respond open-ended
exploratory or structured mathematics questions (see Appendix D).

Figure 4. A screenshot of the VM that students used in Intervention Session 3.

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35

Intervention Session 4 (Session 5 - Linear Transformer Activity). In this session,
students were expected to improve their ability to make connections between slope-
intercept and point-slope forms of linear functions by using slope concept and its
graphical representation. In the session, students used the VM (see Figure 5) that
provides graphical representations correspondent with slope-intercept and point-slope
forrns of linear functions to answer open-ended exploratory or structured questions (see
Appendix D).

Figure 5. A screenshot of the VM that students used in Intervention Session 4.

   

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36

Posttest Session. After completing the four intervention sessions, students
received the posttest (see Appendix C) which has the same set of questions with the
pretest in a different order, and had 45 minutes to complete the test.

Pre- and Posttest Scoring

Responses on the pre- and posttest were scored using the rubric presented in
Table 4. The researcher developed this rubric using the approach taken in his work with
colleagues on MSU’S Teachers for a New Era (TNE) project.

Table 5

General Rubric for Pre— and Posttest Scoring

 

Score Criteria

 

4 Responses that present complete correct mathematical explanations with correct
answers.

 

3 Responses that lack clarity in their explanations and/or include minor calculation
errors.

 

2 Responses that show a chain of mathematical reasoning but include major
conceptual errors or incomplete solutions.

 

1 Responses that involve at least one correct and relevant mathematical statement.

 

0 Responses consisting only of mathematically incorrect, irrelevant, or blank
statements.

 

 

 

 

Based on this general rubric, the researcher developed speciﬁc rubrics for each

question on the pre-and posttest (See Table 6 for a particular rubric).

37

Table 6

Rubric Used Scoring Students ’ Answers on the Pre (Post) Question 8 (5 )

 

Score

Criteria

 

4

Responses that show the steepness of current ramp with correct mathematical
explanations, and include complete correct explanations about how to change the
dimensions of the ramp to obtain a new ramp that has the same steepness with the
current ramp.

 

Responses that provide correct mathematical explanations to ﬁnd the steepness of
current ramp, and to determine the new dimensions of a new ramp with the same
steepness, but include some calculation errors and therefore provide incorrect
results for the steepness of the ramp, and the dimensions of the new ramp

or
Responses that correctly show the dimensions for a new ramp which has the same

steepness by using proportions but include incomplete or unclear explanations
about how he found the dimensions of new ramp.

 

Responses that correctly show the dimensions of a new ramp (that has the same
steepness with current ramp), but did not provide any explanation about how he
found the dimensions of the new ramp

01‘

Responses that show a chain of reasoning (that focuses on the relation between
height and length of the ramp) in their explanations about the dimensions of a new
ramp but involve incomplete explanations and/or provide incorrect values about
the dimensions of the new ramp.

 

Responses that include at least one correct and relevant mathematical statement.

 

 

 

Responses that present mathematically incorrect, irrelevant, or blank statements.

 

To determine the Interrater reliability of the scoring, a second rater who is a

mathematics education doctoral student scored the pre-and posttest responses of 10

randomly selected participants. The Interrater reliability is deﬁned as the Pearson

correlation between two scores from two different raters. The Pearson correlation

between two raters’ scores was calculated to identify the Interater reliability of scoring.

As seen from Table 7, there is a high level of agreement between two scorers.

38

 

Table 7

Interrater Reliability of Scoring for Overall and Subsections of the Pre-and Posttest

 

 

 

 

 

 

 

 

 

Sections Interrater Reliability
Pretest Posttest
Overall 0.93 0.95
Conceptual 0.92 0.95
Procedural 0.94 0.96
Combined Con. and Proc. 0.94 0.95

 

The scoring process revealed problems with one item: Pretest Question 6 (Posttest
Question 8). The purpose of this item was to evaluate students’ understanding of slope on
a non-homogenous system whose axes had different scales. Because of unclear wording

of the item, students’ performance could not be validly assessed on that item. Therefore,

the item was dropped from the data analysis.

39

 

CHAPTER 4
RESULTS

Research Question 1 concentrated on students’ overall performance throughout
the pre- and posttest. Therefore, all students’ gain scores were found in the following
way. First, students’ overall scores on the pretest and posttest were calculated by
summing their scores on the nine questions in the pretest and posttest. Then, students’
overall gain scores were computed by subtracting their overall pretest scores from their
overall posttest scores. Table 8 presents overall mean pretest, posttest, and gain scores for
both OVM and SVM groups.
Table 8

Means for OVM and S VM Groups ' Overall and Gain Scores

 

 

 

 

 

 

 

 

 

 

 

 

Groups Pretest Posttest Gain
M SD M SD M SD
OVM 7.78 4.16 11.48 5.96 3.70 3.59
SVM 8.74 4.46 13.23 5.27 4.49 3.68
Overall 8.26 4.29 12.36 5.63 4.1 3.62

 

OVM and SVM group students had similar overall mean scores on the pretest and

posttest, and revealing no large differences in their gain scores (see Figure 6).

40

 

Figure 6. Comparison of 0 VM and S VM groups ’ overall gain scores.

 

   

 

 

 

15.00 -
10.00 "
.E
a
U
= 5.00 “
a
b
>
O
0.00 "
-5.00 . 1 l
SVM OVM
Group

An independent t-test was conducted to test whether there is signiﬁcant difference
between OVM and SVM students’ overall gains. The data met the following assumptions
of the independent t-test. First, both OVM and SVM group’s overall gain scores have a
roughly normal distribution. Second, the distribution of scores in each group was
homogenous (in other words, both groups had similar variance). Third, the observations
in two groups were independent since students were randomly assigned into the groups.
Finally, t-tests are appropriate for continuous scale data. Although ordinal scale (ﬁom 0
t0 4) was used to assess students’ knowledge in each question, their scores for nine

41

questions on the pretest and posttest were summed. That means each student can have
scores from 0-36 on the pretest and posttest. Therefore, it can be assumed that a
continuous scale was used to measure students’ performance. This is also a common
approach while deciding whether the measurement scale for dependent variable is
continuous or not in educational research.

The t-test showed no signiﬁcant difference between OVM and SVM groups’
overall gain scores [t (46) = 0.744 and p > 0.05].

Procedural and Conceptual Gains

Research Question 2 primarily concerned with students’ performance on the
questions that require using different types of slope knowledge: conceptual, procedural,
and a combination of conceptual and procedural knowledge. To answer this question,
OVM and SVM students’ mean scores and mean gain scores for three subsets of items on
the pre-and posttest were calculated. These three subsets of items draw on students’
conceptual, procedural, and combined conceptual and procedural knowledge of slope,
respectively (see Table 3).

Procedural items. As shown in Table 9, SVM students attained considerably
higher gain scores than OVM group in the subset of procedural knowledge questions.
Table 9
Means for OVM and S VM Groups ' Overall and Gain Scores on the Subset of Procedural

Knowledge Questions

 

 

 

 

 

 

 

 

 

 

 

 

Groups Pretest Posttest Gain
M SD M SD M SD
OVM 2.84 2.26 3.73 2.96 0.89 2.04
SVM 3.3 2.43 5.51 3.04 2.21 2.72
Overall 3.07 2.33 4.62 3.1 1.55 2.47

 

42

 

As seen from Figure 7, there is a substantial difference between OVM and SVM
students’ scores on the subset of procedural knowledge questions.
Figure 7. Comparison of OVM and S VM groups ' gain scores on procedural knowledge

questions.

