Mathematical Thinking: from Cacophony to Consensus

MATHEMATICAL THINKING: FROM CACOPHONY TO CONSENSUS

A dissertation submitted to the Kent State University College and Graduate School of Education, Health, and Human Services in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Sean F. Argyle

August 2012

2012 by Sean F. Argyle Some Rights Reserved http://creativecommons.org/licenses/by-nc-nd/3.0/

A dissertation written by

Sean F. Argyle

B.S., Slippery Rock University, 2005

M.Ed., Slippery Rock University, 2006

Ph.D., Kent State University, 2012

Approved by

______, Co-director, Doctoral Dissertation Committee Michael G. Mikusa

______, Co-director, Doctoral Dissertation Committee Lisa A. Donnelly

______, Member, Doctoral Dissertation Committee Natasha Levinson

Accepted by

______, Director, School of Teaching, Learning Alexa Sandmann and Curriculum Studies

______, Dean, College of Education, Health and Human Services Daniel F. Mahony

ARGYLE, SEAN F., Ph.D., August 2012 Curriculum and Instruction

MATHEMATICAL THINKING: FROM CACOPHONY TO CONSENSUS (199 pp.)

Co-Directors of Dissertation: Michael G. Mikusa, Ph.D. Lisa A. Donnelly, Ph.D.

Various standards have demanded that teachers improve “mathematical thinking,” but definitions are vague – if present at all. What little research on the subject exists is disjointed and dissenting, leading some researchers to lament the possibility of ever coming to an agreement on how to define “mathematical thinking” as a viable construct.

Rather than add one more voice into the cacophony of competing definitions, this dissertation seeks to discuss the results of a conceptual meta-analysis of the term’s use in an appropriately titled journal – Mathematical Thinking and Learning.

ACKNOWLEDGEMENTS

I am grateful to Valerie, who proofread so much of this paper. Surely, it would have been far muddier without her advice. I am, furthermore, indebted to my mother for her

continued modeling of how to proceed through adversity.

TABLE OF CONTENTS

Heading Pg.

ACKNOWLEDGEMENTS...... iv

LIST OF FIGURES...... vii

CHAPTER I: INTRODUCTION...... 1 Guiding Research Questions...... 8 Methodology...... 9 Conceptual Analysis...... 12 Data Analysis...... 16 Procedure...... 21

CHAPTER II: THE MODEL...... 22 “Mathematical Thinking” or “Thinking About Mathematics” ...... 22 Four Views of Mathematical Thinking...... 24 Think about mathematics...... 24 Think when doing mathematics...... 25 Habits of mind...... 26 Think mathematically...... 27 Synthesis of the Four Views...... 27 1. Mathematical thinking could potentially apply to almost anything. . . . 28 2. Mathematical thinking is implicitly tied to the process of mathematics. 30 3. Mathematical thinking requires habituation...... 34 4. Mathematical thinking operates on and outside generalized thinking. . 37 An Overview of the Model...... 39

CHAPTER III: EVERYDAY EXPERIENCE...... 46 Abstraction...... 51 Summary...... 54 Generalization...... 55 Overgeneralization...... 60 Summary...... 65 Reification...... 66 Summary...... 68 Chapter Summary...... 69

TABLE OF CONTENTS (CONT.)

CHAPTER IV: MATHEMATICAL WORLD...... 70 Mathematization...... 72 Summary...... 81 Justification...... 81 Summary...... 87 Chapter Summary...... 88

CHAPTER V: MATHEMATICAL COMMUNITY...... 89 Normalization...... 95 Summary...... 108 Contribution...... 109 Summary...... 114 Chapter Summary...... 115

CHAPTER VI: MATHEMATICAL DISPOSITION...... 116 Internalization...... 119 Summary...... 123 Intuition...... 124 Summary...... 127 Motivation...... 128 Summary...... 132 Aesthetic Evaluation...... 133 Summary...... 135 Chapter Summary...... 136

CHAPTER VII: SENSE-MAKING...... 137 Interpretation...... 141 Representation...... 145 Organization...... 149 Chapter Summary...... 152

CHAPTER VIII: PRACTICAL IMPLICATIONS...... 153 Limitations...... 159 Discussion...... 162 Statistical Thinking...... 163 Curriculum & Instruction...... 169 Conclusion...... 179

REFERENCES...... 180

LIST OF FIGURES Figure Pg.

1. Word Cloud...... 11

2. Model Research Methodologies...... 12

3. Article Category Percentages...... 20

4. Annual Category Counts...... 20

5. A Meta-analytic Model of Mathematical Thinking...... 40

6. Everyday Experience...... 46

7. A Common Cycle of Mathematical Thinking...... 50

8. The Mathematical World...... 70

9. Bonotto’s Mathematization...... 77

10. Rasmussen’s Mathematization...... 77

11. Mathematical Community...... 89

12. A Mathematical Disposition...... 116

13. The Central Node...... 137

14. Sense-making, a Swirl of Information...... 141

15. Cultural Preferences...... 177

vii

CHAPTER I: INTRODUCTION

The genesis of mathematical discovery is a problem which must

inspire the psychologist with the keenest interest. – Henri Poincare

Consider the whole numbers for a moment. A brief reflection might allow the

recognition of the following as properly formatted whole numbers: 0, 1, 7, 17, 70 and

107. With bit more time for experimentation, one might produce the following “whole numbers”: 00, 01, 007, 0017, 070 and 000107 – only to be informed that the members of the second set are improperly formatted. From just these twelve specific examples, one might correctly conjecture the following rule: unless the number is zero, no whole number starts with zero. How is it that the human mind can produce a rule that applies to an infinite collection of mathematical objects from so few examples? Moreover, how does the mind develop such intuitive certainty of the rule in so short a time?

One phrase that has been used to describe this phenomenon and others like it is mathematical thinking. In fact, the Principles and Standards for School Mathematics

(National Council of Teachers of Mathematics, 2000), which has guided American public mathematics education for a decade, uses the phrase several times – highlighting its practical value (p. 4), its place in the learning process (p. 19) and as the primary benefit of certain teaching strategies (p. 359). However, the Principles and Standards for School

Mathematics only defines the term loosely as including, “making conjectures and developing sound deductive arguments” (p. 15). Using the phrase “mathematical thinkers” rather than “mathematical thinking”, the more recently released draft of the

1 2

Common Core Standards (“Common core,” 2010) contains a more detailed definition of

mathematical thinkers as people who:

Make sense of problems and persevere in solving them… Reason abstractly and

quantitatively… Construct viable arguments and critique the reasoning of

others… Model with mathematics… Use appropriate tools strategically… Attend

to precision…Look for and make use of structure…Look for and express

regularity in repeated reasoning (pp. 4-5).

While not inappropriate, both definitions seem to be arbitrarily chosen lists of characteristics that are just as ambiguous in definition. After all, many other disciplines must require practitioners to make conjectures or use appropriate tools. How then, do these listed traits specifically describe mathematical thinking? Additionally, even if these traits are solely the province of mathematics, there is no guarantee that the list is not redundant or overlooking large portions of the process of mathematical thinking.

Similarly, even mathematics education researchers have been known to publish divergent uses of the term “mathematical thinking,” sometimes even failing to define the term at all. For Instance, Silver (2003, as cited in Selden & Selden, 2005) commented that “advanced mathematical thinking” tends to be defined as “whatever the author chooses it to mean” (p. 4). For example, many of those researching mathematical thinking discuss it as if it were a specialized function of the brain. As a case in point,

Schoenfeld’s (1992) position is that “thinking mathematically” involves adopting a certain frame of reference and not simple numeracy (p. 335). At the other extreme,

Sriraman (2009) remarks, “none of the theorists in HEP [Handbook of Educational

Psychology] are willing to explicitly define what thinking is” (p. 179). In other words,

others simply use the phrase as if it were a generic cognitive process that just so happens

to be ascribed to mathematical objects at the time. An example of this would be an

article by Doerr (2006) which never explicitly defines mathematical thinking, although it seems that the implicit definition is any on-task thinking. If the latter case is true, the phrase in question should, perhaps, be replaced by “thinking about mathematics.”

Even if mathematical thinking is a specialized cognitive process, it may not be a universal process or even one that can be directly taught. An example of this might be a divergence between “algebraic thinking” and “geometric thinking.” In fact, an entire issue of the journal Mathematical Thinking and Learning was devoted to “statistical thinking” (Greer, 2000). It is unclear whether these are merely subsets of the more inclusive term, “mathematical thinking,” or if they have only been clustered together through curricular tradition rather than through deeper correspondence. Moreover, mathematical thinking might be utterly idiosyncratic with no strong relationship between any individual teacher and student. In fact, this would be necessarily true if, as Sfard

(1991) postulates, advanced mathematical thinking is “totally inaccessible to our senses”

(p. 3). To put it another way, Sfard (1991) suggests, akin to the Platonic view, that the pinnacle of mathematical thought is constrained to a mathematical world that is related to – though disjoint from – the world in which humans live. Relying on this separatist stance, the penultimate stages of mathematical thinking would, by definition, be disjoint from any sensory impression caused by a teacher lest pedagogy somehow be part of the mathematical world. Assuming this worst-case scenario, research into “mathematical

thinking” could be no more successful than research into Wittgenstein’s famous beetle1.

By Sfard’s (1991) definition, educational research would be unable to penetrate the

mysteries of mathematical thinking by dint of not having direct access to the

mathematical world.

As another potential obstacle to a coherent construct of “mathematical thinking”,

one could consider an international comparison of mathematics performance. Given the

difference in their respective educational systems, Chinese mathematical thinking, for

example, might be distinct from American mathematical thinking. Growing interest in

critical mathematics (Diversity in Mathematics Education Center for Learning and

Teaching, 2007) as well as ethnomathematics (Presmeg, 2007), lends credibility to the

possibility that the incumbent view of mathematical thinking as universal is the result of

ethnocentric monoculturalism. Tall (1991) opines, “There is not one true, absolute way of thinking about mathematics, but diverse culturally developed ways of thinking in which various aspects are relative to the context” (p. 6). If this is the case, the construct worth studying might not be a person’s “mathematical thinking,” which is in this case would be largely a derivative, but rather a person’s “mathematical culture,” which would, in this case, the more independent variable. As a result, the line if inquiry would be better served by sociological research rather than psychological. That is to say, given Tall’s

(1991) perspective, the secret to understanding “mathematical thinking” lies not in study of the individual but in the study of groups of people.

1 Wittgenstein (2003) proposes a thought experiment regarding a world in which every person has a “beetle” in a box, but is forbidden from seeing anyone else’s beetle but his own. Regardless of whether a person’s “beetle” is a frog or a marble or nothing at all, Wittgenstein contends that the word “beetle” merely means “what’s in a person’s box” and nothing more meaningful or explanatory (p. 127).

Another point of contention related to matters of public education is who is capable of mathematical thinking, and to what extent. In other words, do the performance gaps between demographic groups, such as race, affluence or sex, stem from underlying differences in cognition? For example, Gorard and Smith (2004) suggest that standardized testing scores often skew against minority groups. Selden and Selden (2005) muse, “From initial considerations thereof, the term advanced mathematical thinking

[emphasis sic] has been fraught with ambiguity – does the term advanced [emphasis sic] refer to the mathematics, or to the thinking, or to both” (p. 2). Schoenfeld (1992) explains a closely related controversy, in that there appears to be two conceptions of the term “problem solving” in common use, and the dichotomy revolves around whether to apply the term to all mathematical tasks or only challenging ones (p. 337). In this vein, van Oers (2010) makes the claim that:

[W]e actually cannot maintain that very young children (1 to 3 years old) perform

mathematical actions, even when they may carry out actions that we, as

encultured adults, may recognize as mathematical. As long as these actions are

not intentionally and reflectively carried out, we cannot say that children perform

mathematical [emphasis sic] action (p. 28).

Certainly, even Piaget (2000) is hesitant to dub very young children sapient. Such doubts lead Vinner (1991) to state, “We do not believe in ‘mathematics for all’. We do believe in some mathematics for some students. And even this can be achieved only by appropriate pedagogy under appropriate conditions for learning” (p. 81). If not all students are capable of mathematical thinking, then the long-standing tradition of tiered

curricula, particularly in secondary schools might be supportable. On the other side of the dichotomy, Dreyfus (1991) suggests that advanced mathematical thinking is not exclusive to advanced mathematical topics such as real analysis or topology. (p. 26).

Without knowing who is capable of mathematical thinking, elementary or advanced, it is difficult if not impossible to make informed curricular decisions about how to teach it, at what grade level and to which students.

With all these seemingly disparate positions on the term in question, it could be useful to take a deeper look at the ways the construct of “mathematical thinking” has been used in the literature. For mathematics educators, this could provide a more coherent framework for researching student learning, which is a construct of at least equal if not greater fuzziness. For mathematicians, this could provide a common language with which to communicate to their colleagues in education. For policy makers, this could provide the basis for a more robust and concise set of standards. The list of potential beneficiaries is at least as long as the list of disciplines that require a strong mathematical background. In this way, some articles seem to suffer for the utter lack of explicit reference to mathematical thinking. As an example, Davis (2007) describes student understanding with verbs such as believe (p. 399), translate (p. 403) and investigate (p. 405). These verbs have correlates in research on mathematical thinking, but a consumer of research might overlook this article in a keyword search because it is not directly tied to the concept. Conversely, despite being published in Mathematical

Thinking and Learning, Davis’ (2007) reference list has only one instance of the phrase

“mathematical thinking” in an article title. It is possible that Davis (2007) might have

found more useful references if there were a strong construct connecting all the articles.

In fact, perusal of the reference lists of the articles included in the present study reveal that the articles within Mathematical Thinking and Learning cite each other less than once per article, on average. That is to say, for every four articles in the meta-analysis, only three cite any other article in any issue of the same journal. This statistic is even more damning when it is realized that this mean is artificially bolstered by individuals who cite their own earlier works and by special issues that encourage authors to engage in “this issue” type citations. As a brief caveat, this statistic does not include articles which were not considered as relevant to the mathematical thinking construct. However, the fact remains that seventy-six (76) of the one hundred thirty-nine (139) reference lists tracked had no within-journal citations whatsoever. That does not, in my mind, embody the concept of a research “discussion.” The existence of this collection of parallel monologues, rather than a more coherent dialogue, is the primary motivation for this particularly study into the nature of “mathematical thinking” as used in the research literature. Consequently, the following list of questions was generated to guide the present research.

Guiding Research Questions

1. Despite the seeming disunity for an overall definition of “mathematical thinking,” is

there any consensus within the specified literature regarding some facet(s) of the

construct?

a. What are the key obstacles to achieving consensus?

b. Can any compromise be found?

c. What is the source of any uncompromising disagreement?

2. With regard to any consensus found, be it universal or in schools of thought, what is

the nature of “mathematical thinking?”

a. Is “mathematical thinking” a specialized function distinct from other forms of

thinking?

b. What properties and conditions have researchers used to constrain the

construct?

c. How does “mathematical thinking” relate to similar terms such as “statistical

thinking,” “algebraic thinking” or “scientific thinking?”

d. How might this particular definition of “mathematical thinking” influence the

research and practice of pedagogy?

Methodology

Considering that the underlying problem that motivates this study is the panoply of positions regarding the nature of mathematical thinking, it would only expound upon the chaos to present a new definition from scratch – particularly given that this is a dissertation and not a seminal treatise by some long-standing expert. Consequently, a meta-analysis seemed to be the best option to explore the underlying disharmony in the literature. The data will consist of the entire run of articles, book reviews and editorials from the journal Mathematical Thinking and Learning, from the journal’s inception in

1999 (volume 1, issue 1) until the time June of 2011 (volume 13, double issue 1-2).

Given the eponymous state of “mathematical thinking” in this journal, it is a natural choice. Certainly, other journals, such as the Journal of Mathematical Behavior and the

Journal for Research in Mathematics Education were considered, but their contents were not nearly so focused on mathematical thinking. In contrast, the research foci of

Mathematical Thinking and Learning are as follows:

• interdisciplinary studies on mathematical learning, reasoning or thinking, and

their developments at all ages;

• technological advances and their impact on mathematical thinking and learning;

• studies that explore the diverse processes of mathematical reasoning;

• new insights into how mathematical understandings develop across the life span,

including significant transitional periods;

• changing perspectives on the nature of mathematics and their impact on

mathematical thinking and learning in both formal and informal contexts;

• studies that explore the internationalization of mathematics education, together

with other cross-cultural studies of mathematical thinking and learning; and

• studies of innovative instructional practices that foster mathematical learning,

thinking, and development. (Taylor & Francis, 2011)

Each bullet utilizes the construct of “mathematical thinking,” and – more to the

point – even the scope of the journal exhibits multiple variations of the construct,

sometimes even within a single bullet. Not only does Mathematical Thinking and

Learning provide an ample supply of on-topic articles, but the journal also embodies the very discord that is the motivation for the present study.

A cursory review of the key articles in this meta-analysis produced this word cloud (Fig. 1), which serves to illustrate the cacophony of terminology with the journal.

The size of a word represents the frequency with which it occurs. It is, perhaps, worth mentioning that some elements of the journal will be excluded from the data analysis.

Among the excluded items are such informational announcements as obituaries, conference announcements, calls for research, and bureaucratic reorganizations.

Additionally, editorials which were primarily a summary of the key points of the included articles were excluded. Editorials which contained significant exposition were included, though. Book reviews were also included, as they could – to an extent – stand as proxy for the books, which can be classified, with peer-reviewed journals, as a form of research.

Figure 1 – Word Cloud

Conceptual Analysis

One of the most challenging elements of the entire research process, in this case, was selecting a methodology. In many instances, the research questions that guide a study often align neatly with a particular methodology. However, with such an open-ended question as “What is the meaning of mathematical thinking?” no methodology immediately seemed any more conducive than the others. One major obstacle was the nature of the data to be studied. Below is a pie chart (Fig. 2) representing the methodologies used by the most important articles in the meta-analysis.

Model Research Methodologies Quant 4%

Review Mixed 9% 7% Qual 9%

Theoretical / Opinion Teaching 34% Experiment 18%

Case Cobb & Whitenack Glaser & Strauss Study (1996) (1967) 4% 4% 11%

Figure 2 – Model Research Methodologies

Given that much of the literature to be analyzed was of a somewhat philosophical nature, a turn to philosophy for ideas of methodology seemed appropriate. Essentially, the present study rests on a problem that is comparable to one eloquently posed by Quine

(2003a):

There are those who find it soothing to say that [the statements such as “no

bachelor is married”] reduce to [statements of the form “no unmarried man is

married”], the logical truths, by definition; ‘bachelor’, e.g., is defined as

‘unmarried man’. But how do we find that ‘bachelor’ is defined as ‘unmarried

man’? Who defined it thus, and when? Are we to appeal to the nearest dictionary

and accept the lexicographer’s formulation as law? (p.275) [all emphasis sic]

Likewise, in the present study, the problem lies in the definition of “mathematical thinking” in its various forms – which, as has been mentioned, is idiosyncratic or intuitive at best. Surely, one of the myriad of definitions could simply be selected as the “true” definition of the concept, but on what basis? Should we as pedagogues defer to a bureaucrat in the Federal Department of Education as Quine (2003a) would submit to the nearest dictionary? Quine (2003a) asks this question rhetorically, with the assumption that the reader knows the limitations of dictionary definitions for meaning making.

Quine (2003a) argues that the act of defining a word by a synonymous phrase, such as a dictionary does, can muddle both syntactic value – such as the word “bachelor” having fewer letters than “unmarried man” – as well as semantic meaning – such as “bachelor of arts” versus “unmarried man of arts” (p. 277), yet he goes on to suggest that such is the practical reality of life. For example, Quine (2003b) describes a variety of plausible

situations in which the word “rabbit” might not be truly interchangeable with the hypothetically foreign word “gavagai,” but he adds that it would make for a perfectly sensible translation if one used “rabbit” to represent “gavagai” because there is no physical manifestation where one term applies and the other does not (pp. 304-306). In other words, to define something, one should not be concerned with whether the definition gets to the essence of the concept but, instead, with whether the definition allows one to discriminate the defined concept from other concepts.

In this light, the present study will not examine mathematical thinking directly but rather how the phrase “mathematical thinking” is discussed in professional literature. The data will be treated through use of conceptual analysis, which Wilson (1997) suggests is a

process by which the researcher becomes “self-conscious [emphasis sic] about words

which hitherto we had used without thinking – not necessarily used wrongly, but used

unselfconsciously” (p. 14). The goal of conceptual analysis, according to Wilson (1997),

is “concerned with actual and possible uses [emphasis sic] of words” (p. 10) as opposed

to some a priori true meaning. This also allows the present research to utilize

contextually similar phrases, even though they might not be synonymous at a superficial

level. Quine (2003b) observes:

Sometimes we find it to be in the interests of communication to recognize that our

neighbor’s use of some word… differs from ours, and so we translate that word of

his into a different string of phonemes in our idiolect…. We will construe a

neighbor’s word heterophonically now and again if thereby we see our way to

making his message less absurd (pp. 312-313).

Thus, while the researcher is in search of a definition for “mathematical thinking,” it can be useful to include phrases such “mathematical understanding” or really any phrase that describes a non-mechanistic internal process that operates on mathematical objects of some kind. As an example, Zazkis and Leikin (2010) quote a math teacher who was describing the role of his undergraduate education:

“When I think of the relevance of my [Advanced Mathematical Knowledge] in

relation to my teaching, I do see that my university math is helpful for my math

teaching, not necessarily in terms of the specific math content I learned, but more

in terms of the learning of how to handle math. The more math I learn and do, the

more ways I know of handling math” (p. 275, all emphasis sic).

While the teacher did not use the phrase “mathematical thinking,” the concept of

“handling math” is clearly of a similar cut. Moreover, it behooves the present study to – where possible – take the authors of the articles under scrutiny at their word, rather than to discriminate between them by some predetermined definition of mathematical thinking.

Another benefit to studying the usage of the phrase rather than its underlying meaning is that a definition or array of definitions may be generated that is potentially less convoluted and more capable of providing a common vocabulary for the existing research conversation. As Wilson (1997) notes:

This is the real reason, perhaps, why high-flown or tortuous language is to be

avoided: it obscures, not only for your reader but for yourself, the point you are

trying to make. The merit of a simple and lucid style is not just that it is easier to

read: it is that mistakes are more easy to detect, and hence rectify (p. 47).

Simply disregarding the existing vocabulary would only serve to add another

superficially conflicting viewpoint to the cacophony. Instead, the present conceptual

analysis seeks to present one way in which “mathematical thinking” could be utilized in

future research such that previously unnoticed similarities might lead to a semblance of

unity, that key aspects of dissent provide a more concise list of definitions to debate or

that the term is thoroughly discredited as a viable research construct.

Data Analysis

With these ground rules in mind, the data analysis progresses – more or less – in

the procedure of self-dialogue described by Wilson (1997, pp. 94-95). First, those

articles in which the author self-identifies the research focus to be some form of

mathematical thinking as well as those which contain an explicit definition of

mathematical thinking, regardless of stated research focus, are examined to develop an

initial set of model cases. The purpose of this class of articles is to start with the most

overt, intentional uses of the construct of “mathematical thinking” in which the concept

itself is under direct scrutiny of the authors. It is assumed that all of these uses fall within the scope of the construct of “mathematical thinking,” although it is acknowledged that

there may be multiple mutually exclusive sets that define this construct. The model cases

are used to locate these camps, and the benefits and drawbacks of each are considered.

However, it will be left to subsequent research to determine which, if any, is best. As an

example, the first article in the first volume of Mathematical Thinking and Learning

meets this standard. Cobb (1999) states, “My immediate goal is to clarify how we

analyze student’s mathematical reasoning” (p. 5). Certainly, it is worth noticing that

Cobb (1999) used a word with a stronger connotation to logic than mere “thinking” and

that might be a theme that was consistent across the literature or might be placed on a

spectrum opposite more intuitive interpretations of the concept. Another passage from

the article can provide a further example of how the conceptual analysis progresses:

We therefore conjectured that students develop specifically mathematical beliefs

and values that enable them to act as increasingly autonomous members of

classroom mathematical communities as they participate in the negotiation of

sociomathematical norms (Yackel & Cobb, as cited in Cobb, 1999, p. 8).

The use of the verb “develop” in the quoted sentence as an activity performed by the students indicates that Yackel and Cobb believe that mathematical thinking requires a person to actively engage with mathematical objects, rather than be a medium through which mathematics happens to pass. Additionally, the quoted passage admits potentially non-factual beliefs as well as cultural dogmas, in stark contrast to the use of “reasoning” as opposed to “thinking.” This provides evidence that, while Cobb believes logic to be an important part of the process of mathematics, he does not believe logic is the only part.

These concepts of the mathematician as a “developer” of mathematics as well as a

“negotiator” would be well worth exploring in the present study.

While it is the intention of the present research to be as inclusive as possible, certain variations on the construct of “mathematical thinking” may only superficially qualify and should be excluded from consideration. In particular, certain words and

phrases have assumptions built into them through stigma, dogma or traditional use within

technical jargon. Typically, these are words with strong Behaviorist undertones, which

would be inappropriate to include in the present study, as a key tenet of Behaviorism is

that the human mind is not researchable. These words and phrases include: mathematical

literacy, numeracy, achievement and knowledge – although knowledge construction was

allowed. Similarly, some phrases are at least as ill-defined as “mathematical thinking” and provide no further insight. These words include: learning, activity and skill.

There are, of course, many articles which contain some explicit mention of mathematical thinking but failed to meet the earlier standards of model cases; these are examined as borderline cases. The borderline cases provide empirical grounds against which to test the initial criteria for robustness and completeness. At this point, it is sometimes necessary to contrive some invented cases to test a possible combination of criteria that may not exist in the selected literature. As an example, Wilson (1997) forwards the case of hypothetical species of creatures that in all aspects, save for having two heads and having a distinct lineage, is identical to a human and wonders whether or not such a creature would be called “human” (p. 32). Such gray-area articles allow the present research to investigate not just the central, defining features of the construct of

“mathematical thinking” but also its subtleties and various facets which may, in the end, apply to some of the phrases excluded initially.

Some articles provide counterexamples or define mathematical thinking in the negative, which provide important contrary cases. Just as it is important to find examples that are well within the boundaries of the construct of “mathematical thinking”

so, too, is it valuable to find those articles which are far enough outside of them to be

considered distinct. One such example is Ben-Zvi (2000) who explicitly posits that

statistical thinking is not a subset of mathematical thinking. While most articles make no

such claim, it is because of this contrary case that all sub-domains of mathematical

reasoning are considered borderline cases. In this way, contrary cases also provide

evidence of controversial aspects that might otherwise go unexamined, due to being

below the researcher’s perceptual threshold. Any remaining articles can then be analyzed

as potentially overlooked borderline cases or as non-cases of some sort, although

potentially still related to research focus. The non-cases receive no further analysis if no

substantial connection to mathematical thinking is apparent.

The following two figures represent the number of articles in each category.

Figure 2 represents the overall percentages of each classification in the journal as a whole. That the categories fall, roughly, into top quarter, middle half and bottom quarter serves as evidence that the categorizations are more-or-less normally distributed. Figure

3 shows the comparative counts of each category on a per-year basis. It is interesting to notice that while the use of the construct of “mathematical thinking” remains stable as a percentage of total articles – as represented by the sum of model and borderline cases – the balance shifts noticeably towards borderline cases as the years go on. While far from conclusive, these exploratory statistics further reinforce the proposition that the meaning of “mathematical thinking” is becoming increasingly clouded.

Figure 3–Article Category Percentages

Figure 4– Annual Category Counts

Procedure

While conceptual analysis does not have a rigidly defined procedure inherent to it,

the present study will progress in a more-or-less predictable fashion. The first stage will be an in-depth chronological reading of the model articles; during these readings, dialogic commentary and analysis will be created. These research analyses will occur at a resolution of line-level, minimally. After this first stage, the resulting commentaries will be sorted via several filters, which include but are not limited to: similarity of verbiage, evolution of key ideas over time as well as compatibility of definition. A suitable metaphor for this process is the police drama trope of a pin board of evidence connected by strings. Once the commentary categorization is complete, elements within the same category and in closely related categories will be sorted to create a narrative about a particular facet of mathematical thinking. Subsequently, the borderline cases will be examined to search for any inconsistencies in the results generated by the model cases, such as gaps and contradictions. Commentaries and analysis of the borderline cases will likely proceed at a key-point level or be limited to those sections that fall within the scope of the present research but may be more refined as necessary. Borderline cases may also provide much needed empirical examples and vignettes to elaborate points made with the model cases. At this stage, it may be necessary to revisit the model cases as well as examine contrary cases. Analysis at this point will be conducted in a purposive fashion – to seek resolution to key points of cognitive dissonance or confusion. Once the construct of “mathematical thinking” achieves a semblance of stability and viability, the practical implications of the model will be examined.

CHAPTER II: THE MODEL

In this chapter, the model developed through the course of this research will be

presented. However, it is understood that, given the rationale presented in the previous

chapter, that one might be a bit skeptical about the sudden emergence of a singular

coherent model. Therefore, this chapter will open with an exposition on the process of

grappling with the superficially contradictory definitions of mathematical thinking that

exist in the literature. This opening section will attempt to walk the reader through a

similar line of reasoning that led to the juxtaposition of all of the views into one

unexpectedly coherent definition. Following this, there will be a formal presentation of

the model produced through the meta-analysis with some brief explanation of its form.

Finally, this chapter will close on what little information could be gleaned about the

development of mathematical thinking in students from the articles in the present

meta-analysis.

“Mathematical Thinking” or “Thinking about Mathematics”?

Foremost of the concerns of this study is whether or not “mathematical thinking”

is even a sensible construct. Jansen (2008) writes, “Asking students to describe their thinking about mathematics publicly can exacerbate social comparisons, but all adolescents may not be equally reluctant to participate” (p. 71). In this sentence, the phrase “thinking about mathematics” seems to imply, at the very least, that contemplating mathematical objects does not guarantee mathematical thinking; it is, however, also possible that Jansen (2008) does not conceptualize mathematical thinking as a distinct process from generalized thinking. While Jansen (2008) does say:

22 23

The problems emphasize mathematical reasoning and communication and

flexibility among numeric, verbal, symbolic, and graphical representations, as

well as the opportunity for students to make connections among mathematical

ideas and between mathematics and other disciplines (p. 74)

she uses “reasoning” and “mathematical reasoning” interchangeably throughout the

article. Similarly, Carreira (2001) uses the construct extensively, but she describes it only

as a process of meaning-making through metaphor that describes “all human thinking” (p

265).

In this way, the literature varies widely in how it describes the construct, and no

article embodies that more concisely than Kulm (1999), which happens to be a book

review of an anthology entitled Thinking Practices in Mathematics and Science Learning.

Within a single sentence, Kulm (1999) uses both “think about mathematics” as well as

“think when doing mathematics” (p. 315) to describe activities that would both be

considered “mathematical thinking” by the present study. As if these two variations were

not enough, he also uses the phrases “habits of mind” (p. 318) and “think

mathematically” (p. 320) to describe similar activities. Each of these quoted phrases

carries with it potentially very different views about the nature of mathematics and

reality, and these four phrases more-or-less span the opinions to be found within the model cases of the present study. If nothing else, perhaps the present study can help the research discourse develop some internal consistency within the academic language.

Four Views of Mathematical Thinking

The following four views span the majority of perspectives in the literature, and are presented distinctly to highlight the diversity of opinions and usage. Many of the articles do not explicitly adhere to one particular view, nor are articles always internally consistent in their presentation of the construct. To utilize the four perspectives below as a basis for distinguishing schools of thought or separating one research from another would largely be a disservice to the body of literature as a whole. Instead, these four views can provide a basis for understanding the scope of ideas that any formal definition of mathematical thinking must be able to address.

