CONCEPTUALIZING VECTORS IN COLLEGE GEOMETRY: A NEW FRAMEWORK FOR
ANALYSIS OF STUDENT APPROACHES AND DIFFICULTIES
by
Oh Hoon Kwon

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mathematics
2012

ABSTRACT
CONCEPTUALIZING VECTORS IN COLLEGE GEOMETRY: A NEW FRAMEWORK FOR
ANALYSIS OF STUDENT APPROACHES AND DIFFICULTIES
by
Oh Hoon Kwon
This dissertation documents a new way of conceptualizing vectors in college mathematics, especially in geometry. First, I will introduce three problems to show the complexity and subtlety of
the construct of vectors with the classical vector representations. These highlight the need for a
new framework that: (1) differentiates abstraction from a physical embodiment, (2) intertwines
the representational perspective and the cognitive perspective on vectors, and (3) reveals cognitive
development on geometric representations of vectors. These needs ground the development of
the framework that permits a layered view of the construct of vectors. The framework comprises
three layers of progressive refinements: (1) a layer that describes a global distinction between
physical vectors and mathematical vectors by the difference between a physical embodiment and
abstraction, (2) a layer that recounts the difference between the representational perspective and
the cognitive perspective on vectors as the difference between ontological perspective and epistemological perspective, and (3) a layer that identifies ontological and epistemological obstacles
in terms of transitions towards abstraction. Data was gathered from four empirical studies with
ninety-eight total students to find evidence of the three major transition points in the new framework: physical to mathematical coming from the first layer, geometric to symbolic and analytic
to synthetic from the second layer, and the prevalence of the analytic approach over the synthetic
approach while developing abstraction enlightened by the third layer. Limitations of the framework on the distinction of physical and mathematical vectors, directions of cognitive development,
and repetition of transitions suggest further refinements of the framework and the implications for
teaching and learning of vectors.

To my family

iii

ACKNOWLEDGEMENTS

I especially wish to express my gratitude to my advisor, Prof. Brin Keller, who has never stopped
encouraging me in my work, providing me many details and suggestions for research and sharing
her time by reading my super-unorganized manuscripts. I also would like to record my appreciation
to my committee, Prof. Glenda Lappan, Prof. Gerold Ludden, Prof. Zhenfeng Zhou, and Prof.
Jennifer Kaplan for their time, valuable suggestions, and warm encouragements.
As always, my family and friends have been there, providing all sorts of tangible and intangible
support. I would like to give particular thanks to my parents and my parents-in-law for their neverending support. I owe countless things to my son Hyukjin, daughters Sujin and Yoonah. Kiddos!
Daddy is now available! And finally, my most sincere ‘thanks’ goes to my wife Yeaju for her
selfless love and endless support.
Praise God! Go Spartan! Thank you.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTER 1 BACKGROUND, RATIONALE, AND PURPOSE
1.1 Complexity of the Construct of Vectors . . . . . . . . . . . .
1.1.1 Vector as a Translation . . . . . . . . . . . . . . . .
1.1.2 Vector as a Point, Point as a Vector . . . . . . . . . .
1.1.3 Vector Sum Geometrically . . . . . . . . . . . . . .
1.2 Needs for a New Framework . . . . . . . . . . . . . . . . .

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CHAPTER 2 NEW FRAMEWORK FOR THE CONSTRUCT OF VECTORS
2.1 Configuration and its Role . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Construction of the Configuration . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 First Layer towards Mathematical Abstraction . . . . . . . . . . . .
2.2.2 Second Layer for Multiple Perspectives . . . . . . . . . . . . . . .
2.2.3 Third Layer for Transitions towards Abstraction . . . . . . . . . . .
2.3 Features of the Configuration . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 3 EMPIRICAL STUDIES FOR THE NEW FRAMEWORK
3.1 Research Focuses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Method and Participants . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Design and Construction of Survey and Interview . . . . . . . . . . .
3.4 Evolution of Survey Questions . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4 FINDINGS AND DISCUSSIONS . . . . . . . . . . . . . . . . . .
4.1 From Physical to Mathematical Vectors . . . . . . . . . . . . . . . . . . .
4.2 Epistemological and Ontological Perspectives . . . . . . . . . . . . . . . .
4.2.1 Interplay between Epistemological and Ontological Perspectives . .
4.2.2 Prevalence of Analytic Approach to Synthetic Approach . . . . . .
4.3 Obstacles towards Mathematical Abstraction . . . . . . . . . . . . . . . . .
4.3.1 Ontological Obstacles: Transition (D) from Geometric to Symbolic
4.3.2 Epistemological Obstacle: Transition (C) from Analytic to Synthetic
4.3.3 Cognitive Development of Geometric Representations . . . . . . .
4.4 Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 5 LIMITATIONS OF THE FRAMEWORK . . . . . . .
5.1 Categorization of Mathematical Vectors . . . . . . . . . . . . .
5.2 Direction of the Development towards Mathematical Abstraction
5.3 Repeated or Reversed Transitions on the Configuration . . . . .
5.4 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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APPENDIX A

VECTOR REPRESENTATION SURVEY I . . . . . . . . . . . . . . . 73

APPENDIX B

VECTOR REPRESENTATION SURVEY II . . . . . . . . . . . . . . 85

APPENDIX C

VECTOR REPRESENTATION SURVEY III . . . . . . . . . . . . . . 99

APPENDIX D

VECTOR REPRESENTATION SURVEY IV . . . . . . . . . . . . . . 103

APPENDIX E

ADDITIONAL INTERVIEW QUESTIONS . . . . . . . . . . . . . . . 116

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

vi

LIST OF TABLES

3.1

Map from questions to configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2

Questions for Layered View of Configuration . . . . . . . . . . . . . . . . . . . . . . 36

3.3

Example of Interview Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1

A Rainy Day Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2

Prevalence Result from Survey II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3

Origin Result from Survey II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4

Polygon Result from Survey II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1

Resolution of Vectors-as-Arrows vs. Vectors-as-Points Confusion . . . . . . . . . . . 70

vii

LIST OF FIGURES

0.1

Triangle Midpoints Theorem with Analytic Geometry . . . . . . . . . . . . . . . . . .

1

0.2

Triangle Midpoints Theorem with Synthetic Geometry . . . . . . . . . . . . . . . . .

2

0.3

Triangle Midpoints Theorem with Vector Geometry . . . . . . . . . . . . . . . . . . .

2

1.1

Vectors representing a translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2

Action-Effect Approach (Watson, 2004) . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3

Determine the sum of two vectors. (NCTM, 2000) . . . . . . . . . . . . . . . . . . . .

9

1.4

Points are represented as vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5

Vector sum and difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6

Vector Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7

Sum of two vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1

The Configuration of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2

First Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3

Second Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4

Conversion among Registers of Vector Representation . . . . . . . . . . . . . . . . . . 23

2.5

Third Layer with two major transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6

Transition (C) from analytic (procedural) to synthetic (structural) . . . . . . . . . . . . 27

2.7

Transition (D) from geometric to symbolic . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8

Analytic and Synthetic Approaches from the Third Layer . . . . . . . . . . . . . . . . 30

3.1

Modified Cube Figure in Survey II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1

Translation Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2

Cluster Tree of the same translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3

Cluster Tree of the equivalent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

viii

4.4

Construction of a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5

Student response based on vectors with no evidence of symbolic vector addition . . . . 45

4.6

Student Response based on geometric and symbolic vector addition . . . . . . . . . . 46

4.7

Student Response based on geometry without vector addition . . . . . . . . . . . . . . 46

4.8

Robot Arm Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.9

A Student Drawing of Physical Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.10 Cube Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.11 Triangle Midpoints Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.12 Associativity Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.13 Origin Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.14 “Very unique arrangement” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.15 Understanding Structures of Vector Operations . . . . . . . . . . . . . . . . . . . . . 59
4.16 Understanding Addition Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1

Translated Image Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2

Translated Image Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3

Responses to Translated Images Question . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4

Underdeveloped Physical Vectors? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5

Misunderstanding of Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6

Vectors that can explain the correlation concept (Bryant,1984) . . . . . . . . . . . . . 67

5.7

Associativity proved using arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.2 Robot Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.3 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.4 Non-standard Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.5 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
ix

A.6 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.7 A Very Long Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.8 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.9 Triangle Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.10 Point/Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
B.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
B.2 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.3 Robot Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
B.4 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.5 Non-standard Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.6 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
B.7 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B.8 A Very Long Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
B.9 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.10 Triangle Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
B.11 Point/Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
C.1 Polygon Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.2 Translation of Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C.3 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.4 Geometric Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
D.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
D.2 Translation of Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
D.3 Geometric Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.4 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.5 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
x

D.6 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
D.7 A Very Long Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
D.8 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
D.9 Triangle Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
D.10 Point/Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
E.1 Triangle Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
E.2 Point vs. Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xi

PREFACE

During the summer 2009 semester, I had an opportunity to co-teach with my advisor, a college
geometry course for future high school teachers. We developed and taught lessons of classical
Euclidean Plane Geometry by suggesting to our students to choose freely from the methods of
exploration among (1) coordinate geometry, (2) synthetic geometry, and (3) vector geometry. For
example, here is Triangle Midpoints Theorem that can be explored with all three methods above.
Triangle Midpoint Theorem The line segment connecting the midpoints of two sides of a
triangle is parallel to the third side and is congruent to one half of the third side.
Students were asked to prove the theorem using coordinates, similar triangles synthetically, or
vectors. Most students felt comfortable locating a vertex of a triangle at the origin on a coordinate
plane. They tried to use coordinates for the remaining two vertices and midpoints to figure out the
slopes and the lengths of the two line segments connecting points. See Figure 0.1.

A = (a1; a2)

1
1
M = ( a 1; a 2)
2
2

1
1
N = ( (a1 + c1); (a2 + c2))
2
2
C = (c1; c2)

B = (0; 0)

Figure 0.1: Triangle Midpoints Theorem with Analytic Geometry

When students thought synthetically, they also felt comfortable in using triangle similarity. One
could show that

AMN ∼

ABC, and ∠AMN ∼ ∠ABC. This implied MN
=
1

BC and BC = 2MN

by a scale factor. This is a typical approach in the majority of classical Euclidean Plane Geometry
textbooks; so some students tried this approach with the help of books and other resources. See
Figure 0.2.

4AMN ! 4ABC
because \A common, AB = 2AM , AC = 2AN
=
Therefore \AMN ! \ABC
N as corresponding angles of a similar triangle.

A

M

C
B

Figure 0.2: Triangle Midpoints Theorem with Synthetic Geometry

A
1~
b
2
~
b

1
~
a
2

! = !!!
a
AB
!!!
!
b = AC
~
a

1! 1!
b ¡ a
2
2

C
~ ¡~
b a

B 1 !
( b ¡ !)
a
2

Figure 0.3: Triangle Midpoints Theorem with Vector Geometry

2

→
−
−
However, students were very reluctant to use vectors in the proof. By setting up → and b
a
−
−
−
1− 1→ 1− 1→
− →
a
a
a
on each side of a triangle, we can easily find →, b , → − b , and → − b . With a simple
2
2
2
2
− −
−
1 → →
1→ 1−
symbolic calculation using scalar distribution, we have that ( b − a ) = b − →. By compara
2
2
2
ing this result with the the triangle figure, we can see the result implies both geometric meaning
of ‘parallel’ and ‘the same length’ simultaneously. (Figure 0.3). I believed that the proof of the
triangle midpoints theorem was one of the easiest applications of vectors.
Contrary to my belief, students felt this vector proof was much more difficult than coordinate
or synthetic geometric proofs. They drew arrow representations of vectors on the triangle, labeled
−
→
them with symbols such as AB, and tried to translate arrow representations to coordinate representations using coordinates of the initial and terminal points of vectors. Their symbols were mainly
for naming, not for manipulating. What they really tried was measuring slopes of arrows using
coordinates of the initial and terminal points. Again, they apparently just went back to coordinate geometry! They could draw arrows and calculate 2-tuples by subtracting numbers component
wise, but they did not know how to handle arrows with symbolic operations nor how those 2-tuples
differ from the coordinates of points.
Although my observation was just anecdotal evidence from my personal teaching experience,
I saw the students’ ideas of vectors were very scattered and isolated. Students felt uncomfortable
with connecting and converting various representations of vectors to each other, and were reluctant to apply them to geometric problems. This brought me to think again about conceptualizing
vectors in mathematics. If there were a new way of talking about various vectors coherently in
mathematics, it would be very beneficial to both teachers and students. In this dissertation, I tried
to re-conceptualize vectors in mathematics, not just by juxtaposing them with different uses or
looks, but by refining the construct of vectors more carefully.

3

Chapter 1
BACKGROUND, RATIONALE, AND PURPOSE

This dissertation documents a new way of conceptualizing vectors in college mathematics, especially in geometry. In this chapter, I will introduce three problems to describe the complexity and
subtlety of the construct of vectors with the classical vector representations1 . These motivate the
necessity of the new framework that permits: (1) differentiating abstraction from physical embodiment, (2) intertwining the representational perspective and the cognitive perspective on vectors,
and (3) revealing cognitive development on geometric representations. These needs guided me to
deliver a new framework for conceptualizing the construct of vectors.
By building a new framework, I explored a complex construct of vectors in mathematics with
respect to mathematical abstraction, multiple representations, and cognitive development. The
primary goal of the framework is to give a new way to discuss the complexity of vectors, both
conceptual and pedagogical, that students may grapple with in order to understand vectors in geometry effectively. This study will be propitious for students to grasp flexibility on the use of
various vectors.

1.1

Complexity of the Construct of Vectors

The section contains three problems illustrating the complexity and subtlety of vectors that cannot
be identified well with the classical vector representations. These motivate the necessity of a new
framework that provides a better and/or different way to describe the construct of vectors.
1 These

are vector representations frequently used in the textbooks such as arrows, coordinate

forms, etc.

4

1.1.1

Vector as a Translation

When representing a geometric translation with a vector in an arrow form, the complexity of vector
representations comes out in a cognitive sense. This complexity can be described as a difference
between physical vectors based on physical embodiment and mathematical vectors regarding on
mathematical abstraction. Differentiating mathematical abstraction from physical embodiment is
essential to understand the complexity and subtlety of vectors in mathematics.
Students’ experiences with the concept of vectors vary over physics and mathematics, but relatively few studies have focused on an analytic, detailed explanation or examination of learning
of vectors in mathematics compared to physics. Recent critical studies focus on physics education
(Aguirre and Erickson, 1984; Aguirre, 1988; Hestenes et al., 1992; Heller and Huffman, 1995;
Knight, 1995; Savinainen and Scott, 2002; Nguyen and Meltzer, 2003; Flores et al., 2004; Coelho,
2010). These studies covered only algebraic aspects of vectors with simple addition/subtraction
operations and focused only on the interrelationship among physical quantities such as displacement, velocity, acceleration, and force, not vectors themselves. This physical embodiment of
vectors helps students understand vectors initially, but soon block the progression to advanced and
abstract understanding of vectors.
→
−
For example, in Figure 1.1 vector f looks far away from the triangle and nothing to do with
→
−
−
the translation. Vector d pushes the triangle. Vector → pulls the image of the triangle, not the
e
−
−
preimage. Vector → gives an idea of moving vertices on a triangle. Vector → is penetrating, not
a
c
translating the triangle. However, all the vectors represent the same translation of a triangle even
though their locations are quite scattered and different. This difference between physical meaning
and mathematical meaning is from the fact that the motion in mathematics is quite different from
that in physical or realistic contexts. Freudenthal (1983) described the distinction between physical
motion and mathematical motion. Physical motion is something that occurs to an object within
space or plane within time, but mathematical motion should be differentiated from physical motion
in three ways: from the limited object to the total space (plane), from within space (plane) to on
space (plane), from within time to at one blow. When talking about teaching and learning of vectors

5

Figure 1.1: Vectors representing a translation

in mathematics, this distinction is sometimes not regarded as a sharp contrast by the researchers.
Watson et al. (2003) showed that by focusing on the effect rather than the specific actions involved,
several highly sophisticated concepts such as equivalent vectors, vector sums could be taught better.
This study was based on the three worlds of mathematics framework developed in Tall (2004b,a,
2008). See Figure 1.2.
The ‘action-effect’ approach lays the groundwork for students’ cognitive development on vectors. However, to judge from the description of Freudenthal (1983) on the distinction between
physical motion and mathematical motion, this approach has a limitation. As mentioned in Tall
(2004b), this approach does not call attention to a fact that the equivalent vectors cannot be explained by limiting oneself in conceptual-embodied world or proceptual-symbolic world of mathematics.
In the case of vectors, both a totally intuitive sense with physical contexts and a totally abstract
sense with mathematics are entailed in the equivalence relation. The equivalent vector concept is
not just related with the physical world but also related with the formal definition that lives in the
axiomatic-formal world of mathematics. In Figure 1.1, all the vectors are equivalent just as all the
6

2%#! #A-#.3:#8$,(! (#66&86! 4#.#! &8(*! 5&8'75$#'! 43$%! $%#! #A-#.3:#8$,(! ).&7-!
;).&7-!B=?!
!