 

10.00 ‘

Gains

4.00 "

Procedural

2.00 ‘

0.00 "'

   

-2.00 '-

 

 

 

SVM OVM
Group

43

Conceptual items. Although students in OVM group showed lower performance
than SVM students on the procedural knowledge questions, they had higher gain scores
than SVM students for conceptual knowledge questions. As seen in Table 10, OVM
students’ mean gain scores is 1.56 and the mean of SVM students’ gain scores is 0.69.
Table 10
Means for OVM and S VM Groups’ Overall and Gain Scores on the

Subset of Conceptual Knowledge Questions

 

 

 

Groups Pretest Posttest Gain
M SD M SD M SD
OVM 2.38 1.83 3.94 2.41 1.56 2.04

 

SVM 3.04 1.81 3.73 1.69 0.69 2.72
Overall 2.70 1.83 3.83 2.06 1.13 1.54

 

 

 

 

 

 

 

 

 

 

Figure 8 also shows that OVM students obtained considerably higher gain scores

than SVM group on the conceptual knowledge questions.

44

Figure 8. Comparison of 0 VM and S VM groups ' gain scores on conceptual

knowledge questions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.00 ‘

4.00 '
E
'3
g .....
T: 2.00 _ ............
B
a
8
:i
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D

0.00 "

-2.00

SVM OVM
Group

Combined conceptual and procedural items. SVM group students also obtained
higher gain scores than OVM students on the subset of questions that combined
conceptual and procedural knowledge of slope. As presented in Table 11, SVM students’

mean gain score is 1.6 and the mean of OVM students’ gain scores is 0.66.

45

Table l 1

Means for OVM and S VM Groups’ Overall and Gain Scores on the Subset

of Combined Conceptual and Procedural Knowledge Questions

 

 

 

 

 

 

 

Groups Pretest Posttest Gain
Mean SD Mean SD Mean SD
OVM 2.66 1.55 3.32 1.81 0.66 1.09
SVM 2.62 1.52 4.22 1.73 1.6 1.6
Overall 2.64 1.52 3.77 1.81 1.13 1.43

 

 

 

 

 

 

As seen from Figure 9, SVM students had considerably higher gain scores than

OVM group on the combined conceptual and procedural knowledge questions.

46

 

Figure 9. Comparison of OVM and S VM groups ’ gain scores on the combined

conceptual and procedural knowledge questions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.00 - 02
4.00 -
E
Q
o
2.00 -
‘U
0
ﬂ
:5
a
O
Q
0.00 —
-2.oo , ,
SVM OVM

Group

In addition to these descriptive statistics, a Multivariate Analysis of Variance
(MANOVA) test was conducted for OVM and SVM students’ gain scores on conceptual,
procedural, and combined conceptual and procedural knowledge of slope questions.

OVM and SVM students’ gain scores in three types of questions satisﬁed the
followtng assumptions of the MANOVA test: normal distribution of students’ gain scores

in each type of knowledge, equality of variance between groups in three types of
kno‘Wledge, independence of observations between OVM and SVM groups in the study.

47

Moreover, there are two main reasons for using the MANOVA test to determine the
signiﬁcance among three outcomes (students’ gain scores in three types of knowledge).
First, there is a signiﬁcant correlation between OVM and SVM students’ gain scores on
procedural, conceptual and combined conceptual and procedural knowledge questions.
This means that there is some relation (or overlap) between students’ gain scores.
Therefore, using a MANOVA test is more appropriate than an ANOVA test that requires
all outcome variables have no relation with each other. Second, an independent t-test
indicated that there is no signiﬁcant difference between OVM and SVM students’ pretest
scores on conceptual [t (46) = 1.272, p > 0.05], procedural [t (46) = 0.670, p > 0.05], and
combined conceptual and procedural [t (46) = - 0.093, p > 0.05] types of questions.
Because OVM and SVM students’ pretest scores on the three types of questions were not
statistically signiﬁcant, there is no need to control their pretest scores in the MANOVA
test. Thus, using the MANOVA test is more appropriate than the MANCOVA test to
analyze students’ gain scores.

A MANOVA test indicated that there were signiﬁcant differences between OVM
and SVM students’ gain scores in three types of questions. The results of the MANOVA
test can be summarized in the following way. First, SVM students had signiﬁcantly
higher gains than OVM students in the subset of procedural questions [F (1, 46) = 4.409,
p < 0.05, partial n2 = 0.087]. Second, SVM group had signiﬁcantly higher gains than
OVM students in the questions that require using combined conceptual and procedural
knowledge of slope [F (1, 46) = 5.688, p < 0.05, partial n2 = 0.110]. Finally, OVM
students attained signiﬁcantly higher gains than SVM students in the conceptual

questions [F(1, 46) = 13.562, p < 0.05, partial n2 = 0.228].

48

Item by Item Analysis

To identify the learning activities in the intervention sessions that helped students
obtain the most gains on the pre-posttest items, OVM and SVM students’ mean gain
scores on each test item were examined. This analysis also shed light on how OVM and
SVM students’ activities with VMs in the intervention sessions enabled them to show
considerable differences between their gain scores on three types of questions throughout
the pre- and posttest.

Mean gain scores were computed for the OVM and SVM groups on each pre-
posttest item. The OVM group attained largest mean gain score on conceptual question 3,
and the SVM group had the highest mean gain score on procedural question 2. In
addition, these two questions are the items on which both OVM and SVM group showed
the largest differences between their mean gain scores. As seen in Table 12, the
difference between OVM and SVM groups’ mean gain scores are 0.94 and 0.72 on

conceptual question 3 and procedural question 2 respectively.

49

Table 12

Mean Gain Scores Item by Item

 

Question Pretest (Posttest) Groups
Type Q. No.

 

OVM SVM
Con. Q. 1 Pre Q. 1 (Post Q. l) 0.79 0.42
Con Q. 2 Pre Q. 8 (Post Q. 5) 0.5 0.21

 

 

 

Con Q. 3 Pre Q. 9 (Post Q. 3) 0.90 -0.04
Proc. Q. 1 Pre Q. 2 (Post Q. 4) 0.13 0.88
Proc. Q. 2 Pre Q. 3 (Post Q. 2) 0.34 1.06
Proc. Q. 3 Pre Q. 7 (Post Q. 9) 0.40 0.38
Com. Q. 1 Pre Q. 4 (Post Q. 6) 0.19 0.31

 

 

 

 

 

Corn. Q. 2 Pre Q. 5 Post Q. 7) 0.13 0.71

 

Com. Q. 3 Pre Q. 10 (Post Q. 10) 0.34 0.58

 

 

 

 

 

 

Conceptual question 3 (see Figure 10) that OVM group obtained the highest mean
gain score aimed to measure students’ conceptual knowledge of slope by asking them to
compare the speed of two cars based on their distance vs. time graphs.

Procedural question 2 (see Figure 11) that SVM group had the largest mean gain
score entailed students ﬁnding the slope values of linear functions presented in different

algebraic forms in order to assess their procedural knowledge of slope.

50

Figure 10. Conceptual question 3.

 

 

The graph below shows distance versus time relation for car A and car B.

b)

 

 

dl
A
.1." B
E
8
C
is
O
oiiiiiiiii >.
1234 56789

Time (hour)

When t = 2 hours, is the speed of car A greater than, less than, or equal to the
speed of car B? Please explain your answer with reasons.

Do car A and car B ever have the same speed? If so, at what times? Please
explain your answer with reasons.

 

51

 

Figure 11. Procedural question 2.