Think about mathematics. The first view to be discussed really makes no distinction between “thinking” and “mathematical thinking.” Case in point, Ginsburg and Seo (1999) chose a title – “Mathematics in children’s thinking” – that seems to imply that mathematical thinking is merely thinking about mathematical objects. In fact,

Ginsburg and Seo (1999) contend that even the verbal comparison of “more lovely” is a form of mathematical thinking, in that the word “more” can easily be tied to the mathematical concepts of inequality or addition. However, if “lovely” is mathematical thinking, then clearly Ginsburg and Seo consider virtually all thought as being at least a little mathematical. This is unsurprising considering that the world is “deeply mathematical” (p. 115) in their view.

Ginsburg and Seo (1999) try to demonstrate this tenet; they argue that virtually every line of Shakespeare’s Sonnet 18 can be considered thinking about mathematics. As a further example of the extreme end of this view, Ginsburg and Seo (1999) suggest that

a teacher can “think mathematically about the child’s intuitions” (p. 125). Consequently,

either the intuitions are mathematical objects, which is possible under their

practically-everything-is-mathematics doctrine, or mathematical thinking can apply to non-mathematical objects like cognition, in which case there is nothing particularly mathematical about it.

Think when doing mathematics. Doerr and Tripp (1999) explain that they

wanted to study how students “reason about problematic situations” (p. 231).

Specifically, Doerr and Tripp (1999) talk about mathematical thinking as thinking done

while engaged in developing “significant mathematical models” (p. 232). Unlike the

previous view, which applies to virtually any quantitative cognition, this view ties

mathematical thinking to the process of doing mathematics. This begs the question,

“What counts as mathematics?” which will be revisited in more detail in a later section.

Doerr and Tripp (1999) also explain how their construct of “mathematical model” is

related to psychology’s construct of “mental model.” This is valuable in two ways. First,

it serves to provide an explanation of what makes mathematical modeling special – the

focus on system structure, which might provide an answer to what counts as

mathematics. Second, this connection demonstrates how mathematical thinking is a

subset of thinking as a whole. To summarize, this second view espouses that

mathematical thinking is implicitly linked to mathematical processes but is a specialized

subset of general thinking.

Habits of mind. A third viewpoint is that mathematical thinking is really a way of perceiving or processing data. For example, Liu and Niess (2006) cite Carlson (1999) in such a way as seems to imply that mathematical thinking involves a more “generic problem-solving approach” (p. 374). Liu and Niess (2006) go a step further to make an argument very similar to the controversial "ontogeny recapitulates phylogeny" theory from evolutionary biology – specifically that students might learn best if mathematics curriculum emulated the historical development of mathematics. In other words, Liu and

Niess (2006) contend that the way individuals learn mathematics must reflect the way humans, as a species, learned mathematics. Mathematical thinking, in this way, is rather like a sensory organ for perceiving mathematical reality that has evolved over time through the pursuit of mathematics.

In this view, mathematical thinking is almost subconscious – a way of looking at the world rather than a way of interacting with it. Mathematics and mathematical thinking might actually be more separate than implied by the “thinking when doing mathematics” approach. This possibility is further legitimized as Stylianides (2007) suggests that “mathematics as a discipline and children as mathematical learners” (p.

366) may not inherently align. This seems to necessitate that mathematical thinking does not occur solely within the field of mathematics proper or with all the mathematical conventions, such as efficiency. To put it another way, the habits of mind that are associated with mathematical thinking might potentially be social constructions that can either help or hinder mathematical thinking at various stages. For example, even though walking is an activity that is habitual and automatic for an adult such that a walk on a

nature trail is a welcome change of pace, the uneven terrain of a woodland hike might not

be the best place for a toddler to take his first steps. To summarize, the concept of

“habits of mind” necessitates a “habituation” that was not seen in the first two views.

Think mathematically. The fourth view, though a minority perspective in our culture at large, seems to carry the most weight within the mathematics education research community. The key point of this position is that there is something unique about the cognitive process of mathematics. Carreira (2001) cites several works by

Lakoff to posit that all human thinking is metaphorical, and she admits that it would be natural to think, at this point, that metaphorical thinking and mathematical thinking might be equivalent. However, Carreira (2001) suggests that one distinction is that mathematical thinking is a formalized and abstracted subset of metaphorical thinking that comes “after the sublimation of its sensorial content and experiential foundations” (p.

267). Subsequently, she argues that mathematical thinking is a process of coordination and pattern recognition within and across metaphors. In other words, while metaphorical thinking and mathematical thinking intersect and interact, mathematical thinking does something that metaphorical thinking alone does not. This embodies the fourth view of mathematical thinking, namely that it is a special way of thinking that is distinct from general thinking but more than merely a mechanistic application of logic.

Synthesis of the Four Views

The present meta-analysis of the literature seems to indicate four main points of view, between which the vast majority of articles vary. From these views, it is clear that any formal definition of mathematical thinking needs to be both robust and versatile.

However, a deeper examination may highlight valuable points of compatibility between the four perspectives; with such overlap, the definition of mathematical thinking need not be vacuously broad or an unharmonious list of divergent perspectives. Assuming such compatibility exists, any definition of mathematical thinking must account for the following key points of the four views:

1. Mathematical thinking could potentially apply to almost anything.

2. Mathematical thinking is implicitly tied to the process of mathematics.

3. Mathematical thinking requires habituation.

4. Mathematical thinking operates on and outside generalized thinking

At first glance, this list does not appear to have any overt contradictions, which lends credence to the possibility that all views can be sensibly taken into consideration. The next few sections detail each point more thoroughly.

1. Mathematical thinking could potentially apply to almost anything. This key point seems to be more or less compatible with all three of the other views. For example, if mathematical thinking can apply to anything, then perhaps mathematics can as well. Certainly, mathematics has been referred to as the universal language enough times to merit that as a possibility. Similarly, if mathematical thinking applies to everything and mathematical thinking requires habituation, then maybe “habituation” is just a fancy way of saying “learning.” The only possible point of contention would seem to be that mathematical thinking cannot both be applicable to everything and do more than generalized thinking while remaining a proper subset of generalized thinking.

Fortunately, Carreira (2001), who exemplifies the fourth view, elaborates that

metaphorical thinking is how mathematical thinking is “linked to people’s ideas and

perceptions about the world” (p. 262). In other words, Carreira (2001) is not opposed to the possibility that mathematical thinking might apply to anything a human might experience; instead, she posits that generalized thinking and mathematical thinking work together – distinctly but in parallel. Consequently, to achieve harmony between the first

and the fourth view, one must either reject the claim that mathematical thinking is a

subset of generalized thinking or the claim that mathematical thinking does something

that generalized thinking does not. Since the former would contraindicate “mathematical

thinking” as a sensible construct, it will be rejected in favor of the latter. Of the two

options, the insistence on mathematical thinking being subordinate to generalized

thinking is less central to the viewpoint. It would seem that less is lost from the overall

messages of research by sacrificing the concept of a hierarchy; the alternative would be to

discard the fourth view entirely, which would defeat the purpose of an integrated

definition of mathematical thinking.

A significant portion of the literature under review discusses the concept of

“realistic” mathematics. In fact, Sfard (2000) seems to believe that the crux of the

mathematics education debate lies in the perceived value of everyday discourse in the process of learning mathematics. Many researchers are in favor of the idea; Rodd (2000), for one, asserts that mathematical knowledge is grounded in human experience. As an example of this assertion reinforced in other research, Bonotto (2005) suggests that measuring is a potential avenue to better decimal understanding. Straying slightly further away from the quantitative norm, Baroody, Lai, Li and Baroody (2009) give the example

of children understanding that clean laundry can be dirtied then made clean again as a

qualitative “proto-concept” of reversibility that can lead to the mathematical concept of

inversion. However, not all researchers so strongly favor the idea. Edwards, Dubinsky

and McDonald (2005) posit that the defining feature of advanced mathematical thinking

is that it cannot “be accomplished with the use of five senses” (p. 18). While Whitenack,

Knipping and Novinger (2001) note that mathematical thinkers may, indeed, “represent

mathematical objects that are experientially real” (p. 82), they add that a focus on

experiential reality may devalue the standard algorithm. The key to resolving this debate

may be found in Lesh and Lehrer (2003). who explain that they used “real life” situations

in their research because they believe that “premathematized” (p. 113) situations create

artificial restrictions on mathematical thinking. This concept of “mathematization”

becomes important in understanding the second key feature of mathematical thinking.

2. Mathematical thinking is implicitly tied to the process of mathematics.

Earlier, the question of “What counts as mathematics?” was left unanswered. One answer

that occurs in the literature is that it depends on whom you ask. Gravemeijer (1999)

describes mathematical thinking as the process of “mathematizing… reality and their [the

mathematical thinkers’] own mathematical activity” (p. 158). It seems that mathematical

thinking is a evolutionary process that, more-or-less, starts from life experience and ends with mathematics, with “mathematization” serving as the crossover point from generalized thinking into mathematical thinking. The primary point of contention is whether the mathematics is discovered or socially constructed, and Sriraman (2005)

reports that Burton (2004) discovered mathematicians’ beliefs “took on varying

combinations of Platonism and formalism” (p. 174).

A summary of some of these views can be found in Dörfler (2002). He defines

Platonism as the belief that mathematic objects exhibit an “a priori existence outside of

time and space and independent of human thinking” (p. 338). Moreover, Dörfler

contends that a Platonist considers mathematical thinking to be a way of perceiving the

mathematical world. Sinclair (2004) states “it would be difficult to argue that there is an objective perspective – a mathematical reality against which the value of mathematical products can be measured” (p. 274). She adds, in a footnote, that some researchers and mathematicians doubt whether such an independent reality even exists. Dörfler (2002) then defines a second position in which the belief that mathematics “are or arise from structures, patterns, and regularities in the physical world” (p. 338). This is exemplified by the already-mentioned Ginsburg and Seo (1999), who posit that quantitative

information is inherent to the world and that “all the child need do is attend to it and

abstract it from the environment” (p. 119). It is worth noting that Ginsburg and Seo

(1999) admit that a constructivist would disagree with their position. Dörfler (2002)

defines the humanistic point of view as encapsulated by Piaget’s genetic epistemology.

In this viewpoint, mathematical thnking has its genesis in human actions such as

counting. Dörfler (2002) also notes that the discursive stance is that mathematical

thinking is a matter of “learning to use and understand its language, its symbols, its

diagrams, its procedures” (p. 339). For instance, Carreira (2001) claims that the meaning

of one mathematical sign is only sensible in context of all the other mathematical signs.

With such widely different views on the philosophy and nature of mathematics, it seems difficult to juxtapose them into a single coherent definition of mathematical thinking. In fact, Stylianides (2007) posits that mathematical truth depends on the mathematical assumptions. Watson and Mason (2006) specifically distance themselves from assumptions about the reality or social construction of mathematical objects, saying,

“All we need to know is that learners discern differences between and within objects through attending to variation” (p. 102). This post-epistemological position would seem the best way to unify the opposing viewpoints about mathematical objects. While mathematicians may disagree about the reality of the set of mathematical objects, all can be said to agree that humans tend to act as if there are mathematical objects. Lesh, Doerr,

Carmona and Hjalmarson (2003) similarly remark:

Ultimately, both perspectives, the positivist and the postmodernist, assume the

existence of an ontological gap between the object and the subject. From an

epistemological perspective, this gap implies fundamental differences, for

example, between radical constructivists, who believe knowledge is first

constructed internally and then externalized; and socioculturalists, who believe

knowledge is first external and then internalized…. these theories can be used

simultaneously, even though they are contradictory at an epistemological level (p.

213).

It is also worth noting that Liu and Niess (2006) report that, prior to instruction, “students in general held an instrumentalist view and denied this possibility [that math could exist without humans]” (p. 385). That is to say, the belief in mathematics as an objective Truth

seems to be ingrained rather than inherent – all the more reason to postpone debate on its

veracity. Sriraman and Lesh (2007) asked Dienes, a noted researcher of the psychology

of mathematics perhaps best known for his invention of place value blocks, whether mathematics was a human creation, to which he replied:

Well, I think this is a non-problem because we will never solve it. But we can

talk about it [laughing]. Yes. Well, I have an idea that certain things are so

whether we think them or not, so in that sense I am with Plato. I mean 2 + 2 is

always going to be 4 in any system you are likely to design….I really don’t think

it is a problem [laughing]. Playing with new forms of mathematics enables us to

reframe how we look at the universe, and find things we may not have found had

we not been able to reframe it that way. It is a bit of both (p. 69).

In this post-epistemological way, mathematization can be construed as the process of deciding “What counts as mathematics?” regardless of whether or not there exists a mathematical reality and whether or not humans have access to it. Dörfler (2002) contends “at some point or stage, so to say, the learner has to make up his or her mind (a metaphoric expression) whether to consider ‘this’ (i.e., a matrix, a group, a graph, a term) as an object in its own right” (p. 342). Sinclair (2004) argues, “a mathematical result cannot be judged important because it matches some supposed mathematical reality – mathematics is not self-organized” (p. 265). Sinclair (2004) posits that there are

an infinitely many true theorems, and humans clearly do not value them all equally. This

is an excellent point, and may serve to further justify the concept of mathematization

even if one believes in an objectively true mathematical reality.

The question remains: how is that decision of mathematization made? Sfard

(2000) relates an incident in which students in similar courses she taught performed very differently on an identical task, based upon whether they believed they were supposed to engage in “mathematical classroom discourse” or “everyday discourse. The key difference seemed to be that the mathematical discourse required that argumentation be made utilizing mathematical objects, symbols and strategies. It is worth noting that both of the solutions that were presented utilized a recursive algorithm, though they differed.

Combining this aspect of contextual differences in mathematical thinking and the aforementioned unequally valued theorems, one suddenly seems to require a socio-cultural component to encompass mathematical thinking – a component which is at the heart of the third view.

3. Mathematical thinking requires habituation. An inordinate amount of the literature describes what could be a called a mathematical disposition. For example,

Ginsburg and Seo (1999) forward that there is a “mathematical point of view” (p. 114).

Similarly, Liu and Niess (2006) cite Schoenfeld (1994) to include a “mathematical point of view” (p. 374) as part of mathematical thinking. This mathematical disposition is one way by which decisions regarding mathematization can be made. In this vein, Drodge and Reid (2000) cite Maturana (1989b) to define an emotional orientation as a

“preference (emotion of acceptance) for the basic premises that constitute the domain in which he or she operates” (p. 250). This emotional orientation includes “the criteria of accepting explanations, the activities that are considered appropriate and the shared experience and assumptions of a community” (p. 250). This emotional orientation is

idiosyncratic in that the mathematical thinker develops his or her own beliefs about

mathematics; it is simultaneously socio-cultural, in that it seeks to situate the mathematical thinker as part of a collection of mathematicians. Drodge and Reid (2000) provide a few examples of their conception of the mathematical emotional orientation:

1. Behaving as if mathematical objects are real.

2. Behaving and claiming that deductive logic is a natural, inevitable, and

specific way of reasoning.

3. Being able to recognize and produce proofs (although proof is often

defined in a way that is not consistent with practice).

4. Preferring proofs and theorems that are “elegant” (p. 256).

Certainly, any real emotional orientation would be larger and broader than the list here, but this short enumeration serves as an example of the sorts of things a mathematical thinker is, does and feels.

Sinclair (2004), using the phrase “mathematical aesthetic,” talks about a mathematical disposition as involving values as well as emotions. Moreover, Sinclair

(2004) posits that the aesthetics of mathematicians are more convergent than those of artists. This would seem to require a well-established process of enculturation – else the two disciplines would be equally as diverse in the beliefs of their practitioners. As a brief aside, it is acknowledged that the Platonist view of mathematics, which relies on the existence of objective Truth, might cause one to disregard the possibility of a mathematical disposition, as objective Truth would overrule personal preference.

However, Sinclair (2004) argues that the frequent reproving of longstanding

mathematical truths, such as the irrationality of the square root of two, are indicative of

the existence of a mathematical aesthetic. After all, if the only goal of mathematics is

truth, a single proof would be sufficient, yet mathematicians often prove the same

theorem in a dozen different ways. Such behavior surely requires some emotional or

cultural motivation beyond the pursuit of knowledge. Sinclair (2004) argues that

personal preference might push a mathematical thinker towards different proof styles,

depending on the nature of that preference. In other words, consider the choice between

harmony and dissent. A person who finds harmony to be more aesthetically pleasing

might be inclined to utilize direct proof. In contrast, a person who finds dissent to be

more intriguing might instead rely upon reductio ad absurdum. Sinclair (2004) cites

Penrose (1974) to posit that it is the affective response of fascination that drives

mathematical thought among professionals. The natural question to ask now is, how does

this enculturation proceed?

Anderson and Gold (2006) posit that “Knowledge and its practice are also constitutive of identities. One becomes an athlete, musician, reader, writer, or mathematician through participatory practice, as well as learning skills and knowledge”

(p. 264). Said another way, they suggest that to learn mathematics is inseparable with learning to be a mathematician. This process is long enough that Sinclair (2004) argues that even graduate students may not be encultured into the mathematical aesthetic.

Complicating matters, Anderson and Gold (2006) observe, “Children bring their

mathematical orientations, beliefs, social constructs, and knowledge with them into the

classroom” (p. 262). Particularly in an educational research culture where “queer” is a

verb, the word “orientation” contains within it a sense of identity. Haciomeroglue,

Aspinwall and Presmeg (2010) summarize their article by opining that “students’

[mathematical] images are personal and idiosyncratic and can be different from what we

intend to teach” (p. 174). Case in point, Whitenack, Knipping and Novinger (2001) note

that one student described converting an iconic problem to a symbolic problem as making

it “exactly like a math problem” (p. 66). A view like this seems to indicate that the

student had a mechanistic and instrumental belief about the nature of mathematics.

However, the literature seems to indicate that such a view of mathematics is very

limiting, as the fourth view elaborates.

4. Mathematical thinking operates on and outside generalized thinking.

Mathematical thinking is a complex construct, the full depth of which be plumbed later in the present study. For now, let just a few key features serve as an example of this fourth view. The most obvious proof of which would be the metacognitive nature of mathematical thinking. Whitenack, Knipping and Novinger (2001), for example, posit that reflection on more concrete understandings is a hallmark of mathematical thinking.

In this explanation, mathematical thinking operates directly on generalized thinking – a view shared by others. Lesh and Yoon (2004) suggest that mathematical thinking develops when students “repeatedly express, test and revise their own current ways of thinking” (p. 205). To culminate this process, students must then represent their results, and Cai (2005) considers final solution representation to be a “better-organized version of solution strategies” (p. 137), which would surely require a fairly strong metacognitive function. Furthermore, the process might not even end there. Bonotto (2005) posits that

the mathematical thinking of “adult mathematicians” is often “retrospective,” which

“occurs when old notions are recalled to be considered at a higher level and within a

broader context” (p. 318). In this way, mathematical thinking also applies

metacognitively to the products and process of earlier mathematical thinking. This might

even be a primary function of mathematical thinking. Lannin (2005) posits that student

reflection on the viability of their solutions led some students to understand the

“limitations of such ‘local tactics’” (p. 252) of everyday thinking. In this way, it could be said that mathematical thinking abstracts or generalizes thinking.

Abstraction and generality are often prominently positioned when researchers

discuss the difference between thinking and mathematical thinking. For example,

Watson and Mason (2006) cite Mason (2004) to define the learning of mathematics as “a

shift between attending to relationships within and between elements of current

experience (e.g., the doing of individual questions) and perceiving relationships as

properties that might be applicable in other situations” (p. 92). Specifically, Watson and

Mason (2006) posit that “generalizations are then the raw material for mathematical

conceptualization” (p. 95). Likewise, Peled and Segalis (2005) value the ability “to

transfer their abstracted knowledge to new situations” (p. 226). So to paraphrase Watson

and Mason, it might be said that abstraction is, in turn, the raw material for mathematical

generalization. Lobato, Ellis and Muñoz (2003) define abstraction via pattern recognition

and attention to relevant properties, and Baroody, Lai, Li and Baroody (2009) summarize

increasing abstraction as a move from visible, real objects to non-visible, symbolic objects. Consequently, abstraction ties back to the aforementioned concept of

mathematization, which is the cognitive acceptance of the reality of a non-visible,

symbolic object. At this point, the fundamental connectedness of the various views

across so many authors begins to come into focus. Despite superficial differences in

terminology and some potentially eternal debates about the epistemology and ontology of

mathematics and mathematical thinking, there does seem to be a way to combine all four

views into a broader, more well-defined construct of mathematical thinking.

An Overview of the Model

The model proposed in this dissertation (Fig. 5) consists of five nodes and their various interconnections. The five nodes can be distinguished into three groups: the cognitive nodes – the mathematical world and everyday experience; the socio-cultural nodes – mathematical community and mathematical disposition; and, most importantly, the central node of sense-making. The basic premise of the model is that sense-making is the active process of mathematical thinking and the other four nodes provide the data which sense-making digests, as it were. The solid-line arrows are the flow of information through sense-making. In contrast, the dotted-line arrows represent processes which are not inherent to mathematical thinking but which are, in some sense, inseparable from the process. For example, the entirety of the mathematical world is unreasonably vast for any single person to mathematize from scratch. Consequently, the model allows for a certain taking of faith in the mathematical thinking of others in the form of pre-mathematized information. Likewise, it is difficult imagine someone coming to an understanding of mathematics as we currently know it without the help of a

teacher – hence the inclusion of instruction in the model.

Figure 5– A Meta-analytic Model of Mathematical Thinking

It is also important to understand that the terminology of the present model is derived primarily from the actual research language present in the meta-analysis. It is admitted that some of the vocabulary associated with this model is contentious.

However, in order to be as faithful as possible to the original authors, words in the model were chosen based upon their frequency of use and generality. Although specifics of the terms will be discussed in the following chapters, a few moments should be spent demonstrating where a couple of these terms originated. To begin with, the literature

ascribes to mathematical thinking a vast array of features, all of which had to be incorporated in one form or another into the model. In but a single article, Sriraman

(2009) notes that some definitions of mathematical thinking include:

(1) the ability to abstract, generalize and discern mathematical structures; (2) data

management; (3) ability to master principles of logical thinking and inference; (4)

analogical, heuristic thinking and posing related problems; (5) flexibility and

reversibility of mathematical operations ; (6) ability to discover mathematical

principles; (7) decision-making abilities in problem-solving situations; (8) the

ability to visualize problems and/or relations; and (9) distinguish between

empirical and theoretical principles (p. 180).

In the early phases of the meta-analysis, this much data was downright overwhelming.

The initial cold-reading of all the articles simply provided no real insight into a solution that could aggregate all the various characteristics presented by researchers.

It was not until the more formal second read of articles, complete with note-taking, that patterns were able to be highlighted in the data. For example,

Sriraman’s (2009) first point uses the terms “abstract” and “generalize.” Through the use of note-taking and the subsequent categorization of those notes, it became clear that these words were used almost ubiquitously. For example, Baroody, Lai, Li and Baroody

(2009) describe mathematical thinking with the following terms: concrete, abstract, reliable and general (p. 41). Likewise, Battista (2004) cites von Glasersfeld (1995) to define three phases of knowledge construction: action, abstraction and reflection (p. 186).

With three sources all using the word “abstraction,” it was clear that it was a vital part of

the larger construct of mathematical thinking. It was simply a matter of sifting through as

many usages of “abstraction” as could be found and try to determine a consensus on

meaning. Many complex definitions of mathematical thinking had to be negotiated. As

another example, Drodge and Reid (2000) argue that mathematical thinking is “a unified

activity incorporating perceiving, emotioning, reasoning, acting, and being” (p. 251); this

article and others like it added non-cognitive components to mathematical thinking that

dramatically expanded the scope of the construct. Along this line, the concept of “action”

began to repeatedly appear and I inferred that is presence alluded to some physical aspect to thinking that needed to be explored. In a similar fashion as used to select

“abstraction,” the term “everyday experience” came into use to describe this physical component through articles such as Ginsburg, Lin, Ness and Seo (2003) in which the authors posit, “In brief, everyday mathematics is whatever mathematics children acquire in their ordinary physical and social environments. It may be informal and intuitive; it may be based in social and cultural experience; and, to a small extent, it may even be formalized” (p. 236). There were so many definitions coming out of the woodwork at this point, and they were so intertwined that, at this point, I began to wonder if the whole dissertation was futile. However, after all the notes were compiled and categorized, it became clear that the many of the elements in these definitions could fit neatly into the four exterior nodes of the present model. They all shared a certain character of providing data to the thinker. The remaining elements of the definitions, it was then realized, were really describing the process of applying some order to the otherwise chaotic influx of data, and the central node of sense-making was added, providing a site of interaction for

the other four nodes. With sense-making in place, the remaining elements came to be seen as the bridges from one node to another. Below (Table 1) is a log of the initial phase of this coding. At this point in the analysis, many of the terms were not yet fully distinguished from one another or final terminology, but it provides some insight into the process.

Table 1– Research Notes by Category

Research Notes by Category Keywords Count Generic Misc. (mostly research specifics) 109 Perspective/Idiosyncrasy/MEO 86 Teaching 68 Age / Development 64

Important Misc (full definitions, interesting terminology, etc.) 61 Explanation/Justification/Warrant/Proof 53 Mathematical Thinking / Thinking About 53 Research 51 Socio-Cultural 51 Norms/Symbols/Language 48 Organization/Structure/Representation 44

Relevant Features & Properties / Metaphor 41 Sophistication/Formality/Elegance/Clarity 37 Previous Material/Tidy/Dissonance 36 Anticipation/Prediction/Patterns/Conjecture 32 Realistic 30 Abstraction/Generality 24 Communication 24

Interpretation 24 Coherence/Correlation/Platonism 22 Mathematizing 21 Intuition/Insight/Logic/Induction 16 Sense-making 16 Goals 14 Groupthink 14

Reasoning/Learning 14 Reflection/Metacognition 14 Sub-domain 14 Process/Product 12 Visualization/Imagery/Modeling 10 Equity 1 Total 1104

A final important distinction needs to be made about the model, in general. While many models tend to be linear, discrete and developmental, the present model has a continuous, cyclical and cognitive design. Martin, Towers and Pirie (2006) posit that mathematical thinking is not a linear or hierarchical progress “from informal actions to more formal abstractions” (p.150). Similarly, Lesh and Lehrer (2003) argue that mathematical thinking is a continually evolving system that grows in non-linear ways (p.

122). Additionally, Bowers and Nickerson (2001) note that a key feature of the transition to a conceptual orientation were “cyclic patterns of interaction” (p. 2). To summarize,

Lesh, Doerr, Carmona and Hjalmarson (2003) remark that “it tends to be far too simplistic to describe conceptual development using the metaphor of a single ladder-like sequence or stages. Instead, conceptual development generally involves multiple dimensions and interactive, cyclic processes” (p. 218). Consequently, the solid-line arrows in the model are generally paired to indicate that the mathematical thinker is both capable of exerting influence upon and being influenced by the world and people around her. The exception here is generalization, which is a reflexive act and consequently has no converse process. Moreover, given that the model has two axes and their orientation was chosen by fiat, there is no implied hierarchy or order to a given train of mathematical thought, save that it passes through sense-making.

CHAPTER III: EVERYDAY EXPERIENCE

Figure 6 – Everyday Experience In this chapter, the object of discussion is the launching point for the human

voyage into the understanding of mathematics – our daily interactions with the world

around us. While it might be said that mathematics strays from the realm of our everyday

experience, virtually all researchers who consider the point admit that mathematics at least begins in physical interaction with the world. More importantly, Bonotto (2005) argues that:

Immersing students in situations that can be related to their own direct experience

and are more consistent with a sense-making disposition, allows them to deepen

46 47

and broaden their understanding of the scope and usefulness of mathematics as

well as learning ways of thinking mathematically that are supported by

mathematizing situations (p. 317).

In fact, this concept of “real world” immersion has developed into an entire school of thought within mathematics education literature. In the same article, Bonotto (2005) provides a brief definition of this position:

As in the Realistic Mathematics Education (RME) perspective of the Dutch

school of thought, I deem that progressive mathematization should lead to

algorithms, concepts, and notations that are rooted in a learning history that starts

with students’ informal experientially real knowledge (p. 316).

Though it is beyond the scope of the present study to provide an exact count, it is worth noting that RME is mentioned regularly throughout Mathematical Thinking and

Learning; that everyday experience plays some role in mathematical thinking is a founding premise within the literature.

However, researchers do vary in the extent and specific nature of the role they attribute to everyday experience. For example, when using the term realistic, Cobb

(1999) wants to ensure “that the successive classroom practices that emerged in the course of the teaching experiment were commensurable with the activities of proficient data analysts” (p. 12). In other words, Cobb (1999) believes that “real” mathematics is akin to what mathematicians do professionally. In contrast, Gravemeijer (1999) uses

“realistic” such that “the idea is not to motivate students with everyday-life contexts but to look for contexts that are experientially real for the students and can be used as starting

points for progressive mathematization” (p. 158). It would be easy to get lost in trying to

define an adjective like “realistic;” thus, the present model avoids such debate. Instead,

the present model simply assumes that everyday experience does influence mathematical

thinking and focuses on the nature of that influence, without making any grand claims

about what counts as “real” as much as it will refrain from claims about what is

“mathematics” in a subsequent chapter.

It is worth a moment’s pause to caution against overreliance on everyday

experience. McClain (2003) reports having felt that the real world scenario did not

improve her preservice teachers’ understanding of octal arithmetic – they were instead

relying on their existing algorithmic knowledge to bypass the real world component (p.

287). At the opposite extreme, Stylianides (2009) opines that real-world tasks tend to favor pattern identification over more sophisticated forms of mathematical thinking such as proof(p. 282). However, Bonotto (2005) suggests that many of those supposedly more sophisticated tasks tend to be stereotypical, bypassing sense-making; she further notes,

“Rarely will students encounter these activities in this form outside of school” (p. 314).

Moreover, Harel and Sowder (2005) contend:

It is critical to emphasize that one cannot and would not appreciate the efficiency

of the latter [more formal] solution if he or she has not gone, in various

problematic situations, through an elaborated solution…. Hence, although we

desire to label the function solution as more advanced than the elaborated

solution, it may be unlikely that the former could be constructed without the latter

(p. 37).

Even with its limitations, everyday experience is not an expendable part of mathematical

thinking.

Bonotto (2005) argues that mathematical thinking is a two-way street (p. 316). In

other words, for a student to fully appreciate mathematics, he or she must be able to both

construct mathematics from everyday experience and also recognize how mathematics

give insight back into everyday experience. In the present model, the former is considered abstraction – the process by which the mathematical thinker decides which features and properties of the situation are relevant to the mathematical task at hand.

Similarly, the latter is referred to as reification – the process by which the mathematical

thinker applies her mathematical thoughts to the world, either to solve a problem situation

or as an empirical test of her mathematical predictions. It can be easy for teachers to

overlook reification, but it is a vital part of the mathematical thinking process.

D’Ambrosio (1999) cites Oliveira to report that “[Brazilian] Students in the school are

just waiting to reach the legal age so they can get a job. Mathematics has nothing to do

with the ‘world outside,’ where they will get job” (p. 138). In this way, the mathematical

thinking of students can be greatly diminished without some reflection on the application

of mathematics. Also, in this chapter, a third facet of the present model will also be

discussed; generalization is herein defined as the metacognitive ability for the

mathematical thinker to consider his own mathematical thinking and determine whether it

applies to situations other than those that originally generated it. That is to say,

generalization is the process by which the mathematical thinker goes from observation of

cases to prediction of cases. It is worth noting that generalization, as depicted in the

present model, is more specifically part of sense-making than everyday experience.

Moreover, generalization can apply to any of the facets of mathematical thinking, but it will be discussed in this chapter as it is most readily observed in a cycle of abstraction-generalization-reification (Fig. 7) with respect to its use in the literature. This particular cycle is illustrated well by Stylianides (2009) who describes mathematical thinking as:

exploring mathematical relationships to identify and arrange significant facts into

meaningful patterns, using the patterns to formulate conjectures, testing the

conjectures against new evidence and revising the conjectures to formulate new

conjectures that conform with the evidence, and providing informal arguments

that demonstrate the viability of the conjectures (p. 258).