!
!"#$%&$'%()#*"+",-%.)/01-22")*%)3%+4-%,-.+)1%.)*.-0+%
Figure 1.2: Action-Effect Approach (Watson, 2004)
7
C3.6$!,!>&(78$##.!6$7'#8$!4,6!,6D#'!$&!5&:#!$&!$%#!4%3$#!<&,.'E!$&!-7$!%36!%,8'!&8!$%#!

vectors represent the same translation. However, their equivalence is not clearly explained by the
translation ‘action’ with the given triangle and the same ‘effect’ those vectors would bring. Equivalent vectors can be explained by getting students into meta-cognitive level of the three worlds of
mathematics or at least by letting them understand the different use of the similar looking vectors
in different contexts and situations.
For a better observation and description of this complexity and subtlety of vectors, we need a
new framework that integrates and balances these intuitive understanding and abstract understanding of vectors in college mathematics. We want a new framework that (1) refines the notion of
vector representations that distinguish mathematical vectors from physical vectors, and (2) considers the student’s cognitive development in terms of various vector representations.
While talking about vectors as geometric translations, we saw that the cognitive elements play
an important role when a vector representation is fixed such as an arrow form. What if we have
a different form of a representation such as a coordinate form? How does the cognitive elements
work in this case? This question motivated me to think about two different perspectives on the
construct of vectors: representational and cognitive.

1.1.2

Vector as a Point, Point as a Vector

Teaching and learning multiple representations of vectors need to be treated carefully due to the
complexity of the construct of vectors. When dealing with multiple representations, we need to
deliberate two different perspectives: representational and cognitive simultaneously. As we can
see in the following example, the complexity of vectors does not allow students an easier translation/conversion from one representation to the other. A combined view of the representational
perspective and the cognitive perspective helps us understand this translation/conversion from one
representation to the other.
In Figure 1.3, the vector sum in arrow forms is usually accompanied by 2-tuple of numbers.
However, it is not clear if this 2-tuple represents the terminal point of the arrow or the arrow
itself. When geometry meets linear algebra, this problem of multiple representations gets more

8

complicated as in Figure 1.4. The term ‘vectors’, ‘points (vertices)’, and ‘arrows’ are not easily
distinguishable in these two examples. Similar difficulty in distinguishing ‘vectors-as-arrows’ and
‘vectors-as-points’ is introduced by Hillel (2002) as well.

(a, b) + (c, d) = (a + c, b + d)
Figure 1.3: Determine the sum of two vectors. (NCTM, 2000)

According to Hillel, undergraduate linear algebra courses generally include three modes of description for basic objects and operations. They are (a) the abstract mode using the language and
concepts of the general formalized theory; (b) the algebraic mode using the language and concepts
of the more specific theory of Rn ; (c) the geometric mode using the language and concept of 2- and
3-spaces. He distinguished ‘vectors-as-arrows’ and ‘vectors-as-points’ in terms of modes of description, and criticized that in practice, most instructors tended to shift back and forth between the
arrow and point depiction of vectors ‘implicitly’ and ‘unconsciously’. Hillel (2002) offered unique
insight into this problem by putting emphasis on translations/conversions between representations
with the modes of description. However, his argument did not show enough clarity on the subtlety of vectors in a sense that the difference of modes of description cannot capture this ‘implicit’
and ‘unconscious’ changes. ‘Vectors-as-arrows’ and ‘vectors-as-points’ are not just a problem of
modes of description, but also a problem of how students think about representations. The study of
modes of thinking by Sierpinska (2002) reveals an expansion of the discussion. Sierpinska (2002)
classified students’ reasoning in linear algebra courses into three modes: Synthetic-geometric,
9

Find a matrix M so that MA = A , MB = B , MC = C where the points are represented as
vectors (NCTM, 2000).
Figure 1.4: Points are represented as vectors.

analytic-arithmetic, and analytic-structural modes. Sierpinska focused more on students’ thinking
and reasoning about epistemology.
These classifications of vectors motivated us to consider multiple perspectives on vectors: representational and cognitive. In the similar context, one observation that Hillel (2002) made about
‘vectors-as-arrows’ and ‘vectors-as-points’ brought up a refreshing idea. By emphasizing pedagogical and psychological advantages related with physics on ‘vectors-as-arrows’, he has displayed
student’s possible preference of one representation to the other in a specific setting or for a purpose.
However, no indication is provided as to how this preference can change if the setting is different,
such as in synthetic geometry other than linear algebra. Studies on multiple representations of

10

vectors are few and focused on the views from linear algebra (Dorier, 2002; Harel, 1989; Dorier
and Sierpinska, 2001). For this reason, we need more holistic way covering both representational
and cognitive perspectives to talk about vectors in mathematics.
Studies on multiple representations in mathematics education have promoted the importance
of understanding multiple representations and translations/conversions among them. Research in
the mathematics education community posits that students can grasp the meaning of mathematical
concepts by experiencing multiple mathematical representations (Janvier, 1987; Kaput, 1987a).
Keller and Hirsch (1998) emphasized the importance of multiple representations by describing
potential benefits related to the use of representations. Those benefits are “providing multiple concretizations of a concept, selectively emphasizing and de-emphasizing different aspects of complex
concepts, and facilitating cognitive linking of representations.” (Keller and Hirsch, 1998, p.1). In
this context, several standards documents have advocated K-12 curricula that emphasize mathematical connections among representations (NCTM, 1989, 2000; CCSSM, 2010). They suggest
that students use graphical, numerical, and algebraic representations to investigate concepts, problems, and express results. However, these discussions are very focused on the way to talk about
functions, not vectors, and on the way to discuss connections between representations as separate
entities from representations themselves. The construct of vectors is more complex than functions,
so that graphical, numerical, and algebraic representations are not enough to describe this complexity (Pavlopoulou, 1993 as cited in Artigue et al. 2002). One cannot fully use the benefits of
multiple representations until sufficient understanding and sophistication are developed.
The observations bring a need for a unified, inclusive, and multidimensional framework. This
is not about listing all the possible contexts, different perspectives, and juxtaposing levels of sophistication, but about a new way to discuss a combined view of representational and cognitive
perspectives on the complexity of the construct of vectors.

11

1.1.3

Vector Sum Geometrically

When talking about vectors and the interplay between the representational and the cognitive perspectives on them, the problem stems not only from the multiple representations but also from
the translations/conversions among them. What the classical representations cannot provide from
the complexity of vectors is the cognitive development of geometric representations. Specifically,
translations/conversions in terms of the cognitive perspective are portrayed in considerable detail
as cognitive development theories for symbolic representations. The vector sum problem that we
discuss here shows a cognitive obstacle in geometric representations of vectors. This brings us
a deliberation of a framework that can recognize and reveal cognitive development on geometric
representations of vectors.
During the last two decades, the critical problems of translations between and within representations have been discussed, and the importance of moving among multiple representations and
connecting them is also discussed in several studies. Pavlopoulou’s research (as cited in Artigue
et al., 2002) studied translations/conversions from one representation to the other (Duval, 2006).
However, it is very restricted to certain forms of vectors: graphical, table, and symbolic representations (registers). As it has been noted earlier, this translation/conversion of representations is not
just among representations but also on cognitive development.
− −
−
− −
Consider a vector sum → + → + . . . + → or a combination of a sum and a difference → + → +
u1 u2
un
u1 u2
− −
. . . + → − → in Figure 1.5. We only have arrow forms of vectors, not coordinate n-tuple forms.
un v
However, there could be several different ways to approach this calculation, because arrow forms
show a process of a sum as well as a sum as an object. The interpretation of the middle dots is also
a possible obstacle to calculating and proceeding the sum.
Figure 1.6 illustrates these two different approaches of a sum briefly. Addition and subtraction
−
−
− → − →
are binary operations. Students can figure out → + b + → + d as a repetitive binary operation
a
c
process with the triangle method. Students are also able to figure out the sum by connecting
→
−
−
the initial point of → to the terminal point of d focusing the structure without thinking of the
a
intermediate processes. The latter sometimes involves the idea of a particle moving along the

12

→
−
u3

→
−
u2
→
−
u1

→
−
u4

→
−
v

→
−
un

Figure 1.5: Vector sum and difference

arrows.
Two studies (Sfard and Thompson, 1994; Yerushalmy, 1997) articulated these differences of
procedural thinking and structural thinking. They are based on the assumption that students’ ability
to understand mathematical concepts depends on their ability to make translations among several
modes of representations. Tall et al. (1999) analyzed several theories, which describe students’
transitions from viewing mathematical ideas operationally, or as processes, to viewing them structurally, or as objects. These transitions are referred to as “encapsulation” by Dubinsky (1991)
and “reification” by Sfard (1991). The Action-Process-Object-Schema theory (Asiala et al., 1996)
and reification theory (Sfard and Linchevski, 1994) are based on the duality of the mathematical
concepts and on the assumption that process conception precedes object conception. Sfard (1991)
calls process conception operational outlook and object conception structural view. However, these
studies are restricted to discussions of symbolic representations. In terms of geometric representations, Figure 1.7 describes interesting observation about process-object duality (Sfard, 1991; Gray
and Tall, 1993, 2001; Forster, 2000). When students calculate a vector sum with the tip to toe triangle method, the process view of a single vector is described as shifting or moving a particle. As

13

(a) Perform Operations

(b) Represent the Structure
Figure 1.6: Vector Additions

a result of the sum, students can put the resulting vector on the appropriate position, and make the
sum itself as an object in the structure of a triangle. On the other hand, in parallelogram method,
vectors are objects and moving the object with equivalence relation to draw a parallelogram is the
process of sum. This example shows that process and object are preceders and succeedors of each
other and shows further need for cognitive development as a part of the framework.
The work of Sfard (1991) views both operational and structural conceptions as important in
mathematical understanding. A structural conception enables recognition (at a glance) and manipulation as a whole; an operational conception is grounded in actions, processes, and algorithms.
However, this idea of encapsulation or reification, not just in symbolic modes of representation

14

Figure 1.7: Sum of two vectors

but also in geometric modes of representation has not been studied (Meissner, 2001b,a; Meissner
et al., 2006).
The problem of identifying cognitive development in geometric representations of vectors
poses the need for a new framework that can show the representational and the cognitive obstacles
more clearly in terms of the transitions towards mathematical abstraction. 2 This new framework
brings the complexity of vectors to the surface so that one can capture the whole picture of encapsulation or reification both happening in a symbolic way and a geometric way simultaneously in
the construct of vectors.

1.2

Needs for a New Framework

The foregoing three problems on vectors and related observations have demonstrated the need for:
(1) differentiating mathematical abstraction from a physical embodiment, (2) intertwining the representational and the cognitive perspectives, and (3) revealing cognitive development on geometric
representations of the new framework. From those needs, I tried to deliberate a framework that
(1) affords an opportunity to examine the complexity of mathematical vectors as emerging from
physical vectors, (2) includes ontological and epistemological perspectives of vectors, (3) identi2 Representational

obstacles are things that prevent students to translate/convert one form of
representations to the other. Cognitive obstacles are things that avoid students to proceed cognitive
development such as encapsulation or reification.

15

fies pivotal juncture points/transitions towards mathematical abstraction in the construct of vectors,
and (4) permits cognitive development focused on geometric representations.
In preparation for the development of a new framework, I observed the way vector concepts
were described and vector representations were presented in high school textbooks, college textbooks and various research studies. I also observed and listened to the way that mathematics education researchers, mathematicians, graduate students in mathematics and mathematics education,
and undergraduate students describe the vector concepts and present vector representations.
In Chapter 2, I seek to : (1) construct a configuration of vectors as a new framework based
on the process of refinement grounded by the needs in Chapter 1, (2) describe three layers of
progressive refinements that are comprised in the framework, and (3) summarize how this newly
built framework accomplishes the needs that comes from Chapter 1.
In Chapter 3, I describe the method and the research focus to gather student data that would
show evidence of the three major transition points: physical to mathematical, geometric to symbolic, and analytic to synthetic coming from each layer of the framework, and the prevalence of
the analytic approach to the synthetic approach while developing mathematical abstraction. In particular, I describe the design and construction of survey and interview questions to establish these
three layers in the configuration more concretely.
In Chapter 4, I provide evidence of the focal points that I hypothesized in Chapter 2 after
careful analyses of the four consecutive empirical studies. Included from student data are examples
of three transition points and the prevalence of the analytic approach over the synthetic approach
introduced in Chapter 3.
Finally in Chapter 5, I document the extent of my framework and summarize significant evidence away from the framework and suggesting the limitations of my configuration. I also comment on the implications of this study for teaching and learning, and the type of future research
that could be fruitful in helping students engage in learning of vectors.

16

Chapter 2
NEW FRAMEWORK FOR THE CONSTRUCT OF VECTORS

In Chapter 1, we saw the complexity and subtlety of the construct of vectors that cannot be fully
appreciated by understanding the classical representations of vectors. To clarify this complexity
especially in geometry, we needed a new framework that is unified, holistic, inclusive, and multidimensional. In this chapter, I will present a configuration based on series of progressive refinements
of the construct of vectors. The configuration is comprised of three layers of refinements: (1) a
layer that describes a global distinction between physical vectors and mathematical vectors as the
difference between physical embodiment and mathematical abstraction, (2) a layer that expresses
the difference between the representational perspective and the cognitive perspective on vectors
as the difference between the ontological perspective and the epistemological perspective, and
(3) a layer that identifies the ontological and the epistemological obstacles in terms of transitions
towards mathematical abstraction. I will also summarize how these layers in the newly built framework accomplish the needs that come from Chapter 1. With this summary, I will set my goals for
the empirical studies in the next chapter as well.

2.1

Configuration and its Role

Figure 2.1 is the configuration of vectors as a result of progressive refinements of the complex
construct of vectors described in the remainder of this chapter.1
I arranged all the major representations of vectors used in school and college textbooks on
the configuration together in a plane with two axes: ontological and epistemological axes. This
arrangement of representations in each cell is my subjective idea for familiarizing the students
and readers with the configuration of vectors. The way I arranged all the vector representations
1 It is important that the reader consider this configuration figure (Figure 2.1) with the remainder

of Chapter 2 and thus presented first.