 

Please ﬁnd the slope of each of the following equations and provide your explanations.
a) f(x) = —1— - 4x
2
b) f(x) =2x+ 1 —-2x

1 1
f =- -—+— +1
c) (x) x 4 3x
(I) y+0.25=%(x-2)-2(x-2)

e) 3y—2= % (x-l)

 

 

 

These two items with large differences in mean gain scores may help explain how
students’ learning activities on the intervention sessions enabled them to have higher
gains on different types of slope knowledge. Because students showed the largest
increase in their gain scores on these two items, analysis of how they responded to the
questions in the intervention sessions that were most related to these items may provide
considerable evidence about the effects of students’ different learning activities on their
gain scores. Throughout all the intervention sessions, OVM and SVM students engaged
with activities pertaining to pre-and posttest items. Intervention Session 1 and 4 included
questions that were particularly related to conceptual questions 3 and procedural question
2 respectively.

Students’ responses on these two intervention sessions were examined. The

analysis indicated that students’ answers can be categorized in two ways. One way of

52

categorization is assigning students’ responses into different groups based on the type of
expressions they included. Throughout the intervention session 1 and 4, students’
responses mostly included one of the following four types of expressions.

Responses including only statements (Stat. ). This type of response contains only
statements that present no explanations or descriptions of activities with VMs. For
example:

“As the position increase so does the time.”

“The equation is y = 0.5x + 3.9.”

“The runner's speed is dependent on the time he or she is allotted.”

Responses with the descriptions of activities on the VMs (Desc. of acts. on VMs).
On this type of responses, students described the events that occurred on the VMs while
interacting with them. For example:

“y=3.0+2.0 and point slope i get y-l l.0=3.0(x-3.0) i just plugged in numbers until
i got what you were asking for.”

“The girl started ahead of the boy. She really does go fast until boy catched
[caught] up! They were both very close at the end and the girl won by an inch.”

“1 found slope intercept form. I used the points (-2, 3) and plotted it on the graph

then moved the line until 0.5 became my x.”

Responses consisting of explanations with reasons (Exp. ). This type of responses
involves at least one explanation with reasons. These responses often display cause-effect

relations or provide evidences to support their conclusions. For example:

53

“The boy will have a different distance vs. time because he now has a different
feet [foot] per second and the graph will be different because they are at different
positions.”

“Slope-intercept form is extremely easier to deal with because it has a simple

graphing demonstration compared to the point-slope form it is more difﬁcult

because it has a distributing involved. The similarities are when there changes in
the slope and y-intercept.”

“The more steps per second, the faster the person will ﬁnish the race. This is so,

because the more steps an individual does, their rate becomes faster.”

Responses with explanations and descriptions of activities (Exp. & Desc.). Some
of the explanation responses also included descriptions of learning activities with VMs.
Responses in this category often provide descriptions of events occurring on VMs as
evidence to support their conclusions. For example:

“The boy completed the race before because his step size was at 5 the girl at 3.

The boy and girl both started at the same spot but the boy still ﬁnished within 15

seconds and the girl in a little over 20 seconds.”

“The speed is slower, but because the step size is larger it helps them to win the

race. Example- the girls step size was 3 and she ﬁnished 60 (time) before the guy

whose step size was 1.”

“When I observe the graph of the linear equation I can see that the changes 1

create affect the line in both the slope-intercept and the point-slope. When I

change the slope in the slope-intercept form it rises [raises] the line on the y—axis

and can also lower on the y-axis. Looking at the point-slope form I can see that

54

the change I make in the equation moves the line down the y-axis because I am
increasing the number. All the changes I make reﬂect on the x and y-axes
dramatically.”
The percentages for these four types of responses in intervention Session 1 and
Session 4 were presented on Table 13.
Table 13

Categories of Students ’ Responses in Session I and 4 (by %)

 

 

 

 

 

Sessions Group Exp. Desc. Exp. & Stat.
with VMs Desc.
Session 1 OVM 69 39 24 16
SVM 43 48 20 30
Session 4 OVM 63 20 6 24
SVM 39 12 0 56

 

 

 

 

 

 

 

 

Throughout Sessions 1 and 4, OVM students more often provided explanations in
their responses than SVM group. As shown in Table 13, 69% and 63% of OVM students
explained their answers on Session 1 and 4 respectively. In contrast to OVM students,
only 43% and 39% of SVM students made explanations on these two sessions. Similarly,
SVM students’ answers more often included statements than OVM group over the two
sessions. As presented in the table, 30% and 56% of SVM students provided statements
in their responses on Session 1 and 4 respectively, however only 16% and 24% of OVM
students included statements in their answers on the two sessions respectively.

Both OVM and SVM students also differed in the percentage of their responses
between Session 1 and 4. For example, more number of SVM students provided
descriptions of their activities with VMs than OVM students in Session 1. However, this

difference occurred in the opposite way in Session 4. As seen in Table 13, in Session 1,

55

48% of SVM students included descriptions of their activities with VMs whereas 39% of
OVM students provided descriptions in their responses. However, in Session 4, only 12%
of SVM students contained descriptions in contrast to 20% of OVM students. In addition
to these differences, both groups also showed similar changes in their responses between
Session 1 and 4. For example, both OVM and SVM groups had less percentage of
responses including descriptions of activities with VMs on Session 4 than they did on
Session 1. As shown in Table 13, the percentage of OVM and SVM students whose
responses involved descriptions of activities with VMs decreased from 39% to 20% and
from 48% to 12% throughout the Session 1 and 4 respectively.

The other way of categorizing students’ responses is based on whether their
responses include quantitative or qualitative relations. On the intervention session 1 and
4, students’ answers often contained one of the following three types of expressions.

Responses with numeric relations or algebraic expressions (Nam). This type of
responses contained numeric relations among quantities or algebraic equations. For
example:

“The girl step size was half the size of the boy and the boy reached the end of the
race in slightly more then [than] half of the time it took the girl to ﬁnish the race. The
boy's graph was steeper.”

“The girls speed is 2 steps per second. The boys speed is 3 steps per second. So,
the boy covered more distance.”

“y = 0.5x =3 and y-3 = 0.5(x-3) would be the two equations for the line.”

Responses including calculations (Cale). These responses included various types

of calculations. For example:

56

“75-20 = 55- boy's, 20+20 = 40- girl's”

“x = 40-0/20 = 20 20 = x (the girl), x = 100-10/20 = 4.5”

“Boy: 30/20 = 1.5, Girl: 70/20 = 3.5”

Responses involving qualitative relations (Qual.). This type of response included
qualitative relations among variables such as relations between graphs and runners’
speeds or point-slope forms of linear equations. For example:

“When both the boy and girl have the same starting position, the person with the
larger step size is faster and has a steeper graph.”

“The girl and boy both started at 10. The boy had step of 5 and the girl had 3. The
boy wins this race. He is faster than her because he takes larger steps and covers more
ground. The larger the steps the steeper the line.”

“When I change the slope in point-slope form, it changes the steepness of the line
about a point. The line tilts about a ﬁxed point. If I keep increasing it, it slowly turns
about the point till it is nearly a vertical straight line, crosses the Y-axis, and slopes
negatively.”

As in the ﬁrst way of categorization, these three types of responses emerged from
students’ answers in Intervention Session 1 and 4. The percentages for the three types of
responses on these two sessions are presented in Table 14.

