Number Theory in Mathematics Education

Perspectives and Prospects

Rina Zazkis, Stephen R. Campbell

• Publisher: Routledge
• Year: 2012
• ISBN:
  9781136501432
• Edition: 1
• Topic: Didattica
• Subtopic: Didattica generale

This book offers multiple interconnected perspectives on the largely untapped potential of elementary number theory for mathematics education: its formal and cognitive nature, its relation to arithmetic and algebra, its accessibility, its utility and intrinsic merits, to name just a few. Its purpose is to promote explication and critical dialogue about these issues within the international mathematics education community. The studies comprise a variety of pedagogical and research orientations by an international group of researchers that, collectively, make a compelling case for the relevance and importance of number theory in mathematics education in both pre K-16 settings and mathematics teacher education.Topics variously engaged include:
*understanding particular concepts related to numerical structure and number theory;
*elaborating on the historical and psychological relevance of number theory in concept development;
*attaining a smooth transition and extension from pattern recognition to formative principles;
*appreciating the aesthetics of number structure;
*exploring its suitability in terms of making connections leading to aha! insights and reaching toward the learner's affective domain;
*reexamining previously constructed knowledge from a novel angle;
*investigating connections between technique and theory;
*utilizing computers and calculators as pedagogical tools; and
*generally illuminating the role number theory concepts could play in developing mathematical knowledge and reasoning in students and teachers.Overall, the chapters of this book highlight number theory-related topics as a stepping-stone from arithmetic toward generalization and algebraic formalism, and as a means for providing intuitively grounded meanings of numbers, variables, functions, and proofs. Number Theory in Mathematics Education: Perspectives and Prospects is of interest to researchers, teacher educators, and students in the field of mathematics education, and is well suited as a text for upper-level mathematics education courses.

Chapter 1
Number Theory in Mathematics Education Research: Perspectives and Prospects

Rina Zazkis
Stephen R. Campbell
Simon Fraser University

The higher arithmetic presents us with an inexhaustible store of interesting truths—of truths too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remained concealed.

—C. F. Gauss (1847)¹

What makes this book interesting and important is the detailed attention it gives to a subject area exemplifying the essence of mathematics, and one that has yet to receive sufficient attention in mathematics education. A case can be made that greater emphasis be given to number theory in mathematics education from a variety of perspectives: its formal mathematical nature, particularly with respect to the additive and multiplicative structure of the integers and whole numbers; its beauty and mystique; its practical utility (or perceived lack thereof); its importance in the history of mathematics; and perhaps foremost from a pedagogical perspective, its pure simplicity and accessibility. And yet, as Gauss and so many others have long appreciated, the most trivial truths in the higher arithmetic, that is, number theory, reside side by side with many of its most profound and perennial mysteries. This volume of research expands on all of these perspectives.

Since the crash and burn of the “new math” initiatives of the 1960s, it has become customary in mathematics education to argue for teaching the practical utility of mathematics through “real-life” applications. This shift in emphasis from formalization to contextualization has experienced problems of its own. Whereas the new math curriculum left many students with deeply engrained impressions that “math is what you do in math class,” the progressivist agenda has left others wondering “where’s the math?”

We recognize the “practicality” of number theory in more advanced areas of application, such as cryptology and computer science, and that number theory does not readily fit progressivist preconceptions of “everyday life.” Knowledge of number theory will not help in calculating taxes or balancing checkbooks. On the other hand, elementary number theory offers a natural and relatively benign algebraic formalization of integer arithmetic. Significantly, learners engaging in the subject content of elementary number theory readily encounter the very essence of mathematics. Such encounters may engender lifelong fascination rather than aversion. Accordingly, we view the utility of number theory from a different perspective—its utility for teaching and learning mathematics.

Topics from number theory, such as factors, divisors, multiples, and congruences provide natural avenues for developing and solidifying mathematical thinking, for developing enriched appreciation of numerical structure, especially with respect to identifying and recognizing patterns, formulating and testing conjectures, understanding principles and proofs, and justifying the truth of theorems in disciplined and reasoned ways. Considering the importance of this branch of mathematics in the history and philosophy of the discipline, the slim amount of wide-ranging and coordinated mathematics education research in this area is rather surprising—and considering the rich possibilities for research in this area, all the more so.

As the “queen” of mathematics—which is how Gauss referred to it— number theory has yet to establish a place of its own in mathematics education research. In this chapter, we discuss what this research might be and the role number theory could play in mathematics education research, not only as a “queen,” but as “queen and servant.” We start with portraying the presence of number theory in several official documents. Then we provide an overview of recent research and elaborate on possible future directions.