17

Figure 2.1: The Configuration of Vectors

in the appropriate cells is based on the three worlds of mathematics studies that also motivate the
progressive refinements of the construct of vectors.
Tall and Watson placed the notion of vector in the three worlds of mathematics: conceptualembodied world, proceptual-symbolic world, and axiomatic-formal world (Watson and Tall, 2002;
Watson et al., 2003; Watson, 2004; Tall, 2004b). This placement gave an idea of the direction of
cognitive development and a hierarchy of vectors.
This hierarchy and the non-congruent translations/conversions from Pavlopoulou’s work (as
cited in Artigue et al., 2002) show a structure built upon two different developmental directions:
ontological (representational) and epistemological (cognitive).
Pavlopoulou studied translations/conversions among representations for vectors with graphical,

18

table, and symbolic representations. The study showed that in the restricted setting of having three
representations in two and three dimensions, those conversions were not easily accomplished by
students. This study also showed that there were easier directions for translations/conversions.
In the newly built configuration, I used the term, ‘transitions’, instead of ‘conversions’ or ‘translations’. The reason is because I believe that Pavlopoulou’s study shows certain levels of sophistication separated by the representational and the cognitive obstacles in translating/converting representations. Sometimes it is not possible to translate/convert one representation to the other. For
example, not all graphical representations of a function such as stochastic graphs or graphs from
realistic data can be represented by symbols in algebraic equations. I also used the term, ‘geometric representations’, instead of ‘graphical representations’ to avoid the interpretation of the word
‘graphical’ from a graph with the coordinates similar to the case of a function. Vectors in geometry
are not always accompanied by specific coordinate systems. The term geometric vector representations include both ‘graphical’ representations with the coordinate systems and non-‘graphical’
but still iconic/pictorial representations without the coordinate systems. Table representations in
Pavlopoulou’s study are also classified as ‘analytic’ symbolic representations compared to classical
symbolic representations as ‘synthetic’ symbolic representations.
Initially, I tried to build up a hierarchical model that could describe various vector representations along with student conceptualization, or that could predict student thinking or learning
trajectories. However, as I combined the theories of mathematical abstraction, multiple representations, and cognitive development, I found that constructing a hierarchical model was not simple
because vectors are a very complex construct and have nuances in various categorizations such as
examples in Chapter 1. Now, I characterize this new framework as “a map of the territory, a tool
of a certain grain size that we, as teachers, researchers and curriculum developers, can yield as we
organize our thinking about teaching and learning the concept...” (Zandieh, 2000, p.103). This
new role of the framework suggests locations of where to look to see the differences in students’
use of vectors.

19

2.2

Construction of the Configuration

To construct a framework, I revisited the needs that I identified earlier in Chapter 1. Those needs
grounded this development of the framework that allows a layered view to see the complex construct of vectors. Three layers of progressive refinements will be introduced sequentially. They
have different scales of focus from aggregate and global dealing with the difference of physical
and mathematical vectors (first layer) to individual and local dealing with the representational and
the cognitive obstacles (third layer).

2.2.1

First Layer towards Mathematical Abstraction

The first layer describes a global distinction between physical vectors and mathematical vectors
as the difference between physical embodiment and mathematical abstraction. See Figure 2.2.
As we have seen in Chapter 1, the difference between physical motion and mathematical motion
(Freudenthal, 1983), and the idea of the vector equivalence relation with ‘action-effect’ approach
(Watson et al., 2003; Watson, 2004) show the difference between physical embodiment and mathematical abstraction. Roughly speaking, in physical embodiment, not only the vectors and all the
physical settings around the vectors, but also the interaction between the vectors and the physical
settings are important. In mathematical abstraction, vectors are not objects attached to any physical settings. Assuming a difference of mathematical vectors and physical vectors, I first set up
two categories: mathematical vectors and physical vectors. Direction of intended movement is the
direction of the development towards mathematical abstraction wanted from students.
The first layer positions the focus of my study on the understanding of the construct of vectors from a mathematical point of view. Even though this layer is for differentiating mathematical abstraction from physical embodiment, I do not assume the empty intersection of these two
categories. The non-empty intersection will be discussed further in Chapter 5. For now I call
it as a transition ‘point’ from physical vectors to mathematical vectors to emphasize other nonintersecting parts under the assumption of the direction of intended movement. This transition

20

Figure 2.2: First Layer

point serves as the origin of the configuration.
For the next step, I tried to consider multiple perspectives on the direction of intended movement. I could think of two different perspectives of the development: representational and cognitive. These two perspectives come from the ideas of the meaning of multiple representations
(Kaput, 1987a; Lesh et al., 1987; Goldenberg, 1995; Johnson, 1998) and Modes of Description
(Hillel, 2002) vs. Modes of Thinking (Sierpinska, 2002). With the assumption of direction, I
placed these two perspectives as ontological axis and epistemological axis on the plane of the
configuration.

2.2.2

Second Layer for Multiple Perspectives

The second layer describes the difference between the representational and the cognitive perspectives on vectors as the difference between the ontological and the epistemological perspectives.
See Figure 2.3.

21

Figure 2.3: Second Layer

The origin represents an important transition from physical vectors to mathematical vectors.
With this refinement, we can see that ontologically, the origin is a shift of a view from vectors as
representations of physical quantities with physical units to vectors as representations of mathematical objects, i.e., a loss of physical appearance such as units, physical directions attached to
physical objects. We can also see that epistemologically, this origin is a shift related with understanding of mathematical equivalence relations, i.e., a loss of physical embodiment context such
as a translation as a displacement of an object, etc. Now, the first quadrant of the configuration is
for mathematical vectors and the third quadrant is the realm of physical vectors. The focus of the
rest of this dissertation will be on the first quadrant of this layer.
The direction of each axis implies not only one intended movement towards mathematical abstraction, but also changes in two different perspectives: a representational change and a cognitive
change. To talk about translations/conversions from one representation to the other, these two
changes are important.

22

hard
easy

Symbolic form

Table form

easy
hard

easy
Graphical form

hard

Figure 2.4: Conversion among Registers of Vector Representation

Many studies on the multiple representations of functions compare three different representations of functions such as graphical, table, and symbolic representations (Kaput, 1987b,a; Lesh
et al., 1987; Janvier et al., 1993; Goldenberg, 1995). Similarly, Pavlopoulou’s work (as cited in
Artigue et al., 2002) distinguishes between three registers of semiotic representation: the graphical register (arrows), the table register (columns of coordinates), and the symbolic writing register
(axiomatic theory of vector spaces). They talked about translations/conversions only in terms of
the representational change.
From Chapter 1, we also saw that Hillel (2002) and Sierpinska (2002) talked about conversions from one representation to the other without any hierarchy of sophistication. Pavlopoulou’s
work (as cited in Artigue et al., 2002) covered this missing hierarchy of Hillel(2002) and Sierpinska(2002) as an application and verification of Duval (2006)’s theory of semiotic representation
in the context of teaching linear algebra. Pavlopoulou shows that in a restricted setting such as
three registers in two- or three-dimensions, certain conversions are not easily achieved by students
(Figure 2.4).
In the light of above studies, I found needs for a framework that covers both representational
perspective and cognitive perspective, and a framework that assumes a certain type of a hierarchy
despite Pavlopoulou’s the graphical register was not the full substitute for the geometric representations in the configuration.

23

As we have already seen in Chapter 1, this type of the hierarchical structure and the existence
of the levels of sophistication cannot be understood just by the representational perspective or by
the cognitive perspective. To talk about vectors using both perspectives together, I put both axes
on the second layer as a refinement of the first layer. The ontological axis is a directed dimension
that covers changes in existence or being. It is related with explicit change of representations or
notions. Geometric or symbolic representations of vectors are matters of an ‘ontological difference’. Similarly, concrete vs. abstract characteristics are also related with the ontological aspect
of vectors. The epistemological axis is a directed dimension that covers changes in knowledge or
thinking. It is related with understanding of representations. The difference between arithmetic
thinking (using numbers) and algebraic thinking (using letters), the difference between analytic
thinking (procedural) and synthetic thinking (structural) can be described with this axis. Using an
ontological axis and an epistemological axis provides a way to locate and see a vector as a multifaceted mathematical entity in college mathematics and help illustrate a combined view of the
representational and the cognitive perspective on vectors.
While talking about the need for a new framework with a combined view of the representational
and the cognitive perspectives in Chapter 1, I pointed out the observation of student’s possible
preference of one representation to the other in a specific setting or for a purpose (Hillel, 2002). I
also mentioned another possible refinement of the construct of vectors. In the case of vectors, this
observation of the preference can imply more refined layer to see the complexity and subtlety of
the construct of vectors.

2.2.3

Third Layer for Transitions towards Abstraction

The third layer identifies the representational and the cognitive obstacles in terms of transitions
towards abstraction. The prevalence mentioned in the previous discussions could uncover certain
difficulties in understanding the construct of vectors. Those difficulties could be realized as obstacles with respect to the ontological and the epistemological perspectives. See Figure 2.5. In
the domain of mathematical vectors, each axis was hypothesized to have two important transitions

24

that can be identified in the configuration. On the epistemological axis, there are (1) one from
arithmetic to algebraic, and (2) one from analytic (procedural) to synthetic (structural). On the
ontological axis, there are (1) one from geometric to symbolic, and (2) one from concrete to abstract. Among those four transitions, we will only focus on two transitions: analytic to synthetic
and geometric to symbolic in the third layer.

Figure 2.5: Third Layer with two major transitions

The main transition for the discussion of vectors in the mathematical domain is Transition
(A) from physical to mathematical vectors. Along with progressive refinements, the difference
between physical vectors and mathematical vectors is getting clearer with additional layers.
From the first and second layers, we can see what I will call five transition points in the configuration.

Transition (A) from physical to mathematical: Transition (A) is a transition from physical understanding of a motion to mathematical understanding of a motion. Understanding of the vector

25

equivalence relation in arrow forms is critical in this transition. Without understanding of mathematical equivalence relation, one cannot understand the concept of free vectors or mathematical
position vectors. Those vectors are very important for the ontological shift from a geometric representation to a symbolic representation. Transition (A) can be also the transition from vectors
as the representations of physical quantities that have both physical directions and magnitudes to
vectors as mathematical objects. In physics, vectors are attached to physical objects to give information related with physical quantities affecting these physical objects. However in mathematics,
the focus is shifted from the physical objects to the vectors themselves. The vectors do not carry
physical units or directions any more.

Next, I wish to define the following two transitions along the epistemological axis.
Transition (B) from arithmetic to algebraic: Transition (B) is a transition from thinking with
numerical values to thinking with variables. This transition is shown up early in elementary and
middle school mathematics. Even though it is shown up very early, it is important to see this transition is critical to college students when they confront variable thinking in higher-level mathematics
with the context of vectors (Usiskin, 1988).
Transition (C) from analytic (procedural) to synthetic (structural): Transition (C) is a transition from analytic thinking to synthetic thinking. It also includes a transition from procedural
thinking to structural thinking in cognitive development such as reification or process-object encapsulation. In the original research of reification or process-object encapsulation (Sfard, 1991),
this transition is introduced as an ontological shift with more focus on new names and new symbols. However, in the context of vectors, it is not just a change of names or representations but also
a change of thinking. For example, a geometric vector (an arrow) can be studied both analytically
and synthetically. Switching back and forth on the ways of thinking and investigating with geometric vectors are epistemological shifts. See Figure 2.6.

The following two transitions are along the ontological axis.

26

Figure 2.6: Transition (C) from analytic (procedural) to synthetic (structural)

Transition (D) from geometric to symbolic: This transition is a transition of explicit representations from geometric to symbolic. While this transition often entails a simple change of appearance
of vectors, it can give rise to complex issues for vectors. In the high school level, with standard
basis vectors, the coordinates for the terminal points of position vectors will automatically be
the coordinate forms of vectors or become column vector forms. However, in the undergraduate mathematics, this transition includes understanding the difference between a vector space and
an Euclidean coordinate space. The coordinates of the points will not directly correspond to the
representations but the component-wise coefficients of basis vectors on linear combination of basis vectors. This can be one of the most important and hardest transitions in the undergraduate
mathematics. See Figure 2.7.
Transition (E) from concrete to abstract: Transition (E) is a transition from tangible dimensions
such as 2D and 3D to intangible dimension such as 4D and above. Transition (E) is somewhat easier than the rest when restricted to symbolic extensions from 2D or 3D to arbitrary dimension (i.e.,
appending one more component on the representations). However, it can be also a complicated
27

Figure 2.7: Transition (D) from geometric to symbolic

transition in the sense that a definition and the consequence of the definition can be switched. For
example, in 2D or 3D, both geometrical and symbolic representations assume that the magnitudes
−
−
− → − →
of two vectors determine the inner product of two vectors (→ · b = |→|| b | cos θ ). In 4D or above,
a
a
the inner product determines the magnitude of a vector. As such it is a big transition from concrete
concepts to abstract concepts of vectors.
A detail descriptions and extended discussions about the configuration are in an earlier work
(Kwon, 2011). I also want to let readers know that I postponed the empirical studies on the transition (B) and (E). This is because the transition (B) from arithmetic to algebraic occurs early in
elementary and middle school mathematics (Usiskin, 1988), and transition (E) from concrete to
abstract seems a simple extension of the number of entries once transition (B) is done. In this
dissertation, we only deal with the three major transitions (A), (C), and (D).

28

2.3

Features of the Configuration

This new framework has important features that drew the distinction between itself and other
frameworks. These features will be revisited and hypothesized as research focus in the Chapter
3.
First, the configuration shows the difference between physical vectors and mathematical vectors emerged from the first layer. It also illustrates interplay between the ontological and epistemological perspectives in the construct of vectors. These multiple perspectives enable us to talk
about the difference between analytic (procedural) and synthetic (structural) thinking of vectors,
and the difference between geometric and symbolic representations of vectors through the second layer. The configuration also shows the analytic approach and the synthetic approach towards
mathematical abstraction described by the second layer. In terms of the configuration, I can describe the analytic approach as a trend of changing explicit representations along the ontological
axis quickly from geometric representations such as arrows, to symbolic representations such as
coordinate/column vector forms. Epistemological development is postponed and following symbolically after the ontological change. This analytic approach in the configuration can be illustrated
L

as an upside down ‘L’ shape route ( ) towards abstraction. The synthetic approach is defined as
the trend of changing the views/thinking of a geometric representation while maintaining arrow
forms, from analytic (procedural) to synthetic (structural) first. Ontological development is postponed until the achievement of the change in epistemological perspectives such as from analytic
(procedural) to synthetic (structural). This approach in the configuration marked as a reversed ‘L’
shape route (L) towards abstraction. 2 See Figure 2.8.
The transition points on the configuration are identified as representational and cognitive obstacles by the third layer.3 We can see levels in each ontological and epistemological perspective,
2 Analytic/synthetic distinction in Transition (C) is characterized by a focus between procedure

and structure. This is not directly related with the analytic approach and synthetic approach defined
here.
3 As we have mentioned in section 1.1, representational obstacles are things that prevent students to translate/convert one form of representations to the other. Cognitive obstacles are things

29

Figure 2.8: Analytic and Synthetic Approaches from the Third Layer

especially cognitive development along with Action-Process-Object-Schema and/or process-object
duality in geometric representations emphasized in the configuration. Such systematizing transitions among representations of vectors with the ontological and epistemological perspectives is an
important feature of this configuration that was not realized before in other frameworks.
In the next chapter, I will discuss the method for gathering evidence from student data to see
if this layered view of the construct of vectors as a result of progressive refinements is reasonable. Questions and interviews will be designed to investigate student work to find evidence of
these features of the configuration. Those are (1) Transition (A) from physical to mathematical,
(2) Transition (C) from analytic (procedural) to synthetic (structural) and Transition (D) from geometric to symbolic, (3) the prevalence of the analytic approach to the synthetic approach while
that avoid students to proceed cognitive development such as encapsulation or reification.

30

developing abstraction, and (4) process-object duality in geometric representations.