Also to be considered in this chapter is the propensity for students to overgeneralize and how teachers can strive prevent it.

Figure 7 – A Common Cycle of Mathematical Thinking

Abstraction

As a reminder, the present model makes no hierarchical claims about the order or

superiority of any of the sub-processes in mathematical thinking. However, there is some

evidence to suggest that abstraction may be among the first connections to develop with

regard to any given mathematical concept; a way to demonstrate this is by considering

the lamentations of researchers regarding non-mathematical thought among their

participants. Cobb (1999), as a case in point, provides an example of non-mathematical

thinking in which students attend to irrelevant features of data, such as the color of bars

in a graph (pp. 13-14). Emulating a student’s reasoning, Radford (2003) asks

rhetorically, “in the case of the generalization of a sequence of geometric-numeric

objects, do we have to attend to the color of the objects or to their shape or to something

else?” (p. 38). Lesh and Harel (2003) describe a student strategy that focused on

irrelevant data as “foolish,” but they add that even this misstep provided new ideas that

might have eventually been useful. It would seem that this pre-mathematical thinking is

just a wild search for potentially viable ideas. This might explain why Nathan and Kim

(2007) contend that prediction precedes abstraction in younger students, but that

abstraction seems to become more prominent in later years (pp. 205-206). Prior to thinking mathematically, students simply guess and generate predictions. Given the current model, it is not until students take the time to properly abstract can they generate useful and sensible predictions. This haphazard search is strongly reminiscent of the common student practice of guess-and-check, and students engaging in such behavior may be exhibiting a lack of preparation for abstraction.

Yet, this unstructured searching is not necessarily undesirable. Ginsburg and Seo

(1999) declare that mathematical thinking includes: “everyday math, intuitions, spontaneous concepts… invented strategies [and] protoquantities” (pp. 114-115).

Moreover, the first point at which Cobb (1999) refers to student thinking as mathematical is when they select qualitative properties, such as data points being “bunched up,” that seem to be relevant to the problem at hand, which is similar to the concept of protoquantities forwarded by Ginsburg and Seo (1999). Baroody, Lai, Li and Baroody

(2009) cite Resnick (1992, 1994) to highlight the differences as students bridge into mathematical thinking:

• Protoquantitative (reality-bound, context-specific, qualitative) reasoning.

Initially, children reason about nonquantified amounts in a global way. For

example, a toddler may

initially recognize that adding an item to a collection “changes it” and later may

understand that the addition “makes a collection larger.”

• Context-specific quantitative reasoning. Once children develop the means to

represent

collections exactly (e.g., counting) they can reason about specific quantities in a

meaningful context. For instance, given two cookies and then offered one more, a

child might count his new total and discover that one cookie added to two cookies

always makes three cookies altogether (p. 42, all emphasis sic).

Keeping this distinction in mind, Baroody, Lai, Li and Baroody (2009) argue that their research indicates that students may be ready for mathematical training as young as three

years of age, although they add that four year olds were more reliably ready (pp. 51-55).

Watson and Mason (2006) cite Marton and Booth (1997) to posit that “the starting point of making sense of any data is the discernment of variations within it” (p. 92), and this is where the present model contends that abstraction takes effect.

Watson and Mason (2006) clearly believe that pattern recognition is the foundation of mathematical thinking (p. 101), but abstraction need not be so limited. Van den Heuvel-Panhuizen and van den Boogaard (2008) cite Ginsburg et al. (2003) to enumerate the following categories of mathematical thinking: classification, magnitude, enumeration, dynamics, pattern and shape as well as spatial relations (p. 357). All of these, save perhaps classification, are perceptual activities. In other words, abstraction is the process by which sensory information is accepted into the cognitive processes of mathematical thinking. In a similar way, Petrosino, Lehrer and Schauble (2003) argue that mathematical thinkers should decide “how to parse the world into measurable attributes and then [come] to agreement about best ways of measuring them” (p.

133-134).

Watson and Mason (2006) provide some evidence to suggest that at least some of abstraction occurs subconsciously or at least quickly enough that it might as well be; they relate an instance in which all but one problem in a set had the same answer. Watson and

Mason (2006) report that students admitted that “their work shifts from ‘calculating distance’ to ‘verifying that the distance is 3’” (p. 98) once they noticed a pattern. Of those who did notice a pattern, they describe the result as being “jolted into thinking mathematically by being offered points that broke their current sense of pattern” (p. 97).

In this example, students noticed a pattern and had proceeded to make generalizations

and experienced emotional blowback when that generalization was not supported by

subsequent abstraction. The sensation of jolting indicates a lack of cognitive awareness

of this process. The automaticity of abstraction is a noteworthy limitation. McClain

(2003) posits that “empirical testing” is not mathematical thinking, explaining “there was

no understanding to underlie their procedures; merely results of looking for patterns in

solutions” (p. 287). Similarly, Lesh and Harel (2003) provide an example of

non-mathematical thinking that seems to imply that abstraction should involve

exhaustive, or at least fully representative, examination of data. However, Watson and

Mason (2006) suggest that the mathematical data available to students, itself, may be to

blame in such instances. Specifically, Watson and Mason (2006) describe the typical

random selection of textbook problems as being designed to expose students to the

greatest diversity of problem types, but they suggest that students fail to get the repeated

exposure of similar problems that is necessary to observe regularities in the mathematics

(p. 105). Watson and Mason (2006) argue that more deliberately varied problems would

allow students to develop computational fluency while simultaneously allowing students

to view complete sets of problems as mathematical entities rather than merely individual

problems (pp. 106-107).

Summary

Such reports about the limitations of abstraction and the everyday experience

from which it is derived serve as a reminder that no aspect of mathematical thinking is

functional in isolation. Abstraction is simply the noticing of features and properties in a

given object. There is no guarantee that the abstracted elements of experience are salient, useful, function or even accurate. Abstraction is the effort of the mathematical thinker to parse the sea of sensory information into more well-defined chunks. Amateur mathematical thinkers often abstract too little, but in and of itself there is no way of telling when enough information has been abstracted. Specifically, the call to consider more cases, particularly in those cases purposively selected to explore variation along a feature or property, leads naturally to the discussion of another aspect of mathematical thinking, generalization.

Generalization

A common follow-through for the cognitive inertia of abstraction is what the present model refers to as generalization. Again, while generalization is more formally an element of sense-making, its frequent correlation with abstraction makes it more easily understood here than in its own chapter. Lesh and Yoon (2004), for example, state that they assume “mathematical thinking is about interpreting situations mathematically at least as much as it is about processing information that already exists in relevant mathematical forms” (p. 210). The former is abstraction, and the latter is generalization.

In this way, Groth and Bergner (2006) cite Biggs (1999) to posit that increasing sophistication of mathematical thinking involves accounting for increasingly more of the relevant structures of a situation and managing to find a unifying concept that ties it together and allows it to transcend the current situational conditions (pp. 41-42).

Similarly, Bjuland, Cestari and Borgersen (2008) describe mathematical thinking requiring one to “notice abstract mathematical relationships and to be more focused on

conceptual aspects of mathematical objects” (p. 274). This distinction is central in some of the most outspoken views on mathematical thinking. Specifically, Edwards, Dubinsky and McDonald (2005) seem to imply that someone engaged in “advanced mathematical thinking” does not need to use specific examples to justify or confirm their reasoning; guess and check, even if well ordered, is only elementary mathematical thinking to them

(pp. 21-22). In other words, they believe that mathematical thinking distances itself from any specific instance of a concept as it develops. Similarly, Harel and Sowder (2005) clarify that “The phrase way of thinking, on the other hand, refers to what governs one’s ways of understanding, and thus expresses reasoning that is not specific to one particular situation but to a multitude of situations” (p. 31). As another example, Baroody, Lai, Li and Baroody (2009) cite several researchers to similarly define informal mathematical thinking as “localized” (p. 42), and Lannin (2005) cites Mason (1996) to decry “‘local tactics,’ attempting to find a rule to fit a particular instance of the pattern rather than understanding a general relation in the problem situation” (p. 233). That is to say, generalization eventually allows a mathematical thinker to work on hypothetical or purely abstract cases. The word “eventually” is key here – as Peled and Segalis (2005) posit, “When making an analogy between a given structure and some previously encountered structure, one needs to identify the latter as a candidate for analogy and abstract it sufficiently so that the higher order description allows for mapping it to the new structure” (p. 209). Ultimately, Baroody, Lai, Li and Baroody (2009) differentiate generic thinking from mathematical thinking with only the latter being able to predict a mathematical event before it happens (pp. 46-47). While abstraction might allow a

student to notice a pattern, the idea is that generalization allows for the pattern’s

extension. In this way, Bonotto (2005) argues that one mode of mathematical thinking is

that of anticipation – in this specific case, some students noticed a relationship between

two concepts and anticipated that a similar relationship would exist between other

concepts at hand (p. 328). Cobb (1999) opines similarly that “anticipation is at the heart

of data analysis” (p. 11).

Mathematical thinkers are not always immediately capable of this cognitive leap.

Stylianides (2007) notes that students struggled with the idea of multi-addend solutions to a problem involving traveling from one floor to another via an elevator. He quotes one student saying, “The person wants to go to the second floor – he doesn’t want to go all over the building” (p. 371). This is demonstrative of attending only to the particulars of the specific concrete problem. Other students, who had initially proposed the multi-addend solution, remarked that the scenario allowed for any solution that would eventually end at the second floor. These students had generalized beyond the practical limitations of the problem to a mathematical reality that was equally valid. Interestingly, students are capable of, at least the earliest forms of, this leap in elementary school.

Lehrer and Schauble (2000) mention that a fourth grade student “suggested that the model should work with any set of pictures, regardless of whether those pictures were used to make the model in the first place” (p. 65).

Generalization occurs in a variety of forms and may be the most flexible of all the facets of mathematical thinking. Raman and Fernández (2005) believe that the following are key questions in mathematical thinking: “Where do the ideas come from? Why are

certain conventions used and not others? Why do we favor some definitions over others?

What would happen if we defined things differently?” (p. 263). Put another way, Raman

and Fernández (2005) are describing generalization as an act of speculation regarding the

various constructs that the mathematical thinker is attempting to comprehend.

Additionally, it is important to note that only some of these questions would be asked of

everyday experience. Though abstraction and generality often go hand-in-hand in the

literature, they need not. Rasmussen, Zandieh, King and Teppo (2005) provide an

example of how generalization is not solely part of the lower cognitive loop of the model:

The final task in this sequence asked students to come up with a description (in

words and equations) that might help another math or engineering student

understand how to approximate the future number of fish in a pond with the

differential equation dP/dt = f(P), for some unknown expression f(P). This task, of

developing an algorithm, engaged students in the activity of reflecting on and

generalizing their previous work. In this case, students began to consider

situations in which the time increment need not be one unit and for a variety of

types of functions f(P). The procedure needed to be effective across these

different situations, and not for a particular differential equation, initial condition,

or increment of time. Thus, using their previous activity, students engaged in

vertical mathematizing, allowing them to develop generalized formal algorithms

(p. 65, all emphasis sic).

In this scenario, students have been presented with a pre-mathematized formula about fish populations. Once they have properly situated it and justified it, they make it apply

to new functions that have likely already been mathematized themselves. That the

scenario describes a real world situation does not require students to abstract the key

features and properties of the situation at hand. In a sense, they were provided those

abstractions when the formula was presented to them. Ultimately, it is not until they

generate an explanation at the end that students have an opportunity to connect the

mathematics back to everyday experience, through reification. Though, discussion

regarding reification will be postponed a bit longer.

A second form which generalization takes in the literature is that of analogy or

metaphor. Presmeg and Balderas-Cañas (2001) cite various authors to posit that

metaphorical thinking connects mathematical experience to everyday experience as well

as mathematical objects to one another (pp. 291-292). Similarly, Carreira (2001) argues that “metaphor is the selecting, emphasizing, suppressing, and organizing of characteristics of the target topic by suggesting and stressing ideas about it that would normally be applicable [only] to the origin topic” (p. 265). These two definitions of metaphor are virtually identical to the existing purposes of generalization, but the new language suggests that generalization is not merely about transcending a single context.

Rather, “analogy” and “metaphor” imply that generalization is about discovering similarity in otherwise disparate objects – a precursor to the formal structuring and interlinking of mathematical concepts that occurs during mathematization. From this, something new emerges: a mathematical model of the everyday situation. Doerr and

Tripp (1999) define a model as, “a system whose meaning is constituted by the interactions of internal systems, external systems, and the representations that are

distributed across these systems” (p. 233). They add, “Because any given model is of

necessity a model of some other system with its own elements, relations, operations, and

rules, the idea of using analogy in the development of models is to use a familiar system

to understand an unfamiliar system. These models are useful for prediction – one of the

aforementioned main functions of sense-making, and often lead back to reification for empirical verification.

Overgeneralization

As generalization is such a central part of the mathematical thinking process, it is no surprise that it is also the crux of many missteps in the learning process. Typically, these missteps come in the form of generalizing too liberally. That is to say,

mathematical thinkers often overgeneralize. Even in the best case scenario, Olive (1999)

notes that having all the prerequisite understandings for a given concept is not a

guarantee of immediate progress into understanding of said concept (pp. 291-292).

However, that is not necessarily a bad thing. Lobato, Ellis and Muñoz (2003), in fact,

argue that misconceptions and overgeneralizations are part of the mathematical thinking

process (pp. 3-4). The idea that mistakes are beneficial to learning may sound unusual, at

first, but Bowers and Nickerson (2001) provide a quote from a teacher-educator in their

study that recalls that this idea is at least as old as Piaget:

For example, I can ask you what time it is, and you can convey this knowledge.

On the other hand, students may not be ready to receive the information you

impart. This involves assimilation, in which a students [sic] makes sense of

information imparted (p. 12).

That is to say, Piagetian assimilation is the cognitive process by which a person tries to understand new information based upon old information – that is to say generalizing. Of course, Piagetian assimilation is also tied to the concept of accommodation, which is founded on the premise that people do not reconsider their assimilations until they experience a situation which simply cannot be assimilated. This is parallel to overgeneralization requiring counterexamples to be ameliorated. The question then becomes, “Why is it that generalization is so fallible?” Moreover, one wonders how mistakes can lead to deeper understanding.

Harel and Sowder (2005) cite Duroux (1982) to define an “epistemological obstacle” as “a piece of knowledge or a conception that produces responses that are valid within a particular context, and it generates invalid responses outside this context” (p 34).

They add that epistemological obstacles are not immediately overcome by contradictory evidence; Doerr and Tripp (1999) similarly suggest that individuals often need multiple challenges to their initial beliefs in order to accept the line of mathematical reasoning that contradicts those beliefs (pp. 249-250). Esteley, Villarreal and Alagia (2010) note that

“In the literature… [one particular] overgeneralization is known as linear misconception, illusion of proportionality, or linearity… and also proportionality trap” (p. 87). De Bock,

Verschaffel and Janssens (2002) argue that overexposure to a mathematical topic can be one cause of this sort of bias in mathematical thinking; specifically, they posit that the illusion of linearity is an overgeneralization of the commonly occurring relationship of

“more A [implies] more B” (p. 67). To put this in terms of an epistemological obstacle, the “more A [implies] more B” relationship holds true so often in everyday experience as

well as early mathematics instruction that students begin to assume it will always hold true. This overgeneralization can seriously impede the development of multiplicative reasoning, as any middle school math teacher can affirm. Keiser (2004) argues that, prior to instruction, sixth graders conflated sharpness of vertex with magnitude of angle as well as length of rays or area between rays with measure of angle (p. 292). That is to say, the students held the standard “more A [implies] more B” formatted belief that more sharpness implies more of angle. In this way, Greer (2010) suggests that “the tendency toward linear thinking is a ‘universal’ phenomenon, occurring across ages, content domains, and instructional environments” (p. 109), but he also cautions that an overreliance on linear, rather than proportional thinking:

does not [emphasis sic] imply that students deeply understand linear situations

(they may routinely be applying linear methods without understanding the

situation), nor that in the non-linear situations they necessarily have made a

reasoned judgment that linear methods are appropriate when they use them (the

application of linear methods might again just be due to routine behaviour [sic]

instead of a conscious application) (p. 110).

This warning leads one to wonder how to tell the difference. Harel and Sowder (2005), again citing Duroux (1982), seem to believe that advanced mathematical thinking is mathematical thinking which overcomes epistemological obstacles. In other words, it is easy to develop mathematical thinking to a point where overgeneralizations occur, but it is difficult to transition away from the reliance on inductive reasoning to more formal mathematical thought. Unsurprisingly, Edwards, Dubinsky and McDonald (2005)

remark that, “The difficulties for some students began when they were reasoning about concepts that were not physically accessible to them and their intuitions and the definitions conflicted [with earlier understandings]” (p. 17).

There are certainly many other examples in the literature that describe overgeneralization at work. For one, Harel and Sowder (2005) give the example of a situation in which x is both a “variable” and an “unknown” as being difficult for a student to handle simultaneously (pp. 38-39). For another, Stylianides (2007) cites Ball (1993) to suggest sometimes students see the negative sign as an operation on the subsequent integer rather than the negative number as a single entity (p. 367). Likewise, Alibali,

Knuth, Hattikudur, McNeil and Stephens (2007) note, regarding the equals sign,

“Developing a view of the symmetric nature of an equation requires understanding that the equal sign is a mathematical symbol representing a relationship between quantities rather than a signal to perform arithmetic operations” (p. 223). In a similar way, Olive

(1999) notes that earlier research described whole number knowledge as interfering with the acquisition of rational number knowledge but adds that more contemporary research considers the latter a modification of the former (p. 284). Sometimes the tendency to over-generalize is so strong, that students will make assumptions from any data without even considering whether there is any connection to the situation at hand. This may be in part because, as Gravemeijer (1999) describes it, formal mathematics is a “natural extension” (p. 156) of everyday experience. That is to say, the process of learning mathematics relies on the generalization process; thus, students may be inclined to skip directly to it as a sort of cognitive shortcut. Doerr and Tripp (1999) report how some of

the students repeatedly tried to recall information from a previous physics class rather than relying on the raw data available to them; only after being unable to recall an applicable procedure did the students proceed mathematically (pp. 243-244).

Overgeneralization, while a natural part of the process, often becomes tenaciously entangled with the mathematical thinking of students. Harel and Sowder (2005) mention that “relying solely on empirical observations to justify mathematical arguments… [and] over-generalizing mathematical ideas” (p. 33) is not mathematical thinking. In other words, abstraction and generalization do not provide a complete picture of the process.

However, students are notorious for looking in all the wrong places to supplement this fragment of mathematical thinking. Bonotto (2005) explains that she expects “students to approach an unfamiliar problem as a situation to be mathematized, not primarily to apply ready-made solution procedures” (p. 322), but the necessity of enumerating this expectation makes it clear that students are prone to the undesired behavior. Lesh and

Harel (2003) seem to imply that students have an aversion to “messy numbers” such as numbers with repeating decimals – which they add is not mathematical thinking (pp.

170-171). Bowers and Nickerson (2001) cite Thompson (1994) to posit that students can struggle to learn new material based upon an “impoverished image” (p. 5) of an earlier topic. Keiser (2004) suggests that one potential source of this “impoverished” understanding is the tendency for students to be exposed to a single, inflexible definition of any given mathematical object. Keiser (2004) goes on to suggest that such limitations are not fully representative of the mathematics as “depending on their specific mathematical specialty, the meaning may take on a very different emphasis as it is

applied to that field. That is, a geometer may define ‘angle’ in a much different way than

would a topologist” (p. 304). In this way, Stylianides (2009) speaks of a pattern such that

it is “fits the data” (p. 263). This has both implications of intentional belonging-together,

as in a lock and key, as well as an absence of force – unlike when students, to use the

antonym of the aforementioned “messy numbers,” tidy the data. Rephrased, students

trying to understand their experience from a definition that is provided to them will do

everything in their power to put the proverbial square peg in the round hole, and that is

what teachers need to avoid. Moreover, Selden and Selden (2005) note that Resnick et al.

(1989) believed abstract, analytic definitions might not actually influence students with

strong concrete understandings of concepts such as decimals.

Summary

Generalization could easily be considered the defining feature of mathematical

thinking. Without generalization, mathematics would lack the ability to extend beyond

the current cases. It is that liberation from the present situation that gives mathematics its hallmark predictive power for which it is given such high epistemic status among forms of human knowing. However, like all the facets of mathematical thinking, it is useless by itself. For one, generalization is a processing of data rather than a mining of data.

Without the influx of information from other facets of mathematical thinking, there would be nothing to generalize. Be it realizing that the current solution might apply to other real world scenarios or that a proof regarding odd numbers applies to all integers, generalization only grows seeds of knowledge, not plant them. Additionally, without the checks and balances of the other facets, generalization quickly becomes

overgeneralization. The human imagination is capable of wild visions that have no possibility of ever matching reality. This is just one more reason that reification is necessary for mathematical thinking, for without it, students often have no impetus to disbelieve their overgeneralizations.

Reification

As has been shown, students have a tendency to simply abstract and generalize without much concern for the consequences of those actions. Bonotto (2005) cites

Hiebert (1985) to identify the following as features of mathematical thinking: understanding the referents of symbols, knowing when and how to apply procedures as well as gauging the reasonableness of answers (p. 319). Clearly, it is the third feature that is missing at this point, and Lobato, Ellis and Muñoz (2003) posit that mathematical thinking requires opportunities to “develop, test and revise the target knowledge” (p. 31).

A student at this point has access to many possibilities to rectify his or her overgeneralization. Certainly, the student could rely on his or her mathematical intuition in some instances, simply obey mathematical conventions in other instances or, ideally, rely on formal mathematical justification. However, these are all more advanced forms of mathematical thinking that may not have yet developed to a point where the student will feel confident in their meaningfulness. Often, the student must compare his abstracted concepts and generalized models to the real world from which they came in order to assess their reasonableness. Harel and Sowder (2005) use the term “reify” to discuss making a singular solution into a “solution method” (p. 37). Consequently, the present model refers to this reality-checking process as reification. In this way, De Bock,

Verschaffel and Janssens (2002) contend that mathematical thinking is not only a product of past experience but also an adaptation to current conditions. Reification can occur whenever a mathematical thinker applies his or her abstract knowledge back onto the concrete world of everyday experience, but it follows very naturally from generalization.

Gainsburg (2006) cites Gravemeijer to posit “Put simply, mathematical modeling means translating a real-world problem into mathematics, working the math, and translating the results back into the real-world context” (p. 3). As a general rule, Lesh and Lehrer

(2003) posit that Piaget believed that “many of the most important properties of mathematics systems… [are generated by cognitive reorganization when] models fail to fit the experiences they are intended to describe, explain or predict” (p. 120). Similarly,

Lehrer and Schauble (2000) suggest that mathematical thinking is invoked when students

“are asked to invent and revise models” (p. 52). If generalization invents models, then reification is one way by which such models may be revised.

The process of reification tends to be a real challenge for students. Lehrer and

Schauble (2000) observe that fifth grade students – the first age of students that regularly engaged in abstraction – struggled with the fact that it was difficult to generate rules that negotiated complexity of ill-defined sets (p. 72). That is to say, students had the expectation that their common-sense rules would apply universally, and they needed time to explore the level of specificity required in both the rules and the categories in order to get that universality. One thing that can make this part of mathematical thinking particularly difficult is that it needs to be done “prior to acting” (p. 289), as Olive (1999) explains in describing what makes a particular unit of measurement useful to learning.

Simon (2006) similarly declares, “I consider a mathematical understanding to be a

learned anticipation of the logical necessity of a particular pattern or relationship” (p.

364).There is certainly an abundance of both research and anecdotal evidence regarding

the tendency of children to retroactively justify their behavior when confronted with an

adult’s interrogation. Reification, in this way, may be one of the first features of

mathematical thinking that is not well-established before schooling. Once reification is established, it is easily recognized. While researching pre-service teachers, Watson and

Mason (2006) comment that they observed students make “little noises” (p. 98) when they discover the first point at which their expectations prove false. Certainly, it makes sense that pre-service teachers have developed mathematical thinking to the point where reification is extant; without reification, one wonders how it would be possible to design a lesson plan from scratch. One limitation of reification, however, is the very feature that makes it easily observed – it often prompts an emotional reaction. In observing students who were trying to apply their mathematical thinking to the real-world task of assembling

quilts, Lesh and Yoon (2004) note “Carla is sounding tired and frustrated” (p. 221).

Teachers must be very careful not to let the complexity of this stage of mathematical

thinking overwhelm students. More about this potential for emotional interference will be explored in a subsequent chapter on the mathematical disposition.

Summary

All too often, students engage in what would otherwise be fantastic mathematical thinking but miss the mark because they fail to ask the question, “Does this match all of my data?” A simple statement such as “Prime numbers are odd” is just such an

oversight. It is not as if two is a poorly understood number, but it does actually need to be considered when looking at the set of prime numbers. Common algebra mistakes

often involve similarly overlooking the possibility of special numbers such as zero which

are no more obscure. Reification is that fact-checking, but it is more than that.

Reification is also what makes mathematics practical – by finding new real world applications. Be it gear boxes out of fractions or computers out of Boolean algebra, reification makes the abstract world of mathematics imminent in our concrete, everyday experience.

Chapter Summary

The world of everyday experience is inexorably linked to the human pursuit of

mathematics. Even if you believe in orthodox Platonism in which mathematics has an

entirely separate reality from tangible things, you must admit that we humans could have

no window into it without our sensory experiences of time, space and quantity that are

processed through abstraction. Moreover, the human world is just as much an “end” of

mathematics as it is a beginning. There would be no correlation between salary and

mathematical-intensity of a given profession were it not for the seeming fact that math

allows humans to predict their own experiences through reification. Certainly, there are

those mathematicians who decry applied mathematics as somehow inferior to pure

mathematics, and there is a place for such arguments. However, applied mathematics is

clearly mathematics and tends to be far more accessible for the average person to engage

in mathematical thinking.

CHAPTER IV: MATHEMATICAL WORLD

Figure 8 – The Mathematical In their research, Lobato, Ellis and Muñoz (2003) note that two students in the

study could remember the slope formula, but were unable to produce a slope by using it

(p. 14). Such scenes are incredibly common in the math classroom; the frequent, and

often inappropriate, reliance on familiar tools such as the slope-intercept form of a line or the Pythagorean Theorem seems to indicate that students rely on the mathematics that has, in the past, produced correct answers. It would seem that, for them, there is no deeper connection with the mathematical world. In fact, Nasir (2002) argues that much of mathematics can be done without being understood and that this product/process

70 71

dichotomy is a key distinction that is made by contemporary research (p. 214). As an example, Murata (2008) posits that students should “consider mathematics as a subject of systematic relationships” (p. 398).

There are many benefits of such relatedness of mathematical concepts. Perhaps, the greatest of these is the ability for a mathematical thinker to cope with missing information; Bonotto (2005) suggests mathematical thinking can “systematize and consolidate concepts, making clear what is not understood, and filling up the eventual gaps” (p. 335). In a self-study, McGinn and Boote (2003) remark upon an occurrence along these lines, “When we were able to classify a question as belonging to a category of problems, even if we could not remember all the details, the question was much easier” (p. 95). Similarly, Nickerson and Whiteacre (2010) contend, “Once a student understood the relationships among quantities, he or she could use the relationships to find all the values needed” (p. 242). Yet research indicates that students of mathematics often do not produce such robust systems. Schneider and Stern (2009) argue that “Some studies show that children acquire mathematically inter-related concepts independently of each other and do not link them in their mind” (p. 94) However, they add that “Many studies show that children frequently demonstrate understanding of inversion on some tasks but not on others. These patterns of dissociations change over time” (p. 95). The following chapter focuses on those aspects of mathematical thinking that were present in

McGinn and Boote (2003) but absent in Schneider and Stern (2009) – namely organization and deduction.

Harel and Sowder (2005) assert that specific mathematical understandings are

influenced by general mathematical ways of thinking and vice-versa. They give the

specific example of how different definitions of a fraction – unit rate, ratio, etc. – can

change the way one approaches a problem scenario (pp. 39-40). Conversely, it seems

likely that different problem scenarios might lead to the creation of different definitions

of fractions. In this way, sense-making exhibits a two-way relationship with the mathematical world – again making no epistemological or ontological claims about the reality or objectivity of such a world. By “mathematical world” the present model refers to the collection of all mathematical objects and their assorted interconnections. In a simplified way, the mathematical world can be considered as rather like a very large and complex concept map. In the model forwarded by the present study, the process by which sense-making alters the way a mathematical thinker perceives the mathematical world is referred to as mathematizing; the use of existing mathematical knowledge to guide sense-making is referred to as justification.

Mathematization

Mathematization is the process by which sense-making influences the mathematical world or, at least, the mathematical thinker’s perception of it. Without mathematization, it would be difficult to argue that mathematical thinking is distinct from generalized thinking. Sfard (2000) notes that “The tendency to always look for real-life situations and to eschew dealing with ‘distilled’ mathematical content is very much in the spirit of everyday discourses, but it contradicts what is often believed to be the very essence of mathematization” (p. 181). It is this conflict of interest that makes it clear that

the two worlds, mathematical and everyday, are distinct. Consequently, this aspect of the

proposed model may be the most vital to robust mathematical thinking; yet, as will be

shown, it is often underrepresented in both academic and professional mathematics.

Before time is spent exploring the skewed role of mathematization, some time must be

devoted to explaining its normal role in mathematical thinking.

The literature discusses two ways in which mathematization occurs. Bonotto

(2005) provides an excellent synopsis of these two ways. First, Bonotto (2005) defines

horizontal mathematization as “the movement from the situations in which the

mathematics is usually utilized to the underlying mathematical structure and vice versa”

(p. 317). In other words, Bonotto’s definition of horizontal mathematization is a decision

about whether or not a given object is mathematical. This is consistent with such

researchers as Dörfler (2002) who suggests, “If the so-called construction of

mathematical objects is in fact to be interpreted as a decision to treat something as a

unified or reified entity, then this decision has to be taken by each learner” (p. 346). As

an example, Dörfler (2002) argues that matrices, algorithms, modulo equivalence classes,

groups and linear combinations can all be seen as both a collection of objects as well as a

virtual singular object (pp. 344-346). Similarly, Simon (2006) argues that the common occurrence of young students believing in a “bigger half” of an object “do not understand that partitioning a whole into two equal parts creates a new unit whose size, relative to the original unit (whole), is determined; that is, they do not understand that one-half indicates a quantity (amount), not just an arrangement” (p. 361). Cramer and Wyberg

(2009) provide yet another example:

[The] inability to think of a fraction as a single number can lead to

misunderstandings when adding and subtracting fractions. The common error

involving adding numerators and denominators for two fractions might make

sense if you think of the numerator and denominator of a fraction as separate

numbers (p. 226).

Such issues of horizontal mathematization even plague expert mathematicians. Juter

(2006) contends that the invention of calculus hinged on the struggle mathematicians historically had with the concept of infinitesimals (pp. 408-411). A third example of this aspect of mathematization comes from Clements, Wilson and Sarama (2004) who argue that geometrical thinking includes:

The child's ability to make pictures or designs by combining shapes (initially by

trial and error, then by considering attributes, and also progressing to leaving

fewer gaps to full tiling at the upper levels); create, maintain, and operate on a

shape as a mathematical unit with measurable attributes; and compose two or

more other shapes, eventually applying a uniting operation to reconceptualize the

composite shape as a new unit that is conceptualized as independent entity (p.

179).

All-in-all, Bonotto’s (2005) conception of horizontal mathematization is the recognition that a given object is or is not mathematical; those objects which are mathematical are accepted into the collection of mathematical facts.