Number Theory in “Official” Reference Literature

Everyone has some familiarity with numbers. The National Council of Teachers of Mathematics (NCTM, 2000) “Principles and Standards for School Mathematics” (Principles & Standards) emphasizes that historically, numbers have been a cornerstone of the mathematics curriculum. One might risk political incorrectness and reasonably argue that numbers are in fact the cornerstone of the mathematics curriculum. Much emphasis has been placed in k–12 education on number operations, number sense, number systems, and even on number concepts; however, it is now becoming more widely recognized that beyond proficiency with operations, beyond familiarity with numbers, topics from elementary number theory engender and cultivate deep and fundamental understandings of mathematics, especially in areas such as problem solving and reasoning, formulating conjectures, testing generalizations, providing justifications, and proving theorems. Notably, number theory provides “a concrete [italics added] setting for strengthening algebraic and proof-building skills” (CBMS, 2001, p. 126).

Commenting on an NCTM (1981) commission report entitled “Guidelines for the Preparation of Teachers of Mathematics,” Ball (1988) asked: “For primary grade teachers, why no course in number theory?” (p. 43). Evidently, interest in number theory has been slow to take hold in mathematics education. In the “Curriculum and Evaluation Standards for School Mathematics” (NCTM, 1989) number theory is featured only in Standard 6 for grades 5–8. The suggestion is that “the mathematics curriculum should include the study of number systems and number theory so that students can develop and apply number theory concepts (e.g. primes, factors and multiples) in real world and mathematical problem situations” (p. 91). It is further observed that “number theory offers many rich opportunities for explorations that are interesting, enjoyable and useful” (p. 91). The “Principles & Standards” recommend further attention to number theory. First, they echo the recommendation from the former document in the Number and Operations Standard for Grades 6–8, presenting the expectation that students in these grades should “use factors, multiples, prime factorization and relatively prime numbers to solve problems” (p. 214). They add the expectation that in grades 9–12 students should “use number-theory arguments to justify relationships involving whole numbers” (p. 290). If the “Principles & Standards” are any indication of curriculum trends, then we would say that there is an expectation for a stronger presence of number theory in the beginning of the new millennium.

The Conference Board of the Mathematical Sciences (CBMS, 2001) reported that “The Mathematical Education of Teachers” (MET) provides recommendations on what preservice education of mathematics teachers should include. Recognizing the special nature of the mathematical knowledge needed for teaching, these recommendations include:

• Coursework for prospective middle grades teachers should lead them to … understand and be able to explain fundamental ideas of number theory as they apply to middle school mathematics (p. 28).
• To be well prepared to teach high school curricula, mathematics teachers need … understanding of the ways that basic ideas of number theory and algebraic structure underlie rules for operations and expressions, equations and inequalities (p. 40).

The report further elaborates by claiming that middle school teachers

should experience conjecturing and justifying conjectures about even and odd numbers and about prime and composite numbers. They should have a good grasp of the Prime factorization theorem and how it extends to algebra learning. The difficulty of finding the greatest common factor of two numbers can lead students to an appreciation of the efficiency of the Euclidean Algorithm. (p. 30)

Zooming out on the specifics, we acknowledge in this recommendation that number theory topics are considered an asset in gaining experience for justifying conjectures, are considered in the connection to algebra and also as an avenue for the appreciation of mathematical efficiency. We return to these themes in our discussion below of current research in these areas and of some promising prospects for further research.

A recent National Research Council (NRC) report “Adding it up” (Kilpatrick, Swafford, & Findell, 2001) has been charged with synthesizing the rich and diverse research on pre-k–8 mathematical learning and defined its explicit focus on the notion of a number. Research on whole numbers in mathematics education has traditionally focused on number operations, decimal representations, and the like. This traditional research orientation toward the number is reflected in the report, as it devotes a significant amount of space and discussion to these topics. However, mathematical learning in pre-k–8 related to the notion of number is not limited to these topics. It would be reasonable to expect that research on conceptions of number in pre-k–8, that obviously includes conceptions of number in Grades 5–8, would devote some attention to the concepts of number theory. Yet, in devoting much space and time to multiplication and division, the report does not mention concepts related to the multiplicative structure of whole numbers, such as divisibility. There is barely a mention of basic concepts from elementary number theory such as the concept of prime number, or of prime factor. There is no mention whatsoever of prime decomposition/factorization, composite numbers, or of congruence, at least in a number theoretical sense of that latter term. It seems germane to note that a complete absence from any explicit consideration of elementary concepts of number theory was also evidenced in Hiebert’s (1999) review of relationships between research and the NCTM Standards. All in all, this may come as no surprise. Unlike the Standards and MET, which are works that can be seen as “agenda setting documents,” the NRC’s “Adding it up” is primarily a document that attempted, within limits, to review and synthesize available research. The Hiebert article, more of an illustrative review than a comprehensive one, admittedly took a prescriptive stance regarding the role of research in shaping the NCTM standards. As Kilpatrick (2001) pointed out, the NRC report, “Adding it up,” also had a prescriptive component, but one that as we have seen, made no mention of number theory, and barely or nary a mention of its basic concepts. This is surprising in that, despite the pragmatic considerations involved in limiting the focus of such a work, Kilpatrick (2001) noted: (1) the main focus of the “Adding it up” report was on the topic of number in k–8; (2) number is at the heart of the school curriculum in the early grades; (3) the learning of number and related concepts is the most researched topic in the school mathematics curriculum; and (4) the learning of number and related concepts leads to the study of algebra and is strongly connected with other mathematical domains (p. 106).