Table 14

Percentage of Students ’ Responses by Category in Session I and 4 (by %)

 

 

 

 

 

 

 

 

 

 

Sessions Group N um. Calc. Qual.
OVM 9 0 36

5mm“ 1 SVM 23 ll 8
OVM 22 r 39

5mm" 4 SVM 43 1 0

 

57

 

More number of OVM students than SVM group used qualitative relations in their
responses throughout the session 1 and 4. As shown in Table 13, 36% and 39% of OVM
students provided qualitative relations in their answers in Session 1 and 4 respectively
while 8% of and no SVM students included qualitative relations in their responses in
Session 1 and 4 respectively.

While making these two ways of categorization, it has not been taken the
correctness of students’ responses on the intervention sessions into consideration. In other
words, including mathematically correct or incorrect explanations and statements had no
effect on the categorization of students’ answers. There are two main reasons for why
students’ responses were not examined in terms of their mathematical correctness. One
reason is that since some questions entailed students making observations and reﬂecting
their ideas about what they did with VMs, students’ answers to this type of open-ended
questions cannot be evaluated as mathematically correct or incorrect. The other reason is
that sometimes students chose the equations or numeric values themselves under certain
conditions and worked with their own input to answer the questions, and in some cases
students did not explicitly present their initial values in their responses, thus it is difﬁcult

to identify these students’ responses as mathematically correct or incorrect.

58

CHAPTER 5
DISCUSSION

This study examined the effects of using VMs with two different instructional
approaches on students’ knowledge of slope. Analysis of students’ responses on the pre-
and posttest showed that both OVM and SVM groups increased their overall knowledge
of slope from the pre- to posttest with similar amounts of improvement in their overall
knowledge of slope. This suggests that VMs with two different instructions enabled both
groups to improve their knowledge of slope.

Although students in both groups showed similar amount of improvement in their
overall knowledge of slope, SVM students differed from OVM students in their gains on
the subsets of items tapping three types of slope knowledge; conceptual, procedural, and
combined conceptual and procedural knowledge of slope. In particular, the OVM group
had higher gains on the conceptual items; SVM students showed higher gains on the
procedural and combined conceptual and procedural items. It is important to identify the
reasons why the groups differed in the amount of improvement in three types of slope
knowledge.

Analysis of students’ pre-and posttest data indicated that the OVM group students
had signiﬁcantly higher improvement than the SVM group in their conceptual knowledge
of SIOpe. OVM students’ activities with open-ended exploratory questions on the
intervention sessions might have contributed to this higher improvement in the
conceptual knowledge of slope. Throughout the intervention sessions, OVM students
worked on the questions similar to the conceptual knowledge questions on the pre-and

posttest — questions that emphasized providing explanations with reasons and

59

identifying relations among different mathematical ideas. In the intervention sessions, the
OVM group answered open-ended exploratory questions that enabled them to think about
their observations related to their interactions with VMs, identify relations between their
observations and mathematical ideas, and explain their responses with reasons. These
activities might have helped OVM students obtain higher gain scores than SVM students
in the conceptual knowledge questions that require reﬂection, providing responses with
reasons, and making connections among various representations of slope concept.

In contrast to the OVM group, SVM students worked with the questions that
mostly required using mathematics procedures and numeric values throughout the
intervention sessions. These questions rarely entailed students making reﬂections on their
observations or identifying relations between their observations and ideas.

As OVM students worked on the questions requiring explanations and making
connections among various mathematical ideas during the intervention sessions, they
included more explanations and qualitative relations in their responses than SVM group
in these sessions. For example, in Session 1, 69% of OVM students included explanations
in their responses while 43% of SVM group explained their answers with reasons.
Furthermore, 36% of OVM students presented qualitative relations in their responses
whereas only 8% of SVM students included these relations in Session 1. Because more
OVM than SVM students included explanations with reasons and qualitative relations in
their responses over the intervention sessions, OVM students might have had more
potential than SVM students to increase their scores on the conceptual questions.

In contrast to OVM students’ greater performance on the conceptual knowledge

questions, SVM students improved more than the OVM group on the procedural

60

questions. Throughout the intervention sessions, SVM students worked on structured
mathematics questions that led them to engage with more mathematical language, such as
mathematical symbols, formulas, and procedures than the other group. These structured
mathematics questions often presented particular numbers as input, and required students
to ﬁnd speciﬁc numeric values (e. g., what’s the speed of the boy and girl runner in the
ﬁrst 30-second?). Like the procedural questions on the pre- and posttest, structured
mathematics questions required using mathematics symbols, numbers, formulas and
procedures. Also, the analysis of SVM students’ responses in the intervention sessions
indicated that, as expected, they included more numeric expressions and relations,
calculations, and mathematical procedures than OVM students. For example, in Session
1, 23% and 11% of SVM students included numeric expressions, relations and
calculations in their responses, respectively, whereas only 9 % of OVM students used
numeric expressions and none of them included calculations in their responses. Because
SVM students included more numeric expressions and relations, calculations, and
mathematical procedures in their responses than the OVM group throughout the
intervention sessions, they became more familiar with procedural questions than the other
group. Therefore, SVM students’ activities with structured mathematics questions during
the intervention sessions might have helped them attain higher gain scores than the OVM
group on the procedural questions.

In addition to their improvement on the procedural questions, SVM students also
obtained higher gain scores on the combined conceptual and procedural items than the
OVM group. However, this does not mean that instruction for the SVM group is also

effective for improving students’ conceptual knowledge of slope, given that SVM

61

students had low gain scores on the conceptual items over the pre-and posttest. There
might be several reasons for why SVM students did signiﬁcantly better than the other
group students in the questions that require using combined conceptual and procedural
knowledge of slope. These questions might have focused on students’ procedural
knowledge of slope more than their conceptual knowledge. If these combined-type
questions emphasized students’ procedural knowledge of slope more than their
conceptual knowledge of slope, SVM students might have obtained higher gain scores
than the OVM group by using their procedural knowledge.

Another reason might be that combined conceptual and procedural questions
required using ﬁrst procedural knowledge, and then applying conceptual knowledge of
slope. If students possessed the required procedural knowledge, they could move on the
next step that required using their conceptual knowledge of slope. If OVM students had
had insufﬁcient procedural knowledge to solve combined-type questions, they might
have been unable to use their conceptual knowledge successfully in these questions.
Therefore, OVM students might have obtained lower gain scores than the other group in
these questions.

Students’ responses on the intervention sessions also provided some insights
about the reasons of why SVM students had higher gain scores than OVM students on
combined conceptual and procedural items. In session 1, many SVM students included
both explanations and descriptions of their activities with VMs, and their percentage
(20%) was close to the percentage of OVM students (24%) who provided both
explanations and descriptions in their responses. SVM students often used their

descriptions of activities with VMs as examples to explain their answers. This implies

62

that since SVM students had already become familiar with explaining their answers with
reasons throughout the intervention sessions, they might have increased their scores on
the combined conceptual and procedural items over the posttest.

This study also examined students’ responses throughout the intervention sessions
to identify how students’ learning activities with VMs enabled them to show considerable
differences in their gain scores on the conceptual, procedural, and combined conceptual
and procedural questions over the pre-and posttest. During the intervention sessions,
OVM students’ responses involved more explanations with reasons and descriptions of
qualitative relations among variables than the SVM group. It is possible that OVM
students’ engagement with explanations and relations between variables helped them
acquire higher scores than SVM group in the conceptual items that require providing
explanations with reasons and making connections among various representations of
mathematical ideas.