31

Chapter 3
EMPIRICAL STUDIES FOR THE NEW FRAMEWORK

Four empirical studies were used to gather data of ninety-eight students to show evidence of three
major transition points in the new framework: Transition (A) from physical to mathematical coming from the first layer, the prevalence of the analytic approach to the synthetic approach while
developing abstraction enlightened by the second and the third layers, Transition (C) from analytic
to synthetic; and Transition (D) from geometric to symbolic coming from the second and third
layers, and process-object duality in geometric representations. The design and construction of the
surveys as well as interviews reflect the emphasis of my research on gathering evidence in student
work for confirmation of those focuses.

3.1

Research Focuses

This series of empirical studies is highly exploratory and formative. The focus is on evidence in
student data of the following features that are hypothesized by the configuration of the construct
of vectors:
(1) Transition (A) from physical to mathematical in terms of the configuration: Student data
will show evidence of the possible differentiating mathematical abstraction from a physical
embodiment that was described as a global difference between physical and mathematical
vectors in the first layer of refinement. This data will also show evidence of the interplay
between the ontological and epistemological perspectives to help distinguish mathematical
vectors from physical vectors when thinking of the second layer added to the first.
(2) Transition (C) from analytic (procedural) to synthetic (structural) and Transition (D) from
geometric to symbolic: Student data will show evidence of the difference between the representational and the cognitive perspectives, and how those different views combined and

32

interplayed as the ontological and epistemological perspectives of the construct of vectors
in the second layer. The student data also will show evidence of the epistemological obstacle (C) and the ontological obstacle (D), and confirm those obstacles as transition points
towards mathematical abstraction in terms of the second layer together with the third layers.
Process-object duality on geometric representations also will be identified from student data.
(3) The prevalence of the analytic approach to the synthetic approach: Student data will show
evidence of the prevalence of the analytic approach to the synthetic approach as described
in the extended discussion of the second and the third layers while developing mathematical
abstraction. This will also serve as a reasonable clue that of the progression of refinements
to identify obstacles in transitions/conversions on the construct of vectors.
The purpose in collecting data from four different surveys and interviews is (1) to determine if
transition points are evident in student work, (2) to identify the developmental evidence in student
work suggested by the configuration, and (3) to examine likely dominance of the analytic approach
to the synthetic approach towards mathematical abstraction implied by the configuration.

3.2

Method and Participants

Four surveys and interviews were carried out to gather evidence on the important features suggested by the configuration. The results reported in Chapter 4 are a synthesis of the data gathered
with the focus on the three major transitions (A); (C); and (D), and the prevalence of the analytic
approach to the synthetic approach. Multiple administrations were used to: (1) test appropriate
survey questionnaire and interview process, (2) gather deeper knowledge of student background
and idea on vectors, and (3) modify surveys and interviews in order to avoid any confusion derived
from the questions. All data were collected from students located in the midwest public university.
The first empirical study on vectors was done during the beginning of the 2010 Fall Semester.
Five pre-service secondary mathematics teachers from the senior level ‘Reflection and Inquiry in

33

Teaching Practice’ course participated in the survey, and three of them were interviewed after
finishing the surveys.
The second empirical study was conducted during the 2010 Fall Semester. A task-based survey
developed and revised based on the first study was given. This survey asked for student backgrounds on vectors in addition to the survey questions. Twenty-nine students from the senior level
‘Capstone in Mathematics’ course for pre-service secondary mathematics teachers participated.
After the survey, four students were selected for follow up interviews to probe deeper connections
of the configuration and the actual student work.
The third study was carried out in the 2011 Spring Semester. Fifty-eight pre-service elementary
teachers were examined. These pre-service elementary school teachers were taking the ‘Elementary Geometry For Teachers’ course. The main focus of the course was geometry and measurement
needed for K-8 teaching. The textbook was ‘Mathematics for Elementary Teachers’ by Sybilla
Beckmann (Beckmann, 2005). This course consisted of four big topics: Geometry (of 2D, 3D objects), Geometry of Motion and Change, Measurement, More about area and volume. Students in
this course already studied the first three sections of Geometry of Motion and Change that covered
various geometric transformations such as translations, reflections, rotations, and glide reflections.
The fourth study was carried out in the 2011 Summer Semester. Six pre-service secondary
teachers from ‘College Geometry’ course were participated. No interviews were conducted.
After administering each survey (I and II), students were selected for interviews based upon
their response. The selected students signed up for an one-hour block of time for their interviews
on the day and time that was most convenient for them. Interviews were held in a neutral location
away from the students’ classrooms and were audio-recorded for further analysis. Transcribed
interviews were coded and analyzed in order to find evidence in student work. Descriptive statistics
were used for the second study and the third study on which a sufficient number of participants
were available.

34

3.3

Design and Construction of Survey and Interview

The following Table 3.1 shows the overview of my design of the survey questions. Each question is
connected to the focal transition (A), (C), and (D). Survey III is an expansion of ‘Translation’ question in Survey I, II and IV. Survey III and Survey IV also have additional questions not included
in this table that are off the focus of the study. Because Transition (A) has two different perspectives: epistemological and ontological perspectives derived from the second layer, I distinguished
the questions that examine the epistemological and ontological perspective each by marking →
L

and ↑. By marking , and L, I also differentiated the analytic approach related questions from the
synthetic approach related questions. This table also has an indicator V’ for the evolution of the
original question V. The detail evolution is in the following section.

E

I
V

Surveys
II III
V’
V
V

IV
V’
V
V

V
X( )
X( )
X( )
X( )
L
L

X( )
X( )
X(L)
X( )

X(L)

L

X(L)
X(L)
X(L)
X(L)
X(L)
X( )
X(L)

L

L

A
X(→)
X(→)
X(→)
X(→, ↑)
X(→)
X(↑)

L
L
L

Question Name
Translation
Translation of Polygon
Geometric Translation
Rainy Day
Force
Robot Arm
Origin
Basis
Rotation
Polygon
Very Long Sum
Cube
Midpoints
Associativity
Point and Vector

Transitions
B
C
D

V
V
V
V
V
V
V
V
V
V

V
V
V
V
V
V
V
V’
V
V
V’

V
V

V
V’
V’
V
V
V’

Table 3.1: Map from questions to configuration

For the in-depth discussion of the research focus in the next chapter, I chose the following
questions from Table 3.1 and synthesized the results. See Table 3.2.
Interviews were conducted mainly by repeating the survey questions and asking further about

35

Key Features
Physical vs. Mathematical

Related Layers
I

Transitions
A

Epistemological Diff. &
Ontological Diff.
Epistemological Obst. &
Ontological Obst.
Process-Object Duality
in Geometric Representation
Prevalence of
Analytic Approach

II

III

A, C
A, D
(A), C
(A), D
C

II & III

C, D

III

Questions
Translation,
(Translation of Polygon),
(Geometric Translation),
Rainy Day, Robot Arm
Translation, Polygon
Rainy Day, Robot Arm
Polygon, Very Long Sum
Origin, Robot Arm
Very Long Sum
Cube, Midpoints,
Associativity

Table 3.2: Questions for Layered View of Configuration
Which vector representations do want to use to prove the theorem, coordinate/column vecyou

x
− −
→
tor forms such as (a, b, c),  y  or generic vector forms such as AB, →?
u
z
(b) Proceed your proof with the preferred vector representation.
(c) Why have you preferred this representation and not another one?
(d) Prove with the other vector representation that you did not use if you can.
(e) What difficulties are you faced with, for each method? Namely, what forced you to
abandon a representation, if anything like that has happened, or what has prevented you
in reaching a final proof?
Table 3.3: Example of Interview Questions

what the students were thinking. Because questions for examining the prevalence of an approach to
the other in the surveys were only asked to students to choose the representations, I asked students
to proceed and finish the proof in the interview sessions. See Table 3.3. This helped me to see the
deeper thought process of students.

36

3.4

Evolution of Survey Questions

Due to the complexity of the vector construct, a single question can never achieve any clear view
of a single facet. By grouping them with respect to progressive layers of refinements and transition
points related with each layers, I can discuss the construct of vectors in a succinct manner while
still reflecting the complexity and subtlety. Originally, questions were intended to explore all five
transitions in the configuration. However, it was hard to develop appropriate questions to explore
the transition (B) from arithmetic to algebraic vectors. College level mathematics usually assumes
students who finished this developmental transition earlier in their elementary and middle school
mathematics. In case of symbolic vectors, this transition simply means changes from numbers to
letters in entries. Inquiry for Transition (E) was also postponed because the main focus of this
study was vectors in geometry, and the transition was regarded as the one related with symbols in
linear algebra. Further discussion about Transition (E) will be on Chapter 5 as a limitation.
The first study was intended to test appropriate survey questionnaire for investigating the whole
configuration with focus on the three transitions. The follow up interviews informed modifying
survey questions so that questions would more effectively capture student work. This portion of
the study had two goals. One goal was to look into evidence of three important transitions (A),
(C) and (D). The other goal was to identify evidence of two different approaches: analytic and
synthetic approaches towards mathematical abstraction in student work. Locating student work in
the related transitions in the configuration as evidence of developmental obstacles and exploring
different approaches from the data using descriptive statistics were also concerns of this study. See
Table 3.2.
In the second study, background information such as college major, high school courses related
with vectors, college courses related with vectors, etc. was gathered to examine any correlation
between questions and student background information. Twelve vector questions were reviewed
and revised based on the first study. ‘Translation’, ‘Cube’, and ‘Point and Vector’ questions were
modified for a better understanding of the student work and for avoiding confusion while the students attempted surveys. For example, here is the original cube diagonal and its modification. The
37

first survey asked the students to prove that the two vectors shown in the cube were perpendicular.
Some students see two vectors in the cube as they were literally intersecting and perpendicular to
each other. To avoid this confusion, I changed the figure and made the two vectors not intersect.
See Figure 3.1.

Figure 3.1: Modified Cube Figure in Survey II

‘Rainy Day’ question was dropped after survey I, because most responses utilize physics
knowledge more than mathematics knowledge. While developing three layers of refinements, I
emphasized mathematical vectors over physical vectors. By dropping ‘Rainy Day’ question, I
made students concentrate on mathematical vectors with their responses.
‘Force’ question was introduced for the first time in survey II to set up a different physical
situation other than a motion-related situation. This gave additional examples of epistemological
translation (A) other than ‘Translation’ that only covers a motion.
The third study focused on a single specific topic of geometry: a vector as a geometric translation. The first and second studies showed enough evidence of student use of physical vectors in
mathematical context. The difference between physical vectors and mathematical vectors in the
configuration was evident from the student responses. However I sought further refinement in the
data with a population where the differences are most likely to exist. The focus of this study was
how representations were related with cognitive development. By refining the notion of vector

38

representations and by integrating different cognitive development theories, I tried to probe more
about transition (A).
In the fourth study, several questions from the first, second and third studies were selected.
‘Robot Arm’ question was also dropped and some translation questions from survey III were added
instead. ‘Basis’ and ‘Rotation’ questions were dropped because they were heavily dependent on
Linear Algebra knowledge and provided little opportunity to distinguish or differentiate student
thinking relative to framework. ‘Very Long Sum’ question was simplified to only address a sum
structure based upon previous results.
Chapter 4 shows synthesis of results and analyses from the above four studies. This presents a
summary of the collected data, statistical treatments of the data, an interpretation of survey data,
and excerpts of the interviews dedicated to reflecting conceptions to the configuration. Mechanics
of analysis and findings are also discussed.

39

Chapter 4
FINDINGS AND DISCUSSIONS

Collective results from the four empirical studies consisting of written surveys and interviews
are used to look into evidence of focal points that we assumed and hypothesized in process of
refinements. In summary, student data showed evidence of (1) the global difference between mathematical vectors and physical vectors, (2) interplay between the ontological and epistemological
perspectives, (3) the epistemological obstacle defined as Transition (C) and the ontological obstacle defined as Transition (D), (4) the prevalence of the analytic approach to the synthetic approach,
and (5) process-object duality on geometric representations. I present evidence of each following
the refinement process for the construction of the configuration that was described in Chapter 2.

4.1

From Physical to Mathematical Vectors

From Chapter 1, we saw that it is hard to describe the complexity and subtlety of a vector as a
translation only with the classical vector representations. The layered approach to the construct of
vectors enabled us to differentiate mathematical abstraction from physical embodiment in Chapter
2. I hypothesized that there existed a transition from physical vectors to mathematical vectors in
the configuration that could be described from both epistemological and ontological perspectives.
In this section, I begin with student evidence that shows the differentiation of mathematical vectors
from physical vectors is possible as I classified in the configuration.
The global difference between mathematical abstraction and physical embodiment is evident
in student work. Student data showed evidence of the different interpretations between ‘the same
translations’ and ‘the equivalent vectors’. We usually assume that the concept of the vector equivalence relation in physics is the same with that in mathematics, because ‘directions’ and ‘magnitudes’ of vectors are used to verify equivalent relations in both fields. This means that the equivalent vectors are always representing the same translations and vice versa both in physics and
40

−
− −
v
Translation: A translation can be represented by a vector →. T→ (P) = P + → for any
v
v
point P.

(a) List all vectors that do NOT represent the translation of triangle A to triangle B in
the figure.
−
(b) List all vectors that are equivalent to →.
a
Figure 4.1: Translation Question

mathematics. However, student work for ‘Translation’ question (Figure 4.1) shows the difference
of the interpretation between the same translations and the equivalent vectors.
In the second study, the cluster tree diagrams (Figure 4.2 and 4.3) from hierarchical clustering
with Euclidean distance are supposed to show similar categorizations assuming ‘the same translations’ and ‘the equivalent vectors’ are the same concept. The first cluster tree was made from the
student responses to question (a), and the second cluster tree was made from those to (b). They
show two different categorizations in Figure 4.2 and 4.3.
Even though geometric translations are not representing any physical quantities, students seem
to understand them through physical embodiment. They connect geometric translations to physical

41

0
transl_A_vector_negative_d
transl_A_vector_e_plus_f

Translation (a) Dendrogram
Rescaled Distance Cluster Combine
5
10
15
20
25

9
10
2

transl_A_vector_a

1

transl_A_vector_g

7

transl_A_vector_h

8

transl_A_vector_c

3

transl_A_vector_e

5

transl_A_vector_f

6

transl_A_vector_d

4

Y

transl_A_vector_b

Figure 4.2: Cluster Tree of the same translations

movements of geometric objects, and restrict their knowledge on vector equivalence relation to
equivalency of physical quantities.
This result of a quantitative analysis tells us that there is evidence of the difference between
mathematical vectors and physical vectors hypothesized in the configuration. This difference
comes from the students’ ideas of the vector equivalence relation. However, it informed us only of
a cognitive difference assuming the same form of the representation such as arrows. Could there
be an element other than a cognitive element that differentiate mathematical vectors from physical

42

0
5

transl_B_vector_f

6

transl_B_vector_d

4

transl_B_vector_negative_d

9

transl_B_vector_b

2

transl_B_vector_c

3

transl_B_vector_g

7

transl_B_vector_a

1

transl_B_vector_h

Y

transl_B_vector_e

Translation (b) Dendrogram
Rescaled Distance Cluster Combine
5
10
15
20
25

8
Figure 4.3: Cluster Tree of the equivalent vectors

vectors? In response to this inquiry, I introduced another layer of refinement for the construct of
vectors in Chapter 2. With this refinement, we could assume there existed two different perspectives: epistemological and ontological perspectives when we discuss the complexity and subtlety
of the construct of vectors. The difference between mathematical and physical vectors from the
epistemological perspective (cognitive change) was evident in student work. This is a result from
the quantitative analysis above. In the next section, I will seek evidence of this ontological difference (representational change from physical to mathematical) by looking at individual student
43

A Rainy Day: The rain is falling straight down on your head with speed 1 m/s. You start
walking forward with speed of 2 m/s and notice the rain is now hitting your face. What is
the angle that the rain now appears to make with the vertical? Draw a picture that explains
your thinking.
Table 4.1: A Rainy Day Question

data.