Second, Bonotto (2005) defines vertical mathematization as a “reorganization within the mathematical system” (p. 318). Simply put, Bonotto (2005) believes that

vertical mathematization is structuring the relationships between various objects already

determined, via horizontal mathematization, to be mathematical. As an example,

understanding that ½ is a mathematical object is horizontal mathematization, but

understanding that ½ is double ¼ is vertical mathematization. Similarly, Selden and

Selden (2005) posit that many researchers consider mathematical thinking to be a

structuring process; they add that both mathematization and modeling are types of

structuring (pp. 8-9). Lesh and Harel (2003) likewise posit that what makes mathematical thinking special is the “focus on structural characteristics (rather than, for example, physical, biological, or artistic characteristics) of systems” (p. 159). In yet another reference to the structural nature of mathematical thinking, Nickerson and

Whiteacre (2010) note that “Through collectively solving problems… students came to

consider the quantitative structure of a problem independent of the values given in the

problem” (p. 242) Additionally, in an interview, Sriraman and Lesh (2007) quote Dienes

positing:

I suppose it doesn’t matter very much because most mathematics you learn, if you

understand it, will teach you a way of thinking … structural thinking. Thinking in

structures, how structures fit into one another. How do they relate to each other

and so on (p. 63).

This quote from Dienes is very much parallel to Bonotto’s (2005) concept of vertical

mathematization. As an example of the role of vertical mathematization, Nunes, Bryant,

Hallett, Bell and Evans (2009) suggest that with respect to the problem

“a + b = c; c – a = ?” (p. 63, all emphasis sic), the literature contends that student misconceptions either reflect a breakdown of part-whole relationships or a failure to connect addition and subtraction as inverse operations. Both of these options can be described as caused by the lack of inter-concept organization generated during vertical mathematization. Along these lines, Lesh and Lehrer (2003) posit that some of the actions of mathematization are “[to] quantify, [to] dimensionalize, [to] coordinatize” (p.

123).

The directionality of these two facets of mathematization must be taken cautiously as an indication of their independence rather than any hierarchical ordering.

Rasmussen, Zandieh, King and Teppo (2005) offer an alternative definition of horizontal mathematization as “formulating a problem situation in such a way that it is amenable to further mathematical analysis. Thus, horizontal mathematizing might include, but not be limited to, activities such as experimenting, pattern snooping, classifying, conjecturing, and organizing” (p. 54). Drawing from this, they similarly propose that “vertical mathematizing activities serve the purpose of creating new mathematical realities for the students” (p. 54). A moment’s reflection will reveal that while Bonotto (2005) and

Rasmussen, Zandieh, King and Teppo (2005) describe similar activities, the directionality they associate with each act are the reverse of each other. That is to say, Bonotto (2005) considers vertical mathematization to include organization (Fig. 9), but Rasmussen et al.

(2005) would classify that action as horizontal mathematization (Fig. 10).

Consequently, the present model does not differentiate between the two facets of

mathematization; rather, the present model simply acknowledges that mathematization is a multi-faceted process.

Figure 9 – Bonotto’s Mathematization

Figure 10 – Rasmussen’s Mathematization

Battista (2004) provides some depictions of various phases in the development of

mathematization. Battista (2004) first describes disorganized processing of perceptual

information as nonmathematical. That is to say, random guesses are not mathematical.

Batista (2004) is, in this way, contending that mathematical thinking has an element of

intentionality to it that generalized thinking may not. He then defines the beginning level

of mathematical thought as attempting to structure beyond the unit level but being unable

to cope with non-observable information. In short, Batista (2004) believes that

mathematical thinking begins with the attempt to structure, but he acknowledges that

deductive inference does not occur immediately. Battista (2004) posits a second level of

mathematical though as being able to see the overlap between collections of

mathematical objects but still being unable to cope with non-observable information. At

this point, the mathematical thinker is able to negotiate the fact that a given mathematical

concept may not fall neatly into a single mathematical category. This often becomes

crucial for students, such as first being able to understand that a given fraction is a part of

a whole number but can also be considered a whole that can be further subdivided.

Battista (2004) goes on to suggest that the middling levels of mathematical thought

involve being able to infer non-observable information but to not be able to coordinate it.

In other words, by the time which this much mathematization has occurred, a student is

able to determine that needed elements of a given set are missing. However, the limitation is that the mathematical thinker is not yet able to utilize that object’s presence in another set to complete the first set. Finally, Battista (2004) posits that the peak level of mathematical thought is evidenced by the ability to organize the data, including

inferred data, utilizing all categories simultaneously. In summary, Batista (2004)

describes the development of mathematical thinking in terms of increasing completeness

and coherence of the mathematical world.

The one major challenge of mathematization is that it is a very laborious part of

mathematical thinking. There is simply too much mathematics to be actively

mathematized by any one individual, and the benefits of such an exhaustive effort are

dubious – particularly in light of Gödel’s work on the incompleteness of mathematics.

Consequently, mathematical thinkers, out of sheer practicality, simply need to rely on the

work of others. Issues of validating the source aside, the current model refers to this as

pre-mathematized information. As Dörfler (2002) puts it, “In the learning of

mathematics the student is confronted with numerous decisions that have been made by

others” (p. 348). However, not all students find pre-mathematized information to be satisfactory. The following passage is cited in Sfard (2000), which conveys the struggles of even a gifted mathematical thinker:

I thought that mathematics ruled out all hypocrisy, and, in my youthful

ingenuousness, I believed that this must be true also of all sciences which, I was

told, used it.… Imagine how I felt when I realized that no one could explain to me

why minus times minus yields plus.…That this difficulty was not explained to me

was bad enough (it leads to truth, and so must, undoubtedly, be explainable).

What was worse was that it was explained to me by means of reasons that were

obviously unclear to those who employed them.

M. Chabert, whom I pressed hard, was embarrassed. He repeated the very

lesson that I objected to and I read in his face what he thought: “It is but a ritual,

everybody swallows this explanation. Euler and Lagrange, who certainly knew as

much as you do, let it stand. We know you are a smart fellow.…It is clear that you

want to play the role of an awkward person …”

It took me a long time to conclude that my objections to the theorem:

minus times minus is plus simply did not enter M. Chabert’s head, that M. Dupuy

will always answer with a superior smile, and that mathematical luminaries that I

approached with my question would always poke fun at me. I finally told myself

what I tell myself to this day: It must be that minus times minus must be plus.

After all, this rule is used in computing all the time and apparently leads to true

and unassailable outcomes (Stendhal, as cited on p. 158).

Moreover, Ridlon (2009) contends that “low achievers seem unable to ‘understand’ mathematical strategies they are simply ‘shown’” (p. 192). This seems to indicate that the development of mathematical thinking requires a blend of both pre-mathematized, for expedience, and mathematized information, for understanding; though, it would seem that the focus should be more on mathematization, particularly where understanding is not yet robust enough to process pre-mathematized information. This idea will be revisited in a later chapter, where the practical implications of the current model on curriculum and instruction will be discussed.

Summary

In short, mathematization does two things. First, it acts as a gatekeeper to the

mathematical world. That is to say, the process of determining whether something is or

is not mathematical is one of mathematization. Second, mathematization acts as a

cartographer for the mathematical world. Mathematization is, in this way, the perpetual

cognitive activity of categorization of and connection between mathematical concepts.

With every newly admitted fact, connections need to be made or broken and the relative

importance of various concepts needs to be evaluated. Without mathematization, a

mathematical thinker must rely entirely on the pre-mathematized information provided by

others to populate the mathematical world. Of course, without the rest of mathematical

thinking, mathematization would be a Sisyphean task.

Justification

The present model defines justification as the process by which sense-making is influenced by that which the mathematical thinker has previously acknowledged as being part of the mathematical domain. It might seem to the reader that “proof” might be a better choice of terminology. After all, Edwards, Dubinsky and McDonald (2005) define

“advanced mathematical thinking” to be “thinking that requires deductive and rigorous reasoning about mathematical notions that are not entirely accessible to us through our five senses” (p. 17). However, Edwards, Dubinsky and McDonald (2005) suggest that

Newton’s invention of calculus may not have been “deductive” or “rigorous,” but it might still qualify as advanced mathematical thinking (p. 22). Likewise, Liu and Niess

(2006) cite Polya (1954) to note that Euler self-reported that he utilized inductive

evidence to support his mathematical thinking. These high profile examples open the possibility that even the highest levels of mathematical thinking might fail to meet the definition of “proof.” In this way, Fischbein (1999) posits that what makes mathematical thinking special is the “symbiosis between rigor and constructive liberty” (p. 51). He uses the adverbs “freely,” “imaginatively” and “constructively” to describe the process.

Similarly, Olive (1999) explains a progression of understanding from “initial” to “tacit” to “explicit. This seems to indicate that Olive believes mathematical thinking to be a process in which assumptions are made, reflected upon and either changed, removed or legitimized. Moreover, proof may not be developmentally appropriate at all ages. Stein and Burchartz (2006) posit, “Many younger students tend to use actions as part of their reasoning…. If we accept those actions as part of the final argumentation, we get a complete and logically valid argument” (p. 85). The implication here is that, if held to the standard of rigorous proof, children might not be said to engage in this facet of mathematical thinking. Therefore, the current model relies on a broader term than

“proof.” Certainly, there exist many words might be well suited to the role. Liu and

Niess (2006) cite several authors to define mathematical thinking as a, “combination of complicated processes involving guessing, induction, deduction, specification, generalization, analogy, reasoning, and verification” (p. 375). All of these together, though, could be considered attempts to justify oneself, thus the use of the term

“justification.”

It would not be unreasonable to suggest that a model such as presented herein should hold to the pinnacle of mathematical thought, regardless of the practical

limitations of the human condition. That is to say, why should an idealized model of mathematical thinking include anything less than the most perfect aspects of mathematical thinking? After all, Stylianides and Stylianides (2008) posit that “proof is fundamental to doing mathematics – it is the basis of mathematical understanding and is essential for developing, establishing, and communicating mathematical knowledge” (p.

104). Given this stance, only “proof” could truly count as mathematical thinking.

However, it is not clear that “proof” is either the beginning or the end of mathematical thought. Referencing Bell (1976), Rodd (2000) explains, starting with his personal experiences learning mathematics:

Another gap between proving and knowing occurs when a student may be able to

execute a proof but does not have an epistemic (knowledge producing) connection

to the result. For example, there is a standard proof by induction for the formula

for the sum of the first n integers that, as a 17-year-old student, I was able to

reproduce very fluently, but my own production of the proof did not show me that

the proposition was true (neither did it show me, again using Bell’s terms, why or

how it was true!). A similar lack of connection was expressed by a 14-year-old

student recently about the algebra she was doing: “I know what to do but not what

it means!” Going through the motions required by a proof does not guarantee that

the function of those motions is realized; the proof may not warrant (p. 225).

Even Stylianides (2007) agrees, seemingly in opposition to the earlier quotation, that

“One cannot make sense, or examine the validity, of arguments and proofs unless one understands the (stated or unstated) assumptions that underlie them and support their

conclusion” (p. 362). To understand the potential limitation of proof, it is necessary to

examine a passage where Stylianides and Stylianides (2008) refer to their earlier research

to define it:

Proof is a mathematical argument, a connected sequence of assertions for or

against a mathematical claim, with the following characteristics:

1. It uses statements accepted by the classroom community (set of accepted

statements) that are true and available without further justification;

2. It employs forms of reasoning (modes of argumentation) that are valid and

known to, or within, the conceptual reach of, the classroom community; and

3. It is communicated with forms of expression (modes of argument

representation) that are appropriate and known to, or within the conceptual reach

of, the classroom community (as cited on p. 107, all emphasis sic).

In short, Stylianides and Stylianides (2008) consider proof to be a special form of justification that adheres to community norms, allowing the combined works of mathematicians to function as a coherent collective. Consequently, proof may hold little

or no influence over an individual mathematical thinker; Rodd (2000), for example,

posits that students often fail to see the need to prove something that they already believe

(pp. 226-227). Weber (2010) goes so far as to suggest that it is possible to “complete a

major in mathematics without proper understanding of proof” (p. 308). Fortunately, the

literature also provides some insight as to why this may be the case.

To begin with, Weber (2010) notes that there is some debate in the mathematical

community as to what counts as convincing and what counts as proof (p. 312). Rodd

(2000) cites various researchers to define aspects of proof as justification, illumination,

systematization, verification, discovery, explanation and communication; he appends

intellectual challenge to the list; altogether, he categorizes these as proof that, proof why

and proof how (pp. 224-225). In other words, Rodd (2000) suggests that proof has

multiple roles in the mathematical thinking process. Proof can do more than merely

verify truth; it can also clarify the linkages between mathematical concepts – codifying

one thinker’s mathematization such that another might adopt it. Stylianides (2009)

describes proof in a similar way, citing de Villiers (1998, 1999) to posit that proof can be

used to communicate mathematical knowledge or to organize an axiomatic system (pp.

268-269). However, Raman and Fernández (2005) note that Jensen (2003) noticed that many math teachers had no experience of proof as explanatory (p. 260). If such is the case, it would make sense that there is a disconnect between proof and belief for many mathematical thinkers. More on the role of belief will be discussed in a later chapter on the mathematical disposition. Proof would not be the epitome of mathematical thinking but merely part of the absorption of pre-mathematized information. Izsák (2004) seems to imply that once mathematical thinking is completed, the mathematical objects are

“beyond justification” (p. 58). Yet, if proof is considered to be the culmination of mathematical thought, then it would explain why Harel and Sowder (2005) observe that when a teacher presents a proof of a theorem, students cease to question and reason about it (p. 45). The influence a teacher has over the development of a student’s beliefs will be discussed in more detail in a subsequent chapter regarding the mathematical community.

However, in general, Sriraman (2005) reports that Burton (2004) considers such

instruction as presenting the discipline as “dead” (p. 172).

Sadly, the literature makes it clear that students are hesitant to generate their own justification. Lubienski (2002) contends:

More lower SES students consistently said that having a variety of ideas proposed

in discussions confused them. In general, the confusion centered around feeling

unable to discern which of the various ideas proposed in a discussion were

sensible, or as Dawn put it, “I get confused, ’cause you don’t know if this is right

or this is right ’cause they don’t agree.” Most lower SES students said they

preferred more teacher direction; they wished I would just “show how to do it” or

“tell the answer” (p. 114).

Such student comments only serve to reinforce the bi-directionality of the current model.

Without a sense of how to mathematize information and argumentations presented by the community, these students subsequently reject the concept of individual justification entirely. This is consistent with Weber (2010) who cites several authors to posit that

“students at all levels have serious difficulties with many aspects of proof, including the construction and evaluation of proof and understanding the role that proof plays in mathematics” (pp 306-307). Lesh and Harel (2003) provide a good description of the ideal interplay between mathematization and justification when they posit that mathematical thinking involves the ability to test and refine ideas as well as to invent mathematical constructs (p. 175), which just goes to show that the self-reflective nature of generalization does not apply only to abstractions. As Gravemeijer (2004) suggests:

The RME-guided reinvention heuristic is connected with mathematizing; the

students invent by mathematizing. The idea is that the students not only

mathematize contextual problems - to make them accessible for a mathematical

approach-but also mathematize their own mathematical activity, which brings

their mathematical activity at a higher level (p. 115).

However, as generalization was already detailed in the previous chapter, it is sufficient to simply reiterate – generalization can apply to any aspect of mathematical thinking that is processed by sense-making.

Summary

Justification, while not entirely unique to mathematical thinking, does consist of some forms not seen elsewhere and is, likewise, considered of more importance than in most other domains of human reasoning. Rigorous mathematical proof gives humans such a profound sense of access to universal truth that has led individuals such as

Bertrand Russell and Alfred North Whitehead to try to use it as the basis for all human understanding. Yet, proof tends to be self-limiting as the works of Kurt Gödel and

Raymond Smullyan have used proof to demonstrate the shortcomings of proof.

Consequently, one must understand that mathematical thinking is not limited entirely to syllogistic forms of argumentation. Informal and empirical forms of justification can be just as important, especially for someone who is not a professional mathematician. Any form of mathematical justification is ultimately limited by the norms obeyed by and beliefs held by those producing and receiving the arguments.

Chapter Summary

The mathematical world is, for many, the be-all and end-all for mathematics and, consequently, mathematical thinking. It is no surprise that mathematicians would want to emphasize the features of mathematics that are the most strikingly different from other domains of thought. However, not everyone agrees on what belongs to the mathematical world, and not everyone values a given proof equally. Mathematization and justification can, then, only provide structure and meaning insofar as they are applying the rules and principles invented elsewhere. If the mathematical world were the entirety of mathematical thinking, students might well be right in their claims that there is no use for it.

CHAPTER V: MATHEMATICAL COMMUNITY

Figure 11 – Mathematical Community In this chapter, and the subsequent one on the mathematical disposition, the present model begins to stray from the traditional portrayal of mathematics. Dörfler

(2002) contends that the dominant view of mathematics is such that math is believed to be a cognitive process, though he singles out Sfard (1991) as a significant exception (p.

340). Likewise, D’Ambrosio asks rhetorically, “Where have mathematicians' imagination, sensibility, and inner life gone?” answering “The reaction I usually hear to these comments is, ‘But mathematics is the quintessence of rationalism’” (p. 140). Here, it is clear that many believe that mathematics is somehow transcendently above such

89 90

human foibles, likely the result of the long-incumbent Platonism. Perhaps that is the case, but Mathematical Thinking and Learning is rife with examples of researchers arguing to the contrary; from its very first issue, the journal has included a significant socio-cultural perspective. Cobb (1999) immediately focuses on the “increasing emphasis” on socio-cultural perspectives with respect to mathematics, stating “My immediate goal is to clarify how we analyze students' mathematical reasoning as acts of participation in the mathematical practices established by the classroom community” (p.

5). Cobb (1999) adds that it is natural, though false, to assume mathematical reasoning is a solitary cognitive process; he relates his own research experiences:

In my own case, for example, my colleagues and I initially intended to analyze

students' mathematical reasoning in purely psychological terms when we began

working intensively in classrooms 12 years ago…. Incidents that occurred at the

beginning of the first teaching experiment, which was conducted with 7-year-old

students in a second-grade classroom in the United States, led us to question our

sole reliance on an individualistic, psychological orientation…. as a consequence

of their prior experiences in school, the students assumed that their role was to

infer the responses that the teacher had in mind all along rather than to articulate

their own interpretations (p. 7).

In other words, the students had been indoctrinated into the belief that there is a “right” answer to mathematical questions. Consequently, mathematics, for these students, is a game of trying to read the teacher’s mind rather than to make use of their own. Similarly,

Drodge and Reid (2000) posit that “emotions play a positive and central role in

mathematics and that it is through social discourse that people come to be

mathematicians” (p. 249). As time goes on, the socio-cultural perspective in research

persists (Bowers & Nickerson, 2001; Whitenack, Knipping & Novinger, 2001 – just to

name two in the next volume).

The modus operandi of this dissertation is to accept, at least as mostly true, the

various research traditions that exist in the literature and to juxtapose them.

Consequently, the socio-cultural views already expressed as well as those yet to be

discussed significantly altered the present model’s final form. However, it is understood

that some may not yet be convinced of the salience of the socio-cultural view. I admit to having experienced significant cognitive dissonance when I first began to encounter articles that expressed such view; and, because of my own struggles with facing my preconceived notions about mathematics, I think that it is valuable spend a bit of time explaining how I came to accommodate my scheme for mathematical thinking in light of this new perspective.

Two book reviews by Sriraman (2005, 2009) provided valuable footholds for conceptual analysis. Sriraman (2005) opines that Burton’s (2004) book, Mathematicians as enquirers: Learning about learning mathematics, is convincing about the socio-cultural nature of mathematics, but that the book fails to situate “mathematics within a larger linguistic framework” (p. 177). This leads naturally to the question of what exactly is the nature of the mathematical community and how does it relate to other communities? Sriraman (2009), in reviewing the Handbook of Educational Psychology

(2nd edition), notes that Bishop (1988) defines mathematical intelligence in socio-cultural

terms as “(1) playing, (2) designing, (3) locating, (4) explaining, (5) counting, and (6)

measuring” (p. 180). This leads to a second important conceptual inquiry – specifically,

how can actions such as counting be considered social and, furthermore, how do such

social actions lead to the development of mathematical thinking?

The answer to the first question is, unfortunately, not well answered in the

literature and is, as a result, not directly addressed by the proposed model. Just as it is

difficult to precisely define the nature of mathematics such that all would agree, so too is

it unfeasible to look at any group of people and say that these individuals are the

mathematical community. The most straightforward answer is, perhaps: whoever the

mathematical thinker believes the mathematical community to be. For example, Civil

and Bernier (2006) forward that “Parents do bring other ways of doing mathematics,

some of which we (the authors) have not seen before” (p. 314). Clearly, Civil and

Bernier (2006) consider parents of students in mathematics classes to be part of the

mathematical community, but there are surely those who would disagree. There is likely a connection between who a mathematical thinker believes to be part of the mathematical community and the beliefs that thinker holds about the nature of mathematics. For

example, McCrone (2005) cites Cobb and Yackel (1996) to posit that “The individual’s

beliefs about school and mathematics and that person’s mathematical activities also

contribute to the mathematics knowledge that is constructed” (p. 114). However, it is

beyond the scope of the present study to plumb those depths.

That being said, the literature does provide some insight into the complexity of

the question. Stylianides’s (2007) aforementioned elevator problem that likens integer

addition to going up and down in an elevator (see: p. 57) seems to be an indication that no real world scenario can be fully encapsulated by a finite number of words. Students assumed the man in the problem was either in a rush or wanting to visit friends along the way, based on their real world experience. This dichotomy in student assumptions rings of the socio-cultural nature of mathematical thinking; it would be easy to imagine that a child who had been chastised for needlessly pressing elevator buttons might be averse to a solution that necessitates pressing all of the buttons several times. In this way, totally non-mathematical cultural expectations might influence mathematical thinking.

Conversely, individuals not normally considered part of the mathematical community can and do try to influence the development of mathematical thinking in children. Van den Heuvel-Panhuizen and van den Boogaard (2008) cite Anderson (1997) to posit that “the study revealed that parents who are neither mathematics educators nor teachers can engage their [preschool] children in mathematical talk and activity” (p. 347).

In a related vein, Anderson and Gold (2006) provide two excellent examples of this behavior – a student learning a strategy for solving jigsaw puzzles from watching her mother and a father asking his child questions such as “How many more blocks till we get the shop?” (p. 270). Sadly, not all such influences have a positive impact on the development of mathematical thinking. Remillard and Jackson (2006) suggest that enculturation can be so pervasive that it passes from one generation to the next; in the case of rapidity of solution, they quote one grandmother recalling the work of her granddaughter, Aisha:

She would try to do all these little sticks…and it would take her forever, you

know, to get the multiplication. I’m like, hold up Aisha, multiplication is memory

…. 3 times 5 will be 15 from now and forever (p. 252).

In some ways, almost anything can stimulate mathematical thinking in a community of people. Nasir (2002) reports that in high school basketball, “a player’s individual stats profile was of great importance” (p. 232). Nasir provides some evidence that the desire to have good basketball statistics was a common discussion point. Thus, this first line of conceptual analysis begins to demonstrate the need for the human element of individuality that will be discussed in the following chapter.

The second line of conceptual inquiry generated by Sriraman’s (2009) book review, in a complementary fashion, offers the beginning of a justification for the need to include interpersonal relationships in the overall structure of mathematical thinking.

Nasir (2002) suggests that “opposition identities” are an indicator of the two-way relationship of culture and mathematics (p. 214). In other words, some students are aware of the mathematical community’s desire for a certain degree of conformity and consciously choose to reject those notions and actively avoid any acceptable behavior.

Such students provide examples, albeit pathologically, of the two ways in which the mathematical thinker interacts with the mathematical community. Turning this example inside-out, as it were, one might envision the eager-to-please student who might, in stark contrast to the oppositional student presented by Nasir (2002), obey the rules set forth by the teacher with such blind obedience that, again, mathematical thinking might be strangled by impulsive reliance of formulas like a flower surrounded by hardy weeds.

Whether the community is fellow sixth graders working in a group, the

classroom mathematics teacher, or the attendees of the most recent professional research

conference, a mathematical thinker must account for certain socio-cultural standards, which the present model refers to as normalization. As has been demonstrated by the dichotomy of the defiant student and the obedient student, normalization requires attaining a complex, dynamic and fragile balance between accepting and rejecting the behaviors and declarations of community members. Additionally, a mathematical thinker has the potential to reciprocally influence the community, which the present model refers to as contribution. The remainder of this chapter will focus on these facets of mathematical thinking.

Normalization

Dörfler (2002) contends that the word “mathematics” is like words such as “love” in that they are intangible, but he adds that it is also different from words like “love” in that “mathematics” is not as readily understood by people (pp. 341-342). The point here

is that, if mathematical thinking did not have a social component, it would be as

immediately understandable to all mathematical thinkers in some visceral way. Referring

again to Wittgenstein’s beetle, if math were a purely individualistic activity, there would be some sort of metaphorical box into which the mathematical thinker could peek. As a contrasting example, a human has no need of others to understand feelings of hunger or fear. Such concepts are immediately available to a person. The same cannot be said of

mathematics, at least not without indoctrination. Common misconceptions, such as the

illusion of linearity, could not possibly be so prevalent if humans had, to put it crudely, a

sixth sense for mathematics. Moreover, even among mathematicians, there is some room for miscommunication, as Stylianides (2007) notes:

“Every rectangle is a trapezoid.” Depending on what one takes the definition of a

trapezoid to be—“a quadrilateral with at least one pair of parallel sides” versus “a

quadrilateral with exactly one pair of parallel sides”—the statement is true or

false, respectively. In each case, the statement can be verified or refuted using

different proving strategies such as the development of a direct proof or the

construction of a counterexample, respectively (p. 380).

Even among experts, there is no inherent consensus. As Stylianides (2007) argues, your mathematical knowledge can be utterly dependent upon how you were inducted into the mathematical community.

While mathematics may be a formal, purely logical language, Bjuland, Cestari

and Borgersen (2008) cite Lemke (2003) to argue that mathematics cannot be learned

save through natural language (p. 273). Even supposing mathematics was a perfect

language, the idea is that no human could ever learn it perfectly because of the limitations

of their native tongues. In this way, Novick (2004) cites Dreyfus and Eisenberg (1990) to

argue that some “diagrams rely on ‘conventions, notations, generalizations, and

abstractions without which the diagram is unintelligible’" (p. 311). In other words,

mathematical thinking, even were it capable of being developed in isolation, would not

be recognizable to an outside observer without requiring some socio-cultural element,

and normalization can, alternatively, be considered the mathematical community’s efforts

to make the mathematical thinking of a given individual sensible.

Han and Ginsburg (2001) argue that “language may provide a cognitive tool that

can play an important role in shaping mathematics learning and thinking” (p. 202). Han

and Ginsburg (2001) go on to explain how the Chinese counting words are more

internally consistent and more closely linked to base-10 place value than English counting words; moreover, they add that Chinese mathematical terms tend to be compound Chinese words, which provide more meaning “on the ‘face of the words’” (p.

204) than the English tradition of utilizing Greek or Latin compound words for terminology. As another example, Moschkovich (2002) argues that learning the language of mathematics sometimes involves overriding everyday meanings of words:

One example is the word prime, which can have different meanings, depending on

whether it is used in “prime number,” “prime time,” or “prime rib.” In Spanish,

primo also has multiple meanings; it can mean “cousin” or “prime number,” as in

the phrase “número primo” (p. 194).

Certainly, the same could also be said of the relationship between “x” as a letter, as a variable, as an unknown and as an operation. Likewise, normalization is in effect every time a student must adapt to some new conventional representation. Even such elementary concepts as that a number line gets bigger as one progresses to the right is a social convention. Along similar lines, Mitchell (2001) notes that:

students sometimes made a substitution of a word or phrase in the original

problem statement. The substitutions discussed here appeared insignificant

because the natural language meaning overlapped with that of the replaced word

or phrase, but they had major consequences mathematically because they affected

the mathematical structure that modeled the resulting problem statement (p. 30).

Mitchell (2001) gives an example of this act of this “wordwalking” behavior:

Terri: [Reading] Pam’s car has a gas tank that holds 15 gallons. The car gets 20

miles to the gallon. How many tanks of gas will it take for a trip of 900 miles?

…

Terri: [To her partner, from notes and memory] OK. Pam’s car holds- um- fif-

Pam’s gas tank holds 15 gallons of gas- of course, and- gets 20 miles to the

gallon? How much gas will it take for 900- 900 miles- a 900-mile trip? (p. 35, all

emphasis sic)

Even without the complications of natural language, Mitchell (2001) alludes to Gödel’s incompleteness theorem to argue that mathematics is not as unambiguous as people would like to believe (pp. 43-44).

In its most distilled form, this argument harkens to the well-known Sapir-Whorf hypothesis, which is a belief that human perception is directly influenced by human language (Swoyer, 2003). Several of the model articles make similar claims. For one,

Drodge and Reid (2000) pronounce, “Just as it is an error to see reasoning and emotioning as separate, languaging and emotioning must be seen as two views of one process of cognition” (p. 253). This declaration ties together not just individual cognition and social norms but also identity. Drodge and Reid (2000) basically argue that just as it is fallacious to consider a person’s mind without their body, so too is it fallacious to consider a communicative act without both a speaker’s intent and the listener’s

expectations. Bowers and Nickerson (2001) take this idea a step further and refer to both a “collective orientation” and an “individual orientation” and suggest that learning mathematics may involve the negotiation between the two (pp. 2-5). Another argument akin to the Sapir-Whorf hypothesis comes from Sfard (2000), who suggests:

The student who arrives in a mathematics classroom is supposed to learn

participation in a discourse that, so far, was inaccessible to him or her, and this

means, among others, getting used to acting according to a new set of

meta-discursive rules. The new discursive behaviors of the learner develop

gradually as a result of classroom interactions (p. 171).

In other words, Sfard (2000) believes that learning mathematics is a process of learning to communicate mathematics. In this way, Izsák (2004) reports that the norms employed by the classroom teacher in his study were different than other teachers have used, which

“led to possibly different standards for classroom mathematical practices becoming taken-as-shared” (p. 42). Moreover, Martin, Towers and Pirie (2006) cite Sawyer

(2001a) to argue that an improvisational performance, such as in jazz, only seems to progress naturally because each action serves to guide and limit the possible next choices based upon the rules of dialog (p. 159). This might explain why mathematics is so often seen as a universal language; its rules, in many cases, uniquely determine an acceptable next action. However, Stroup, Ares and Hurford (2005) caution:

As long as we hold on to notions of a [math as a] “universal language” as a gold

standard, there is little or no room for linguistic diversity to matter in and of itself,

100

or for the generative potential of varied expressive forms of language and

participation to be realized (p. 200).

A final example of this linguistic perspective can be seen in the work of Lesh and Lehrer

(2003) who explain:

[W]hereas some cognitive theories speak about these interpretive or descriptive

systems as if they resided totally within the minds of students, models and

modeling perspectives recognize that to have sufficient power for dealing with

realistically complex problem-solving situations, relevant conceptual systems

usually must be expressed using a variety of interacting media that may range

from spoken language, to written symbols, to diagrams, to experience-based

metaphors, to computer-based simulations (p. 111).

The complexity of cognizing from so many sources may explain why mathematics has historically been so difficult to learn. Bjuland, Cestari and Borgersen (2008) cite Duval

(2006) to posit that “Mathematical objects are not accessible by perception or by instruments. They are only accessible through signs and semiotic representations” (p.

274), and Maher (1999) notes that Davis believed a significant part of mathematical thinking was negotiating the inherent ambiguities of these mathematical symbols (p. 86).