In contrast to the OVM group, SVM students’ responses over the intervention
sessions involved more numeric relations, expressions, and calculations. Because the
SVM group mostly focused on mathematical relations, expressions, and procedures while
working with VMs, they might have become more experienced than the OVM group in
solving the procedural questions that mostly require using formal mathematics language
(i.e., algebraic expressions and procedures). This experience with numeric expressions
and mathematical procedures across the intervention sessions may explain, in part, the
SVM group’s increased scores on the procedural and combined conceptual and

procedural items on the posttest.

63

Analysis of students’ responses on the intervention sessions also indicated that
intervention questions helped identify how students interact with the VMs. OVM and
SVM students used the VMs for different purposes to respond the questions. OVM
students often used VMs to explain their answers by describing the events on the VMs as
examples for their claims. In contrast, SVM students mostly interacted with VMs to ﬁnd
particular algebraic equations, graphs or numeric values. Thus, questions in the
intervention sessions directly inﬂuenced how students interacted with the VMs.

Throughout the intervention sessions, both groups provided the descriptions of
their activities with the VMs in their responses. The amount of VM activity description,
however, varied across sessions. For example, in Session 1, 39% of OVM and 48% of
SVM students described their activities with VMs in their answers while in session 4,
only 20% of OVM and 12% of SVM students talked about their activities with VMs.
Although students were explicitly asked to use VMs to answer the questions at the
beginning of each intervention session, in some sessions they rarely referred to their
activities with VMs. However, it should be noted that if the students do not mention the
events on the VMs or their activities with VMs in their responses, this does not mean that
they did not use the VMs. It is possible that students might have used VMs to answer the
questions even though they did not refer to their activities with VMs in their responses.
However, intervention questions in Session 1 are more effective than the questions in
Session 4 in encouraging students to use VMs in their answers. Because students could
respond the questions in Session 4 without using VMs, they might have answered these
questions with little or no interaction with VMs. Therefore, instructional tasks should be

designed in a way that encourages and supports students’ learning activities with VMs.

64

Once instructional activities effectively support students’ interactions with VMs, students
may realize that they need to use VMs to provide successful responses to given
instructional questions. Only stating that students in a learning setting should use VMs to
answer the questions might not be enough to initiate and foster students’ effective
interactions with the VMs.

Throughout the four intervention sessions, two different instructional approaches
have been used with the same set of VMs. In one approach, the SVM group worked on
structured mathematics questions. While studying these questions, SVM students were
exposed to more mathematics symbols, expressions, and procedures than the other group.
In this type of questions, SVM students often needed to begin with given numeric values
and apply some mathematical procedures or formulas to answer the questions. In the
other instructional approach, OVM students mostly focused on exploratory learning
activities (e. g., interpreting the activities on the VMs, making connections among their
observations, and supporting their answers with reasons). These two instructional
approaches were chosen for the study for two main reasons. First, these approaches at
some level reﬂect traditional versus reform-based approaches to mathematics instruction.
Second, these two approaches can also be considered to be an extension of discussion
about whether conceptual or procedural type of knowledge has the most importance in
students’ mathematics learning among mathematics education researchers.

Because the two instructional approaches used in this study have deep roots in
both practical and theoretical areas of mathematics education, this study attempted to
show how using the same technology with these two approaches affected students’

mathematics learning. The two different approaches helped students improve different

65

types of mathematics knowledge. They had also inﬂuences on students’ interactions with
VMs. This shows how technologies with different instructional approaches affect both
our knowledge and our learning activities with them. This also conﬁrms the claim that
technologies do not only inﬂuence our knowledge but also our learning activities with
them (Pea, 1987).

Clark (1983) recommended that researchers should explore the effects of different
instructional methods within various media on learning instead of studying the impact of
one speciﬁc media on learning. This study followed Clark’s suggestion by exploring the
effects of a single medium—a set of VMs—with two different instructions on students’
knowledge of slope.

Strengths and Limitations

In the study, students’ written responses were automatically recorded by the
computer throughout the intervention sessions. Students never wasted time by writing
their answers on the paper while working with VMs. In other words, students worked
with VMs without having any distraction, i.e. using paper and pencil. All students
completed their activities on the computer during the intervention sessions. This might
have helped students deeply focus on their learning activities, and had more opportunities
to improve their knowledge of slope by using VMs.

Forty-eight students completed all session of the study. The number of students
taking MTH 1825 in Fall 2008 semester was about 800. Having 48 participants out of
nearly 800 students in the study considerably increased the chance to identify the

differences between OVM and SVM students’ knowledge of slope.

66

The students who participated in the study were taking a remedial mathematics
course. Because the effects of VMs on students’ learning of slope were particularly
examined, all participants completed the study before working on the slope concept in
their remedial courses. Therefore, students’ learning regarding slope concept from their
remedial courses did have no inﬂuence on their knowledge about slope while they were
participating to the study. However, it is always possible that students could learn about
SIOpe in their other mathematics or related courses. In these courses, instructors might
have slightly mentioned about the slope concept to help students learn other relevant
mathematics concepts or ideas. This study was designed to help students complete all
sessions before they began learn slope concept in their remedial mathematics courses in
order to reduce the effects of students’ learning about slope outside of the study.

Like every study, this study had several limitations that need to be taken into
consideration when interpreting its results and contributions. First, although each
participant used additional time to become familiar with the features of VMs, some
participants might still have not become comfortable in using the VMs, and therefore
they might have not used the VMs effectively to answer the intervention questions.
Furthermore, some participants might have misunderstood some tools of the VMs. This
might have led them to incorrectly (or inadequately) use some tools of the VMs while
responding to the questions. Therefore, they might have had some incorrect or incomplete
knowledge about the SIOpe concept.

Another limitation of the study is that students interacted with VMs for a small
amount of time. Students worked with VMs for only four 30-45 minute sessions during

the two weeks. They had no access to use VMs outside of the study sessions. Therefore,

67

students might not have had adequate time to show considerable changes in their slope
knowledge.

A third limitation of the study was the selection of the participants. Students were
encouraged to participate in the study with the recommendation of their instructors and a
small amount of money offered by the researcher. Students who accepted the offer
participated in the study. Therefore, the participants of the study were not randomly
selected from the actual population of MTH 1825 students. The lack of random selection
of students limits the generalization of this study’s ﬁndings to a larger population of
students.

The other limitation of the study was regarding the data collection. During the
intervention sessions, some participants provided short responses. They wrote one or two
sentences, they did not elaborate their answers even though the researcher prompted them
and encourage them to present their answers with details. Also, when some participants
were using the VMs, the lntemet was suddenly disconnected. This occurred when
students completed their activity and were saving their answers. Because they lost the
work, they had to return the activity and complete it again.

Conclusions

This study reported the effects of using VMs with two different instructional
approaches on students’ knowledge of slope. These two different instructions were (a)
providing open-ended exploratory questions that need to be answered by using a set of
VMs, and (b) asking students to solve structured mathematics questions while working
with the VMs. The results of the study indicated that each type of instruction helped

students improve different types of slope knowledge. In particular, the instruction that

68

focused on mathematical procedures and numerical values resulted in greater
improvement on tasks requiring procedural and combined conceptual and procedural
knowledge of slope. However the instruction that emphasized exploratory learning
activities such as observing, analyzing, and interpreting helped students develop their
conceptual knowledge of slope.

These ﬁndings suggest that teachers can alternatively use VMs with different
instructions to help their students improve different types of mathematics knowledge. For
example, when students have difﬁculties in using mathematical procedures, teachers can
use VMs with procedural based instructional activities. Teachers can also integrate VMs
into their instructional activities that focus on exploratory learning activities to help
students improve their conceptual knowledge. As Stein et al. (2000) pointed out, different
instructional tasks can be used to improve students’ learning of various types of
mathematics knowledge. Therefore, VMs with different instructional approaches can help
students increase their mathematics knowledge.