4.2

Epistemological and Ontological Perspectives

Student data showed evidence of the difference between mathematical vectors and physical vectors
in the previous section. However, it only showed the epistemological difference. In this section, I
examine student data to see if there is an ontological difference between physical and mathematical vectors. In Chapter 2, this ontological difference between physical vectors and mathematical
vectors was described and hypothesized as a change from vectors accompanied by physical units
for representing physical quantities to vectors as self-standing mathematical objects. Translating
physical directions and magnitudes to mathematical directions and magnitudes, on a coordinate
system or on a coordinate free Euclidean (Affine) space, or possible dropping of physical units
from vectors were those examples.

Figure 4.4: Construction of a Diagram

44

Figure 4.5: Student response based on vectors with no evidence of symbolic vector addition

Student work from the ‘Rainy Day’ question showed evidence of this ontological difference
between mathematical and physical vectors. This difference was not shown as a clear cut of two
opposite examples that have no intersection. Instead, it was realized as a gradual change in spectrum of responses. Because the ‘Rainy Day’ question itself was designed to see specific instances
of student work, I provided representative and qualitative evidence of student work.
Figure 4.4 shows student’s work that exhibits no constructions from the vector addition. This
student provided only the descriptions of moving directions and speeds in a physical situation.
Each numbering of the situation shows the unfinished effort moving towards abstraction. The
student in Figure 4.5 showed evidence of a different situation that the student knew the solution
vector should be from the triangle vector sum. We are not sure if that is from the actual symbolic
operation of the vector addition. Student work in Figure 4.6 shows evidence of another situation.
This student responded with both geometric and symbolic vector sum, and showed an example
of vectors as self-standing objects in mathematics by translating relative speed of raindrops from
− −
the walking speed to → + →. These three examples illustrate a gradual change in spectrum of rev1 v2
sponses from physical vectors to mathematical vectors with a loss of physical appearance attached
to physical objects and with developing symbolic calculations.
In other responses to ‘Rainy Day’ question, I saw several cases regarding arrow forms as line
segments and not dealing with any vector operations. Figure 4.7 shows one of those examples. The
construction of a triangle figure was independent of the triangle method of vector addition and the

45

Figure 4.6: Student Response based on geometric and symbolic vector addition

Figure 4.7: Student Response based on geometry without vector addition

student just used vectors as sides of the triangle. This example shows an epistemological change
hidden on arrow forms of vectors. From the above example, we can see that when talking about
the gradual ontological change in spectrum of student responses, we should consider the hidden
epistemological change too. To discuss the interplay between the epistemological perspective and
the ontological perspective is therefore reasonable.

4.2.1

Interplay between Epistemological and Ontological Perspectives

The previous responses shows evidence of the difference between the epistemological and ontological perspectives with the difference between epistemological changes and ontological changes
on the second layer of refinement. However, we also noticed that it was really hard to see evidence
of the difference solely from one perspective. This is because the ontological perspective does not
always assume the representational change i.e., change in explicit forms such as arrows to coordinate forms. In practice, physics uses very similar forms of representations with mathematics.

46

Changes within a representation with respect to ontological perspective are always hard to see, and
this is why we need to put an emphasis on interplay between the epistemological and ontological
perspectives. Figure 4.7 is an example that shows interplay between a cognitive change (epistemological change) and a representational change (ontological development towards mathematical
abstraction).
The interplay between the ontological and epistemological perspectives is also evident in the
following example. In the ‘Robot Arm’ question, numerical information given in the question was
not consistent with the pictures if one read the picture physically (Figure 4.8).
Robot Arm: A robot arm has three joints as shown. The bottom part
−
can be represented by a vector → = (10, 15, 20), the middle part can
a
→
−
be represented by a vector b = (5, −5, 15), and the top part can be
−
represented by → = (20, −5, −10). The units are in centimeters(cm).
c

c¯
b¯

=(5 -5 15)

a¯
O

−
Find the vector → that tells the location of the hand.
p
Figure 4.8: Robot Arm Question

a™

-OA≥

a£) B(b¡

b™

b£)

One of the responses with the correct 3-tuple answer said, “It’s a little difficult to reconcile this
−
diagram with the given vector values. (i.e., → seems to go up, so why does z = −10 ?)” It was
c
about the ontological perspective (comparison of representations) together with the epistemolog-

b™ b£)-(a¡ perspective (interpretation of directions and coordinate axes).
ical a™ a£)

−
In physical view, vector → is
c

b£-a£)
-a¡ b™-a™ going up so that the z-value should be positive not negative assuming x, y, z-coordinate system of a

a¡)¤ +(b™-a™√)¤ +(b£-a£)¤

47

physical space where positive z represents a space above xy horizontal plane. In mathematics, the
coordinate system is separate from physical locations or directions. Positive directions of axes can
be anywhere. This student work shows the difference between mathematical and physical vectors
from the first layer view as previously discussed. This difference can be discussed as well in more
detail as above with the second layer that provides multiple perspectives and their interplay.

Figure 4.9: A Student Drawing of Physical Axes

The above discussions about the epistemological and ontological perspectives and their interplay on the construct of vectors are accompanied by a direction towards mathematical abstraction.
The second layer of refinement presumes decomposing this direction into two perspectives as well.
In the configuration, the ontological axis is a directed dimension that covers changes in existence
or being. It is about explicit change of representations or notions. The epistemological axis is
a directed dimension that covers changes in knowledge or thinking. It is about understanding of
representations. The direction of each axis implies not only the intended movement towards mathematical abstraction, but also certain degrees of the developmental movement. We hypothesized
this from the prevalence of the analytic approach to the synthetic approach as described in Chapter
3. In terms of the configuration, I defined the analytic approach as a trend of changing explicit representations along the ontological axis from geometric representations to symbolic representations
quickly as possible. Because this prevalence of the analytic approach to the synthetic approach
could suggest a refinement to see the complexity and subtlety of the construct of vectors, I looked

48

Analytic Approach
Cube:
Midpoints:
Associativity:

Synthetic Approach

No response

41% (12)
55.5% (16)
72.4% (21)

31% (9)
34.5% (10)
24.1% (7)

28% (8)
10% (3)
3.4% (1)

Table 4.2: Prevalence Result from Survey II

into this from the student data.

4.2.2

Prevalence of Analytic Approach to Synthetic Approach

Studies on prevalence are natural in a sense that we set up the developmental direction towards
mathematical abstraction by differentiating mathematical vectors from physical vectors through
the first layer and identified two perspectives to see the development in detail through the second
layer. Moreover the result of prevalence can give a motivation of further refinement because the
prevalence may be coming from harder obstacles to overcome in one developmental direction.
This evidence of the prevalence in this section will motivate us to expect the next refinement in
section 4.3.
I categorized students’ responses into two approaches: analytic and synthetic as defined in
Chapter 2. Students tended to use particular representations more and confine their understanding
and using vectors in one approach rather than having flexibility of using both. This tendency was
identified in the responses as the prevalence of the analytic approach to the synthetic approach.
Because this prevalence is studied and regarded as a trend, and not a specific student’s preference,
I use the collective data of twenty-nine students rather than concentrating on specific cases. ‘Cube’,
‘Triangle Midpoints’, ‘Associativity’ are questions specially designed to look into the prevalence
of the analytic approach to the synthetic approach. See Table 4.2. The results show that the
prevalence of the analytic approach to the synthetic approach that we hypothesized in Chapter 2 is
evident in student work.
In the ‘Cube’ question, the choice of (i) or (ii) was categorized as the choice of the analytic
approach and (iii), (iv) as the synthetic approach. The result of ‘Cube’ question implies more

49

Cube: In the following figure, we want to show that the two vectors shown in the cube
are perpendicular using the inner product also known as the dot product. The sides of the
cube can be assumed to be of length 1.

(a) Which form of vectors, do you think will be most useful? Circle your answer.
 
x
−
→
−
(i) (a, b, c) (ii)  y  (iii) PQ (iv) → (v) Other:
u
z
Figure 4.10: Cube Question

students chose the analytic approach than the synthetic approach. One can argue that the cube
structure itself encourages students to use the analytic approach because mutually orthogonal sides
of the cube can be three axes associated with a space. However, when mutual orthogonality is not
guaranteed such as in ‘Triangle Midpoints’ (Figure 4.11), the prevalence of the analytic approach
to the synthetic approach still shows up.‘Triangle Midpoints’ question is actually accessible when
students use the synthetic approach and the structure of vector additions. This is because positioning the origin of the coordinates onto the appropriate location of the triangle makes the question
harder due to no evidence of a right angle. Therefore, it is worth using this quesion as a way to see
the prevalence together with the ‘Cube’ question.
All the interview results also supported the prevalence of the analytic approach to the synthetic
approach. One student described the reason as his experience on vectors in high school mathematics. He said that because the focus of the curricula was on algebra, it made the coordinate form

50

Triangle Midpoints: In the following triangle
want to show that BC is parallel to DE.

ABC, AB = 2AD and AC = 2AE. We

(a) Which form of vectors, do you think will be most useful? Circle your answer.
(i) (a, b)

(ii)

x
y

−
→
(iii) PQ

−
(iv) →
u

(v) Other:

Figure 4.11: Triangle Midpoints Question

and the analytic approach “more recognizable.” ‘Associativity’ gave a similar prevalence of the
analytic approach to the synthetic approach. See Table 4.2.
Associativity: We want to prove the following property of vector addition:
−
→ −
−
− → − −
(→ + b ) + → = → + ( b + →).
a
c
a
c
(a) Which form of vectors, do you think will be most useful? Circle your answer. You
don’t actually need to prove this.
x
(i) (a, b) (ii)
(iii) directed line segments (iv) Other:
y
(b) Why have you preferred this representation and not another one?
Figure 4.12: Associativity Question

In the follow up interviews from the first and the second studies, only one student out of eight
was able to prove this associativity with arrow forms of vectors when asked to proceed the proof.
51

Even though it would be tempting to say because it is much easier to prove associativity using
coordinate/column vector forms, we should be alerted from this example with possible obstacles
that prevent student’s flexible use of various vectors. These obstacles were hypothesized to be
evident in student work through the third layer of refinement.
Another interview result describes that the prevalence against the synthetic approach is because
it is not as ‘concrete’ as the analytic approach. This ‘concreteness’ was from the fact that when
using analytic approach, students can utilize number substitutions in entries. An interesting observation is that this dominance is spread over all the data and not just the data from those three
intended questions. This dominance is in disgreement with Hillel (2002) that geometric vectors are
more concrete than symbolic vectors. This prevalence together with Hillel’s argument shows that
even if we have the same arrow forms of vectors, up to the setting (cognitively different settings or
obstacles), the prevalence of the coordinate/column vector forms to arrow forms can change.
In this section, it was evident in student work that (1) there are the epistemological and the
ontological perspectives in the construct of vectors, (2) interplay between those two is essential
to talk about the construct of vectors, and (3) there is a prevalence of the analytic approach to
synthetic approach that might imply certain obstacles. In terms of the configuration, we need a
refinement of the construct of vectors that can see these obstacles as transitions. The third layer of
refinement will identify transitions in the next section.

4.3

Obstacles towards Mathematical Abstraction

In Chapter 1, we saw that when talking about interplay between the representational and the cognitive perspectives on the construct of vectors, the problem stems not only from the multiple representations but also from the translations/conversions among them. We can also recall that the third
layer was the layer that identifies the representational and the cognitive obstacles in terms of transitions towards mathematical abstraction in Chapter 2. In this section, I want to look for evidence
from student work that shows these ontological and epistemological obstacles. These obstacles
will play roles as transitions in the configuration of vectors. By doing so, we can especially see
52

cognitive development along with Action-Process-Object-Schema and/or process-object duality in
geometric representations emphasized in the configuration.

4.3.1

Ontological Obstacles: Transition (D) from Geometric to Symbolic

As we explored in section 4.2, it was hard to extract a clear ontological part from a combined view
when we talked about the difference between mathematical vectors and physical vectors. ‘Robot
Arm’ question still gives a blurry idea of the ontological part different from the epistemological
part about matching up the coordinates with the axes. The coordinate axes that a student drew in
Figure 4.9 were not for coordinate representations of arrow vectors, but for the coordinates of the
terminal points of vectors when their initial points were located at the origin. The following quote
from a student response supports that there is an obstacle related with the interpretation of physical
world mathematically. It is still cognitive, but the physical settings change to mathematical settings
in an explicit way.
R: “You said, unsure, cannot reconcile image & vector coordinates. What was happening here?”
S: “[...] Vectors are being used to describe the parts of the arm in here. [...] I was
trying to see [...] if I draw the space out for three axes, where should this point be,
where should the point be for this vector. I don’t know [...] not sure if the pictures are
not actually representing the vector listed here or if I just couldn’t map it out right in
my head or on the graph I drew out, but I just couldn’t determine [...]”
R: “In the picture, the origin was the intersection of those three axes. What do you
mean by coordinates? For example, (10, 15, 20) in your case would be the coordinate
→
−
−
of the terminal point of →. What about b ? (5, −5, 15)?”
a
S: “I think that also represents the point well..though [...] At that time, I was setting
all the vectors out to have their initial points be the origin, and then their end points be
what was given for the vector.”

53

one correct
Position Vector:
Free Vector:

two correct

three correct

non correct

total

3.4% (1)
31% (9)

0% (0)
3.4% (1)

86.2% (25)
37.9% (11)

10.3% (3)
27.6% (8)

100% (29)
100% (29)

Table 4.3: Origin Result from Survey II

R: “[...] What was the hardest part? What was the obstacle?”
S: “[...] coordinates are not matched up with my axes. [...]”
Taking into account the response above, we can see evidence of the ontological obstacles in:
(1) interpreting and relating the location and direction of a vector without the reference of the
physical world corresponding to Transition (A), and (2) identifying the locations of vectors and
the directions of vectors when they are given in coordinate forms corresponding to Transition (D).
We can also identify these ontological obstacles in the responses for the ‘Origin’ question.
‘Origin’ is related with transition (D) from geometric vectors to symbolic vectors (Figure 4.13).
As we expected from Chapter 2 on Transition (D), confusion between the coordinates of points
and the coordinate forms of vectors still exists and is evident in student work. See Table 4.3.
Students are usually familiar with converting an arrow form of a vector to a coordinate form
from their basic vector study in physics and high school mathematics. However, those conversions
are limited to position vectors with (0, 0) as the origin in most cases. I could see from student
work that Transition (D) was still a big obstacle to some college students even if they already
knew how to convert one from the other. The variable location of the origin and the coordinate
forms of free vectors are still not familiar to them. See Table 4.3. In terms of our discussion on
the layered view of the construct of vectors, this is where the interplay comes in. Transition (D)
as an ontological obstacle is evident from the data (Table 4.3), but to talk about why, we need
to think from the epistemological perspective. Converting from geometric to symbolic was more
successful in the case of position vectors than free vectors as expected. This is because students are
more familiar with the external coordinate form of vectors when producing the coordinate forms
using the difference in the coordinates for initial and terminal points. They do not feel comfortable

54

Origin: For the three different origins: O, C, and A, complete the table by finding the
−
→
position vector of A and the vector AB.

Origin
O
C
A

−
→
AB

Position Vector of A
(
(
(

,
,
,

)
)
)

(
(
(

,
,
,

)
)
)

Figure 4.13: Origin Question

with the internal coordinate form for vectors that can be calculated without the coordinates for
the origin, initial and terminal points, but counting the grids from the initial points to the terminal
points. The latter focuses more on vectors themselves, but the former focuses more on additional
information such as coordinates of points in vectors. Knowledge of vector equivalent relation and
the concept of position/free vector constituted to this obstacle. This means that the epistemological
perspective can give ideas of the reason to the ontological obstacles.
In Chapter 2, we hypothesized epistemological obstacles in the cognitive development towards
mathematical abstraction. In the previous section, we talked about the prevalence of the analytic

55

approach to the synthetic approach. For this reason, it is no less dubious to connect that to the
obstacles along the synthetic approach, especially to cognitive obstacles.