This process takes time and experience to develop; Whitenack, Knipping and Novinger

(2001) suggest that a conventional symbolic solution to a problem was not meaningful to the students who had not been exposed to it previously whereas more empirical solutions were (pp. 79-80). Rodd (2000) defines mathematical thinking as including “the ability to follow perspicuous deductive arguments and to have the sensibility that such arguments

101

are paradigmatically convincing” (p. 231), yet research shows students are not always capable of following through on their beliefs.

Lannin (2005) reports that students had agreed, as a class, that “empirical justification” was not very good reasoning, but that students still used it frequently (p.

251). As an example, Lannin (2005) relates an interchange between two students. In the interchange, Dirk is a student who had just moments before exhibited conceptual understanding of the task at hand, and Brett is a fellow student that Dirk is trying to help.

Rather than explain his solution, Dirk encourages Brett to fiddle with the formula in the spreadsheet, seemingly at random to Brett.

(Dirk erases Brett’s formula and enters = A6 * 4.)

Brett: I don’t get why you’re doing that.

(Dirk presses return and the value in the formula cell changes to 420 as the

reference cell, A6, has 105 in it.)

Brett: Okay, I got 420.

Dirk: Then you know you did it wrong. Now do one.

Brett: One

[The exchange continues in this instrumental fashion for a while]

Brett: I got it.

Dirk: You did? Now, what is it?

Brett: It is right.

Dirk: Now press in 3 and see if it is right.

Brett: It is right. I am good.

102

Dirk: You are good. We are good as a team! (Lannin, 2005, pp. 244-245)

So, while the teacher has been able to elicit strong justification from Dirk, it was not instilled as the acceptable norm for Dirk’s mathematical thinking. (Lannin, 2005, p. 244).

It is clear that the only criterion for success being used by the two boys is the “empirical justification” that the computation provides the expected answer. In this way, the teacher plays an important role in the development of normalization in mathematical thinkers.

This scenario could have provided a teachable moment in which the teacher could question why Dirk did not use the same justification with Brett as had been provided to the teacher. Similarly, the teacher could also have presented the question of how one knows that the expected answer is actually correct, be it wrong through typographical error on behalf of the publisher or some other cause. In either case, this interchange would have been an excellent opportunity to reinforce the socio-mathematical norm that correctness is determined by justification rather than by appeal to an answer key, a teacher or even the smart kid next to you who already has an answer.

Such teacher reinforcements of socio-mathematical norms can have profound effects on the mathematical thinking of students. Ridlon (2009) includes an interview of twin girls in sixth grade classes. “Tess” was in a traditional class and “Pat” was in a problem-centered learning class. The conversation that follows, involving the twins answering questions asked jointly by their mother and the interviewer, clearly relates the powerful impact of classroom norms on mathematical thinking:

Mother: What does it mean when you do math in your sixth-grade class?

103

Tess: To me it means problems, problems, problems that I can’t do. Everything is

done with rules and hard steps. You block everything out and watch very

carefully. (Very thoughtful) It means doing what the teacher says.

Mother: She has a low grade because she has to listen to every word the teacher

says. She doesn’t have anyone to talk to, not even the other students. And she

doesn’t always understand what the teacher says.

Interviewer: Pat, what about you? What does “doing” math mean to you?

Pat: What I do is think of a plan. To make it clear. I look at the question to see

how to do it the easiest way. I think about all the ways to do stuff. I’d look at

everyone’s answer to see what they did. But I still trust my own answer if I

thought it was right. I’d let people challenge me. I know if I’m right because I

figure it out for myself.

Mother: What goes through your mind when you see a math problem?

Pat: Well, I would try to figure it out the best I could using what I knew. I’d use

some of the quick ways the teacher or the other kids showed me how to do it. If

we were in groups, I’d talk to my partner or ask for help. One of the groups can

always figure it out every time. Sometimes the teacher showed us a short-cut (p.

219).

The girls in this vignette have the same home, even the same DNA, yet their mathematical thinking diverges significantly after just a short time in different mathematical communities. That is the power of normalization in action, and that is why

104

it and the mathematical community are indispensible parts of the present model of

mathematical thinking.

With such an understanding of normalization, McClain (2003) explains that her

instruction of pre-service teachers focused on highlighting the limitations of a purely

symbolic representation of quantity by utilizing questions that required an enactive or

iconic representation to answer meaningfully, noting, “Had I started with the symbolic

forms, I conjectured that those preservice teachers who needed the drawings would

attempt to decipher the symbols devoid of any real understanding” (p. 291). These

preservice teachers had been so indoctrinated by previous educational experiences to

answering in certain, normalized ways that they overlooked what McClain (2003)

considers as being the deeper, more important mathematics. The literature makes clear

that the preservice teachers described by McClain are only the tip of the proverbial

iceberg when it comes to egregious misconceptions about the nature of mathematics. Liu

and Niess (2006) report that nearly half of students considered mathematics to be about

deriving answers and applying formulas prior to instruction, and they add that the few

students who mentioned logic and reasoning tended to seem as if they were parroting

rhetoric (p. 383). Similarly, Weber (2010) cites a variety of research that indicates

students have many non-mathematical beliefs such as that symbolic proof is superior to

narrative proof or that who writes the proof makes a difference (p. 310).

That students have misconceptions is not unexpected. Surely, every discipline must cope with misinformation, but the important research question is the origin of such beliefs. In other words, are such misconceptions the result of a lack of normalization or

105

of a malignancy in the process? The literature, unfortunately, points the finger at the latter option. De Bock, Verschaffel and Janssens (2002) cite Greer (1997) to suggest that students may be encultured to not fully engage their mathematical thinking when being assessed at school (p. 70). For example, Peled and Segalis (2005) cite Resnick and

Omanson (1987) to suggest that students can be in a “symbol manipulation mode”

(p.209) when working with manipulatives, nullifying their effectiveness as learning tools.

To make matters worse, De Bock, Verschaffel and Janssens (2002) go on to suggest that schools may train students to associate certain problem formats with particular solution procedures (p. 70). Harel and Sowder (2005) argue that key word instruction, such as that “all together” implies addition, are limiting and encourage a computational perspective of mathematics (pp. 42-43). As McCrone (2005) observes:

Whether implicitly or explicitly stated, the teacher has authority in setting the

rules for interaction and for requiring use of appropriate terminology. Students

may, at times, negotiate the specifics of a rule, such as requesting time to check an

idea with a classmate prior to sharing with the whole class. However, students

most often are left to interpret acceptable speech acts through trial and error,

based on the teacher’s actions (p. 115).

However, not all blame should fall to teachers. There is significant evidence that such misinformation is inherited by teachers from a source of greater authority – the textbook.

Alibali, Knuth, Hattikudur, McNeil and Stephens (2007) cite McNeil et al. (2006) to contend that one possible explanation for the difficulty students have learning the meaning of the equal sign is that:

106

middle school curricular materials often present the equal sign in contexts that

support an operational interpretation, such as the “operations equals answer”

problem structure (e.g., 38 + 27 = ), which is compatible with the interpretation

that the equal sign means “find the total” or “put the answer” (p. 235).

In other words, even the best teaching may be overwhelmed by apparently contrary evidence presented in textbooks. Citing Cramer and Post (1993), De Bock, Verschaffel and Janssens (2002) explain that the missing-value problem format is far more commonly used by textbooks for linear reasoning rather than polynomial reasoning. To make matters worse, it would be perfectly natural, due to limited content knowledge or lack of preparation time or any number of other reasons, for a teacher to present material almost verbatim from the textbooks. In fact, Asquith, Stephens, Knuth and Alibali (2007) cite

Nathan and Koedinger (2000) to contend that math teachers often acquire the belief “that symbolic reasoning precedes verbal reasoning…. [because of] the influence of traditional textbook organization, in which ‘word problems’ are presented after comparable symbolic problems” (p. 252). While Peled and Segalis (2005) do argue that “traditional algorithms can be harnessed to promote conceptual knowledge” (p. 208), they acknowledge that these are not the ideal conditions. The mathematical community, intentionally or not, presents a great deal of non-mathematical norms to students learning to be mathematical thinkers. Sfard (2000) suggests that a key obstacle to students coming to professional mathematical discourse is their lack of opportunity to perceive mathematics as a “consistent,” “coherent” and “harmonious system," which stems from the incomplete and disjoint presentation of school mathematics, in Sfard’s eyes (p. 182).

107

As a point of reference, Sfard (2000) posits that mathematical discourse has “rules

that govern the content of the exchange, and… [meta-rules] which regulate the flow of the exchange” (p. 161); the term “meta-rules” is similar to that of norms, though Sfard

(2000) distinguishes between the two. She defines meta-rules such that they:

manifest their presence in our instinctive choice to attend to particular aspects of

symbolic displays (e.g., the degree of a variable in algebraic expressions) and

ignore others (e.g., the shape of the letters in which the expressions are written)

and in our ability to decide whether a given description can count as a proper

mathematical definition, whether a given solution can be regarded as complete

and satisfactory from mathematical point of view, and whether the given

argument can count as a final and definite confirmation of what is being claimed

(p. 167).

As an example of meta-rules, Sfard (2000) enumerates the following: “The belief in proof as a formal derivation to form axioms, and the mathematicians’ prerogative to establish axiomatic systems in any way they wish, provided the systems are free of contradiction, belong to this category” (p. 173). To combat the current ineffectual learning environment described in the previous paragraph, Sfard (2000) suggests reducing the overall number of meta-rules and being very explicit about those which we, as a community, select for preservation (pp. 184-185). In other words, successful mathematics instruction needs to help students become normalized into the mathematics community while avoiding such pitfalls of the hidden curriculum; and, to do that, we need to try to significantly reduce the overwhelming presence of non-mathematical norms in the mathematical zeitgeist. As

108

an aside, it is worth noting that Bessant (2001) expresses concern regarding in what manner such beliefs are defined, by whom and whether they change over time (p. 318).

This concern is given credence as Stylianou and Silver (2004) explain that picture-as-proof was once an acceptable mathematical practice until it “fell into disfavor in the nineteenth century when it proved misleading in several cases” (p. 354).

Summary

The simple fact of the matter is that mathematics, like any human endeavor, requires the establishment of some sort of standard by which communication can proceed. Without the conventions of a decimal number system and Arabic numerals, the symbol “10” could as easily represent two as ten or even be outright meaningless. In this trivial sense, this is normalization in effect. However, the present model contends that obedience to cultural expectations goes far beyond mere language and grammar.

Moreover, the present model contends that mathematical thinking can be both helped and hindered by such expectations and that such expectations often arise unintentionally.

Were normalization to proceed unchecked by other facets of mathematical thinking, such as justification, mathematics would quickly devolve into the rote application of formulas and graphs and would likely also fragment into multiple disciplines. Only the generalization that such rules are often selected arbitrarily is one capable of unifying branches of mathematics that rely on mutually exclusive assumptions. In this way, the influence of the community on the individual is significant, but not absolute.

109

Contribution

Of course, there is no mathematical community without the individuals who comprise it. Moreover, though mathematics certainly can be conducted in solitude, Lesh and his collaborators frequently contend that the purpose of mathematics is often to make something sharable, reusable or modifiable (Lesh & Lehrer, 2003; Lesh & Harel, 2003;

Lesh & Yoon, 2004). Doerr and Tripp (1999) posit that the social acts of “sharing and refining ideas” may be an extension of the creation of models in the process of generalization (p. 232). Doerr and Tripp (1999) suggest that an individual’s internal model and the model in use by a given mathematical community can be related but are not necessarily identical – adding that the communicative act tends to alter both of the models (p. 233). A commonplace example of this in practice is the spontaneous generation of informal terminology that often becomes a standard in a classroom setting.

For example, a student might use the term “the two cars axiom” to refer to a numbered axiom in a geometry textbook useful for determining the distance between two cars travelling perpendicularly away from each other. That student’s internal model depends on the concrete context of the two cars. In turn, that student’s use of the impromptu term might influence the mental models of other students in the class. In fact, Martin, Towers and Pirie (2006) seem to suggest that such improvisational interaction helps the participants to escape creative doldrums because it inherently requires altering one’s own thinking and doing to accommodate the group (p. 173). Specifically, Doerr and Tripp

(1999) argue that a key factor in how such communicative acts may influence mathematical thinking of various individuals or groups is in noticing discrepancies

110

between the perceptions of the members of the dialog; they give the example of one

student asking clarification question of another (p. 244). Bjuland, Cestari and Borgersen

(2008) even go so far as to cite several researchers who consider the simple act of

pointing to be a key aspect of mathematical discourse during problem solving.

The idea of discourse also provides an important opportunity to elaborate on the

difficulty of formally defining who is a member of a given mathematical community.

Martin, Towers and Pirie (2006) report that “we found it impossible to develop clear

mappings of the growth of understanding for individual learners” (p. 152). In contrast,

they suggest that the development of the group at large was easier to observe. Part of the

problem, for Martin, Towers and Pirie (2006) is that the model on which their research is

based can apply equally well to individuals as to groups (p. 155). This leads Martin,

Towers and Pirie (2006) to cite Davis and Simmt (2003) in order to provide an

explanation of the validity of this idea in the form of the construct of distributed

cognition. Drodge and Reid (2000) similarly cite Oatley (1996) to tie into the construct

of distributed cognition (p. 265). Basically, the idea of distributed cognition is that a

group of individuals cooperatively engaged in mathematical thinking can be considered a

single gestalt entity, thus causing the aforementioned difficulty in delineating the

mathematical thinker from the mathematical community. Martin, Towers and Pirie

(2006) define the word “coaction” as a way to describe the behavior of such a gestalt

entity, which can be seen as “as a means to describe a particular kind of mathematical action, one that whilst obviously in execution is still being carried out by an individual, is also dependent and contingent upon the actions of the others in the group” (p. 156).

111

Martin, Towers and Pirie (2006) provide the example that, at times, “The students complete one another’s sentences, but more than that they seem to be speaking with one voice” (p. 164). In this way, Leonard (2001) relates an instance in which a low-achieving student with good spatial skills, Karen, was able to help the higher-achieving students in her heterogeneous group who were, in turn, able to provide confirmation that her suggestion was a good one despite Karen’s misgivings. The individuals, kept separate, might never have completed the problem. However, the students’ complimentary skill sets worked in conjunction so the group, as an entity, could proceed.

Such instances of collaboration can also serve to reinforce the role of normalization, sometimes even to the detriment of the mathematical thinker. Rasmussen,

Zandieh, King and Teppo (2005) argue that “the need for notation and symbolism arises in part as a means to record reasoning and serves as an impetus to further ... students’ mathematical development” (p. 57). Grades can be just such an impetus, and Jansen

(2008) contends that “discussions provide teachers with an opportunity to engage in informal assessments of their students’ thinking” (p. 70). Reinforcement on the importance of normalization is something that often seems to be missing. Civil and

Bernier (2006) quote a mother recalling her frustration being taught mathematics with the expectation of blind obedience to conventional methods:

I have trouble seeing well, why would you use this kind of math for this, says

who? You say that because we want to know the difference, you subtract, or you

want to know how much it is together, then you add, so then you say that if you

want to find a ratio, you cross multiply, says you! And when you want to find the

112

proportion you divide!? I know this is when you lose people; it’s when you lose

me! How do you know when to do what for what? It’s very confusing sometimes

(p. 321)

Without the understanding that a mathematical thinker has something to offer back to the

community, normalization can seem very oppressive. Oppressive norms may not be the formal intent of the mathematical community, but they often accrue over time through the hidden curriculum. Watson and Chick (2001) emphasize the need for norms about contribution, such as the need to express doubt, the occasional need to abandon ideas, and the positive role disagreement can play in group work (p. 169). Bjuland, Cestari and

Borgersen (2008) likewise posit that posing questions and making suggestions are important parts of mathematical thinking (p. 272). Unfortunately, Gutiérrez (2002) laments:

[E]ven proponents of equity issues tend to frame their arguments in ways that

suggest that benefits move from mathematics to persons and not the other way

around. The assumption is that certain people will gain from having mathematics

in their lives, as opposed to the field of mathematics will gain from having these

people in its field (p. 147).

This trend may be result of the Platonic view of mathematics as somehow perfect and beyond the taint of us mere mortals, and it may be exacerbated by the de-emphasis on the history of mathematics in the mathematics curriculum. Perhaps if it was more widely known how influential minority groups have been to the development of mathematics

113

have been, people might see the converse flow of benefit. Along these lines, Verschaffel et al. (1999) describe an ideal classroom culture:

1. Stimulating pupils both during the small-group activities and the whole-class

discussions to articulate and to reflect on their personal beliefs, misconceptions,

problem-solving strategies, and feelings with respect to mathematical application

problems.

2. Debating on new norms about what counts as a good mathematical problem

(e.g., "Many problems can be interpreted and solved in many different appropriate

ways"), a good response (e.g., "Sometimes it is better to respond with a rough

estimation than with a precise numerical answer"), or a good solution procedure

(e.g., "In case of uncertainty even an expert problem solver may count on his

fingers").

3. (Re)defining the role of the teacher and the pupils in the mathematics

classroom (e.g., "Don't expect the teacher to decide autonomously which of the

generated solutions is the optimal one; this decision is taken by the whole class as

a community of practice after an evaluation of the pros and cons of all distinct

candidate solutions") (pp. 203-204).

Unfortunately, this can be difficult for even the best teachers to accomplish; there is often a lot of socio-cultural inertia to overcome in the development of such a classroom.

McCrone (2005) provides such an example, in a section entitled Episode 2 which regards student discussion of a different though potentially equivalent solution to a problem:

114

Bethany: In some ways it’s up to the teacher…whether you want to accept the 6

and the 1, or just the 6, or none at all.

Mrs. M: So the teacher would make that decision?

Bethany: Yes.

Mrs. M: Do I make that decision when I…you put it up and I say “yes, I accept

it,” or “no, I don’t accept it”? Or does it happen somewhere else?

Bethany: I think it (decision making) happens somewhere else …. In some ways,

it’s the teacher’s decision whether they want to count the problem (p. 123).

Clearly, the teacher, Mrs. M, believed she was encouraging a norm of negotiation, but the student, Bethany, was operating on more deeply entrenched norms of deference to authority. McCrone (2005) notes, “The discussion in Episode 2 clearly identifies at least one student’s expectation for the teacher to determine the validity of a mathematical solution” (p. 123). McCrone adds, “Mrs. Miller [the teacher] noted her surprise at

Bethany’s confession because Mrs. Miller felt she had been clear with the students about her expectations that all members of the classroom were responsible for determining the validity of solutions presented” (p. 124). In this way, contribution may be the most difficult to elicit aspect of mathematical thinking, given the nature of the American public educational system.

Summary

Lest mathematics become stagnant, it is vital that mathematical thinkers continue to generate new mathematical understandings and applications. If such discoveries are kept private or, worse, never made, then mathematics would become little more than a

115

static body of facts to be memorized. Sadly, the means by which to perpetuate mathematical thinking, contribution, is vastly undervalued by most math teachers. It would not surprise me if research that follows this dissertation reveals that contribution is the least common element of mathematical thinking in the k-12 classroom. As has been shown, even teachers who try to encourage the act in their students are not always successful.

Chapter Summary

To put it bluntly, mathematics would likely not be recognizable to us as such were it not for the strong and ubiquitous socio-cultural conventions that direct it, collectively referred to herein as the process of normalization. Likewise, there would not be nearly so much known of the mathematical world were people not actively working together either in parallel or in series. That is to say, engaging in acts of contribution. In these ways, it becomes clear that the mathematical thinker is not solely a rational thinking machine that processes quantitative data, but also a living entity in an ecosystem of other mathematical thinkers. Mathematical thinking may not focus primarily on social and cultural elements, but both social and cultural elements are significant components thereof.

CHAPTER VI: MATHEMATICAL DISPOSITION

Figure 12 – A Mathematical Disposition Sinclair (2004) posits that mathematical thinking includes an aesthetic component because some mathematical ideas “could not be derived by logical steps alone” (p. 264).

This has been explored, to some extent, by the discussion of social norms imposed by the mathematical community. However, many of those impositions still have a fairly rational underpinning. Sriraman (2005) suggests that several mathematicians have described mathematical thinking as composed of: “(a) the intuitive ability of mathematicians to somehow choose or construct a viable combination of thought patterns to produce new results and (b) the aesthetic appeal of mathematics” (p. 172). These two assertions seem

116 117

to indicate that mathematical thinking is neither an entirely conscious process nor entirely a rational one, and many of the authors in the meta-analysis discuss such possibilities.

For example, Watson and Mason (2006) argue that mathematics is “subject to interpretations arising from past experience and social, cultural, environmental, and mathematical dispositions and practices” (p. 92). Similarly, Bisanz, Watchorn, Piatt and

Sherman (2009) cite Siegler and Stern (1998) to suggest that children sometimes engage in subconscious mathematical thinking (p. 12).

The present model suggests that the development of a mathematical disposition –a set of personally held beliefs, interpretations and frames of reference – is an indispensible or, perhaps more rightly, an inextricable part of the process of learning to be a mathematical thinker. If the reader has any doubt that people develop such a disposition, Remillard and Jackson (2006) report that the fear and confusion associated with school mathematics often persists well into adulthood (p. 243). In contrast, Presmeg and Balderas-Cañas (2001) note that the successful problem solvers in their study relied on their “mathematical self-confidence” (p. 311). This dichotomy presents clear evidence that such development does occur, and the present model suggests that the relationship is bi-directional with sense-making. As a mathematical thinker makes sense of mathematical data, some of it becomes engrained within the psyche of the mathematical thinker – a process herein referred to as internalization. In other words, internalization can be thought of as the process of developing a mathematical identity or perspective.

118

It should be noted that this use of the term “internalization” probably the most contentious vocabulary choice in the whole model. It is, herein, used consistently with the Vygotskian use of the word. However, not all authors in the meta-analysis used

“internalize” this way. Batista (2004), for example, uses “internalize” in a cognitive fashion rather than a socio-cultural fashion:

For arrays of squares, student's operating on internalized mental models of the

arrays need to draw squares to correctly enumerate them; students operating on

interiorized mental models can correctly enumerate squares ... without drawing

the missing squares (p. 193).

In Batista’s (2004) usage, the word “internalize” is used dichotomously with “interiorize” to delineate whether students merely remember the layout visually or have a more conceptual, mathematical understanding. This terminology choice, then, might alienate some from the use of the present model, but Simon (2006) suggests that the research dichotomy between socio-cultural perspectives and cognitive perspectives is false – particularly that ambiguous terms such as “internalize” obscure the overlap between the two paradigms (p. 364). Both paradigms use the term to describe a sort of permanent absorption. The key differences lie in the types of things absorbed and, to a lesser extent, the completeness of the absorption. In the cognitive model, the absorption is more about content knowledge. In the socio-cultural model, the absorption is more about norms. The present model allows for both to be internalized. Rather than distinguish between what is absorbed, the present model distinguishes between that which is “known” which is the process of mathematization and that which is “believed”

119

which is internalization. This distinction is made because of research, such as that

conducted by Weber (2010), which demonstrates that there is sometimes a disparity

between what a mathematical thinker believes personally and what that same person

believes mathematically.

In the opposite direction as internalization, existing elements of a mathematical

thinker’s disposition often creep back into the process of sense-making, be it subtly or

overtly. This subconscious influence most readily comes in the form of affective

response but also includes non-logical insights. That is to say, a sense of frustration

regarding lack of progress towards a solution and a hunch regarding the viability of the

present solution method could push a mathematical thinker towards abandoning a

problem or persevering, respectively. It need not be emotional to be subconscious

influence, simply not actively reasoned. Sinclair posits that a mathematical disposition

contains the particular aspects of interest, pleasure and insight (p. 262). The present

model refers to such “insight” as intuition and considers it to be the sum of the

subconscious influences on sense-making. Also discussed in this chapter will be the

mathematical disposition’s influence via interest towards certain everyday experiences,

motivation, and regarding certain elements or features of the mathematical world via

pleasure, aesthetic evaluation.

Internalization

Sriraman (2005) reports that Burton (2004) demonstrated that mathematicians

come to the profession by a variety of “trajectories,” going so far as to insist that “These trajectories contradict the myth that mathematicians are ‘born’ into the profession. (p.

120

174). So long as mathematicians are believed to be a fluke of nature or nurture, it is perfectly understandable why so many people are convinced that being a mathematical thinker is utterly beyond them. Such research shows that mathematical thinking can blossom in a variety of situations and that it is a gradual process. In this way, Nasir

(2002) posits, “identity is not purely an individual’s property, nor can it be completely attributed to social settings” (p. 219). That is to say, a person does not typically develop in isolation from others; while that person may be a unique collection of experiences, most of those experiences are shared and influenced by other members of a shared culture. The process of internalization is the process by which a mathematical thinker comes to base his or her entire worldview on certain experiences considered to be trustworthy or consistent.

While the mathematical community may reinforce certain ideas and viewpoints through normalization, ultimately, only the individual can choose whether to believe something. In this way, McClain (2003) contends that a computational orientation towards mathematics will not include a personal need to understand mathematics conceptually. In other words, a mathematical thinker who has internalized that mathematics is simply a process of obeying rules of symbolic manipulation is not likely to grasp the importance of subsequent experience designed around the belief that mathematics involves conceptual understanding, possibly even outright rejecting such a possibility. Yet, Rodd (2000) opines that a mathematical thinker is a “needer” of proof (p.

234). So it is not clear whether both extremes of the conceptual/computational dichotomy can be admitted under the same definition of mathematical thinking.

121

As an example of this conflict in action, Lehrer and Schauble (2000) report on the vexing interference of pre-existing beliefs, such as that “many of the [4th grade] children were apparently operating under the assumption that the more ingredients a model had, the better” (p. 64). The students in the study by Lehrer and Schauble (2000) were trying to develop a system for categorizing pictures, and tried to make as many ways as possible to discriminate one picture from the next. To these students, the underlying mathematical concepts of organization, efficiency and the composition of sets were irrelevant to the point of conceptual non-existence. Their mathematical thinking was obstructed by the internalization that “more is better” despite that such is not the case much of the time. In this way, it can be very difficult to internalize new beliefs without also admitting that one’s previous worldview was flawed. Consequently, Nasir (2002) seems to believe that agency – the sense of control over one’s own life – is a key factor to identity development (p. 220). In other words, a mathematical thinker may come to a computational view of mathematics as a result of an underlying belief that his or her only avenue of participation is computation. Referring back to the students categorizing pictures, perhaps the case was simply that they had internalized that they were capable of generating rules and chose to showcase that ability rather than admit there might be some other ability as yet unavailable to them. That is to say, perhaps they were thinking mathematically in the only way that they felt they could achieve success, and the teacher-introduced concepts only served to diminish their sense of success. Given the information available, it is impossible to determine the precise nature of the preconceived

122

ideas that were obstructing further mathematical thinking, but it is clear that there was

such an obstruction in effect.

With the understanding that students will develop some profoundly influential

beliefs about mathematics if left unattended, it is important to reflect on those beliefs that

the community would like a mathematical thinker to internalize. Only through such

deliberate reflection can curriculum and instruction be harnessed to promote an

efficacious disposition. Moreover, such efficacious pedagogy is clearly lacking if, as

Sinclair (2004) remarks, even graduate students may not be fully encultured into the

mathematical aesthetic (p. 268). Liu and Niess (2006) cite Schoenfeld (1994) to include

a “mathematical point of view” (p. 374) as part of mathematical thinking, and Sinclair

(2004) cites several authors to suggest that, while there is not a universal consensus on

the matter, the aesthetics of mathematicians are more convergent than those of artists (p.

266). In the meta-analysis, the most thorough review of such beliefs comes from Drodge and Reid (2000), who use the term “emotional orientation” to describe what the present model refers to as mathematical disposition. Some of the elements of their definition were mentioned in an earlier chapter (see: p. 34) and will not be repeated here; but, fortunately, Drodge and Reid (2000) provide a wealth of other facets of an ideal mathematical disposition:

• Feeling that isomorphic structures are significant…

• Explaining using numerical features of a situation…

• Requiring more than examples for a convincing…

• Desiring precision…

123

• Desiring explanation in the face of lack of understanding…

• Valuing patterns…

• Wise restraint, or sticking to a conjecture unless given sufficient reason to

change one’s position…

• Trusting in the certainty of calculations and other deductive processes…

• Accepting the fallibility of human instantiations of deductive processes…

• Wanting to make connections (p. 263).

Certainly, not that all of these emotional states are necessarily mathematical. “Wise restraint” for example, seems like a hallmark of many kinds of thinking, as do “desiring precision” and “wanting to make connections.” In fact, Drodge and Reid (2000) admit that it is not always clear what emotions are distinctively mathematical (p. 261).

However, it is clear that mathematical thinkers have emotions and subconscious beliefs that are connected to mathematics.

Summary

The line between “true” and merely “correct” is difficult to make, and it is likely that every individual places that line slightly differently. The difference, though, can said to be a matter of personal conviction. Developing such conviction is the domain of internalization. It is difficult to imagine that anyone would invest the time and energy to develop their own ability to think mathematically let alone the countless more hours of investigation necessary to professionally wield their mathematical thinking unless they had something truly valuable at stake. While other elements of mathematical thinking develop meaning in an epistemological or ontological sense, internalization is about

124

developing meaning in a teleological or ethical sense - not merely what exists but what

matters. Of course, without the other elements of mathematical thinking, there is nothing

to value.

Intuition

In analyzing a reform known as the “The Problem-Centered Learning (PCL)

Instructional Approach,” Ridlon (2009) suggests:

This brings us to another reason that PCL might have been effective for second

graders. The PCL approach seeks to promote an environment where students

believe they are empowered to make sense of mathematics for themselves. Studies

have suggested that students benefit from greater engagement with mathematical

content when they are permitted to use their own insights to make meaning rather

than being forced to follow teacher-presented procedures (p. 194, all emphasis

sic).

In this way, Sfard (2000) defines mathematical insight as the ability to extricate oneself from the limitation of mathematical norms in order to come to a solution (pp. 172-173).

Put another way, simple obedience to normative pressures may impinge upon the ability for mathematical thinking to generate new knowledge of, and relationships within, mathematics. Novick (2004) explains how intuition can occur in both the novice as well as the master of mathematical thinking:

Students may be deemed to have implicit, but not explicit, knowledge of a

concept or procedure if they can express accurate knowledge of it on some type of

(nonverbal) performance assessment, but they do not have verbal access to that

125

knowledge. Implicit knowledge may be most prevalent at low levels of expertise

or competence and at very high levels (p. 311)

At one end of the spectrum, students may be able to cognize the task but be unable to frame the desired result into something that can be communicated due to some limitation of experience. That is to say, just because they understand the material at an intuitive level does not mean that they can utilize it consciously. At the other end of the spectrum, experienced mathematicians might have developed such an advanced degree of automaticity that they do not even need to process the mathematical data at a conscious level. In either case, the mathematical thinker might feel that they have insight into a given problem.

Raman and Fernández (2005) indicate a belief that mathematical thinking stems from following through on hunches and guesses “with the autonomy and confidence that is inherent to solving problems on one’s own” (p. 261). It would be easy to deny the legitimacy of Raman and Fernández (2005) on the grounds that their definition of mathematical thinking includes often maligned features of uncertainty and emotion. Yet,

Raman and Fernández (2005) are not alone in their beliefs. Goldin (2000) argues that affect is just one of many ways in which humans represent information, adding:

Expert problem solvers, such as research mathematicians and scientists, often

show signs of intense affect—words, gestures, and facial expressions—during

problem-solving activity. In intimate conversation, they will sometimes

acknowledge not only intense feelings and great passion but also times of terrible

pain and frustration in connection with their mathematical work (p. 211).

126

Another author from the model cases, Liu, explains that he became “convinced that an inductive attitude toward mathematical thinking is imperative in mathematics; that is, deduction and logic do not describe the whole domain of doing mathematics” (Liu &

Niess, 2006, p. 381). So, controversial or not, the idiosyncrasy of an individual’s disposition must play some role in mathematical thinking for so many to acknowledge it.