This study also revealed that instructional activities should support students’
effective interactions with VMs to facilitate their learning of mathematics concepts. As
pointed out earlier, only asking students to use the VM to answer a set of questions might
not be enough to initiate and foster productive interactions with VMs in their learning.
While engaging with instructional activities, students should use VMs to answer the
questions. Therefore, teachers need to revise and redesign their instructional activities in
a way that enables their students to effectively interact with VMs while learning

mathematics concepts.

69

In the research literature, few studies focused on the impact of VMs on
mathematics learning and teaching. This study examined the effects of using VMs with
two different instructional approaches on students’ learning of a particular mathematics
concept to ﬁll this gap in the literature. Based on the ﬁndings of this research study,
future research can deeply examine the use of VMs with a variety of instructional

approaches to improve students’ learning of different mathematics concepts.

70

Appendix A

Information about Recruitment of Students
The researcher arranged meetings with the instructors of MTH 1825 in the ﬁrst week of
the semester and provided information about the educational beneﬁts of the study for the
students.
The instructors of MTH 1825 were given a paper that provides brief information about
the study to make announcement for the study in their courses. The researcher also gave
sign-up papers to the instructors.
Instructors made the announcement for the study in their courses and distributed sign-up
papers among their students. Studens who were interested in participate to the study
wrote their contact information (name, phone number, e-mail address) and available
times to participate the study on these sign-up papers. At the end of the course session,
each instructor collected these sign-up papers and gave them to the researcher. Then the
researcher contacted with the students in the sign-up papers via e-mail or phone and
arranged a meeting time and place with students.
The researcher hired a college student as a recruiter and informed the recruited about the
design of the study. The researcher and recruiter put ﬂyers of the study on announcement
boards in classroom halls and dormitories. Flyers included brief information about the
study; the rules of the participation, the amount of money students would receive at the
end of their participation, and contact information for the researcher (e-mail address and

phone number).

71

The researcher also prepared small cards that include short information about the study.
The recruiter distributed these cards among the students in classroom halls and
dormitories. The recruiter also put an announcement for the study on a web site which is
popular among college students in the campus.

Students who had the information about the study from ﬂyers, small cards, or web
contacted with the researcher via e-mail or phone. Then the researcher arranged a
meeting time and place with students. In each meeting, the researcher met at least 2 at
most 5 students. The researcher reserved rooms having wireless connection as meeting
places for the study.

In the ﬁrst meeting with students, the researcher thanked for their interest to the study and
provided detailed information about the study and responded students’ questions. Then
each student was given a consent form and provided time to read the form. Students who
accepted participate to the study signed the consent form and gave it to the researcher.
Having assigned the consent form, students completed the pretest in the ﬁrst session of

the study, and scheduled a time with the researcher for the next session of the study.

72

Appendix B

Background Questionnaire

Gender: M F
Age:

1. How many hours per week do you study mathematics on average?

2. How many math courses did you take in the last year of your high school? Please

write the name of courses.

3. How many math courses have you taken so far at college? Please write the name of

COLII’SCS.

4. Which speciﬁc topics in algebra are most difficult for you? Why are these topics
difﬁcult to understand for you?

73

5. Have you ever been used any technology (e.g. mathematics software, math
simulations, applets, virtual (computer) manipulatives,. . .etc.) in your math courses or

outside the school?

a. If you have used any technology, please write the name of the type of

technology.

b. If you used technology in your courses, please write your grade level and the

name of course(s) that you used technology.

c. Please write 2-3 sentences about your experiences with each type of

technology that you used in your courses or outside the school.

6. Do you believe that using educational mathematics software, applets, simulations, or
virtual (computer) manipulatives (in school or outside the school) is helpful for

students’ learning algebra? Why? Please explain your answer with 2-3 sentences.

74

Appendix C
Pretest (Posttest)

Please show all your work clearly!

Note: Use a pen for all your work. If you make a mistake, you can lightly cross out your
work. You have 45 minutes to complete this test.

Pretest Question 1(Posttest Question 1)

a) Please write a deﬁnition of slope with your own words.

b) If you choose any three points on the coordinate plane, can you ﬁnd a straight line

that passes all these three points? Please explain your answer with reasons.

c) What do you think about the conditions that make three points on the same straight

line? Please explain your answer with reasons.

Pretest Question 2 (Posttest Question 4)

 

x -4 -2 4

 

f(x) 3 2 -1

 

 

 

 

 

 

a) Please ﬁnd the slope of the f(x), and explain your answer with reasons.

b) Please write an equation for f(x), and explain your answer.

75

Pretest Question 3 (Posttest Question 2)

Please ﬁnd the slope of each of the following equations and provide your explanations.

f) f(x)= %-4x
g) f(x)=2x+ 1 —2x

1 l
h f =- -—+— +1
) (x) x 4 3x
i) y+0.25= %(x-2)-2(x-2)

j) 3y—2=§(x-1)

 

Pretest Question 4 (Posttest Question 6)

Note: This question was adapted from Moschkovich (1996) study.

If you change f(x) = x to the f(x) = %- 5x, what can you tell about the changes in the

graphs of f(x)?

a)

b)

Does the graph of new f(x) steeper than the graph of previous f(x)? Why? Please

explain your answer.

Is it always true that when the numeric value of slope is bigger, the graph of the line

always becomes steeper? Please explain your answer with reasons.

Is there any relation between the slope value and the steepness of the line? Please

explain your answer with reasons.

76

Pretest Question 5 (Posttest Question 7)
Note: This question was adapted from Stump (1997) study.

Suppose that you are measuring the diameters and the circumferences of several
aluminum disks. If you plot a graph of your data with diameter on the x-axis (in inch),
and circumference on the y-axis (in inch), what can you say about the graph you
obtained? Can you ﬁnd the slope of your graph? Describe the meaning of the slope in the
context of the problem. Please explain your answer, and show all your work.

(Note: The Circumference of a circle whose diameter is R is equal to Ru, and assume that
1t = 3.14)

Circumference (inch)

 

v

Diameter (inch)

77

 

Pretest Question 6 (Posttest Question 8)

Note: This question was adapted from Zaslavsky et al. (2002) study.

y

A

........

...........

"""F

lllllllllllllll

""1""1

IIIIIIIIIIIIIIIIIII

L---------
t-----
-8

 

V

a) What is the slope of the function f(x)? How did you ﬁnd it? Please explain

your answer.

b) Please write an equation for the line on the graph above.

78

Pretest Question 7 (Posttest Question 9)

a) Please draw three straight lines whose slopes are -2, 0, 1 respectively on the

following Cartesian plane.