4.3.2

Epistemological Obstacle: Transition (C) from Analytic to Synthetic

The existence of an epistemological obstacle is evident in student work for ‘Polygon’ and ‘A very
long sum’ questions. This obstacle prevents students to continue calculating the binary vector sum.
This is Transition (C) from analytic to synthetic where the obstacle is along cognitive development
on geometric vectors in the configuration. For ‘Polygon’ question, students’ responses told us that
question (a), (b), and (d) based on a triangle, a parallelogram, a rectangle figure were easier for
students, but question (c) with a pentagon was harder to solve than other questions (Table 4.4).
From Table 4.4, it is clear that students were not familiar with vector sum on a weird shape
or a polygon with more than four sides. It is interesting to see a response that shows drawing
parallelogram sum, stopping, and saying ‘very unique arrangement’ in the regular pentagon. This
student described his experience during the interview as follows.

Figure 4.14: “Very unique arrangement”

56

Polygons: label each vector and write an equation of the relations.

question
correct
not correct
total

(a)

(b)

(c)

72.4% 65.5% 48.3%
27.6% 34.5% 51.7%
100 %

100 % 100 %

(d)
69%
31%
100 %

Table 4.4: Polygon Result from Survey II

57

R: “Why did you draw these middle line segments? What are they?”
S: “I think I was trying to do the vector addition and [...] I couldn’t really find, based
on the method I was trying, couldn’t find the way to express the relationship together
from a polygon or from a pentagon. [...] Normally I’ve never seen vectors arranged in
that kind of relationship. I’ve seen them in the triangle, [...] usually in many of these,
a parallelogram, a four sided figure, but nothing like this one.”
From the interview with the student above, I could see that the student drew a parallelogram
to figure out the sum. Thinking the sizes and the directions of arrows compared to thinking the
structure that vectors lie on can be regarded as procedural thinking, because the sum was a binary
operation and we needed those information for a binary operation. It is evident that there is an
obstacle that prevents students using synthetic vectors or structural thinking. What we expect from
a structural thinking is also evident from an interview quote for question (c) in the ‘Polygons’
question, “If I started here, I kind of saw that...if I just kind of follow my nose around this thing,
I’m going to end up with where I started. Okay. Sum of all these is zero." This student saw
that sum as a whole, not separated operations. Seeing a structure as a whole, not as collection
of separated operations could be evidence of a successful transition from analytic (procedural) to
synthetic (structural). We can notice this more in the following example.
Student responses to ‘A Very Long Sum’ also show that there is an obstacle to continue the
sum procedurally. This obstacle is visually clear in a sense that the middle dots in the figure block
students to continue the binary sum operations. Only 27.6% (8) of 29 students gave a correct
answer. One of students who responded with structural thinking of the vector sum in terms of flow
with moving a nose around the figure. Usually a good interpretation of middle dots and/or skipping
dots imply a transition from process to object or from procedural to structural thinking in symbolic
representation (0.9 . . . = 1).
Proceeding from what has been observed above, Transition (C) from analytic (procedural) to
synthetic (structural) is evident in student work as a cognitive obstacle. In the next discussion, we
will also see that process-object duality and Action-Process-Object-Schema development in terms
58

Figure 4.15: Understanding Structures of Vector Operations

of geometric representations in a similar way to symbolic representations.

4.3.3

Cognitive Development of Geometric Representations

So far we have seen evidence in student work for three important transition points: (A) Physical to
Mathematical, (D) Geometric to Symbolic, and (C) Analytic to Synthetic with the progressive refinements of the construct of vectors. In the configuration, geometric representations and symbolic
representations are displayed as parallel to each other along epistemological axis. Transition (C)
from analytic (procedural) to synthetic (structural) is not just for symbolic representations but also
for geometric representations. This is to reveal cognitive development in geometric representations
in a similar fashion to symbolic representations. Addition of two numbers is one of the famous examples in process-object duality theory. For example, 1+3 may be understood as adding 3 to 1. On
the other hand, 3 + 5 may also be understood as a sum without calculation (Sfard, 1991; Gray and

59

Tall, 1993, 2001). These basic examples are all about symbolic representations. How geometric
representations of vectors correspond to the cognitive development of process-object duality is the
main concern. From process to object, or from procedural to structural transitions in the construct
of vectors are expected to be evident in student work simultaneously in both representations.
− −
−
A Very Long Sum: Find → + → + . . . + →.
u1 u2
un

→
−
u3

→
−
u2
→
−
u1

→
−
u4

→
−
un

Figure 4.16: Understanding Addition Structures

Figure 4.16 shows the modified version of ‘Very Long Sum’. I had two students whose responses were totally different. One student used the strategy of the structure sum with drawing an
arrow as the result of the sum. The student connected two end points of the picture and gave the
direction of the vector. The other did not give the answer in arrow forms. Instead, he provided the
following work.
→
− −
Let u, u1 be points such that → = uu1 .
u1
−
− −→
Let u1 , u2 be points such that → = u1 u2 .
u2
···
−−
− −−→
Let un−1 , un be points such that → = un−1 un .
un
→
− −
− −
Then → + → + . . . + → = uun
u1 u2
un

60

The student clearly noticed that the dots in the figure correspond to dots in the symbolic manipulation. To get over the obstacle of ‘· · · ’ in the figure, this student used the corresponding ‘· · · ’
in the symbolic representations. By changing a reduced symbol to a vector connecting two points,
he continue to convert symbolic representations to geometric representations. It seemed the student already knew the fact that symbolic calculation could be done without assuming pictures, for
→ → −
− −
→ → − −
→
example, a symbolic canceling such as AB + BC = A + C = AC.
B B
It seems reasonable to assume that cognitive development in both representations can help
students see more coherent relationship between geometric representations and symbolic representations. This example together with ‘Polygon’ extends the coverage of cognitive development
theories such as process-object duality from symbolic world to geometric world.

4.4

Summary of Findings

In this chapter, we saw that the following are evident in student work when discussing the complexity/subtlety of vectors:
(1) the difference between physical and mathematical vectors,
(2) the multiple perspectives: ontological and epistemological, and interplay between those two,
(3) the prevalence of the analytic approach to the synthetic approach,
(4) an epistemological obstacle defined as Transition (C) and an ontological obstacle defined as
Transition (D),
(5) process-object duality on geometric representations of vectors.
What stands out most from these empirical studies is this new framework is very helpful when
talking about the complexity and subtlety of the construct of vectors. However, these tentative
conclusions with the configuration await further refinement and correction in the light of further
research.

61

Chapter 5
LIMITATIONS OF THE FRAMEWORK

This chapter described unexpected evidence from the three progressive refinements of the construct of vectors. These limitations include: (1) the categorization of mathematical vectors while
the first layer differentiating mathematical vectors from physical vectors, (2) the direction of the
development towards abstraction while the second layer recounting the ontological and epistemological perspectives and decomposing the direction of development into two, and (3) the transitions
on the configuration while the third layer identifying ontological and epistemological obstacles in
terms of transitions towards abstraction. Specifically, the limitations described in this chapter are:
(1) non-empty intersection between mathematical vectors and physical vectors, (2) unreasonable
levels of sophistication reflected in the direction towards abstraction, and (3) problems in repetition
of transitions and revered transitions in different contexts. These limitations suggest further refinement and correction of the framework and the implications for teaching and learning of vectors.

5.1

Categorization of Mathematical Vectors

What is mathematical and what is physical? This question stimulates serious rethinking about
differentiating mathematical vectors from physical vectors done in the first refinement. The major difference was described as different understanding of the coordinate systems including axes,
points on the space, and the space itself. What we call a motion was quite different in mathematics
compared to a motion in physics according to Freudenthal (1983). He insisted that physical motion
is something that occurs to an object within space or plane within time, but mathematical motion
should be differentiated from physical motion in three ways: from the limited object to the total
space (plane), from within space (plane) to on space (plane), from within time to at one blow. This
difference is noteworthy for students to develop their vector understanding in mathematics. However, this difference is not clear cut in a sense that we still have a non-empty intersection between
62

Translated Image: Circle one that describes the image of a translation by an arrow in
the following figure.

Figure 5.1: Translated Image Question

(1)
.

(2)

(3) None of the left.
(Explain or draw your own answer.)
Figure 5.2: Translated Image Choices

mathematical vectors and physical vectors. Transition (A) in the configuration does not mean a
short time jump. It can be a long time development. For example, when a question from the survey
III asked about the definition of the geometric translation (Figure 5.1), the assumption behind the
question was that understanding a geometric translation as ‘moving entire plane’ is ‘mathematical’
not ‘physical’. In the discussion related to the complexity and the first refinement in Chapter 1 and
2, this assumption was regarded as true. If students focused on the object, they would choose the
picture that moves only a triangle as expected from a physical motion (Freudenthal, 1983).
Most of students gave the correct answer (2) in Figure 5.2 as anticipated. However, this answer
(2) could originate from physical meaning of ‘moving an object’ if the students assumed the translation as ‘moving entire plane’, and ‘entire plane’ as an object. Although we have the mathematical

63

(1)

(2)

(3)

Figure 5.3: Responses to Translated Images Question

terms such as a ‘map’ or ‘generalized function’ for the rigorous definition of the translation, ‘moving entire plane’ is also usually assumed to be a correct description of a geometric translation in
mathematics Beckmann (2005). Lakoff and Nunez (2001) also described the similar nature of
mathematics that is inseparable from physical embodiment. In addition to the above discussion,
a few more responses from the ‘Translated Image’ question showed different interpretations on
a geometric translation and its representation. The response (1) in Figure 5.2 is exactly what I
expected under the assumption of students’ conception of geometric translation as ‘moving an object’. However, some students did not regard an arrow as a mathematical representation reflecting
both direction and magnitude (Figure 5.3). Rather, they regarded an arrow as a non-mathematical
diagram or a visual symbol that represented the direction of movement. Some thought the arrow
should not intersect the geometric object because geometric objects might hinder the direction.
These responses could be an example of under-developed notion of physical vectors that needs
to be studied further both in mathematics education and physics education. Because we set up
the two different perspectives: ontological and epistemological in the second refinement of the
construct of vectors, looking into how those perspectives work in physics could be interesting as
well. The question could be about physical vectors in the second, third, and forth quadrants on the
configuration differentiated from mathematical vectors. See Figure 5.4.
In summary, the categorization of mathematical vectors and physical vectors while the first
layer differentiating mathematical vectors from physical vectors needs more refinements and reconsideration. In the next section, I will discuss the limitation from the direction of the develop64

Figure 5.4: Underdeveloped Physical Vectors?

ment towards mathematical abstraction in terms of the above two perspectives about the direction.

5.2

Direction of the Development towards Mathematical Abstraction

The direction and the dimension of the development introduce levels of sophistication reflected
in the direction towards abstraction while the second layer intertwining the ontological and epistemological perspectives and decomposing the direction into two. Moving up or moving to the
right suggest more abstract and higher levels of sophistication in the configuration. However, there
is no definite evidence that this direction always follows the direction that student understanding
follows. For example, we saw that a structural sum is harder to achieve than a procedural binary
sum. However, some students used non-mathematical, even non-physical ways of thinking such as
thinking arrows as routes of displacements and solved the structure sum easily. I identified those
students who interpreted vectors in arrow forms, as routes of displacements, not as representing

65

displacements themselves. In fact, displacement vectors never specify the routes in physics. They
include the information about the initial, terminal locations, and the distances between them, not
the actual routes. These displacements can be of points within space (one by one), of objects within
space (objects as collections of points), and of a space itself (as a collection of points) in physics as
well. Depending on the attachment of objects to a vector, and students’ understanding of points and
spaces, students’ interpretations can vary from ‘route’ of a displacement to a ‘displacement’ itself.
All the above interpretations about ‘displacement’ are helpful in certain mathematical situations,
although not all of them are with a mathematical rigor. To solve ‘Very long Sum’ questions, some
students regarded arrows connected in a ‘tip to toe’ way as routes of displacements that a particle
moves along. This enables students to perform the structure sum without thinking the intermediate
steps of series of binary operations. However, this is not mathematically true or even not physically
true. Students who uses only arrows as routes of particle movements for vector sums can misunderstand the vector sum when translated to symbolic representations. For example, there were
students who marked the final destination of the consecutive displacements in connected routes as
the answer for the vector sum. See Figure 5.5.

Figure 5.5: Misunderstanding of Arrows

This example shows (1) a higher level thinking does not always mean mathematically rigorous,
66

(2) a lower level thinking can help higher level thinking when it used carefully, and (3) directions
of the ontological and epistemological developments are not a reflection of strict hierarchy or levels
of sophistication.
Here is another example of (3) that does not support the levels of sophistication in the configuration. Bryant (1984) introduced geometric ideas of vectors and their statistical and probabilistic
analogs. In science and engineering, vectors depend heavily upon physical vectors with physical
meanings such as forces, velocities, moments, momentum, etc. However, this new application of
vectors emphasizes not only the efficiency or convenience of calculations but also the fundamental
idea behind the concept. The formula for the correlation coefficient looks very complicated and
hard to remember. However, once we regard data as vectors, the correlation coefficient is just the
cosine value of the angle between two arrows representing z − µz and x − µx for two data sets x
and z. See Figure 5.6.