Sinclair (2004) quotes Aristotle, Russell and Hardy – famous mathematicians all – to demonstrate that the concept of mathematical beauty is by no means new (p. 263).

Moreover, Sinclair (2004) provides some examples in which a feeling of being on the right track or other subconscious affect, such as pleasure or excitement, influenced mathematicians at work (pp. 270-271). Sinclair (2004) contends that mathematical thinkers have an aesthetic reaction, which she dubs intuition, to certain structures, such as parallel lines or symmetry; Sinclair further argues that it is the subsequent confirmation of intuition that leads mathematicians to believe that they are “glimpsing the truth” when engaged in mathematical thinking (p. 273). In this way, Goldin (2000) posits that curiosity is a signal that “divergent” and “exploratory” strategies, particularly to define the problem, are necessary for continued progress in the given task (p. 214). Conversely,

Goldin (2000) states that the sensation of frustration is a signal to the mathematical thinker that his or her current line of reasoning may be unproductive (p. 215).

Despite such ardent proponents of the role of intuition as Sinclair (2004), not all researchers look upon intuition favorably or even as part of mathematical thinking.

Presmeg and Balderas-Cañas (2001) cite Krutetskii (1976) to posit that “all mathematical thinking relies on logic” (p. 290). Similarly, Edwards, Dubinsky and McDonald (2005)

127

talk about intuition as being an obstruction to advanced mathematical thinking; they

describe one student, saying, “Her intuitions collided with her understanding of the

mathematical definitions and her intuitions prevailed” (p. 17). The compromise between

these two seemingly irreconcilable views can be found in Sriraman (2005) who, in

reviewing Burton (2004), portrays intuition as important to mathematical thinking but

“sometimes horribly fallible” (p. 175). In this way, Tirosh and Stavy (1999) contend that student overgeneralizations “are determined by the specific, external characteristics of the task that activate the intuitive rule and not necessarily by students' ideas about the specific content or concepts to which the task relates (e.g., temperature, angle)” (p. 181).

It seems unlikely that even Sinclair (2004) would contend that intuition is infallible, and thus it would seem that intuition can be an impulse that stimulates sense-making but cannot replace it. In this way, Battista (2004) seems to use “intuitive” synonymously with “informal,” saying that mathematical thinking has “levels of sophistication that students pass through in moving from their intuitive ideas and ways of reasoning to goal states of learning” (p. 188).

Summary

There might be the temptation to simply decry intuition as non-mathematical, and are likely those who would. However, to do so would be to insist that mathematics emerges from the aether fully-formed. Such a belief is dogmatically Platonist, and

Sriraman (2005) reports that Burton (2004) discovered mathematicians’ beliefs “took on varying combinations of Platonism and formalism” (p. 174). That is to say, generally speaking, mathematicians do acknowledge that there is a human element to mathematics.

128

Instead, it is more viable to view intuition as the spark plug in an internal combustion engine. The car does not drive without the propulsion of the explosion in the piston cylinder, much as mathematics does not reach its pinnacle without formal proof.

However, the explosion would never initiate without the spark plug. In the same way, it could be said that formal mathematics might not be derived without the occasional gut feeling or half-formed visualization. Of course, the spark only gets you so far. Likewise, intuition is often wrong, and it is the role of the rest of mathematical thinking to point out that fact.

Motivation

Goldin (2000) posits that some emotions, such as curiosity and puzzlement, are important motivating forces to the beginning of the mathematical thinking process (p.

212). Sinclair (2004) goes so far as to cite Weil (1992) in order to describe the process of fully realized intuition as – and this is no exaggeration – better than sex (p. 273). Even supposing that Sinclair’s (2004) statement is hyperbolic, the underlying impetus to seek similar life experiences must be significant to even merit the comparison. Additionally,

Haciomeroglue, Aspinwall and Presmeg (2010) contend that ability and preference for types of mathematical thinking vary independently (p. 159). In other words, these motive forces are not merely in effect for those areas of mathematics where a student excels.

However, Goldin (2000) continues by adding that the intensification of these initial feelings of bewilderment can lead to frustration, anxiety and despair which ultimately lead to long-term feelings of hating mathematics (p. 213). Jansen (2008) cites Lampert,

Rittenhouse and Crumbaugh (1996) to suggest that “Being corrected during classroom

129

discussion felt, for some students, like a personal attack and affected how they felt about themselves and their classmates” (p. 72). To make matters worse, Sfard (2000) notes that a learner not only has the potential to feel these emotions but rather is likely to experience exasperation while learning mathematics (p. 158), and Presmeg and Balderas-Cañas

(2001) report that one participant clearly exhibited affect without mentioning it. In other words, it would be very easy for a student to become demoralized in the process of learning mathematics and not realize it until those unpleasant feelings had become entrenched. As an example, Leonard (2001) describes two groups working on the same task, one homogenous and the other heterogeneous. One key difference between the two groups was that the heterogeneous group had a member that projected confidence from the onset, whereas the homogeneous group had to struggle with expectation of failure before proceeding (pp.185-187). Civil and Bernier (2006) go further, arguing that such demoralization can even become intergenerational:

[parental] encouragement to succeed and take mathematics classes might be

tainted by parents’ personal experiences with mathematics or even lack of formal

mathematical preparation. Helping children with their mathematics homework

may cause some parents stress or feelings of inadequacy due to perceptions of

their own ability (p. 316).

In this instance, the parents may not have even realized how much they had been conditioned to be perturbed by mathematics until they had been re-exposed to it as adults.

Goldin (2000) additionally contends that students often develop defense mechanisms to avoid exposure to the unpleasant emotions that can follow extended

130

frustration. Such avoidance strategies could very quickly inhibit the ability of the student to reach the later stages of mathematical thinking that Goldin (2000) and others suggest produce strong positive emotions. Lest the reader think that education should strive to eliminate negative emotional experiences, it is worth mentioning that Stein and Burchartz

(2006) suggest, “Younger students need many such experiences [of identifying a dead-end] before their sometimes-complete logical analyses lead them to a deeper insight” (p. 81). Watson and Chick (2001) likewise posit:

A change in understanding is often preceded by doubt. Students may feel doubt

about their current perceptions in the light of new evidence or consideration of

some point that had been overlooked previously. This can lead students to a

reexamination of current understanding, a revision of their knowledge, and,

ideally, an improvement in cognitive functioning (p. 137).

Moreover, Goldin (2000) notes that teacher encouragement can help counteract frustration but that a student receives the most pleasure “when it is the solver’s own method that succeeds, not one scripted by authority” (p. 216). To borrow a phrase from educational psychology, Goldin (2000) seems to be saying that mathematical thinking is most well suited to thinkers with an internal locus-of-control.

In the light of a student’s need for a sense of control, it is unsurprising that Nasir

(2002) insists that goal setting and identity formation are “central to learning” (p. 215).

For example, Inoue (2008) suggests, “If students solve mathematical problems by recognizing the pragmatic meaningfulness of pursuing the goal, they may no longer solve the problem to simply get the right answer” (pp. 38-39). Similarly, Lesh and Harel

131

(2003) talk a little about needing to have a goal in mind in order to solve problems,

saying “problem solving has been defined as ‘getting from givens to goals when the path

is not immediately obvious or it is blocked’” (p. 160). In this way, Cai (2005) declares,

“In solving a problem, a solver first needs to formulate a representation of the problem,

based on individual interpretation of the problem conditions. From the representation, the

solver determines goals to achieve” (p. 137). From Cai’s (2005) statement, it might seem

that goal-setting influences sense-making rather than everyday experience. Certainly, the present model allows for some blurring of boundaries between processes; one of the proposed model’s key features is that it is a continuous rather than a discrete model.

However, Nasir (2002) argues that mathematics learning might best proceed when the students doing their mathematical thinking “in the service of a nonmathematical goal” (p.

242). Consequently, a mathematical thinker’s disposition would be directly

influencing – and perhaps influenced by – everyday experience. Such a process is not

inherent to mathematical thinking, hence its parenthetical status in the model using a

dotted rather than a solid line, but it seems likely that such motivational aspects as

goal-setting do regularly occur. Anderson and Gold (2006) relate an instance in which a

three-year-old student intentionally cheated by double-counting in a board game when it

was to his advantage but did not cheat when there was no benefit; they contend that while

the motivation was to win – clearly nonmathematical – the boy was clearly utilizing his

understanding of mathematics to gain a competitive edge (pp. 271-275). Turner,

Gutiérrez, Simic-Muller and Diéz-Palomar (2009) give another example of

non-mathematical goals of identity, calculating the amount of food and water necessary

132

to safely cross the US-Mexican border by foot. Turner et al. (2009) contend that it was a way that mathematics could lead to broader identity development for immigrant students, saying, “In this way, their mathematical investigations contributed to a deeper understanding of the perilous conditions of the journey (i.e., a recognition of how difficult it would be to carry eight pounds of water)” (p. 145). More abstractly, Sinclair

(2004) defines one mathematical affective activity as “playing” in which mathematicians ignore any mathematical goals and simply combine mathematical objects “looking for appealing structures, patterns and combinations” (p. 272).

Summary

Regardless of whether you believe mathematical thinking is a matter of uncovering the truth of the world or merely an elaborately abstract semiotic game, you are likely to agree that the process is not always easy. It often takes monumental motivation to overcome the various obstacles to mathematical thinking, and what such motivation does is encourage the mathematical thinker to explore the world of everyday experience for new data to sift and mull. These motivations can be intensely personal or quite common. The Platonist belief in the ability to discover absolute truth certainly seems more motivating than the formalist game-playing stance, which might explain why the former seems to be more common among mathematicians. Sadly, all the motivation in the world is useless without the effort that follows, which is why, again, this facet of mathematical thinking is useless without all the others.

133

Aesthetic Evaluation

Sinclair (2004) seems to imply that words such as “elegance, harmony, and order”

(p. 263) are words describing mathematical beauty. As a case in point, Presmeg and

Balderas-Cañas (2001) argue that “the strength of the logic underlies the ease and elegance of solution” (p. 291). Harel and Sowder (2005) consider “look[ing] for a simpler problem” to be a mathematical way of thinking (p. 32), and it is would be a natural extension, should the strategy prove to be successful, for a student to come to see simpler solutions as somehow superior. Bonotto (2005), for example, considers a student remark of “That would be too much mathematics!” (p. 337) to be an indication that the student had come to realize the impracticality of exact calculation in certain real-world scenarios. However, Harel and Sowder (2005) argue that students need to practice reasoning in order to learn about broader mathematical ideals, such as efficiency (pp.

40-41).

Efficiency only begins to describe the many ways in which mathematical thinkers prioritize some mathematics over others. McClain (2003) describes better mathematical thinking as “increasingly sophisticated” as well as “increasingly efficient” (p. 286).

Battista (2004) posits that the “levels of sophistication” (p. 186) of mathematical thinking range from informal to formal. Cobb (1999) clearly considers multiplicative reasoning to be more sophisticated than additive reasoning (p. 19). Earlier in the article, Cobb (1999) mentions that unequal data sets are problematic for additive reasoning (p. 17), which would seem to indicate that the phrase “more sophisticated” really means “applicable in more situations.” Along these lines, Bonotto (2005) posits that formal mathematics

134

emerges as a “gradual growth” from informal mathematics (p. 315). However, such evaluative words as “formal” tend to be tied to the researchers’ beliefs about the nature of mathematics, which seems to be a strong reinforcement of the idea that these evaluations are related to but not inherently part of mathematical thinking. As an example, Ginsburg and Seo (1999) define “formalized” as synonymous with “symbolic” (p. 126). This likely is a consequence of their earlier statement that the mathematical perspective is distinct from “personal meaning” – they go so far as to claim that the world is “deeply mathematical in structure” (p. 115). While it is not necessary to rehash the arguments for the existence of socio-cultural aspects of mathematical thinking at this point (see: p. 33), it would seem that Ginsburg and Seo (1999) demonstrate just how deeply mathematical thinkers may internalize the community norm that mathematics is objective and real.

Similarly, Sriraman (2005) mentions a subculture within the mathematical community, the Bourbakists, that “to this day demand[s] a very high degree of rigor in submitted articles” (p. 178). In this way, the mathematical disposition can strongly influence what a mathematical thinker considers to be “good” mathematics, yet it seems only natural that members of a community would want their fellow members to adhere to the standards.

Lehrer and Schauble (2000) describe an instructional sequence that involved a scenario of a stupid robot following the fourth grade students’ rules to help students realize their lack of clarity:

A lot of times we try to do this in factories. There are all kinds of nuts and bolts to

be sorted. If you hire people to do this, it’s very expensive and it’s boring.

Industries try to program robots to do this. But the robot is very, very stupid. …

135

You should be able to take these descriptions and think not about everything you

know but what they really say. Almost as if you were writing a computer

program for a robot. Remember, you’re much smarter than any robot (p. 62).

While some students understood the lack of clarity, others could not see past their own

ability to follow the vague directions (pp. 62-63). This is just another way in which the teacher of mathematical thinking is not only conveying facts, but a frame of reference in which to see the world. That is to say, mathematicians value clarity and precision of language, though mathematics itself does not differentiate between a concise definition and a prolix one. Clearly, the mathematical thinker needs to come to the understanding that what is convincing to him or her, may not be convincing to anyone else; this is what separates aesthetic evaluation from mathematization. While all the various aspects of a mathematical disposition may be internally consistent, without cross reference to community standards, even the process of abstraction may be significantly limited.

Summary

Be it the fixation on the Mandelbrot set or the Fibonacci numbers, certain mathematical objects are clearly held in high esteem than others. This, quite simply, is aesthetic evaluation in action. There is, likewise, nothing inherent to mathematics that makes prime numbers more interesting than any other subset of the natural numbers.

Interest is a human evaluation. However, our fascination with prime numbers drives our organization of natural numbers, going so far as to ascribe to prime factorization the lofty adjective fundamental. Humans value rarity, and prime numbers are seemingly rare.

136

Even in this post-Cantor age when we understand the prime numbers have the same cardinality as the natural numbers, there is a certain reverence for the prime numbers – nostalgia, perhaps. The important note is that mathematical thinking can proceed equally well given the choice between two viable options, yet people do tend to develop preferences. That is all that this element of the model acknowledges.

Chapter Summary

Given a finite number of data points, there are an infinite number of patterns that

“fit” the data. The inclusion of the mathematical disposition is merely a pedagogical version of this mathematical truth. People differ, of that there is no question. However, it is sometimes easy to overlook how these differences may influence mathematical thinking through moments of intuition, particularly if you want to ascribe some sort of objectivity to mathematics. The mathematical disposition includes values and beliefs and, to a lesser extent, the mathematical facts that comprise them. The elements of a person’s mathematical disposition might not always seem mathematical to the outside observer, but that does not preclude them from influencing that person’s mathematical thinking. Perhaps mathematical thinking could proceed without such a disposition, but it certainly does not do so in humans. We far too often engage in the internalization of illogical thoughts.

CHAPTER VII: SENSE-MAKING

Figure 13 – The Central Node The previous four chapters have detailed the present model’s four nodes as sources of raw mathematical data and have provided accounts of how various researchers have described them, focusing on the underlying similarities between portrayals.

However, there has been some discussion, herein, about the limitations of each facet and how other facets can be compensatory. The mathematical thinker is constantly in a metaphorical maelstrom of often conflicting data; the true cornerstone of mathematical thinking as presented by this model is the perpetual process of organizing, interpreting and representing this data – sense-making. Certainly, other terms exist. For example,

137 138

Lesh and Harel (2003) use the phrase “mathematical judgments” (p. 161). The word

judgment, of course, makes one think of a judge. What is it that a judge does? A judge

listens to all the available data and considers it with respect to a set of rules. It could be

that a judge makes sense of evidence. Given the inclusion of intuition and other less

rational processes, the more generic term sense-making seemed more appropriate.

Moreover, variations on the phrase “make sense of” are far more common in the literature. For example, Lesh and Lehrer (2003) seem to use “mathematical models” and

“sense-making systems” interchangeably. While few articles within the present meta-analysis focus primarily on the phenomenon of sense-making – Bonotto’s 2005 article entitled “How Informal Out-of-School Mathematics Can Help Students Make

Sense of Formal In-School Mathematics: The Case of Multiplying by Decimal Numbers” being a notable exception that does actively study sense-making – many of the articles casually or formally allude to the concept.

Just as a short jaunt through the journal’s timeline, here are a selected group of examples that demonstrate the near-constant presence of sense-making throughout the entire run of the journal. In a book review, Kulm (1999) notes that one chapter, “explores the extreme conditions necessary for achieving some of the characteristics of thinking practice rather than imposing meaning on students” (p. 318). The implication here is clearly that students should generate their own meanings. In a similar vein, Whitenack,

Knipping and Novinger (2001) discuss the ways in which students generate their own meanings, noting “Sunita and Marna participated more fully in the activity of unpacking candies to reason with collections in sensible ways” (p. 79). Here, the researchers seem

139

to imply that the students who participated to a lesser degree, and were consequently not

thinking as mathematically as their peers, either failed to reason or did so in insensible

ways. Likewise, Dörfler (2002) seems to imply that mathematical thinking ultimately

boils down to looking at collections of existing objects as new “virtual” wholes – only

some of which “will prove useful and sensible” (p. 343). Moreover, Nasir (2002)

contends that increased training in mathematics may lead to increased ability to make

sense of the world using that mathematics (pp. 233-234). Lesh and Harel (2003), as

another example, argue:

For model-eliciting activities, what is most problematic is that students must make

productive symbolic descriptions of meaningful situations. That is, descriptions

and explanations (or constructions) are not just relatively insignificant

accompaniments to “answers.” They are the most critical components of

conceptual tools that need to be produced (p. 159).

Having reached this point in the meta-analysis, it became clear that the concept of sense-making was more than just ubiquitous; it was foundational. In fact, conceptual

analysis of the opening sentence of Batista (2004) indicates that he uses sense-making

and mathematical thinking almost interchangeably (p. 185). It would be possible to swap

the locations of the two constructs, and the meaning of the sentence would largely be

unchanged. The only difference seems to be specificity – this idea will be revisited in the

following chapter on the practical implications of the present model. Moreover, the

meta-analysis also indicates that the concept of sense-making is not limited to the

140

included articles. Stylianides (2007) cites several authors outside the meta-analysis who favor the concept of sense-making:

First, given the growing appreciation of the idea that doing and knowing

mathematics is a sense-making activity (e.g., Fennema & Romberg, 1999; Hiebert

& Carpenter, 1992; Mason, Burton, & Stacey, 1982; National Council of

Teachers of Mathematics, 2000), explicitness on the role of assumptions can

allow children to understand and examine critically the conclusions that they

accept based on the grounds that support them (p. 362).

If there is one strong consensus throughout the reviewed literature, it is that the quintessential feature of mathematical thinking is sense-making. In a way, everything else in the present model is simply a data-source for the process of sense-making.

Through generalization, which was described in an earlier chapter (see: p. 55), sense-making even applies reflexively to itself.

The question then becomes, what is the nature of sense-making? In one of the rare cases of a direct citation from one model article to another, Watson and Mason

(2006) cite Lesh and Yoon (2004, p. 210) to define sense-making as “identifying objects, relations, operations, transformations, patterns, regularities and quantifying, dimensionalizing, coordinating, and systematizing them, using problem-solving strategies, often organized heuristically” (as cited on pp. 103-104). Likewise, Zahner and

Corter (2010) contend:

that the process of probability problem solving can be resolved into stages that

usually but not always occur sequentially, with repeated shuttling between some

141

of the stages. The stages, in typical order of occurrence, are: text comprehension,

mathematical problem representation, strategy formulation and selection, and

execution of the strategy (p. 195).

To summarize these two quotations, the present model considers sense-making to be the trifecta of interpretation, representation, and organization of data. However, given that these processes tend to occur simultaneously, the model does not generally differentiate between them – much as it does not differentiate between vertical and horizontal mathematization. That being said, the remainder of this chapter will be devoted to exemplifying and justifying the selection of these facets of sense-making.

Figure 14 – Sense-making, a Swirl of Information Interpretation

From the very inception of the journal, the concept of interpretation has been woven throughout the articles in Mathematical Thinking and Learning, with Cobb (1999) mentioning that students have “mathematical interpretations” (p. 7) in the very first issue.

More recently, Lesh and Harel (2003) add, in a footnote, that problem solving “involves finding ways to interpret these situations mathematically” (p. 160). Much of the process

142

of interpretation has been lightly touched upon in the previous chapters because, as

Carreira (2001) argues utilizing semiotic terminology from Peirce (1931), “meaning is always the product of an interpretation” (p. 263). In fact, Lesh and Harel (2003) argue that both Piaget and Van Hiele “considered mathematical thinking to be about seeing (or interpreting) situations at least as much as it is about doing algorithmic procedures” (p.

177). Likewise, Bonotto (2005) explains that the first stage of her expectation of student learning involves “the mathematical decoding of the message contained in the artifact [a product label]” (p. 324). In more detail, Lesh, Doerr, Carmona and Hjalmarson (2003) argue:

“Construction” does not capture many of the most important activities that

students need to engage in when learning mathematically significant conceptual

systems. For example, complex notational systems (such as the Cartesian system)

and symbol systems (such as the language of functions, limits, and continuity) do

not in themselves need to be constructed by each individual (although certain

meanings associated with them may well need to be constructed)…. If such

representational systems are given or “told” to students, the central activity that

students need to engage in is the unpacking of the meaning of the system and the

flexible use of the system in ways that enable them to make sense of their

experiences (pp. 215-216).

Lesh and Yoon (2004) provide an example of students thinking mathematically when they changed their definitions of part and whole to suit the task at hand (pp. 223-224).

The task, which involved a geometrically patterned quilt, was finally solved by the

143

students when they stopped viewing all of the shapes as individual wholes made up of the

common part of inches of fabric but instead realized that the all of the shapes could be

viewed as compositions of the same basic diamond shape. The students had to “unpack”

the scenario to find a meaningful unit of interpretation by which all their subsequent

interpretations could be made As another example, Lobato, Ellis and Muñoz (2003) report that many students in their study referred to slope as “what it goes up by” (p. 11); this phrasing would surely make the concept of negative slope nonsensical. It would similarly require re-interpreting slope as something equivalent to “what it goes up or down by” in order to make any sense of negative slopes.

In this way, D’Ambrosio (1999) opines, “The capabilities of drawing conclusions from data, making inferences, proposing hypotheses, and drawing conclusions from the results of calculations are as important as simply reading data” (pp. 133-134). The ability

to interpret mathematical data can become even more vital as students encounter higher

level mathematics; Garfield and Chance (2000) cite Gal and Garfield (1997) to posit that:

Statistical reasoning may be defined as the way people reason with statistical

ideas and make sense of statistical information. This involves making

interpretations based on sets of data, representations of data, or statistical

summaries of data. Students need to be able to combine ideas about data and

chance, which leads to making inferences and interpreting statistical results (p.

101).

In other words, without the sense-making process of interpretation, statistics could

provide no predictive power, and interpretation can consequently be seen as laying the

144

foundations for both generalization and reification. Interpretation of the data provided by

the other four nodes of mathematical thinking really sets the stage for all subsequent

mathematical thinking. Inoue (2008) refers to his earlier research to describe the way

students initiate problem solving:

For example, in solving the problem that asks if they could get to a metropolitan

airport by car on time, many students solved the problem as if there was no traffic

jam or road construction on the road. However, diverse reasons were found to

underlie the seemingly “unrealistic” answers. Some students were aware of the

real life factors but intentionally ignored them based on the understanding that the

problem solving should not reflect the real life factors. Other students did so

because they interpreted the problem situation differently from the common-sense

understanding (e.g., it is possible to drive alternative routes; you can increase the

speed in the middle by expecting a traffic jam later, etc.). These types of answers

are not totally unrealistic but could be seen to reflect students’ “realistic” efforts

to solve the problems based on their personal understanding of the meanings

involved in the problem solving (p. 37).

Each of the choices a student makes in interpreting the nature of a given problem situation can drastically alter the solution process. While it is abstraction that identifies potentially useful features and properties of a given real world scenario, it is interpretation that decides which features and properties are useful for further mathematical thinking. McGinn and Boote (2003) self-report:

145

Through years of experience in mathematics classrooms, we had embodied

particular techniques and strategies that were conceptual resources for our

solution attempts. For example, after working thousands of algebraic word

problems, we had developed proficiency translating sentences into algebraic

statements (p. 83).

The metaphor of mathematics as a language is, as such, reliant upon the process of interpretation. Even as an objective formal language, which the present model neither confirms nor denies, mathematics is not inherently sensible to the human mind as we typically communicate with it. Fischbein (1999) provides an example of the interpretive nature of mathematical thinking, which he describes as “the metaphorical structure of mathematical language” (p. 55). He goes on to explain that interpreting a mathematical set as “a container” is a better metaphor than “a collection,” but that student errors indicate use of the “collection” metaphor. That is to say, it would seem that one way that the interpretive process of mathematical thinking could be seen is as a process of determining the fitness of a given metaphor. Of course, to determine the fitness of a metaphor, it must first be generated, which leads nicely into discussion of the next element of sense-making, representation.

Representation

Cai (2005) cites a wide variety of articles to argue that mathematics is predominantly tied to matters of representation (p. 136). Maher (1999) similarly reports that noted mathematics education research Robert Davis believed mathematical thinking involves “how representations are constructed within one’s mind” (p. 89). McClain

146

(2003) goes farther, arguing that her study indicates that students have “a need to have visual representations of all the candies [used as a manipulative in the scenario] to be able to perform even the simplest transaction to developed notational schemes that parallel efficient algorithms for addition and subtraction of three-digit numbers” (p. 301). Such a result demonstrates just how vital representation is to the development of understanding.

Tying into the previous section, these representations can be seen as an attempt to generate a model or working metaphor, and such representations take on a variety of forms. As an example, Stylianides (2007) describes a student who seemed to conceptualize infinity as a lot of zeros in a number, “Lucy wrote the number

8,000,000,000,000,000,000,000,000 on the board, commenting that she wanted to add more zeroes to the number but there was not enough space in her notebook” (p. 372).

Izsák (2004) also suggests that determination of the “primitive models” for mathematics, specifically multiplication, is a contentious subject within the literature, with researchers typically jockeying between repeated addition and rectangular arrays (pp. 40-41). In other words, not only is representation a necessary component for mathematical understanding, but it also plays a significant enough role that researchers actively study its influence. One explanation for this may come from Murata (2008) who contends, “If a certain representation is consistently used with instruction, this representation will become a part of students’ mathematical thinking and the foundation for their future understanding” (p. 376).

It is worth mentioning that Stylianou and Silver (2004) report that students are not reluctant to create visual representation but do not typically use them for exploration of

147

the task (p. 372). Specifically, Stylianou and Silver (2004) note that “although novices

were able to anticipate the drawing of a diagram, they knew very little about how to make

diagrams a helpful tool” (p. 380). In fact, McClain (2003) goes so far as to suggest that

the non-presence of drawings can indicate a lack of conceptual understanding (p. 287).

Lest the reader think that representation is purely visual, it should be noted that Martin,

Towers and Pirie (2006) clarify, “By ‘images’ the theory means any ideas the learner

may have about the topic, any ‘mental’ representations, not just visual or pictorial ones”

(p. 151). As an example, Cobb (1999) explains that he wanted students in his

intervention to “view data sets as entities that are distributed within a space of possible

value” (p. 10). Rather than being visual, this representation is spatial. Certainly, other

configurations are conceiveable.

A third camp in the research literature seeks a more inclusive attitude; Bjuland,

Cestari and Borgersen (2008) cite Behr, Lesh and Post (1987) to claim that mathematical thinking involves translating between multiple mathematical representations (p. 274).

Izsák (2004) notes, “students can calculate the area of a three-by-five rectangle by counting 15 individual unit squares, adding three groups of five unit squares, adding five groups of three unit squares, or recalling 3 x 5 = 15” (p. 43); he goes on to explain that each of these options has different useful features that are applicable to different situations. Put simply, this group of researchers believes that every mathematical representation by necessity excludes certain features of the real scenario, thus requiring several representations to create a reasonably accurate portrayal of it. In this vein, Berk,

Taber, Gorowara and Petzl (2009) cite Blöte et al. (2001) to suggest that expertise in

148

mathematical thinking with respect to arithmetic is to “choose from among several

different but equally valid procedures in a way that exploits characteristics of the

numbers involved to reduce computational demands” (p. 114). As an example of this in

action, Bisanz, Watchorn, Piatt and Sherman (2009) contend:

The evidence that even some 3 and 4-year-olds may show sensitivity to inversion

raises the possibility, however, that older students in other studies may not have

been discovering the principle of inversion so much as how to apply it to

problems presented symbolically (p. 20).

In this instance, the students do not yet have the expertise to choose between representations. The students must struggle to adapt their real world infant experiences to the demands of the formal mathematical symbolism; this is in contrast to the assumption

that the students simply do not grasp the mathematics. Mitchell (2001) believes

similarly, “I do not see the relation between natural language and mathematical

representations as being a sharp dichotomy. Rather, on the dimension of ambiguity, I see

mathematics as the extreme end of a process of disambiguation” (p. 45).

Representation, like all the facets of sense-making, is also a point at which

idiosyncrasies of the mathematical thinker can become apparent. In their research,

Haciomeroglue, Aspinwall and Presmeg (2010) utilize two assessment instruments – one

for determining a student’s preference for visual reasoning and a second instrument for

determining that student’s capacity for visual reasoning. These scores formed the basis

of their categorization of students. One student, for example, was categorized as an

analytic reasoner because, “Amy has the ability to process visually but prefers not to do

149

so” (p. 157). Subsequently, Haciomeroglue, Aspinwall and Presmeg (2010) note that the

analytic thinker in their study assumed discontinuity of a piecewise continuous function

simply because it was calculated in two pieces; they add that interpreting graphs posed a

perpetual problem for this student and that her explanations were about determination of

equations. Cai (2005) provides a plausible rationale for this behavior, positing that

representation is based on “individual interpretation of the problem conditions” (p. 137).

Conversely, Haciomeroglue, Aspinwall and Presmeg (2010) note that the visual thinker

did not take advantage of the initial condition f(0)=0 and was, consequently, unable to

precisely identify the minimum point on a graph. In this way, Haciomeroglue, Aspinwall

and Presmeg (2010) reiterate “that visualization plays a significant role for which analytic

thinking alone cannot substitute in students’ thinking” (p. 173).

Organization

Stylianides (2009) posits that mathematical thinking is “the process of making sense and establishing mathematical knowledge: identifying patterns, making conjectures, providing non-proof arguments and providing proofs” (p. 259). While the final two items are part of justification, the first two in that list are what the present model would define as organization for the purpose of sense-making. Pattern recognition requires sequencing and often re-sequencing data, in search of the pattern that in some way seems to the mathematical thinker as if it “should be” rather than merely “could be” – perhaps drawing on an aesthetic evaluation regarding symmetry or prime numbers or efficiency. As an example of this in the literature, Doerr and Tripp (1999) report studying “situations designed to elicit students' thinking about the underlying

150

mathematical structure and patterns” (p. 237). Similarly, Watson and Mason (2006) explain:

What intrigues us, given the very different backgrounds, mathematical

knowledge, goals, and social contexts of the groups with whom we have worked,

is that there is an almost universal global response to the exercise, an articulation

of taxi-cab circles. The structure of the local responses, contributing to the global

response, varies according to the sequence in which the individual tasks are

undertaken. It is the structure of the exercise as a whole, not the individual items,

that promotes individual disturbance and common mathematical sense making (p.

97).