 

 

x
>
. . . . 4 . . . .
. . . . . . . .
. . . _ . . . .
. . _ _ . . . .
n '''' - ..... c ..... r'--lr'"'--"--L--'-Lalnl-lllh
. . . . 3 . . . .
. . . . . . . .
. . . . . . . .
. . . . . . _ .
n ..... — ..... c ..... —'--' ‘ ---- ..... —----L'-"L
. . . . L2 . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
r'-"- """ . ..... - """ I--'--'--'L-'-IL-"IF
. . . . .I. . . . .
. . . . . . . .
y . . . . . . . .
. . . . . . . . .
h . p p _ p p p . V
A . q q — a — 1 _ ‘
. 3. 2. I... 1.1. 2. 3. 4.
. . . . . .. .. ..
. . . . . . . .
. . . . . . . .
. IIIII . ttttt . uuuuu . 000000 .It. It... IIIII . IIIII . titan.
. . . . . . . . .
. . . . . . . .
. . . . . _ . .
. . _ . L . . . .
. lllll . IIIII . ttttt .itll llllt.iItIJtlitJilttJ
. . . . 0. . _ . .
. . . . . . . .
. . . . . _ . .
. . _ _ . . . .
. IIIII . IIIII . lllll . lllll TIIII.tIIIJtittJIIIIJ
. . . . Q... . . . .
. . . . . . . .
. . . _ . . . .
. . . _ . . . .
.IitiLIiItLtoIiL ttttttttt LIIIILIttILtiIib
x.

79

2

l

b) Please draw two different straight lines that have the same slope which is -—

on the following Cartesian plane.

“y

 

I'

I

I

I
L----

""""""""""""""

urn---.-

""""""""""""""

-
---——-----_--------------
-----

‘ '''''''''''''''

---‘d-----

I .........

L----J----J-----

I
I
I
I
L
I
I
I
I
L
I
I
I
I

.4

 

V

80

Pretest Question 8 (Posttest Question 5)

Note: This question was adapted from Lobato & Siebert (2002) study.

Suppose that a customer needs a wheelchair ramp that reaches to his doorstep, which is 4
ft high (point to ground and door). This particular ramp will not reach since it is only 3 ft
high. How can you change the dimensions so that you have a new ramp that is the same

steepness as this ramp but it reaches the door? Please show your work with details.

6ft

 

10ft

 

 

15ft

81

Pretest Question 9 (Posttest Question 3)

Note: This question was adapted from Stump (1997) study.

The graph below shows distance versus time relation for car A and car B.

 

 

d ‘l
A
B
13
L5.
8
§ so
Q
l l I l l l J l l g
0 I I l I r I I I r r t
1 2 3 4 5 6 7 8 9
Time (hour)

c) When t = 2 hours, is the speed of car A greater than, less than, or equal to the

speed of car B? Please explain your answer with reasons.

d) Do car A and car B ever have the same speed? If so, at what times? Please

explain your answer with reasons.

82

Pretest Question 10 (Posttest Question 10)

Which of the following statements are true? Please circle your choice and give your
explanation with reasons.

‘ We can always consider slope of a linear function as a measure for the rate of change in
the function.
True / False

Explanation: .............................................

The x-intercept of the graph of a linear ﬁmction is always dependent on the slope of the
linear function.

(Note: The x-intercept of a graph is the point at which the graph crosses the x-axis.)

True / False

Explanation: .............................................

If the slope of a linear function is positive, then the y-intercept of the graph of the linear
function must be positive.

(Note: The y-intercept of a graph is the point at which the graph crosses the y-axis.)

True / False

Explanation: .............................................

If the y-intercept of a linear function is negative, then the slope of the linear function
must be negative.
(Note: The y-intercept of a graph is the point at which the graph crosses the y-axis.)

True / False

Explanation: .............................................

83

If two lines are intersecting each other on the Cartesian plane, one line always has
positive slope and the other line always has negative slope.
True / False

Explanation: .............................................

If the graph of a line gets steeper, the absolute value of its slope becomes bigger.
True / False

Explanation: .............................................

 

If we choose any two points on a straight line, we always ﬁnd the same slope value.
True / False

Explanation: .............................................

The slope of a linear function does not always indicate a ratio of changes on y-values
over changes on x-values.
True / False

Explanation: .............................................

84

Appendix D

Intervention Session Questions

Exploratory Questions for Distance vs. Time Activity (0 VM Group)

Please use the VM on the leﬁ to answer the following questions.

Question 1
Note: This question was adapted from http://standards.nctm.org/.

Set different values for a position and step size for the boy and girl to start a new race

and run the simulation.

a) Please describe your observation about the race (e. g. who is faster, who completed
the race before,. . .etc.) and the related position vs. time graph (e. g. whose graph is

steeper, whether both graphs start to same position,. . .etc.)

b) Now, make changes on your settings (e.g., change the value of the size of the step
size for the boy, and change the girl’s starting position).How will the change you
made inﬂuence the race and the related position vs. time graph? Please explain your

answer with details.

Question 2

a) Set different values for the step size for the boy and girl runner (with the same
starting position) each time and run the simulation. Based on your observations on
the race and the related position vs. time graph, please describe the relation between
the step size of a runner and the speed of the runner. For example, if a runner is

using a larger step size, how does this affect his/her speed in the race?

b) How do the different step sizes affect the position vs. time graphs of runners? Please

explain the relation between the step size and the position vs. time graphs of runners.

85

 

Question 3

The following graph shows position vs. time graphs of the boy and girl runner. The blue
line shows girl’s graph, and red line shows boy’s graph.

 

Based on your observations on this graph, please answer the following questions.

a) What can you tell about the step size of the boy and girl runner? Does the girl have
bigger step size than the boy, or do they have the same step size? Please explain your

answer.

b) What can you tell about the speed of the boy and girl runner? Do they have the same

speed or different throughout the race? Please explain your answer.

c) As you see in the graph, two lines intersect. What can you tell about the speed of the
both runners when their position vs. time graph crosses each other? Do they have the

same speed? Please explain your answer.

86

Structured Mathematics Questions for Distance vs. Time Activity (S VM Group)

Please use the VM on the left to solve the following questions.

Question 1
Note: This question was adapted from http://standards.nctm.org/.

Please set the starting position and step size for both runners on the VM in the

following way.
Boy
Starting position: 10 Step size: 3
Girl
Starting position: 40 Step size: 2

a) Then, run the simulation. Now, write a story that describes the trip. For example,
" T he girl is going really fast. She catches up to and passes the boy, who is going
slow. " or "The girl started way behind the boy, who was already halfway to the tree
by the time she got going. She went really fast and caught up to him more and more.

F inally, at 75 she passed him and kept going really fast and got to the tree ﬁrst. "

b) Based on your descriptions of the trip in your story at a), does the speed of the boy

equal to the girl all the time in the race? Please explain your answer with reasons.

c) What’s the speed of the boy and girl runner in the ﬁrst 20 seconds of race?

Please show your work.

87

 

Question 2

Set starting position 10 for the boy and girl runner, and make step size 1 for the boy

runner, and 3 for the girl runner.

8)

b)

CD

What’s the speed of the boy and girl runner in the ﬁrst 20 seconds of the race? Please
show your work.

What’s the speed of the boy and girl runner throughout the race? Please show your
work.

Does each of the runners have the constant speed throughout the race? Please
explain your answer.

What is the relationship between the step size of a runner and the speed of the

runner? Please explain your answer.

88

Question 3

The following graph shows position vs. time graphs of the boy and girl runner. The blue line
shows girl’s graph, and red line shows boy’s graph.

 

Based on your observations on this graph, please answer the following questions.

a) Does the girl have higher speed than the boy in the ﬁrst 30 seconds of the race?
What’s the speed of the boy and girl runner in the ﬁrst 30 seconds? Please show
your work.

b) As you see in the graph, the graphs of two runners intersect each other. Do both
runners have the same speed at 30th second? Please explain your answer.

c) What is the speed of the boy and girl runner over the race? Does each of them
have constant speed throughout the race? Please explain your answers.

d) What is the relationship between the speed of a runner and the position vs. time

graph of the runner? Please explain your answer.

89

Exploratory Questions for Geoboard Activity (0 VM Group)

Please use the VM on the left to answer the following questions.