Figure 5.6: Vectors that can explain the correlation concept (Bryant,1984)

This is one of the applications of vectors that shows the significance of ambiguity and connections in mathematics independent from physical meanings (Bryant, 1984). Students can learn that
arrows are not just for two- or three-dimensional vectors. However, this brings a serious reconsideration of common belief that geometric vectors are more concrete than symbolic vectors (Hillel,
2002). In Bryant (1984)’s example, two arrows connected together means any n-dimensional vec-

67

a

D

A, B, C

CD

b

C

b`+`a

b

a`+`b

tors. This is conflicting with what I described in the configuration that higher dimensional vectors

A

B

were more abstract and ware represented only as coordinate/column vector forms. The example

a

of associativity when we talked about the prevalence of the analytic approach to the synthetic ap-

proach also gives us a similar insight that geometric representations can be more abstract than
BC≥=AC≥, b ≤+a ≤=AD≥+DC≥=AC≥
symbolic representations. A proof using arrows gives one that works in any dimension. See Figure
5.7

D

c ≤=CD≥

c

a`+b`+`c

b`+`c

a ≤+( b ≤+c ≤ )

A
a

.

b ≤+a ≤

a`+b`
B

C
b

Figure 5.7: Associativity proved using arrows

These examples show that a dimension of a vector space is not actually an element that distinguishes the difference or levels of sophistication in the current form of refinements for the construct
of vectors. This implies Transition (E) in the configuration could be on a different axis. Further
refinements in the future would form more axes and transitions along the axes. For this possible extension of dimensions to the configuration, I rather see the new configuration as a map or snapshot
of multi-dimensional construct, not as a full developmental hierarchy or a simple two-dimensional

+c ≤=a with respect to multiple perspectives.
≤+(b ≤+c ≤ )

model for levels of sophistication. Further refinements should be able to detect more transitions

a + 0 ≤= 0 ≤+a ≤=a ≤
≤

68

5.3

Repeated or Reversed Transitions on the Configuration

The three transitions, (A) from physical to mathematical, (C) from analytic to synthetic, and (D)
from geometric to synthetic on the configuration were evident in student work when we saw ontological and epistemological obstacles in terms of transitions towards abstraction with the third
layer. When assuming the direction of the development towards abstraction, we never discussed
the possibility of the repetition of transitions and the reversed transitions in different contexts. In
Chapter 1, one of the difficulty that students could have on the construct of vectors with the classical vector representations was confusion between vectors-as-points and vectors-as-arrows (Hillel,
2002). This confusion is mostly related to Transition (D) from geometric to symbolic vectors on
the configuration as discussed earlier. However, a complete understanding of the arguments in
Hillel (2002) requires the interpretation of directions of each axis on the configuration because
the confusion is caused by reversing or repeating the developmental directions, that is, repeating
or reversing Transition (D). For example, representations of vectors for a real vector space Rn
used in linear algebra sometimes are not stemmed from physics or geometry. Moreover the use
of Cartesian coordinates is different from that in geometry. In linear algebra, symbolic representations of vectors were developed first and the Cartesian coordinates are an aid to translate the
language of symbolic representations of vectors to geometric language. ‘Point’ is another name of
n-tuple of numbers, not a geometric object. However, the direction towards mathematical abstraction implicitly and initially reflected historic development of vectors in mathematics curriculum
and corresponded to the student’s learning trajectory. The learning trajectory of vectors introduced
in high school mathematics is beginning with geometric representations of vectors. See Table 5.1.
This means Transition (D) from geometric to symbolic vectors can be reversed or repeated
in different contexts such as contexts before linear algebra or after linear algebra, with or without
bases, in Rn or in En , etc. Some students can succeed in linear algebra with the textbooks reflecting
this reversed transition and felt uncomfortable with geometric representations that assumed to be
easier in the configuration. This means a reversed or a repeated transition in a different context
could be easier than the original transition. The third layer of refinement basically follows the
69

Vectors in Linear Algebra
producing vectors-as-points,
vectors-as-arrows confusion
Symbolic vector (a1 , a2 ) in R2

Start

Corresponding Development
in the configuration emphasizing
Geometry
A directed line segment in E2
Position Vector in E2
Terminal point (a1 , a2 )
of position vector in E2
Corresponding (a1 , a2 ) in R2
with different coordinates for R2

Point (dot) in coordinates for R2

End

Point (dot) in R2
with new coordinates for R2
Coordinates for E2 and those for R2
are different.

Emphasis

Symbolic to Geometric

Geometric to Symbolic and then to Geometric

Table 5.1: Resolution of Vectors-as-Arrows vs. Vectors-as-Points Confusion

original direction of development towards abstraction. Further refinements of the construct are
required to unfold these repeated or reversed transitions for more careful description such as in
Table 5.1 of the construct of vectors for teaching and learning .

5.4

Epilogue

Collectively, the three progressive refinements that were described in this dissertation help push
the understanding of vectors in an important direction: They encourage students to see vectors as
a rich trove of intuitive and also abstract representations with complex connections with multiple perspectives, rather than an assemblage consisting merely of different symbols, notations and
procedures. They also suggest to teachers to prepare their own refinement needed in specific curriculum and instruction of vectors. It emphasizes the APOS cycles, and domain-specific instruction
of vectors such as vectors in physical world with physical motions; vectors in geometry with ab70

stract understanding of coordinate systems, points, and motions; and vectors in linear algebra with
a vector space Rn . The teacher and student should consider both ontological and epistemological
perspectives to understand the construct of vectors. They should also be careful with the learning
trajectories that can go backward from the direction of cognitive development in the configuration.
In the curriculum of school mathematics, in standards, and in assessment, connecting geometry
with the other content areas such as algebra and probability will be important more and more
(Bryant, 1984). Study of vector representations is therefore worth continuing. I want to initiate
studies on the analysis of student work with this new framework while continuing to modify and
refine this work.

71

APPENDIX

72

Appendix A
VECTOR REPRESENTATION SURVEY I

−
− −
v
1. Translation: A translation can be represented by a vector →. T→ (P) = P + → for any point
v
v
P.

Figure A.1: Translation

(a) List all vectors that do NOT represent the translation of triangle A to triangle B in the
figure.

−
(b) Circle all vectors that are equivalent to →.
a

73

2. A Rainy Day: The rain is falling straight down on your head with speed 1 m/s. You start
walking forward with speed of 2 m/s and notice the rain is now hitting your face. What is
the angle that the rain now appears to make with the vertical? Draw a picture that explains
your thinking.

74

3. Robot Arm: A robot arm has three joints as shown. The bottom part can be represented by
→
−
−
a vector → = (10, 15, 20), the middle part can be represented by a vector b = (5, −5, 15),
a
−
and the top part can be represented by → = (20, −5, −10). The units are in centimeters(cm).
c

3

c¯
b¯

b¯=(5 -5 15)

-5 -10)
a¯

d≤

O

Figure A.2: Robot Arm

A(a¡

−
Find the vector → that tells the location of the hand.
p

a™

a£) B(b¡

b™

b£)

AB≥=OB≥-OA≥
AB≥=(b¡ b™ b£)-(a¡ a™ a£)
AB≥=(b¡-a¡ b™-a™ b£-a£)

AB≥|="√(b¡-a¡)¤ +(b™-a™√)¤ +(b£-a£)¤

AB≥

) B(3 -1 6)

A(3 -1 2) B(0 5 75
-1)

4. Origin: For the three different origins: O, C, and A, complete the table by finding the posi−
→
tion vector of A and the vector AB.

Origin

−
→
AB

Position Vector of A

O

(

,

)

(

,

)

C

(

,

)

(

,

)

A

(

,

)

(

,

)

Figure A.3: Origin

76

− −
−
5. Non-standard Basis: In a standard basis of {→, →} for R2 , the vector → can be repree1 e2
u
− −
sented by the ordered pair (1, 2). The set {→, →} also forms a basis for R2 . Find an ordered
e u
1

−
− −
pair (a, b) for the vector → using the basis {→, →}.
v
e1 u

Figure A.4: Non-standard Basis

77

− − −
6. Rotation: Which, if any, of the following vectors →, →, → is the result of “the 90◦ rotation
v w z
−
about the origin” of →? You may or may not have multiple answers. Give an explanation of
u
your thinking.

Figure A.5: Rotation

78

7. Polygons: For each polygon and drawn vectors, label each vector and write an equation that
can describe the relations among them. You may reuse labels.

Figure A.6: Polygons

79

− −
− −
8. A Very Long Sum: Find → + → + . . . + → − →.
u1 u2
un v

→
−
u3

→
−
u2
→
−
u1

→
−
u4

→
−
v
→
−
un

Figure A.7: A Very Long Sum

80

9. Cube: Prove that the two vectors shown in the cube are perpendicular. The sides of the cube
can be assumed to be of length 1. Hint: You may wish to show something about the inner
product also known as the dot product.

Figure A.8: Cube

81

10. Triangle Midpoints: In the following triangle

ABC, AB = 2AD and AC = 2AE. We want

to show that BC is parallel to DE.

Figure A.9: Triangle Midpoints

(a) Which form of vectors, do you think will be most useful? Circle your answer.

(i) (a, b)



 x 
(ii)  
y

−
→
(iii) PQ

(b) What vectors might be useful to show BC DE?

(c) What would you try to show to get that BC DE?

82

−
(iv) →
u

(v) Other:

11. Associativity: We want to prove the following property of vector operation:
−
→ −
−
− → − −
(→ + b ) + → = → + ( b + →).
a
c
a
c

Which form of vectors, do you think will be most useful? Circle your answer. You don’t
actually need to prove this.

(i) (a, b)



 x 
(ii)  
y

(iv) Other:

83

(iii) directed line segments

12. Point/Vector: For each case, find the midpoint M of AB using vectors. Explain with your
vector notation.

Figure A.10: Point/Vector

84

Appendix B
VECTOR REPRESENTATION SURVEY II

1. Majors: You are a (i) Mathematics major with secondary education focus,
(ii) Mathematics major with elementary education focus,

(iii) Mathematics minor,

.

(iv) Other:

2. High School: In which high school courses did you see vectors? Select all that apply.
(i) Physics

(ii) Algebra I

(v) Trigonometry

(iii) Algebra II

(iv) Geometry

(vi) Precalculus

(viii) Integrated Mathematics I, II, III, IV

(vii) Calculus

(ix) Other:

3. College: In which college courses did you see vectors? Select all that apply.
(i) Physics

(ii) Linear Algebra

(iv) Higher Geometry
(vii) Differential Equations

(iii) Multivariable Calculus

(v) Abstract Algebra

(vi) Real Analysis

(viii) Complex Analysis

(ix) Axiomatic Geometry

(x) Other:

4. Concept Images: How would you describe what a vector is to a high school student?

5. Scale: How important are vectors for high school mathematics?
Unimportant

Of Little Importance

Moderate Important

6. Scale: How can you describe your understanding of vectors?

85

Important

Very Important

Very Poor

Poor

Barely Acceptable

Good

Very Good

− −
−
7. Translation: A translation can be represented by a vector →. T→ (P) = P + → for any point
v
v
v
P.

Figure B.1: Translation

(a) List all vectors that do NOT represent the translation of triangle A to triangle B in the
figure.

−
(b) Circle all vectors that are equivalent to →.
a

(a) List all vectors that do NOT represent the translation of triangle A to triangle B in the
figure.

86

−
(b) List all vectors that are equivalent to →.
a

87

8. Force: The following vectors represent forces applied on a physical object. A force ‘a’
pushes the tip of the block.

Figure B.2: Force

(a) List all that would produce the same result.

(b) List all vectors that are equivalent to ‘a’.

(c) List all forces that are equivalent to ‘a’.

88

9. Robot Arm: A robot arm has three joints as shown. The bottom part can be represented by
→
−
−
a vector → = (10, 15, 20), the middle part can be represented by a vector b = (5, −5, 15),
a
−
and the top part can be represented by → = (20, −5, −10). The units are in centimeters(cm).
c

3

c¯
b¯

b¯=(5 -5 15)

-5 -10)
a¯

d≤

O

Figure B.3: Robot Arm

A(a¡

−
Find the vector → that tells the location of the hand.
p

a™

a£) B(b¡

b™

b£)

AB≥=OB≥-OA≥
AB≥=(b¡ b™ b£)-(a¡ a™ a£)
AB≥=(b¡-a¡ b™-a™ b£-a£)

AB≥|="√(b¡-a¡)¤ +(b™-a™√)¤ +(b£-a£)¤

AB≥

) B(3 -1 6)

A(3 -1 2) B(0 5 89
-1)

10. Origin: For the three different origins: O, C, and A, complete the table by finding the posi−
→
tion vector of A and the vector AB.

Origin

−
→
AB

Position Vector of A

O

(

,

)

(

,

)

C

(

,

)

(

,

)

A

(

,

)

(

,

)

Figure B.4: Origin

90

− −
−
11. Non-standard Basis: In a standard basis of {→, →} for R2 , the vector → can be repree1 e2
u
− −
sented by the ordered pair (1, 2). The set {→, →} also forms a basis for R2 . Find an ordered
e u
1

−
− −
pair (a, b) for the vector → using the basis {→, →}.
v
e1 u

Figure B.5: Non-standard Basis

91

− − −
12. Rotation: Which, if any, of the following vectors →, →, → is the result of “the 90◦ rotation
v w z
−
about the origin” of →? You may or may not have multiple answers. Give an explanation of
u
your thinking.

Figure B.6: Rotation

92

13. Polygons: For each polygon and drawn vectors, label each vector and write an equation that
can describe the relations among them. You may reuse labels.

Figure B.7: Polygons

93

− −
− −
14. A Very Long Sum: Find → + → + . . . + → − →.
u1 u2
un v

→
−
u3

→
−
u2
→
−
u1

→
−
u4

→
−
v
→
−
un

Figure B.8: A Very Long Sum

94

15. Cube: Prove that the two vectors shown in the cube are perpendicular. The sides of the cube
can be assumed to be of length 1. Hint: You may wish to show something about the inner
product also known as the dot product.

Figure B.9: Cube

95

16. Triangle Midpoints: In the following triangle

ABC, AB = 2AD and AC = 2AE. We want

to show that BC is parallel to DE.

Figure B.10: Triangle Midpoints

(a) Which form of vectors, do you think will be most useful? Circle your answer.

(i) (a, b)



 x 
(ii)  
y

−
→
(iii) PQ

−
(iv) →
u

(v) Other:

(b) List two vectors in any form that might be useful so that you would want to use.

(c) What would you try to show to get that BC DE?

96

17. Associativity: We want to prove the following property of vector addition:
−
→ −
−
− → − −
(→ + b ) + → = → + ( b + →).
a
c
a
c
(a) Which form of vectors, do you think will be most useful? Circle your answer. You
don’t actually need to prove this.

(i) (a, b)



 x 
(ii)  
y

(iii) directed line segments

(iv) Other:

(b) Why have you preferred this representation and not another one?

97

18. Point/Vector: For each case, find the midpoint M of AB using vectors. Explain with your
vector notation.

Figure B.11: Point/Vector

98

Appendix C
VECTOR REPRESENTATION SURVEY III

Student Understand of Geometric Translation
Translation from CMP
1. The diagram below shows a polygon and its image under a translation. Draw a line segment
from each vertex of polygon to its image. Describe the relationship among the line segments you
drew.

Figure C.1: Polygon Diagram

2. Complete this definition of a translation: A translation matches any two points X and Y on a
figure to image points X and Y so that...

3. Ella specified a translation of a polygon by drawing an arrow. How would you interpret Ella’s
drawing? Perform the translation Ella specified.

99

Figure C.2: Translation of Polygon

4. When you translate a figure, you can imagine that you are translating the entire plane and brining
the figure along for the ride. When you perform a translation, are any points in the plane in the
same location after the translation? That is, are there any fixed points? Explain your answer.

100

Translation from Beckmann
1. A translation (or slide) of a plane by a given distance in a given direction is the end result of
moving each point in the plane the given distance in the given direction. The solid arrow represents
the distance and direction of the translation. In the following figure, triangle B is the image of
triangle A under a given translation.

Figure C.3: Translation

(a) List all arrows that do NOT represent the translation that was specified above.

−
(b) List all arrows that represent the same translation with the arrow →.
a

101

2. Circle one that describes the image of a translation by an arrow in the following figure.

,

, None of the left.(Explain or draw your own
answer.)
Figure C.4: Geometric Translation

102

Appendix D
VECTOR REPRESENTATION SURVEY IV

1. Majors: You are a (i) Mathematics major with secondary education focus,
(ii) Mathematics major with elementary education focus,

(iii) Mathematics minor,

.

(iv) Other:

2. High School: In which high school courses did you see vectors? Select all that apply.
(i) Physics

(ii) Algebra I

(v) Trigonometry

(iii) Algebra II

(iv) Geometry

(vi) Precalculus

(viii) Integrated Mathematics I, II, III, IV

(vii) Calculus

(ix) Other:

3. College: In which college courses did you see vectors? Select all that apply.
(i) Physics

(ii) Linear Algebra (309)

(iv) Higher Geometry (330)

(iii) Multivariable Calculus (133, 234)

(v) Abstract Algebra (310)

(vii) Differential Equations (235, 340)

(vi) Real Analysis (320)

(viii) Complex Analysis (425)

(ix) Axiomatic

Geometry (432)
(x) Other:
4. Concept Images: How would you describe what a vector is to your future students?
5. Scale: How important are vectors for school mathematics?
Unimportant

Of Little Importance

Moderate Important

Important

Very Important

6. Scale: How can you describe your understanding of vectors?
Very Poor

Poor

Barely Acceptable

103

Good

Very Good

−
− −
v
7. Translation: A translation can be represented by a vector →. T→ (P) = P + → for any point
v
v
P.

Figure D.1: Translation

(a) List all vectors that do NOT represent the translation that maps triangle A to triangle B
in the figure.