In this instance, the students’ sense-making was bolstered by having the mathematics organized in advance. Consequently, the students simply had to organize the data chronologically, which comes very naturally, in order to produce meaning. When the data is not so self-revealing, the mathematical thinker is required to supply the cognitive effort to reorganize it. In contrast, Lesh and Yoon (2004) note several instances where students became confused by their poor record keeping. In this way, Nasir (2002) describes the mathematical thinking underpinning a game of dominoes such that one player “thought not only about his own play and that immediately following his but also about how that play could affect the flow of pieces and his subsequent opportunities to score” (p. 226). The player essentially would try all the possible combinations such that he could determine which was best; this is a good metaphor for the process of sense-making. As a brief aside, making a conjecture is similar to pattern

151

recognition – except rather than organizing data, it is a matter of organizing premises and

correlations. Altogether, it becomes very clear that without some sort of organization,

mathematical thinking could not proceed. Healy and Hoyles (1999) declare:

For us, mathematical progress is not characterized by the replacement of one way

of knowing by another that supposedly is "higher" or more abstract; rather, it is

characterized by the development and interlinking of different forms of reasoning

that can develop alongside and in combination with one another (p. 60).

The organization of sense-making, in this fashion, lays the foundation for the

generalization and mathematization that so often follow. After all, as Tirosh and Stavy

(1999) argue, “our cognitive system tacitly assumes that if a certain relation exists

between two objects in respect to a certain quantity, the same relation will hold for other quantities as well” (p. 191). Organization, like all the aspects of sense-making, tends to be idiosyncratic. Lehrer and Schauble (2000) use the phrase “a mathematics of classification” (p. 52) – not “the mathematics” but “a mathematics.” That is to say, depending on which objects you choose to organize and by which criteria, you can generate very different mathematics. Again, this is why the present model makes no claims about the particular objects that may be in or out of the mathematical world. In this way, Lesh and Lehrer (2003) argue that Vygotskyan tenets imply that teaching mathematics is “to help students extend, revise, reorganize, refine, modify, or adapt constructs…that they DO [sic] have – not simply to find or create constructs they do not have” (p. 121).

152

Chapter Summary

To summarize, the process of mathematical thinking falls apart without

significant effort invested in sense-making. The common malady of students answering word problems in such a way as to suggest that they would cut a person in half is the perfect example of the ridiculous disconnects that can occur without sense-making.

Central to the process of mathematical thinking are the deliberate and careful acts of interpretation, representation and organization.

CHAPTER VIII: PRACTICAL IMPLICATIONS

If this dissertation contributes nothing else, it supports the notion that it is possible

to create a definition of mathematical thinking that encompasses all current perspectives

in the research discussion. It is admitted that some fine points of epistemology and

ontology are still very much open for debate; but, as has been discussed, these talking

points do not preclude a definition of mathematical thinking that everyone can use such

that the research discussion becomes more fruitful. All-in-all, the model that has been

presented herein should be seen not as a final answer to the question, “What is

mathematical thinking” but rather the development of a lingua franca for subsequent

research.

With that caveat in mind, the results of this conceptual meta-analysis does establish some paradigmatic assumptions about the nature of mathematical thinking. The most fundamental assertion of the present model is that mathematical thinking does not function in discrete stages. While a researcher might point to a given behavior and say

“That means this person is doing some mathematization,” it is impossible, at this point, to rule out the possibility that normalization, for example, is also in effect. That is not to say that this model refutes the possibility that certain configurations of mathematical thinking may be identifiable and consistent, particularly with respect to the developmental process. After all, the frequency with abstraction, generalization and reification occur in sequence was mentioned in Chapter III.

While some age-related data was available within the analyzed literature, there was a general lack of discussion of the development of mathematical thinking in this

153 154

journal. The present model is, as a result, a completed picture of the cognitive structure.

Beyond long standing hallmarks of cognitive psychology, such as Piagetian stages, there

is little in this particular body of literature that explains how any one facet of

mathematical thinking develops. Fischbein (1999), for example, acknowledges the

prominence of Piaget in conceptions of mathematical thinking (pp. 48-49). Although,

Battista (2004) posits that the precise combination of levels through which a student

passes to attain mathematical thinking is idiosyncratic, but he also notes that these levels

can be ranked and used as a “major landmark” (p. 188) in research.

The development of mathematical thinking can be thought of as similar to the

development of the ability to walk. Humans have a biological capacity for bipedal

locomotion, though it certainly varies from walker to walker based upon physiological

and experiential idiosyncrasies. Some people may be natural-born sprinters while others are better suited to long-distance jogs. To extend the metaphor, what I have produced in the present study is develop a picture of walking. The key point is that, generally speaking, people walk. In the same way, people are mathematical thinkers, although one might be more algebraic and the other more graphical. Torbeyns, De Smedt, Stassens,

Ghesquière and Verschaffel (2009) list some limitations on mathematical thinking:

A first possible explanation relates to the children’s limited general cognitive

resources—whether these limitations are conceived in traditional Piagetian terms

(e.g., the lack of reversibility in their logico-mathematical thinking; Piaget, 1965),

in terms of their general processing potentials (such as speed of processing,

attention, and working memory; Demetriou, 2004), or in terms of their

155

metacognitive capacities (Flavell, 1982)—or to their more restricted specific

attentional resources due to the fact that they were still struggling with the

mastery of the DS strategy, as Siegler’s (1998) Strategy Choice and Discovery

Simulation (SCADS) model of cognitive strategy change would predict (p. 88).

As with muscles that atrophy in a sedentary person, so too does mathematical thinking depend on use to grow and thrive. Similarly, just as a person crawls then toddles then walks, so too does the mathematical thinker not attain any one facet of mathematical thought all in one session. Stylianides (2007) cites Davis (1992) to describe the cognitive impact of mathematical thinking as a “residue” (p. 376). This seems to imply mathematical thinking is a process of accretion that produces immeasurably small increases to mathematical knowledge. The development of mathematical thinking, in these ways, is primarily a process of increased facility and automaticity.

In time, a mathematical thinker is able to strive for increasingly complex mathematical thoughts as the cognitive load of any given element of mathematical decreases. In this way, Chiu (2001) cites Anderson (1987) to describe increasing expertise as a condensing of knowledge through transitivity, “So, A→B→C→D becomes

A→D” (p. 96). The cache of mathematics immediately accessible to the individual from memory grows with time, as well. This choice is informed by Murata (2008), who utilizes the model of Tharp and Gallimore (1988) to contend that advanced levels of mathematical thinking involve increased automaticity and reflectiveness (pp. 377-378).

Stylianou and Silver (2004) also suggest that experts have a greater level of automaticity in their mathematical thinking than novices (p. 380). As a brief caveat, the word

156

“individual” is used very loosely here. Lesh and Yoon (2004) claim to be interested in

“problem solving rather than problem solvers… Therefore, it is somewhat irrelevant to us

whether ‘the problem solver’ is a group or an individual” (p. 213). In other words, the

present model does not assume that the mathematical thinker is a single person. It could

just as easily be a small group or a whole class working together as a gestalt entity.

However, in terms of development, the literature focuses on individuals.

As little discussion as there is in this collection of articles regarding development,

there is some. Watson, Callingham and Kelly (2007), as a starting point, describe pre-mathematical thought as being idiosyncratic:

For the Lollies protocol, students were likely to explain outcomes… in terms of

their favorite numbers, of the position of lollies in the container, or of the sizes of

their hands. Similarly for the Spinners Task… explanations for observed

outcomes from trials were likely to be based on egocentric or anthropomorphic

beliefs, for example, suggesting “nine” black outcomes “because I’m turning nine

this year.” (pp. 93-94)

Baroody, Lai, Li and Baroody (2009) cite several researchers to posit that the earliest signs of mathematical thinking occur around the age of three, but they add that children’s thinking tends to remain unreliable and non-logical until roughly five years of age (p. 43).

As a caveat to this claim, they cite Bower (1974) to add that children as young as five months have some perceptual sense of quantitative concepts such as uniqueness and plurality. Similarly, Baroody and Lai (2007) report that children as young as four were capable of grasping the inverse relationship between addition and subtraction, but they

157

add that it is not until the age of six that a significant percentage of children show this inversion capability (pp. 162-163). Stylianides (2007) also agrees that young children can think mathematically, but he cites Bruner (1960) to classify their thinking as

“rudimentary” because of its “local” nature (p. 379). Likewise, Van den

Heuvel-Panhuizen and van den Boogaard (2008) cite Scribner and Cole (1973) in such a way as to suggest that young children may be limited to informal mathematical thinking

(p. 343). With a slightly older age of participant, Lehrer and Schauble (2000) report that the first and second grade students “never really grasped the idea of a model – especially the separation of a model from its referent” (p. 55). Moreover, Lehrer and Schauble

(2000) note that the first and second grade students were “reluctant to reconsider or revise their first groupings upon reflection or the introduction of new cases” (p. 56). In short, first and second grade students, in that study, were able to abstract and mathematize but struggled to generalize, reify, and justify. In fact, Lehrer and Schauble (2000) append that “holistic judgments remained more compelling to many of the children, who found discussion of descriptors less useful” (p. 57).

Other researchers set the minimum bar for successful mathematical thinking much higher. For example, Izsák (2004) cites various research that target fourth grade as the earliest at which students can typically understand rectangular arrays without having already understood it (p. 40). More extremely, Edwards, Dubinsky and McDonald

(2005) define “advanced mathematical thinking” as “the phenomenon that seems to first occur during a mathematics student’s experience in undergraduate mathematics when he or she first begins to deal with abstract concepts and deductive proof” (p. 16). However,

158

Selden and Selden (2005) cite Tall (1991) who “emphasized that ‘advanced mathematical

thinking does not begin after high school’ and that ‘this thinking must begin in the first

grade’” (as cited in p. 4). In fact, Selden and Selden (2005) acknowledge that researchers

are occasionally surprised by the complexity of a student’s mathematical thinking; they

cite an example when one researcher offhandedly admitted as such of an eighth grade

student being researched, saying, “I wish some of my [university] students were able to

reason that well” (p. 6). These results only go to reinforce that the underlying cognitive

elements are generally available to school-age children, even if they are not yet productive as modes of thought. However, productivity is not the only worthwhile goal.

Sriraman and Lesh (2007) quote Dienes opining, in an interview, “Children do not need to reach a certain developmental stage to experience the joy, or the thrill of thinking mathematically and experiencing the process of doing mathematics” (p. 61). Does age improve mathematical thinking? Surely. Vlahović-Štetić, Pavlin-Bernardić and Rajter

(2010) report that, “The younger students (aged 15 to 16) were successful 99.6% in linear and 10% in non-linear problems, while the older students (aged 18 to 19) were successful

100% in linear and 57.1% in non-linear problems” (p. 63). However, it would be doing a great disservice to students to say that they are incapable of mathematical thinking as instances of precocious mathematical thinking has already been provided. To summarize, the present model does not provide a developmental trajectory for mathematical thinking but, instead, acknowledges that mathematical thinking becomes more complex and complete as the individual matures – idiosyncratically approaching the model as time passes with some consistency across large groups of indivudals.

159

Limitations

The present study provides an excellent framework for the continued discussion

of mathematical thinking amongst educational researchers and practicing educators and

even policy makers. However, it is only the earliest sapling of a tree that will hopefully grow quite large and complex. At this point, there is not even a protocol for determining what facets of mathematical thinking may be in effect during a given classroom scene.

Likewise, while the potential to use the present model as a diagnostic tool to help students does exist, it is not yet clear what mathematical thinking should look like in a given population. In this way, the most obvious vein of subsequent research into mathematical thinking would be determining the relative proportions of each facet of the present model at various stages of development as a mathematical thinker and in assorted populations. It is entirely possible that the composition of mathematical thinking conducive to a high-school student in a high socio-economic status family would be different than to an average elementary student from a poor, urban family. As such,

Lubienski (2002) considers:

It might seem reasonable to think that open discussions, in which a variety of

methods and ideas are considered, and open-ended problems that can be solved a

variety of ways (including by drawing from one’s own experiences), would

communicate to all students that their ways of thinking and communicating are

valued. However, could the very nature of a classroom culture that expects

students to share, puzzle over, and judge opposing ideas conflict with the beliefs

or preferred practices of some groups of students? (pp.108-109).

160

The very ambiguity that allows a coherent and inclusive model to be formed also proceeds to point where researchers have made assumptions about how to proceed with curriculum research. That is to say, it is possible that some of our reform efforts have overcompensated for perceived flaws. For example, Ginsburg and Seo (1999) caution that constructivism may be ignoring vital aspects of mathematical thinking; Ginsburg and Seo (1999) believe it is not sufficient to encourage students to construct mathematical knowledge but that it is also necessary to direct them towards formal mathematics (p. 127). The present study simply is not capable of determining whether such is the case.

A second major limitation of this study is the selection of literature to be analyzed. The present study, being a meta-analysis, cannot make claims about aspects of mathematical thinking that were not discussed in Mathematical Thinking and Learning.

That is to say, some aspects of mathematical thinking have been researched more than others. Certainly, it would seem that generalization has gotten a great deal of attention thus far. However, in trying to assemble other sections, such as on abstraction or internalization, there was a less complete picture, as is evidenced by their shorter length and relative lack of specificity. In this way, Goldin (2000) posits that there is not sufficient theory on affect in mathematics education, particularly on: ”(a) beliefs and belief structures, (b) attitudes, and (c) emotional States… also (d) values, ethics, and morals” (p. 210). In several cases, it became clear that certain processes do occur but not how. For example, Martin, Towers and Pirie (2006) cite Becker (2000) to discuss the evolution of collective understanding that seems to follow “obviously better” ideas; they

161

declare “In mathematics, ‘better’ is likely to be defined as an idea that appears to advance

the group toward a solution to the problem, the drawing on a concept that seems

appropriate and useful in the present situation” (p. 174). The process by which students

track their progress towards the completion of a given problem is left unexplored; and,

given the wealth of anecdotal evidence regarding how easily students get off-task, there is some question as to the reliability of this mechanism.

In retrospect, there are many avenues of research that would follow logically as immediate next steps to the present study. For example, it would be very helpful to do a detailed analysis of collections of curricula that were contrasted against each other. If curriculum A produces higher achievement than curriculum B, does curriculum A actually cover a broader or more efficacious mixture of mathematical thinking? The meta-analysis conducted in this dissertation also seems to indicate that research is needed that regards the perceived membership in the mathematical community. That the community exists is not in doubt, but do parents have a different perception of membership in the mathematical community than mathematicians do, for example?

Alternatively, there is a need to expose the hidden sociomathematical norms that have become so pervasive in the mathematical community, especially in K-12 math classrooms. Where along the way do talented math students get exposed to the concept that symbolism is better than visualization? Could a simple insistence by a teacher that the pronunciation of a number is “one hundred eighty-five” as opposed to “one-hundred and eighty-five” degrade the ability of a student to comprehend place value? That such norms lead students to skew their mathematical thinking is similarly not questioned.

162

Rather, the present model does not explain which norms seem to have the most impact and whether they have predictable influence over students. Likewise, this model assumes that students objectify, dimensionalize and categorize their everyday experience, but the present model is unable to explain on what basis those perceptual filters are applied. To some extent, this dissertation could lead into questions of cognitive psychology as well as sociology just as much as pedagogy. The important thing to remember is that any research articles that choose to adopt the model of mathematical thinking presented herein will be able to be used together as part of a larger research discourse, given that this dissertation defines the construct where they overlap.

Discussion

Beyond the basic assertions regarding the value of the term “mathematical thinking” laid out in the early chapters of this dissertation, some time must be spent expounding the particular benefits of the present definition beyond those to the research discussion. Generally speaking, there are two other main benefits of the present model – philosophical and pedagogical. The philosophical benefit of this definition of mathematical thinking comes from a corollary increase in the understanding of the nature of mathematics. That is to say, by understanding how humans cognitively engage with

mathematical objects, we gain some insight into the nature of those objects. An alien

archeologist, to make a somewhat tongue-in-cheek analogy, might better comprehend the

nature of a treadmill by watching an exercise video in which someone walked on it. In

this way, time will be spent in this section considering the question, “Is statistics part of

mathematics?” by utilizing the present model to analyze the related question, “How is

163

statistical thinking related to mathematical thinking?” The pedagogical benefits of the present model are numerous, and several will be discussed. Among the pedagogical points that will be presented is an attempt to explain the cultural differences in mathematics education between the United States and China, which are often highlighted in international studies.

Statistical Thinking

Given that the most recent issue of Mathematical Thinking and Learning in the meta-analysis (v. 13, i. 1-2) happens to be a special issue on “informal statistical inference,” it would seem natural to assume that statistics is a sub-field of mathematics, much as astrophysics is a sub-field of physics. However, Ben-Zvi (2000) cites Garfield and Gal (1999) to posit that “Today’s leading statistical educators see statistics and mathematics and statistical and mathematical reasoning as quite distinct” (p. 129). More recently, Gil and Ben-Zvi (2011) enumerate mathematics as among “other disciplines” than statistics, including science and psychology (p. 87). Similarly, Langrall, Nisbet,

Mooney and Jansem (2011) differentiate between statistical knowledge and mathematical knowledge (p. 51). In fact, Makar, Bakker and Ben-Zvi (2011) cite a variety of research to argue, “Unlike pure mathematics, where the context is set aside to focus on the underlying abstract structure, in statistics, data cannot be meaningfully analyzed without paying careful consideration to their context” (p. 155). From this small selection of data from the meta-analysis, it would seem that the initial assumption about the relationship between statistics and mathematics might very well be incorrect. However, Makar,

Bakker and Ben-Zvi (2011) are clearly operating out of a definition of mathematical

164

thinking that denigrates everyday experience and a definition of statistical thinking that does not include deductive justification. As has been mentioned (see: p. 29), there do exist community members who seek to obfuscate the role of everyday experience in mathematical thinking, particularly at the advanced levels. However, the present model also makes it very clear that those norms are misrepresentative of the scope of mathematical thinking. If the present model is accepted as the definition of mathematical thinking and, by implication, of mathematics, then the distinction made by Makar,

Bakker and Ben-Zvi (2011) is prime for rebuttal. With careful study, it becomes clear that virtually every facet of the present definition of mathematical thinking finds parallel in the literature’s discussion of statistical thinking.

Beginning with one of the aforementioned articles that seeks to delineate between mathematics and statistics, Langrall, Nisbet, Mooney and Jansem (2011) cite a variety of research to describe statistical thinking as the interplay between statistics and everyday experience (p. 50). This depiction is consistent with the cognitive axis of the present model and only becomes more apparent as the researchers explain their results. For example, Langrall, Nisbet, Mooney and Jansem (2011) remark, “What we did observe were conversations in which students… identified data that would be useful to examine”

(p. 60). This phenomenon could easily be described as part of the abstraction process presented in the current definition of mathematical thinking. In this way, Dierdorp,

Bakker, Eijkelhof and van Maanen (2011) contend, “Learning about correlation and regression seemed meaningful to students because they recognized its role in authentic situations” (p. 147). Here the act of reification is in play. Langrall, Nisbet, Mooney and

165

Jansem (2011) also note that students engage in both formal and informal acts of

justification (pp. 60-61); this is likewise parallel to the present model’s definition, down

to the very name of the act. Moreover, the statistics that has been discussed thus far is in

application of rather than generation of methods of data analysis. When one considers

that common statistical methods such as Student’s t-test have their basis in

well-established mathematics such as the distance formula, there is also the

understanding that formal deductive justification undergirding statistics, not simply

inductive reasoning.

Another of the articles that asserts a fundamental distinction between mathematics

and statistics can be used to highlight their very similarities. Makar, Bakker and Ben-Zvi

(2011) cite Marak and Rubin (2009) to define statistical inference as:

1. A statement of generalization “beyond the data,”

2. Use of data as evidence to support this generalization, and

3. Probabilistic (non-deterministic) language that expresses some uncertainty

about the generalization (p. 154).

This definition provides a strong basis for claiming that statistical thinking, like mathematical thinking, is a process of sense making; it even defines “generalization” in

very similar terms as the present model. Gil and Ben-Zvi (2011) similarly declare that:

Informal Inferential Statistical Reasoning (IIR) [emphasis sic] refers to the

cognitive activities involved in informally drawing conclusions (generalizations)

from data (samples) about a wider universe (the population), while attending to

the strength and limitations of the sampling and the drawn inferences (p. 88).

166

Granted, generalization is only one piece of sense-making, and sense-making itself is not

named. However, Madden (2011) cites Ben-Zvi and Garfield (2004) to define statistical reasoning as:

The way people reason with statistical ideas and make sense of statistical

information. This involves making interpretations based on sets of data,

representations of data, or statistical summaries of data. Statistical reasoning may

involve connecting one concept to another (e.g., center and spread), or it may

combine ideas about data and chance. Reasoning means understanding and being

able to explain statistical processes and being able to fully interpret statistical

results [emphasis added] (as cited on, p. 110).

By simply replacing the word “statistical” with “mathematical” and “reasoning” with

“thinking,” the above could potentially be a definition of mathematical thinking. It is very near, indeed, to the present definition of sense-making, mentioning interpretation and representation by name. Likewise, Mooney (2002) defines “statistical thinking” as

“describing, organizing and reducing, representing, and analyzing and interpreting data”

(p. 25). Jones et al. (2000) cite Shaughnessy et al. (1996) to congruently define statistical thinking as consisting of “describing, organizing and reducing, representing, and analyzing and interpreting data” (p. 271). Incessently, the concepts of the sense-making processes of organization, representation and interpretation are mentioned. It seems futile to argue that the sense-making aspects of mathematical thinking are absent in statistical thinking.

167

Even the socio-cultural aspects of mathematical thinking can be found if one looks past the surface of the statements of one of the very authors who denounces the overlap between the two disciplines. Ben-Zvi (2000) argues:

Thinking of statistics as a liberal art balances its essential technical expertise with

its flexible and broadly applicable mode of thinking and reasoning about data,

variation, and chance. Similar to statistical thinking, the liberal arts, especially in

their philosophical tradition, encourage skeptical, analytical thinking

unconstrained by a priori standards and recognize that any conclusions are subject

to continuing challenge (p. 129).

Two points immediately come to mind from this passage. First, the idea of the

negotiability of statistics is clearly parallel to the present model’s assertion that mathematics is also produced by negotiation between mathematical thinkers. Secondly, and more insidiously, Ben-Zvi asserts that statistical thinking is objective and without a priori standards, yet clearly standards such as an acceptable limit for statistics do exist.

For example, Hoyles, Bakker, Kent and Noss (2007) note that:

One of the SPC rules of thumb is that if there are eight consecutive data points

above (or below) the mean, there is probably a special cause, that is, a

“non-random” effect (such an occurrence has a probability of (1/2)8 =1/256

0.39%). Similar rules of thumb for signals of special causes are six points in∼ a row

going up (or down), or cyclic patterns (p. 340).

As has been discussed, mathematicians often make similar claims about mathematics and likewise tend to overlook their own normative conventions. Both contribution and

168

normalization are evident here. Consequently, it is unsurprising that Pfannkuch (2011)

contends, “Statistics… demands both social and data-based argumentation skills” (p. 27).

Overall, it would seem that the present model of mathematical thinking can

certainly also match the literature’s portrayal of statistical thinking. It would be

tempting, at this point, to simply declare that statistical thinking is the same as or part of

mathematical thinking. However, it seems likely that this reasoning could also

demonstrate the potential fit of this model to, say, scientific thinking. While one might

be easily convinced that statistical thinking is a subset of mathematical thinking, one

might also be less inclined to say the same of scientific thinking. This inclination to

disbelieve the same basic argument in a different case is cause for a moment of reflection.

It may be that the philosophical implication of this dissertation is not that statistics is

subsumed by mathematics but simply that they take the same form cognitively.

Likewise, science and mathematics may not be part of the same system but actually two

cognitive systems of isomorphic form. It is worth noting that the present model explicitly

declined to enumerate who was a member of the mathematical community and which

facts are considered part of the mathematical world. In this way, the key differences

between the types of thinking that undergird science, statistics and mathematics might simply be selection of community members and norms. This revelation opens the door for philosophical debate regarding where one discipline ends and the next begins. For

example, “algebraic thinking” does not rely on inductive reasoning in the same way as

“statistical thinking” does. Algebra is clearly considered to be within the scope of

mathematics. Is inductive reasoning enough to separate statistics from mathematics?

169

Moreover, algebra may not formally incorporate inductive reasoning, but surely some

inductive reasoning must occur in order to notice patterns in the first place. Where to

draw the line becomes blurry very quickly and becomes almost inscrutable when one

starts trying to juxtapose subsets of mathematics Euclidean and non-Euclidean

geometries which differ only in their axiomatic assumptions. The present model makes it

clear that there is a need to study what facts, beliefs and assumptions are elements of

which human endeavor and whether or not there is any clear way to separate them. To

take it a step further, the model makes it reasonable to ask, “When two communities share some members, how can their related forms of thinking be said to be truly

distinct?” Only future philosophizing and research can answer that.

Curriculum & Instruction

The model of mathematical thinking presented in this dissertation has also has

clear implications for the teaching of mathematics. If nothing else, the present model can

be used to advocate that mathematics instruction needs to include far more than

pre-mathematized information and rote calculation. Bonotto (2005) forwards that it is

“the teacher’s role…to stimulate and facilitate mathematical discussion, and also to

legitimize certain aspects of mathematical activity and implicitly sanction others through

discussion” (pp. 333-334). For example, Lobato, Ellis and Muñoz (2003) suggest that the

nature of teaching mathematics is one of focusing student attention on relevant data (pp.

29-30). In this way, Olive (1999) notes that pieces of the instructional technology he

used in his research were added over time such that students only had access to certain

parts of the software at certain stages of learning; these changes were done intentionally

170

to “shape the children’s possible constructions” (p. 288). Additionally, Bowers and

Nickerson (2001) note that the teacher is often trying to highlight salient properties and

features across mathematical objects for students (p. 15). Another way in which, to use

Bonotto’s term, teachers can legitimize certain activities is through grading. Weber

(2010) relates a student’s depiction of the power of this sort of normative pressure:

P27: I probably, in the beginning, when I was taking [the transition-to-proof

course], I would have proved it like that, and then my professor probably would

have murdered my answer. He would have said that my answer only proves from

4 to 26. Technically you only prove it from 4 to 26, so I mean I probably would

have done that initially, but I don’t think it proves it for all (p. 329).

Without the social influence of the professor, this student may not have developed such

an understanding of the generality of proof. In this way, Lesh and Lehrer (2003) contend

that the artificial requirement for students to externalize their mathematical thinking – so

that teachers have something to assess – also allows students to begin to examine their

own thinking in ways they might not otherwise do (p. 118). The teacher can do more

than merely provide opportunities for desirable behavior, though. Citing Schoenfeld

(1987), Liu and Niess (2006) note that Liu tried to serve as a role model for

metacognitive behavior to his students; specifically, Liu worked out problems from

scratch to emulate a mathematician at the students’ level (pp. 378-379).

Another depiction of the role of the teacher supported by the present model can be seen in Watson and Mason (2006), who posit that one role of the teacher is to encourage students to explore new – rather than rely on old – mathematical ideas to solve problems

171

(p. 104). Given that the present model stresses the tendency for students to over-generalize, it makes sense that teachers should focus on avoiding its development in students. Speaking of development, it is worth reiterating that the present model is designed with the understanding that the development of mathematical thinking is not so much attainment of new kinds of thinking but rather the increased fluency, accuracy and automaticity of existing cognitive structures. In this frame, Haciomeroglue, Aspinwall and Presmeg (2010) suggest that one role of the teacher can be to reduce the cognitive load of problems, thus allowing students to focus on specific facets of the process while the teacher provides others (p. 153). Unfortunately, Selden and Selden (2005) suggest that mathematical thinking is “usually not taught explicitly, and in current school curricula, may not be considered by teachers as part of their responsibility” (p. 1).

Moreover, Sfard (2000) contends that if teachers refrain from modeling the professional mathematical discourse, students will never attain it (p. 184).

At this point, the present model begins to provide a way to provide actionable critiques of the existing curricula of public schools. Rather than random shots in the dark, the present model can be used to highlight specific deficits or surpluses of instruction that might be the cause of many of our pedagogical problems in mathematics education. In other words, rather than simply trying whole new curriculum packages to ameliorate low student achievement, the present model provides the basis for alterations on a smaller, more surgical level to the existing curricular materials. While there is certainly a need for subsequent research to verify the viability of this claim, it is possible to provide some specific examples of how the model might be used in this way.

172

Essentially, it is possible to point out examples of curriculum that ignore or distort any

given facet of mathematical thinking. At the very core of the matter, Reed (2001)

reviews an anthology entitled Making Sense of Word Problems and summarizes it thusly:

The premise of the Making Sense book is that by including only standard

problems in the curriculum, curriculum designers encourage a “suspension of

sense making.” Students soon learn that any word problem has an answer that can

be produced by the straightforward application of addition, subtraction,

multiplication, or division, and that the numbers in the problem are both necessary

and sufficient for its solution. This weakness in the curriculum encourages

students to routinely apply arithmetic operations without judging whether they are

appropriate for modeling the situation (p. 88).

Bonotto (2005) goes further, adding that student-teachers often rate nonsensical, though

mathematically correct, answers better than sensible answers “grounded in context-based considerations” (p. 314). Similarly, Verschaffel et al. (1999) cite a wide variety of research to depict school curriculum and instruction as virtually without opportunities for metacognition; they note that standardized word problems and trite or erroneous portrayals of mathematics are particularly harmful to the development of mathematical thinking in students (pp. 197-198).

To make matters worse, this “suspension of sense making” applies to more than one part of this most important nexus of mathematical thinking. Lannin (2005) suggests

“This difficulty [in validating generalizations] appears to be due to the traditional focus on finding particular instances of a situation rather than determining a general relation in

173

the early grade levels” (p. 235). Furthermore, Presmeg and Balderas-Cañas (2001) posit that public school pressure to adopt symbolic solutions inhibited student visualization even into college (p. 307). Additionally, Lehrer and Schauble (2000) note that fourth grade students maintained an organizational structure for data provided by the teacher for the duration of the project, and it is clear that the researchers were hoping for the students to develop more organizational idiosyncracy (pp. 63-64).

The present model can find pedagogical problems in other nodes as well. In terms of mathematization, Harel and Sowder (2005) argue that focusing instruction on basic-to-advanced sequences, such one-step algebraic solutions with integer coefficients preceding those with decimal coefficients, do not allow the student to reason why the basics are basic (p. 43). Likewise, Bonotto (2005) hypothesizes that student misconceptions regarding decimals occur because teachers do not allow students to

“reflect on decimal number properties and relationships” (p. 320). In terms of justification, Stylianides (2007) argues that the formality of textbook definitions often precludes discussion of underlying mathematical assumptions (p. 381). Stylianides

(2009), in a subsequent analysis of textbooks, reports that less than half of all analyzed tasks provide what he considers to be reasoning-and-proving opportunities. Even in terms of developing a mathematical disposition, Sinclair (2004) suggests,

Students may, in fact, share some aesthetic tendencies with mathematicians, but

may not know how to use them in the context of mathematical inquiry. The

emphasis that school mathematics places on propositional, logical reasoning

174

might actually discourage students from recognizing and trusting the generative

type of aesthetic responses that operate in inquiry (p. 282).

Certainly, it would be possible to go on listing the faults of the educational system, but it is sufficient to stop here in order to demonstrate the underlying point of the usefulness of the present model. To use the words of Lesh and Lehrer (2003), curricular materials in mathematics education “typically represent only a remarkably narrow and shallow subset of those needed for success beyond school” (p. 116).

At this point, it is a good opportunity to highlight a limitation of the present model, namely the outside factors that influence mathematics education. Battista (2004) suggests “because teachers are sent mixed signals by professional recommendations, students, parents and high-stakes testing, they[teachers] are often confused about goals, assessment, and even the very nature of the mathematics they should teach” (p. 190).