To be included in the NRC “Adding it up” report, research studies had to be deemed relevant, sound, generalizable, and convergent (Kilpatrick, 2001, p. 109). A number of research studies in number theory may have conformed well to the first two criteria, but meeting the last two criteria, especially the last, requires that such studies be part of a “collection of studies … in the sense that the findings stood up across different groups of students and teachers, were obtained using different data-gathering methods, and fit well within a larger network of evidence that made good common and theoretical sense” (Kilpatrick, 2001, p. 109). Unfortunately, much like preservice teachers’ understandings of mathematics in general, research in teaching and learning number theory has been sparse and disconnected. This does not imply that research in this area has been nonexistent, but it does point to why research in this area is not represented in the “Adding it up” report.

Taken together, these studies reflect, in our opinion, an unfortunate reality—research related to or underlying elementary number theory in mathematics education has been slim to a degree of insignificance. If similar reviews are to be conducted in the next decade, will the situation be different?

Number Theory and Mathematics Education Research: Looking Back and Looking Forward

Number theory received some attention in the “new math” era (e.g., Barnett, 1961; Peck, 1961; SMSG, 1960a, 1960b, 1962). There was a minor flurry of educational interest in number theory with the advent of the microcomputer, as topics from number theory lent themselves well to programming (e.g., Anderson, 1982; Jackson, 1990; Owens, 1990). Research is continuing to flourish in this area (e.g., Abramovich, 2000; Abramovich & Brantlinger, 2004; Sinclair, Zazkis, & Liljedahl, 2003).

In the past few decades, although most often few and far between, there have been some relatively isolated studies conducted on topics ranging from the study of patterns, as a precursor to or under the rubric of algebra or pre-algebra (e.g., Healy & Hoyles, 2000; Orton & Orton, 1999), to the role of number theory in reasoning, proof, and problem solving (King, 1973; Lester & Mau, 1993; Zazkis, 1999b). Some of these studies along with others are treated in more detail later, when we turn our focus to ways in which the queen of mathematics can, and often does, play the role of servant in mathematics education research.

There has been, at least since Ball (1988), an increasingly strong interest in number theory as a means of teaching and learning mathematics in the professional development of teachers (e.g., Campbell & Zazkis, 2002). In contrast to the relative dearth of mathematics education research in the area of number theory per se, however, there has been an ongoing vibrant treatment of the utility of number theory in teaching mathematics in the professional literature, as a quick glance at Mathematics Teacher (e.g., Fischer, 1986; Johnson, 2001; Kappel, 1976; Lefton, 1991; Prielipp, 1973), Arithmetic Teacher (e.g., Bezuszka, 1985; Esty, 1991; Lappan & Winter, 1980; Rogers, 1970), School Science and Mathematics (e.g., Prielipp, 1970; Woods & Flowers, 1976), and Mathematics and Computer Education (e.g., Anderson, 1982; Joyner & Haggard, 1990), well illustrate.

In looking back on number theory in mathematics education, we would be remiss not to mention the ongoing and enthusiastic attention given to number theory in the popular press under the rubric of recreational mathematics, notably through mathematicians like John Conway and Richard Guy (1997), and, of course, the philosopher Martin Gardner for many years, through his famous Scientific American column, “Mathematical Recreations.”

We observed a couple of years ago that mathematics education research related to number theory is slim, especially when considering the central role that number theory has ...

Table of contents

• Cover
• Title
• Copyright
• Contents
• List of Contributors
• Preface
• 1 Number Theory in Mathematics Education Research: Perspectives and Prospects
• 2 Understanding Elementary Number Theory in Relation to Arithmetic and Algebra
• 3 What Makes an Example Exemplary: Pedagogical and Didactical Issues in Appreciating Multiplicative Structures
• 4 For the Beauty of Number Theory
• 5 On the Path to Number Theory: Repeating Patterns as a Gateway
• 6 Learning by Teaching: The Case of Sieve of Eratosthenes and One Elementary School Teacher
• 7 Learning Elementary Number Theory Through a Chain of Discovery: Preservice Teachers’ Encounter with Pentominoes
• 8 The Number-Theoretic Experience of 12- to 15-Year-Olds in a Calculator Environment: The Intertwining Coemergence of Technique and Theory
• 9 Learning Number Theory Concepts Via Geometrical Interactive Computerized Setting
• 10 Overlooking Number Patterns in Algorithmic Problem Solving
• 11 Revisiting Algebra in a Number Theoretical Setting
• Author Index
• Subject Index