−
(b) List all vectors that are equivalent to →.
a

104

8. Triangle: Assume you have a triangle

ABC. Find each of the following.

− −
→ →
(a) AB + BC
− −
→ → −
→
(b) AB + BC + CA

9. Rectangle: Assume you have a rectangle

ABCD. Find each of the following.

− −
→ →
(a) AB + CB
− −
→ → −
→
(b) AB + BC + CA
− −
→ → −
→ −
→
(c) AB + BC + CD + DA

105

10. Translation of Polygon: Ella specified a translation of a polygon by drawing an arrow. How
would you interpret Ella’s drawing? Perform the translation Ella specified.

Figure D.2: Translation of Polygon

106

11. Geometric Translation: Circle one that describes the image of a translation by an arrow in
the following figure.

(1)

,

(2)

(3) None of the above. (Explain or draw your own answer.)
Figure D.3: Geometric Translation

107

,

12. Force: The following vectors represent forces applied on a physical object. A force ‘a’
pushes the tip of the block.

Figure D.4: Force

(a) List all vectors that are equivalent to ‘a’.

(b) List all forces that are equivalent to ‘a’.

108

13. Origin: For the three different origins: O, C, and A, complete the table by finding the posi−
→
tion vector of A and the vector AB.

Origin

−
→
AB

Position Vector of A

O

(

,

)

(

,

)

C

(

,

)

(

,

)

A

(

,

)

(

,

)

Figure D.5: Origin

109

14. Polygons: For each polygon and drawn vectors, label each vector and write an equation that
can describe the relations among them. You may reuse labels.

Figure D.6: Polygons

110

− −
−
15. A Very Long Sum: Find → + → + . . . + →.
u1 u2
un

→
−
u3

→
−
u2
→
−
u1

→
−
u4

→
−
un

Figure D.7: A Very Long Sum

111

16. Cube: In the following figure, we want to show that the two vectors shown in the cube are
perpendicular using the inner product also known as the dot product. The sides of the cube
can be assumed to be of length 1.

Figure D.8: Cube

(a) Which form of vectors, do you think will be most useful? Circle your answer.

(i) (a, b, c)



 x 
 
(ii)  y 
 
 
z

−
→
(iii) PQ

−
(iv) →
u

(v) Other:

(b) Which form of vectors would be the next choice if the form you choose in (a) does not
work?

112

17. Triangle Midpoints: In the following triangle

ABC, AB = 2AD and AC = 2AE. We want

to show that BC is parallel to DE.

Figure D.9: Triangle Midpoints

(a) Which form of vectors, do you think will be most useful? Circle your answer.

(i) (a, b)



 x 
(ii)  
y

−
→
(iii) PQ

−
(iv) →
u

(v) Other:

(b) Which form of vectors would be the next choice if the form you choose in (a) does not
work?

(c) List two vectors in any form that might be useful so that you would want to use.

(d) What would you try to show to get that BC DE?

113

18. Associativity: We want to prove the following property of vector addition:
−
→ −
−
− → − −
(→ + b ) + → = → + ( b + →).
a
c
a
c
(a) Which form of vectors, do you think will be most useful? Circle your answer. You
don’t actually need to prove this.

(i) (a, b)



 x 
(ii)  
y

−
→
(iii) PQ

−
(iv) →
u

(v) Other:

(b) Why have you preferred this representation and not another one?

114

19. Point/Vector: For each case, find the midpoint M of AB using vectors. Explain with your
vector notation.

Figure D.10: Point/Vector

115

Appendix E
ADDITIONAL INTERVIEW QUESTIONS

1. Triangle: Assume that you have a triangle

ABC. Find each of the following. Explain.

− −
→ →
(a) AB + BC
− −
→ → −
→
(b) AB + BC + CA
2. Vector Space: Only one of the following statements is true. Which one? Explain.
(i) The vector space V = {0} consisting of a zero vector has the basis {0}.
(ii) The vector space V = {0} consisting of a zero vector has the basis an empty set 0.
/
(iii) The vector space V = {0} consisting of a zero vector has no basis.
3. Triangle Midpoints: In the following triangle

ABC, AB = 2AD and AC = 2AE. We want

to show that BC is parallel to DE.

Figure E.1: Triangle Midpoints

(a) Which  of vectors, do you think will be most useful? Circle your answer. • (a, b)
 form
−
→
 x 
−
• 
• PQ
•→
u
• Other:
y
116

(b) Using your vector form, what would you try to show to get that BC DE?
(c) Proceed your proof with the preferred vector representation.
(d) Why have you preferred this representation and not another one?
(e) Prove with the other vector representation that you did not use if you can.
(f) What difficulties are you faced with, for each method? Namely, what forced you to
abandon a representation, if anything like that has happened, or what has prevented
you in reaching a final proof?

4. Associativity: We want to prove the following property of vector operation:
−
→ −
−
− → − −
(→ + b ) + → = → + ( b + →).
a
c
a
c
(a) Which form of vectors, do you think will be most useful? Circle your answer.




 x 
• (a, b) •   • directed line segments • Other:
y
(b) Proceed your proof with the preferred vector representation.
(c) Why have you preferred this representation and not another one?
(d) Prove with the other vector representation that you did not use if you can.
(e) What dimension are you working on with your vector representation?
(f) What difficulties are you faced with, for each method? Namely, what forced you to
abandon a representation, if anything like that has happened, or what has prevented
you in reaching a final proof?

117

g¯=

G

a¯+b¯+c¯
3

5. Point vs.Vector: Prove that the midpoints of the two diagonals in the figure coincide using
vectors.

AG

DF

b¯

ABC

a¯ b¯ c¯

D

C

A

B
H

G

E

F

Figure E.2: Point vs. Vector

(a) Which vector representations do you want to use to prove the theorem, coordinate/column
 

a¯

 x 
− −
→
b¯
3 BA≥ c),  
b¯
vector forms such as (a, b,a¯  y  or generic vector forms such as AB, →?
u
 
 
z

(b) Proceed your proof with the preferred vector representation.
(c) Why have you preferred this representation and not another one?
a¯ b¯

A

3:1

(d) Prove with the other vector representation that you did not use if you can.
P

a¯
(e) What difficulties are you faced with, for each method? Namely, what forced you to
P
abandon a representation, if p¯
anything like that has happened, or what has prevented
B

you in reaching O final proof?
a

b¯

118

6. Additional Questions: Any questions from test that the researcher may want to revisit with
selected students will be included in addition.

119

BIBLIOGRAPHY

120

BIBLIOGRAPHY

Aguirre, J. and Erickson, G. (1984). Students’ conceptions about the vector characteristics of three
physics concepts. Journal of Research in Science Teaching, 21(5):439–457.
Aguirre, J. M. (1988). Student preconceptions about vector kinematics. The Physics Teacher,
26(4):212–216.
Artigue, M., Chartier, G., and Orier, J.-L. (2002). Presentation of other research works. In Dorier,
J.-L., editor, On the Teaching of Linear Algebra, volume 23 of Mathematics Education Library,
pages 247–271. Springer Netherlands. 10.1007/0-306-47224-49 .
Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., Thomas, K., Kaput, J., Schoenfeld, A. H., and Dubinsky, E. (1996). A framework for research and curriculum development in
undergraduate mathematics education. Research in collegiate mathematics education II, 1:35.
Beckmann, S. (2005). Mathematics for elementary teachers. Addison-Wesley.
Brousseau, G., Balacheff, N., Cooper, M., Sutherland, R., and Warfield, V. (1997). Theory of
didactical situations in mathematics: Didactique des mathématiques, 1970-1990. Kluwer academic publishers.
Bryant, P. (1984). Geometry, statistics, probability: Variations on a common theme. The American
Statistician, 38(1):38–48. ArticleType: research-article / Full publication date: Feb., 1984 /
Copyright c 1984 American Statistical Association.
CCSSM (2010). Common core state standards for mathematics. Retrieved June, 2:2010.
Chevallard, Y. (1988). On didactic transposition theory: Some introductory notes. In International
Symposium on Research and Development in Mathematics, Bratislava, Czechoslavakia.
Coelho, R. L. (2010). On the concept of force: How understanding its history can improve physics
teaching. Science & Education, 19(1):91–113.
Dorier, J. (2002). On the Teaching of Linear Algebra.
Dorier, J. and Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. The
Teaching and Learning of Mathematics at University Level, pages 255–273.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. Advanced mathematical thinking, pages 95–126.
Dubinsky, E. and Tall, D. (1991). Advanced mathematical thinking and the computer. Advanced
mathematical thinking, pages 231–248.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
Educational Studies in Mathematics, 61(1):103–131.

121

Edwards, B. S., Dubinsky, E., and McDonald, M. A. (2005). Advanced mathematical thinking.
Mathematical Thinking and Learning, 7(1):15.
Flores, S., Kanim, S. E., and Kautz, C. H. (2004). Student use of vectors in introductory mechanics.
American Journal of Physics, 72(4):460.
Forster, P. (2000). Process and object interpretations of vector magnitude mediated by use of the
graphics calculator. Mathematics Education Research Journal, 12(3):269–285.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Springer.
Goldenberg, E. P. (1995). Multiple representations: A vehicle for understanding understanding.
Software goes to school: Teaching for understanding with new technologies, page 155–171.
Gray, E. and Tall, D. (1993). Success and failure in mathematics: the flexible meaning of symbols
as process and concept. Mathematics Teaching, 142:6–10.
Gray, E. and Tall, D. (2001). Relationships between embodied objects and symbolic procepts: an
explanatory theory of success and failure in mathematics. In PME CONFERENCE, volume 3,
pages 3–65. Citeseer.
Gueudet, G. (2004). Geometric thinking in a n-space.
Gueudet-Chartier, G. (2004). Should we teach linear algebra through geometry? Linear algebra
and its applications, 379:491–501.
Gueudet-Chartier, G. (2006). Using geometry to teach and learn linear algebra. Research in
Collegiate Mathematics Education, 6:171–195.
Harel, G. (1989). Learning and teaching linear algebra: Difficulties and an alternative approach to
visualizing concepts and processes. Focus on Learning Problems in Mathematics.
Harel, G. and Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its
development. Mathematical Thinking and Learning, 7(1):27–50.
Heid, M. K., Harel, G., Ferrini-Mundy, J., and Graham, K. (2000). Advanced mathematical thinking: Implications of various perspectives on advanced mathematical thinking for mathematics
education reform. In Proceedings of the 22nd Annual Meeting of the North American Chapter of
the International Group for the Psychology of Mathematics Education, volume 1, pages 33–37.
Heller, P. and Huffman, D. (1995). Interpreting the force concept inventory. A Reply to Hestenes
and Halloun–In: The Physics Teacher, 33:507–511.
Hestenes, D., Wells, M., and Swackhamer, G. (1992). Force concept inventory. The physics
teacher, 30(3):141–151.
Hillel, J. (2002). Modes of description and the problem of representation in linear algebra. In On
the Teaching of Linear Algebra, pages 191–207.
Janvier, C. (1987). Problems of representation in the teaching and learning of mathematics. L.
Erlbaum Associates.
122

Janvier, C., Girardon, C., and Morand, J. C. (1993). Mathematical symbols and representations.
High school mathematics, page 79.
Johnson, S. (1998). What’s in a representation, why do we care, and what does it mean? examining
evidence from psychology. Automation in Construction, 8(1):15–24.
Kaput, J. J. (1987a). Representation systems and mathematics. Problems of representation in the
teaching and learning of mathematics, pages 19–26.
Kaput, J. J. (1987b). Towards a theory of symbol use in mathematics. Problems of representation
in the teaching and learning of mathematics, pages 159–195.
Keller, B. A. and Hirsch, C. R. (1998). Student preferences for representations of functions. International Journal of Mathematical Education in Science and Technology, 29(1):1–17.
Knight, R. D. (1995). The vector knowledge of beginning physics students. Physics Teacher,
33:74–74.
Kwon, O. H. (2011). Student approaches and difficulties in understanding and use of vectors.
In Brown, S., Larsen, S., Marrongelle, K., and Oertman, M., editors, Proceedings of the 14th
Annual Conference on Research in Undergraduate Mathematics Education, volume 1, pages
187 – 202. SIGMMA RUME.
Lakoff, G. and Nunez, R. (2001). Where Mathematics Comes from: How the Embodied Mind
Brings Mathematics into Being. Basic Books, reprint edition.
Lesh, R., Post, T., and Behr, M. (1987). Representations and translations among representations
in mathematics learning and problem solving. Problems of representation in the teaching and
learning of mathematics, pages 33–40.
Meissner, H. (2001a). Encapsulation of a process in geometry. PME CONFERENCE, 3:3–359.
Meissner, H. (2001b). Procepts in Geometry. European Research in Mathematics Education II,
page 58.
Meissner, H., Tall, D., et al. (2006). Proceptual Thinking in Geometry. RETIREMENT AS PROCESS AND CONCEPT, page 165.
NCTM (1989). Curriculum and evaluation standards for school mathematics. NCTM.
NCTM (2000). Principles and standards for school mathematics. Reston, Va.: National Council of
Teachers of Mathematics.
Nguyen, N.-L. and Meltzer, D. E. (2003). Initial understanding of vector concepts among students
in introductory physics courses. American Journal of Physics, 71(6):630–638.
Rasmussen, C., Zandieh, M., King, K., and Teppo, A. (2005). Advancing mathematical activity: A
practice-oriented view of advanced mathematical thinking. Mathematical thinking and learning,
7(1):51–73.

123

Rider, R. L. (2004). The Effects of Multi-Representational Methods on Students’ Knowledge of
Function Concepts in Developmental College Mathematics. PhD thesis.
Robert, A. (2002). Level of conceptualization and secondary school math education. In On the
Teaching of Linear Algebra, pages 125–131.
Savinainen, A. and Scott, P. (2002). The force concept inventory. Physics Education, 37:45–52.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and
objects as different sides of the same coin. Educational studies in mathematics, 22(1):1–36.
Sfard, A. and Linchevski, L. (1994). The gains and the pitfalls of reification—The case of algebra.
Educational studies in mathematics, 26(2):191–228.
Sfard, A. and Thompson, P. (1994). Problems of reification: Representations and mathematical
objects. In Proceedings of the XVI Annual Meeting of the International Group for the Psychology
of Mathematics Education–North America, pages 1–32.
Sierpinska, A. (2002). On some aspects of students’ thinking in linear algebra. In On the Teaching
of Linear Algebra, pages 209–246.
Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and
proof. Handbook of research on mathematics teaching and learning, pages 495–5.
Tall, D. (2004a). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24(1):29–32. ArticleType: research-article / Full publication date: Mar., 2004 / Copyright c 2004 FLM Publishing Association.
Tall, D. (2004b). Thinking through three worlds of mathematics. In Proceedings of the 28th
Conference of the International, volume 4, pages 281–288.
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2):5–24.
Tall, D., Thomas, M., Davis, G., Gray, E., and Simpson, A. (1999). What is the object of the
encapsulation of a process? The Journal of Mathematical Behavior, 18(2):223–241.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra,
K-12, 8:19.
Watson, A. (2004). Effect as a pivot between actions and symbols: the case of vector. PhD thesis,
University of Warwick.
Watson, A., Spyrou, P., and Tall, D. O. (2003). The relationship between physical embodiment and
mathematical symbolism: The concept of vector. The Mediterranean Journal of Mathematics
Education, 1(2):73–97.
Watson, A. and Tall, D. (2002). Embodied action, effect and symbol in mathematical growth. In
PME CONFERENCE, volume 4, pages 4–369.

124

Yerushalmy, M. (1997). Designing representations: Reasoning about functions of two variables.
Journal for Research in Mathematics Education, 28(4):431–466.
Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept
of derivative. Research in collegiate mathematics education IV, 8:103–126.

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