Specifically, Asquith, Stephens, Knuth and Alibali (2007) note that “teachers more often cited knowledge of notation for implicit multiplication and procedural knowledge about evaluating expressions as key to student success than students’ deeper understandings about variables” (p. 264). Certainly, instruction is parenthetically depicted in the present model, but Batista’s statement would seem to suggest that part of improving the mathematical thinking of students involves clarifying the nature of mathematical thinking in math teachers. This situation has likely arisen out of the aforementioned difficulty in specifically defining who is a member of the mathematical community and which social norms are actually mathematical in nature. It is also worth mentioning that Bowers and

Nickerson (2001) quote more than one of their participants tying their teaching strategy

175

to how they were taught themselves (pp. 7-8). In short, the present model can depict the

mathematical thinking of an individual or a group, but it does not do well in portraying

the generational shifts in mathematics as it is transmitted through both formal and

informal teaching.

That being said, the present model is useful for examining snapshots of

mathematical culture. In the meta-analysis, there are two articles by Cai (2000, 2005)

that provide cross-cultural studies of mathematics education contrasting the United States

with China. As an example of the explanatory power of the present model, a moment

should be spent summarizing Cai’s two national profiles of mathematical achievement

and thinking. Cai (2000) is a cross-national study comparing US and Chinese students.

Specifically, he is extending his research into what he had previously found – that overall there is no difference at complex levels of problem solving; but, on specific types of tasks, one nation typically did better than the other (pp. 310-311). Cai (2000) reports that his data indicates Chinese students do better overall on mathematics assessments but that

American students do better on open-ended tasks (pp. 315-316). Moreover, Cai (2000) argues that in every case, Chinese students were far more likely to use a symbolic solution than American students, who preferred concrete solutions; Cai (2000) goes on to suggest that these preferences seem to match preferred teaching styles (pp. 332-333).

As a follow-up to this finding, Cai (2005) posits that teaching is a cultural activity and that, consequently, the second article “is to investigate U.S. and Chinese teachers’ conceptions and constructions of representations in mathematics instruction” (p. 136).

Cai (2005) reports that Chinese lesson plans were very homogenous and detailed, being

176

roughly three times as long as US lesson plans on average, and that US lesson plans did

not have such unification or depth. Moreover, Cai depicts Chinese lesson goals as

revolving around “understanding” and “computing.” Several of the US goals use similar

language, although “find” seems to be more common than “compute.” Additionally, US

goals include some enactive components such as “leveling 2 columns of cubes” (p. 143).

Cai (2005) notes that only US plans included an “evening-out” process and that only the

Chinese lesson plans included multiple data sets with unequal membership. It would

seem that when US teachers say “understand” they mean “have a physical representation”

whereas when Chinese teachers say it, they mean “can use it in a problem.” While this

may merely be a translation issue, it seems far more likely that there is a cultural

difference in what it means to “do mathematics.” Cai (2005) summarizes, “The Chinese

curriculum focuses more on understanding the concept of arithmetic average as a

computational algorithm than on understanding the concept of arithmetic average as a

representative of a data set; however, the two U.S. NCTM Standards-based curricula focus more on the latter exposition of the concept” (p. 153).

The present model of mathematical thinking would explain these cultural differences in different socio-mathematical norms (Fig. 15). While American students are encouraged to engage in a loop of abstraction, generalization and reification, the

Chinese students seem to be engaged in a pattern of pre-mathematized information leading to justification and arguably to contribution, in a trivial sense. While each set of cultural preference has both benefits and drawbacks, it seems likely that any sort of standardized math assessment as are commonly used today would be biased towards a

177

symbolic solution method, such as the Chinese teachers prefer. Additionally,

Charlambous, Delaney, Hsu and Mesa (2010) suggest that one of the many factors influencing the high mathematical achievement in East-Asian countries is that their textbooks tend to be more cognitively demanding and use more formal definitions of mathematical objects, notably the unit rate fraction rather than the part-whole fraction used elsewhere. Yet, at the same time, over-emphasis on symbolic solution precludes real world application, to an extent. Cai (2005) reports that “It is clear that the two problems requiring an appropriate interpretation and using the mean in a statistical context are more difficult for Chinese teachers than for U.S. teachers” (p. 150).

Figure 15 – Cultural Preferences

178

In a way, the two national profiles are complimentary, and each could be

bolstered by the strengths of the other. Healy and Hoyles (1999) argue that “designing

activities that assist students in connecting visual and symbolic reasoning is…. one of

considerable importance” (p. 83). However, it is also apparent that the Chinese concept

of mathematical thinking is, at least superficially, more closely aligned with the

traditional American understanding of mathematical thinking. Thus, it is unsurprising

that Schorr and Koellner-Clark (2003) cite several pieces of research that seem to

indicate that most instruction, including those that are superficially reform-oriented, typically revolve around “memorizing procedures and using them to compute right answers” (p. 192). Overcoming this dichotomy can be difficult; Healy and Hoyles (1999) explain the disconnect people experience between different forms of reasoning:

For the few students who focus on the visual to uncover spatial patterns, there is a

tendency to see this as the task; so, once completed, there is little point in any

symbolic description. Other students who focus only on symbolic aspects and

ignore visual properties in favor of manipulating letters and numbers, although

judged as successes in our classrooms, are in danger of losing the opportunity to

exploit the visual to explain or justify their symbolic constructions or to develop

the capacity to move flexibly between representations (p. 83).

In one direction, Inoue (2008) reports “that incorporating familiar situations into problem descriptions results in an increased awareness of the realistic constraints and a decreased usage of the calculational approach, but it does not ensure that the students go beyond that” (p. 61). In the other direction, Juter (2006) argues, “It is important that students get

179

examples that they cannot solve without the sharpness of a strict definition; otherwise, they see no reason to learn a strict definition” (p. 427). Altogether, the present model is able to highlight under-represented aspects of mathematical thinking in the respective curricula of both countries and could be used to create more balanced instructional sequences in both cases.

Conclusion

Despite the seeming disunity for an overall definition of “mathematical thinking,” there does actually seem to be a great deal of consensus in the selected literature regarding its nature. The most obvious obstacle in realizing this unity was seeing beyond the superficial differences in terminology. Effort also had to be put forth to find common ground between Platonist and formalist epistemologies, which are also the source of many of the lingering areas of doubt. By compensating for these issues, it was possible to realize a great deal about mathematical thinking. First is that “mathematical thinking” a specialized function distinct from generalized thinking. However, it remains difficult to delineate exactly where mathematical thinking ends and where scientific thinking, for example, begins. Second is that “mathematical thinking” is best seen as a continuous, cyclical process of cognition in which a person strives to make sense of a vast sea of sensory data, map the mathematical world, attend to social convention all while coping with the individual differences in belief of every mathematical thinker. Ultimately, there is much yet to be learned about mathematical thinking, but now, perhaps, there is a potential basis for a strong paradigm to emerge from the Kuhnian pre-science in which such research has been mired.

REFERENCES

Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A

longitudinal examination of middle school students' understanding of the equal sign

and equivalent equations. Mathematical Thinking and Learning, 9(3), 221-247.

doi:10.1080/10986060701360902

Anderson, D. D., & Gold, E. (2006). Home to school: Numeracy practices and

mathematical identities. Mathematical Thinking and Learning, 8(3), 261-286.

doi:10.1207/s15327833mtl0803_4

Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school

mathematics teachers' knowledge of students' understanding of core algebraic

concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3),

249-272. doi:10.1080/10986060701360910

Baroody, A. J., & Lai, M. (2007). Preschoolers' understanding of the addition—

subtraction inverse principle: A Taiwanese sample. Mathematical Thinking and

Learning, 9(2), 131-171. doi:10.1080/10986060701341407

Baroody, A. J., Lai, M., Li, X., & Baroody, A. E. (2009). Preschoolers' understanding of

subtraction-related principles. Mathematical Thinking and Learning, 11(1), 41-60.

doi:10.1080/10986060802583956

Battista, M. T. (2004). Applying cognition-based assessment to elementary school

students' development of understanding of area and volume measurement.

Mathematical Thinking and Learning, 6(2), 185-204.

181 182

Ben-Zvi, D. (2000). Toward understanding the role of technological tools in statistical

learning. Mathematical Thinking and Learning, 2(1), 127-155.

Berk, D., Taber, S. B., Gorowara, C. C., & Poetzl, C. (2009). Developing prospective

elementary teachers' flexibility in the domain of proportional reasoning.

Mathematical Thinking and Learning, 11(3), 113-135.

doi:10.1080/10986060903022714

Bessant, K. C. (2001). A commentary on Drodge and Reid's "Embodied cognition and the

mathematical emotional orientation". Mathematical Thinking and Learning, 3(4),

315-321.

Bisanz, J., Watchorn, R. P. D., Piatt, C., & Sherman, J. (2009). On “Understanding”

children's developing use of inversion. Mathematical Thinking and Learning, 11(1),

10-24. doi:10.1080/10986060802583907

Bjuland, R., Cestari, M. L., & Borgersen, H. E. (2008). The interplay between gesture

and discourse as mediating devices in collaborative mathematical reasoning: A

multimodal approach. Mathematical Thinking and Learning, 10(3), 271-292.

doi:10.1080/10986060802216169

Bonotto, C. (2005). How informal out-of-school mathematics can help students make

sense of formal in-school mathematics: The case of multiplying by decimal numbers.

Mathematical Thinking and Learning, 7(4), 313-344.

doi:10.1207/s15327833mtl0704_3

Bowers, J. S., & Nickerson, S. (2001). Identifying cyclic patterns of interaction to study

individual and collective learning. Mathematical Thinking and Learning, 3(1), 1-28.

183

Cai, J. (2000). Mathematical thinking involved in U.S. and Chinese students' solving of

process-constrained and process-open problems. Mathematical Thinking and

Learning, 2(4), 309-340.

Cai, J. (2005). U.S. and Chinese teachers' constructing, knowing, and evaluating

representations to teach mathematics. Mathematical Thinking and Learning, 7(2),

135-169. doi:10.1207/s15327833mtl0702_3

Carreira, S. (2001). Where there's a model, there's a metaphor: Metaphorical thinking in

students' understanding of a mathematical model. Mathematical Thinking and

Learning, 3(4), 261-287.

Charalambous, C. Y., Delaney, S., Hsu, H., & Mesa, V. (2010). A comparative analysis

of the addition and subtraction of fractions in textbooks from three countries.

Mathematical Thinking and Learning, 12(2), 117-151.

doi:10.1080/10986060903460070

Chiu, M. M. (2001). Using metaphors to understand and solve arithmetic problems:

Novices and experts working with negative numbers. Mathematical Thinking and

Learning, 3(2), 93-124.

Civil, M., & Bernier, E. (2006). Exploring images of parental participation in

mathematics education: Challenges and possibilities. Mathematical Thinking and

Learning, 8(3), 309-330. doi:10.1207/s15327833mtl0803_6

Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young children's composition of

geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6(2),

163-184.

184

Cobb, P. (1999). Individual and collective mathematical development: The case of

statistical data analysis. Mathematical Thinking and Learning, 1(1), 5-43.

Common core state standards for mathematics. (2010, March 10). Retrieved from

http://www.corestandards.org/Files/K12MathStandards.pdf

Cramer, K., & Wyberg, T. (2009). Efficacy of different concrete models for teaching the

part-whole construct for fractions. Mathematical Thinking and Learning, 11(4),

226-257. doi:10.1080/10986060903246479

D’Ambrosio, U (1999). Literacy, matheracy, and technocracy: A trivium for today.

Mathematical Thinking and Learning, 1(2), 131-153. doi:

10.1207/s15327833mtl0102_3

Davis, J. D. (2007). Real-world contexts, multiple representations, student-invented

terminology, and y-intercept. Mathematical Thinking and Learning, 9(4), 387-418.

doi:10.1080/10986060701533839

De Bock, D., Verschaffel, L., & Janssens, D. (2002). The effects of different problem

presentations and formulations on the illusion of linearity in secondary school

students. Mathematical Thinking and Learning, 4(1), 65-89.

Dierdorp, A., Bakker, A., Eijkelhof, H., & van Maanen, J. (2011). Authentic practices as

contexts for learning to draw inferences beyond correlated data. Mathematical

Thinking and Learning, 13(1-2), 132-151.

Diversity in Mathematics Education Center for Learning and Teaching (2007). Culture,

race, power and mathematics education. In F.K. Lester, Jr. (Ed.), Second

185

handbook of research on mathematics teaching and learning (pp. 405-433).

Charlotte, NC: National Council of Teachers of Mathematics.

Doerr, H. (2006). Examining the tasks of teaching when using students' mathematical

thinking. Educational Studies in Mathematics, 62(1), 3-24.

doi:10.1007/s10649-006-4437-9.

Doerr, H. M., & Tripp, J. S. (1999). Understanding how students develop mathematical

models. Mathematical Thinking and Learning, 1(3), 231.

Dörfler, W. (2002). Formation of mathematical objects as decision making. Mathematical

Thinking and Learning, 4(4), 337-350.

Dreyfus, T. (1991). Advanced mathematical thinking processes. In, D. Tall (Ed.),

Advanced mathematical thinking (pp. 25-41). Norwell, MA: Kluwer Academic

Publishers.

Drodge, E. N., & Reid, D. A. (2000). Embodied cognition and the mathematical

emotional orientation. Mathematical Thinking and Learning, 2(4), 249-267.

Edwards, B. S., Dubinsky, E., & McDonald, M. A. (2005). Advanced mathematical

thinking. Mathematical Thinking and Learning, 7(1), 15-25.

Ginsburg, H. P., Lin, C., Ness, D., & Seo, K. (2003). Young American and Chinese

children's everyday mathematical activity. Mathematical Thinking and Learning,

5(4), 235.

Esteley, C. B., Villarreal, M., & Alagia, H. R. (2010). The overgeneralization of linear

models among university students' mathematical productions: A long-term study.

186

Mathematical Thinking and Learning, 12(1), 86-108.

doi:10.1080/10986060903465988

Fischbein, E. (1999). Psychology and mathematics education. Mathematical Thinking

and Learning, 1(1), 47.

Gainsburg, J. (2006). The mathematical modeling of structural engineers. Mathematical

Thinking and Learning, 8(1), 3-36. doi:10.1207/s15327833mtl0801_2

Garfield, J., & Chance, B. (2000). Assessment in statistics education: Issues and

challenges. Mathematical Thinking and Learning, 2(1), 99-125.

Gil, E., & Ben-Zvi, D. (2011) Explanations and context in the emergence of students’

informal inferential reasoning. Mathematical Thinking and Learning, 13(1-2),

87-108.

Ginsburg, H. P., & Seo, K. (1999). Mathematics in children's thinking. Mathematical

Thinking and Learning, 1(2), 113.

Goldin, G. A. (2000). Affective pathways and representation in mathematical problem

solving. Mathematical Thinking and Learning, 2(3), 209-219.

Gorard, S., & Smith, E. (2004). An international comparison of equity in education

systems. Comparative Education, 40(1), 15-28.

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal

mathematics. Mathematical Thinking and Learning, 1(2), 155.

Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in

reform mathematics education. Mathematical Thinking and Learning, 6(2), 105-128.

187

Greer, B. (2000). Statistical thinking and learning. Mathematical Thinking and Learning,

2(1), 1-9.

Greer, B. (2010). Overview of the papers: Why is linear thinking so dominant?

Mathematical Thinking and Learning, 12(1), 109-115.

doi:10.1080/10986060903465996

Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers' conceptual and

procedural knowledge of mean, median, and mode. Mathematical Thinking and

Learning, 8(1), 37-63. doi:10.1207/s15327833mtl0801_3

Gutierrez, R. (2002). Enabling the practice of mathematics teachers in context: Toward a

new equity research agenda. Mathematical Thinking and Learning, 4(2), 145-187.

Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of

calculus students' understanding of derivative graphs. Mathematical Thinking and

Learning, 12(2), 152-176. doi:10.1080/10986060903480300

Han, Y., & Ginsburg, H. P. (2001). Chinese and English mathematics language: The

relation between linguistic clarity and mathematics performance. Mathematical

Thinking and Learning, 3(2), 201-220.

Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature

and its development. Mathematical Thinking and Learning, 7(1), 27-50.

Healy, L., & Hoyles, C. (1999). Visual and symbolic reasoning in mathematics: Making

connections with computers? Mathematical Thinking and Learning, 1(1), 59.

188

Hoyles, C., Bakker, A., Kent, P., & Noss, R. (2007). Attributing meanings to

representations of data: The case of statistical process control. Mathematical

Thinking and Learning, 9(4), 331-360. doi:10.1080/10986060701533326

Husserl, E. (1999). The essential Husserl: Basic writings in transcendental

phenomenology (D. Welton, Ed.). Bloomington, IN: Indiana University Press.

Inoue, N. (2008). Minimalism as a guiding principle: Linking mathematical learning to

everyday knowledge. Mathematical Thinking and Learning, 10(1), 36-67.

doi:10.1080/10986060701820293

Izsák, A. (2004). Teaching and learning two-digit multiplication: Coordinating analyses

of classroom practices and individual student learning. Mathematical Thinking and

Learning, 6(1), 37-79.

Jansen, A. (2008). An investigation of relationships between seventh-grade students'

beliefs and their participation during mathematics discussions in two classrooms.

Mathematical Thinking and Learning, 10(1), 68-100.

doi:10.1080/10986060701820327

Jones, G. A., Thornton, C. A., Langrall, C. W., Mooney, E. S., Perry, B., & Putt, I. J.

(2000). A framework for characterizing children's statistical thinking. Mathematical

Thinking and Learning, 2(4), 269-307.

Juter, K. (2006). Limits of functions as they developed through time and as students learn

them today. Mathematical Thinking and Learning, 8(4), 407-431.

doi:10.1207/s15327833mtl0804_3

189

Keiser, J. M. (2004). Struggles with developing the concept of angle: Comparing

sixth-grade students' discourse to the history of the angle concept. Mathematical

Thinking and Learning, 6(3), 285-306.

Kulm, G. (1999). Thinking practices in mathematics and science learning (book review).

Mathematical Thinking and Learning, 1(4), 315.

Langrall, C., Nisbet, S., Mooney, E., & Jansem, S. (2011) The role of context expertise

when comparing data. Mathematical Thinking and Learning, 13(1-2), 47-67.

Lannin, J. K. (2005). Generalization and justification: The challenge of introducing

algebraic reasoning through patterning activities. Mathematical Thinking and

Learning, 7(3), 231-258. doi:10.1207/s15327833mtl0703_3

Lehrer, R., & Schauble, L. (2000). Inventing data structures for representational

purposes: Elementary grade students' classification models. Mathematical Thinking

and Learning, 2(1), 51-74.

Leonard, J. (2001). How group composition influenced the achievement of sixth-grade

mathematics students. Mathematical Thinking and Learning, 3(2), 175-200.

Lesh, R., Doerr, H. M., Carmona, G., & Hjalmarson, M. (2003). Beyond constructivism.

Mathematical Thinking and Learning, 5(2), 211-233.

Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual

development. Mathematical Thinking and Learning, 5(2), 157-189.

Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of

students and teachers. Mathematical Thinking and Learning, 5(2), 109-129.

190

Lesh, R., & Yoon, C. (2004). Evolving communities of mind --- in which development

involves several interacting and simultaneous developing strands. Mathematical

Thinking and Learning, 6(2), 205-226.

Liu, P. & Niess, M. L. (2006). An exploratory study of college students' views of

mathematical thinking in a historical approach calculus course. Mathematical

Thinking and Learning, 8(4), 373-406. doi:10.1207/s15327833mtl0804_2

Lobato, J., Ellis, A. B., & Muñoz, R. (2003). How "focusing phenomena" in the

instructional environment support individual students' generalizations. Mathematical

Thinking and Learning, 5(1), 1-36.

Lubienski, S. T. (2002). Research, reform, and equity in U.S. mathematics education.

Mathematical Thinking and Learning, 4(2), 103-125.

Madden, S. (2011). Statistically, technologically, and contextually provocative tasks:

Supporting teachers’ informal inferential reasoning. Mathematical Thinking and

Learning, 13(1-2), 109-131.

Maher, C. A. (1999). Mathematical thinking and learning: A perspective on the work of

Robert B. Davis. Mathematical Thinking and Learning, 1(1), 85.

Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical

inference. Mathematical Thinking and Learning, 13(1-2), 152-173.

Martin, L., Towers, J., & Pirie, S. (2006). Collective mathematical understanding as

improvisation. Mathematical Thinking and Learning, 8(2), 149-183.

doi:10.1207/s15327833mtl0802_3

191

Maso, I. (2008). Phenomenology and ethnography. In P. Atkinson (Ed.), Handbook of

Ethnography (pp. 136-144). Thousand Oaks, CA: SAGE Publications Ltd.

Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case

of tape diagrams. Mathematical Thinking and Learning, 10(4), 374-406.

doi:10.1080/10986060802291642

McClain, K. (2003). Supporting preservice teachers' understanding of place value and

multidigit arithmetic. Mathematical Thinking and Learning, 5(4), 281.

McCrone, S. S. (2005). The development of mathematical discussions: An investigation

in a fifth-grade classroom. Mathematical Thinking and Learning, 7(2), 111-133.

doi:10.1207/s15327833mtl0702_2

McGinn, M. K., & Boote, D. N. (2003). A first-person perspective on problem solving in

a history of mathematics course. Mathematical Thinking and Learning, 5(1), 71-107.

Mitchell, J. M. (2001). Interactions between natural language and mathematical

structures: The case of "wordwalking". Mathematical Thinking and Learning, 3(1),

29-52.

Mooney, E. S. (2002). A framework for characterizing middle school students' statistical

thinking. Mathematical Thinking and Learning, 4(1), 23-63.

Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual

mathematics learners. Mathematical Thinking and Learning, 4(2), 189-212.

Nasir, N. S. (2002). Identity, goals, and learning: Mathematics in cultural practice.

Mathematical Thinking and Learning, 4(2), 213-247.

192

Nathan, M. J., & Kim, S. (2007). Pattern generalization with graphs and words: A

cross-sectional and longitudinal analysis of middle school students' representational

fluency. Mathematical Thinking and Learning, 9(3), 193-219.

doi:10.1080/10986060701360886

National Council of Teachers of Mathematics. (2000). Professional standards for school

mathematics. Reston, VA: author.

Nickerson, S. D., & Whitacre, I. (2010). A local instruction theory for the development of

number sense. Mathematical Thinking and Learning, 12(3), 227-252.

doi:10.1080/10986061003689618

Novick, L. R. (2004). Diagram literacy in preservice math teachers, computer science

majors, and typical undergraduates: The case of matrices, networks, and hierarchies.

Mathematical Thinking and Learning, 6(3), 307-342.

Nunes, T., Bryant, P., Hallett, D., Bell, D., & Evans, D. (2009). Teaching children about

the inverse relation between addition and subtraction. Mathematical Thinking and

Learning, 11(1), 61-78. doi:10.1080/10986060802583980

Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization

hypothesis. Mathematical Thinking and Learning, 1(4), 279.

Peled, I., & Segalis, B. (2005). It's not too late to conceptualize: Constructing a

generalized subtraction schema by abstracting and connecting procedures.

Mathematical Thinking and Learning, 7(3), 207-230.

doi:10.1207/s15327833mtl0703_2

193

Petrosino, A. J., Lehrer, R., & Schauble, L. (2003). Structuring error and experimental

variation as distribution in the fourth grade. Mathematical Thinking and Learning,

5(2), 131-156.

Pfannkuch, M. (2011). The role of context in developing informal statistical inferential

reasoning: A classroom study. Mathematical Thinking and Learning, 13(1-2), 27-46.

Piaget, J., & Inhelder, B. (2000). The psychology of the child. New York: Basic Books.

Poincare, H. (2004). Science and method. USA: Barnes & Noble Publishing, Inc.

Presmeg, N. (2007). The role of culture in teaching and learning mathematics. In F.K.

Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and

learning (pp. 435-458). Charlotte, NC: National Council of Teachers of

Mathematics.

Presmeg, N. C., & Balderas-Cañas, P. E. (2001). Visualization and affect in nonroutine

problem solving. Mathematical Thinking and Learning, 3(4), 289-313

Quine, W.V.O. (2003a). Two dogmas of empiricism. In J. Baillie (Ed.), Contemporary

analytic philosophy (2nd ed.) (pp.272-289). Upper Saddle River, NJ: Pearson

Education, Inc.

Quine, W.V.O. (2003b). Ontological relativity. In J. Baillie (Ed.), Contemporary analytic

philosophy (2nd ed.) (pp.302-323). Upper Saddle River, NJ: Pearson Education,

Inc.

Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural

approach to students' types of generalization. Mathematical Thinking and Learning,

5(1), 37-70.

194

Raman, M., & Fernández, E. (2005). Book review. Mathematical Thinking and Learning,

7(3), 259-263. doi:10.1207/s15327833mtl0703_4

Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical

activity: A practice-oriented view of advanced mathematical thinking. Mathematical

Thinking and Learning, 7(1), 51-73.

Reed, S. K. (2001). Making sense of word problems (book review). Mathematical

Thinking and Learning, 3(1), 87–91.

Remillard, J. T., & Jackson, K. (2006). Old math, new math: Parents' experiences with

standards-based reform. Mathematical Thinking and Learning, 8(3), 231-259.

doi:10.1207/s15327833mtl0803_3

Ridlon, C. L. (2009). Learning mathematics via a problem-centered approach: A

two-year study. Mathematical Thinking and Learning, 11(4), 188-225.

doi:10.1080/10986060903225614

Rodd, M. M. (2000). On mathematical warrants: Proof does not always warrant, and a

warrant may be other than a proof. Mathematical Thinking and Learning, 2(3),

221-244.

Schneider, M., & Stern, E. (2009). The inverse relation of addition and subtraction: A

knowledge integration perspective. Mathematical Thinking and Learning, 11(1),

92-101. doi:10.1080/10986060802584012

Schoenfeld, A. (1992). Learning to think mathematically: Problem solving,

metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook

195

of Research on Mathematics Teaching and Learning (pp 334-370). Reston, VA:

National Council of Teachers of Mathematics.

Schorr, R. Y., & Koellner-Clark, K. (2003). Using a modeling approach to analyze the

ways in which teachers consider new ways to teach mathematics. Mathematical

Thinking and Learning, 5(2), 191-210.

Selden, A., & Selden, J. (2005). Perspectives on advanced mathematical thinking.

Mathematical Thinking and Learning, 7(1), 1-13.

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, 1-36.

Sfard, A. (2000). On reform movement and the limits of mathematical discourse.

Mathematical Thinking and Learning, 2(3), 157-189.

Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for

investigating and establishing learning goals. Mathematical Thinking and Learning,

8(4), 359-371. doi:10.1207/s15327833mtl0804_1

Sinclair, N. (2004). The roles of the aesthetic in mathematical inquiry. Mathematical

Thinking and Learning, 6(3), 261-284.

Sriraman, B. (2005). Book review. Mathematical Thinking and Learning, 7(2), 171-180.

doi:10.1207/s15327833mtl0702_4

Sriraman, B. (2009). Educational psychology and mathematics education: Potential for a

new symbiosis? Mathematical Thinking and Learning, 11(3), 177-182.

doi:10.1080/10986060903013861

196

Sriraman, B., & Lesh, R. (2007). A conversation with Zoltan P. Dienes. Mathematical

Thinking and Learning, 9(1), 59-75. doi:10.1207/s15327833mtl0901_5

Stein, M., & Burchartz, B. (2006). The invisible wall project: Reasoning and problem

solving processes of primary and lower secondary students. Mathematical Thinking

and Learning, 8(1), 65-90. doi:10.1207/s15327833mtl0801_4

Stroup, W. M., Ares, N. M., & Hurford, A. C. (2005). A dialectic analysis of

generativity: Issues of network-supported design in mathematics and science.

Mathematical Thinking and Learning, 7(3), 181-206.

doi:10.1207/s15327833mtl0703_1

Stylianides, A. J. (2007). Introducing young children to the role of assumptions in

proving. Mathematical Thinking and Learning, 9(4), 361-385.

doi:10.1080/10986060701533805

Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks.

Mathematical Thinking and Learning, 11(4), 258-288.

doi:10.1080/10986060903253954

Stylianides, G. J., & Stylianides, A. J. (2008). Proof in school mathematics: Insights from

psychological research into students' ability for deductive reasoning. Mathematical

Thinking and Learning, 10(2), 103-133. doi:10.1080/10986060701854425

Stylianou, D. A., & Silver, E. A. (2004). The role of visual representations in advanced

mathematical problem solving: An examination of expert-novice similarities and

differences. Mathematical Thinking and Learning, 6(4), 353-387.

197

Swoyer, C. (2003). Stanford encyclopedia of philosophy. In Stanford University.

Retrieved from http://plato.stanford.edu/entries/relativism/supplement2.html

Tall, D. (1991). The psychology of advanced mathematical thinking. In, D. Tall (Ed.),

Advanced mathematical thinking (pp. 3-21). Norwell, MA: Kluwer Academic

Publishers.

Taylor & Francis (2011). Journal details: Mathematical Thinking and Learning. Retrieved

from http://www.tandf.co.uk/journals/titles/10986065.asp

Tirosh, D., & Stavy, R. (1999). Intuitive rules and comparison tasks. Mathematical

Thinking and Learning, 1(3), 179.

Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P., & Verschaffel, L. (2009).

Solving subtraction problems by means of indirect addition. Mathematical Thinking

and Learning, 11(1), 79-91. doi:10.1080/10986060802583998

Turner, E. E., Gutiérrez, M. V., Simic-Muller, K., & Díez-Palomar, J. (2009).

“Everything is math in the whole world”: Integrating critical and community

knowledge in authentic mathematical investigations with elementary Latina/o

students. Mathematical Thinking and Learning, 11(3), 136-157.

doi:10.1080/10986060903013382 van den Heuvel-Panhuizen, M., & van den Boogaard, S. (2008). Picture books as an

impetus for kindergartners' mathematical thinking. Mathematical Thinking and

Learning, 10(4), 341-373. doi:10.1080/10986060802425539 van Oers, B. (2010). Emergent mathematical thinking in the context of play. Educational

Studies in Mathematics, 74(1), 23-37. doi:10.1007/s10649-009-9225-x.

198

Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., &

Ratinckx, E. (1999). Learning to solve mathematical application problems: A design

experiment with fifth graders. Mathematical Thinking and Learning, 1(3), 195.

Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics.

In, D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Norwell, MA:

Kluwer Academic Publishers.

Vlahović-Štetić, V., Pavlin-Bernardić, N., & Rajter, M. (2010). Illusion of linearity in

geometry: Effect in multiple-choice problems. Mathematical Thinking and Learning,

12(1), 54-67. doi:10.1080/10986060903465871

Watson, J. M., Callingham, R. A., & Kelly, B. A. (2007). Students' appreciation of

expectation and variation as a foundation for statistical understanding. Mathematical

Thinking and Learning, 9(2), 83-130. doi:10.1080/10986060701341332

Watson, J. M., & Chick, H. L. (2001). Factors influencing the outcomes of collaborative

mathematical problem solving: An introduction. Mathematical Thinking and

Learning, 3(2), 125-173.

Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object:

Using variation to structure sense-making. Mathematical Thinking and Learning,

8(2), 91-111. doi:10.1207/s15327833mtl0802_1

Weber, K. (2010). Mathematics majors’ perceptions of conviction, validity and proof.

Mathematical Thinking and Learning, 12(4), 306-336.

199

Whitenack, J., Knipping, N., & Novinger, S. (2001). Coordinating theories of learning to

account for second-grade children's arithmetical understandings. Mathematical

Thinking and Learning, 3(1), 53-85.

Wilson, J. (1997). Thinking with concepts. USA: Cambridge University Press.

Wittgenstein, L. (2003). Selections from Philosophical Investigations. In J. Baillie (Ed.),

Contemporary analytic philosophy (2nd ed.) (pp.110-130). Upper Saddle River,

NJ: Pearson Education, Inc.

Zahner, D., & Corter, J. E. (2010). The process of probability problem solving: Use of

external visual representations. Mathematical Thinking and Learning, 12(2),

177-204. doi:10.1080/10986061003654240

Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice:

Perceptions of secondary mathematics teachers. Mathematical Thinking and

Learning, 12(4), 263-281.