Question 1

Choose any two points on the Geoboard in the following way: Point 1 can be any point
above the x-axis, and Point 2 can be any point below the x-axis. When you connect these
two points, what can you tell about their slope? Can you ﬁnd the value of the line’s

slope? How? Please use the VM, and explain your answer.

Question 2
Draw two lines with the same slope on the Geoboard. What are the differences and

similarities between these two lines? Please use the VM to explain your answers.

Question 3
Please construct two different right triangles on the Geoboard. Then, answer the
following questions by using the VM.
a) What do you notice about the lengths of the sides of right triangles? Please explain
your answer.
b) Can you ﬁnd slope of the hypotenuse (please see note below) in each of these two

right triangles? Please explain your answer.

hypotenuse

 

_1

Note: Hypotenuse is the longest side on a right triangle.

c) Describe the relationship between the slope of hypotenuse and the lengths of other

two sides in a right triangle. Please explain your answer.

90

 

Structured Mathematics Questions for Geoboard Activity (S VM Group)

Please use the VM to answer the following questions.

Question 1

By using the VM, please draw a line that connects the following two points (1, -1) and

(-l, 3).

a) Does this line have a slope? Is the slope of the line positive or negative? Why? Please
explain your answer.

b) Can you ﬁnd the slope value of the line that connects (1,-l) and (-l, 3)? How can you
ﬁnd it? Please explain your answer.

c) Is it possible that you can ﬁnd a different line that connects (l , -1) and (-1, 3)? Please

explain your answer.

Question 2

Can you ﬁnd two different lines that have a slope of - 1/2? After you draw these two lines
on the Geoboard, what are the differences and similarities between these two lines?

Please explain your answer.

91

 

Question 3
Please construct a right triangle by connecting (-1, 3), (-l, -3), and (2, -3) on the
Geoboard. Then answer the following questions by using the VM.

a) What is the slope of the hypotenuse (please see the note below) in the triangle that

you just constructed? Please explain your answer.

hypotenuse

 

"L

Note: Hypotenuse is the longest side on a right triangle.

b) What is the relationship between the slope of hypotenuse and the lengths of other two

sides in your right triangle? Please explain your answer.
c) Please construct a right triangle to ﬁnd the slope of a line that connects (0, 3) and (-4,

-3). Is the slope positive or negative? What’s the slope of the line? Please explain

your answer.

92

Exploratory Questions for Graphs Activity (0 VM Group)

Please explore the graphs of various linear equations by using the VM on the left side of
the computer screen. Click and drag the sliders under the graph to change the slope and
y-intercept of the line shown.

Please use the VM on the left to answer to the following questions.

Question 1

What do you observe on the graph when you change the value of slope in the

equation? Please use the VM to answer the question, and explain your answer.

Question 2

How does the graph of a linear function change when its slope value takes positive versus

negative values? Please use the VM to answer the question, and explain your answer.

Question 3

How does changing the values of y-intercept affect the graph of a linear function? Please

use the VM to answer the question, and explain your answer.

Question 4

Are there any relations between the values of y-intercept and slope on the linear

equations? Please use the VM to answer the question, and explain your answer.

93

 

Structured Mathematics Questions for Graphs Activity (S VM Group)

On the VM, please adjust the green and brown sliders to graph the equations to answer

the following questions.

Question 1

a) Please ﬁrst draw the graph of y = 1.0x + 1.0 and then draw the graph of y = 2.0x +
1.0 by using the VM. What did you observe when you move from the graph of y
=1.0x + 1.0 to the graph of y = 2.0x + 1.0? Please explain your answer.

b) Please ﬁrst draw the graph of y = 2.0x + 1.0 and then draw the graph of y = -2.0x +
1.0 by using the VM. What changes happened when you move from the graph of y =
2.0x + 1.0 to the graph of y = —2.0x + 1.0? Please explain your answer.

c) What is the relationship between slope value of linear functions and their graphs?

Please explain your answer.

Question 2

a) Please ﬁrst draw the graph of y = 2.0x + 1.0 and then draw the graph of y = 2.0x +
3.0 by using the VM. What did you observe when you move from the graph of y =
2.0x + 1.0 to the graph of y = 2.0x + 3.0? Please explain your answer.

b) Please ﬁrst draw the graph of y = 2.0x + 3.0 and then draw the graph of y = 2.0x -1.0
by using the VM. What changes happened when you move from the graph of y = 2.0x
+ 3.0 to the graph of y = 2.0x - 1.0? Please explain your answer.

c) What is the relationship between y-intercept value of linear functions and their
graphs? Please explain your answer with reasons (Note: The y-intercept of a graph is

the point at which the graph crosses the y-axis).

94

Question 3

Please draw the graph of y = - 1.0x + 2.0 by using the VM.

a) Can you ﬁnd another linear equation that has the same graph as the graph of y = -

1.0x + 2.0? Please explain your answer.

b) Is the graph of y = 2x same as the graph of y -6 = 2(x-3)? Please explain your

answer.

c) Is it always true that we can ﬁnd two different forms of linear equations that have the

same graph? Please explain your answer.

95

 

Exploratory Questions for Linear Transformation Activity (0 VM Group)

The VM on the left side provides point-slope and slope-intercept forms of linear
equations. You can make changes on the values of slope, x-intercept, and y-intercept by

changing the numeric values on the boxes with the aid of arrows.
Please use the VM on the left to answer the questions below.

Question 1

What do you observe on the graph of a linear equation when you make changes on the
slope values in the point-slope form of the linear equation? Please explain your answer

with details.

Question 2
a) What are the differences and similarities between slope-intercept and point-slope

forms of linear equations? Please explain your answer.

b) Does the value of slope or y-intercept change from moving slope-intercept form to

point-slope form of a linear equation? Please explain your answer.

Question 3

a) Choose a line (such as, a line that passes from (0, 2), and whose slope is 3). Then,
please ﬁnd two different algebraic forms (slope-intercept and point-slope) for the

line that you chose. Pease explain your answer.
b) Which form (slope-intercept or point-slope) of linear ﬁrnctions that you formed ﬁrst

to ﬁnd the algebraic forms of the line in your answer at a)? Please explain your

answer.

96

Structured Mathematics Questions for Linear Transformation Activity (S VM Group)

The VM on the left side provides point-slope and slope-intercept forms of linear
equations. You can make changes on the values of slope, x-intercept, and y-intercept by

changing the numeric values on the boxes with the aid of arrows.
Please use the VM on the left to answer the following questions.
Question 1

a) By using the VM, please ﬁnd the equation of a line that passes through (-2, 3) and
has a slope of 0.5?

b) Please ﬁnd the slope-intercept form and point-slope form of the equation that you

found at a).

c) In your answer at a), which form of the equations (slope-intercept or point-slope)
that you found first? How did you ﬁnd your ﬁrst equation? Please explain your

answer.

Question 2

Please compare these two algebraic forms (that you found in Question 1) with each
other. What changes or does not change while moving from slope-intercept formula to
point-slope formula of a linear equation? What are the differences between these two

algebraic forms? Please explain your answer.

97

Question 3

a) Please ﬁnd two different linear equations for the line that passes through (-1, 3) and

whose slope is -2. Show all your work.

b) Please ﬁnd two different linear equations for the line that passes through (-1, 0) and
(l, 2). Show all your work.

c) Which form (slope-intercept or point-slope) of liner fImctions that you used ﬁrst to
ﬁnd the algebraic form of the lines in your answer at a) and b)? Please explain your

answer.

98

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