Integrating Research on the Graphical Representation of Functions

Thomas A. Romberg, Elizabeth Fennema, Thomas P. Carpenter

• Publisher: Routledge
• Year: 2012
• Print ISBN: 9780805811346
• eBook ISBN: 9781136476365
• Topic: Education
• Subtopic: Education General

This volume focuses on the important mathematical idea of functions that, with the technology of computers and calculators, can be dynamically represented in ways that have not been possible previously. The book's editors contend that as result of recent technological developments combined with the integrated knowledge available from research on teaching, instruction, students' thinking, and assessment, curriculum developers, researchers, and teacher educators are faced with an unprecedented opportunity for making dramatic changes.

The book presents content considerations that occur when the mathematics of graphs and functions relate to curriculum. It also examines content in a carefully considered integration of research that conveys where the field stands and where it might go. Drawing heavily on their own work, the chapter authors reconceptualize research in their specific areas so that this knowledge is integrated with the others' strands. This model for synthesizing research can serve as a paradigm for how research in mathematics education can -- and probably should -- proceed.

1	Toward a Common Research Perspective

Thomas A. Romberg

Thomas P. Carpenter

Fennema, Elizabeth

University of Wisconsin-Madison

In the last few years there has been significant progress in integrating programs of research in school mathematics. This has occurred through the cooperation of scholars working on content analysis, student learning, teaching, curriculum, and assessment in particular mathematical domains. Furthermore, the products of such integration have demonstrated considerable potential for influencing curricular reform. In this volume a different approach to scholarly integration has been followed. The impact of technology on the way mathematical functions are represented provides a common focus for the research discussed here.

This volume was conceived as an initial attempt to create a scholarly community with a common approach to the study of functions and their graphs. There is general consensus that functions are among the most powerful and useful notions in all mathematics. Nevertheless, the learning and teaching of functions are understudied in comparison to other areas of mathematical instruction. Leinhardt, Zaslavsky, and Stein (1990), based on their review of research, found that “actual studies of teaching [functions] at either the elementary or secondary level are quite rare and, in general, unconnected to the knowledge that a student develops” (p. 54).

The common approach to teaching functions in American schools follows the historical evolution of the ideas in the domain. Although the formal discussion of functions is a 19th century development, its roots go back centuries. Mesopotamians in the 17th century BC constructed tables of related values for two variables (e.g., taxation and flow of goods) and in some cases were able to express, in a primitive manner, that relationship (Säljö, 1991). Their efforts are similar to “guess my rule” exercises common in elementary texts. The concept of variable, with the problem of how to represent the relationship between two variables, underlies this domain.

The ancient Greeks by the 3rd century BC used ratios and proportions to express a wide variety of relationships, such as the corresponding sides of similar triangles, the radius of a circle and its circumference, and weights and lengths on a balance beam. They were able to develop considerable mathematics based on this way of considering relationships. Given that all ratios [y:x] can be considered as simple linear functions of the form y = mx (where m is the constant of proportionality) ratios, proportions, and proportional reasoning are still considered as necessary background skills to the study of functions even if these connections are rarely pointed out in contemporary texts.

In spite of the mathematical insights these ancients were able to develop they were hampered by lack of a suitable notational system. This was remedied in the 14th century AD with the importation of the Hindu-Arabic numerals and algebraic notation from the Mideast at the end of the era of the Crusades. The use of a letter (e.g., X) to represent a variable and of an equation to relate two variables now made it possible to express in a compact form a vast number of relationships. Since that time the core of work on relations and functions has grown as a consequence of the introduction of this powerful notational system. A second effective way of representing the relationship between two variables was invented by René Descartes in the 17th century: By picturing related sets of values on a coordinate system, it becomes possible to visualize the relationship between the variables via a graph.

In summary, to express the relationship between two variables humans have invented tables of related values, algebraic expressions, and graphs on a coordinate system. For many years students have been taught both how to construct such representations and how to interpret them. However, it is basically the algebraic representations and the subsequent methods of manipulating those representations that have been emphasized in most mathematics courses in traditional systems — from beginning algebra to calculus and beyond.

Today, the impact of technology on the way mathematical functions can be represented and manipulated is forcing scholars to reconsider the way functions are used and taught. The algebraic instructional emphasis is being challenged in most of the chapters in this book. Technology makes it possible to deal with functions in new ways and to explore new ideas in curriculum and classroom practice. The abstractness of the algebraic expressions and the variety of transformations of such expressions have proved to be difficult for many students to fathom. In the past, graphical representations of special functions were often employed to motivate the use of an algebraic expression. However, graphs often were difficult and cumbersome for teachers and students to create or manipulate. Today, with the advent of the computer and graphing calculator, these representations not only are easy to create, but they too can be transformed in a variety of ways. The belief is that emphasizing graphical representations will make functions easier to learn and use for most students. The chapters in this volume represent an attempt by the authors to substantiate the basis of this belief by examining (1) the shift to graphical representations of functions using new technological tools and (2) the impact of this shift on content, learning, teaching, and assessment in mathematics classrooms.

The development of a common research perspective with respect to the teaching and learning of any mathematical domain is relatively recent (Romberg & Carpenter, 1986). It has become apparent that a more unified program of research is needed if we are to acquire an understanding of teaching and learning in schools that will inform curriculum development and assessment (Carpenter & Peterson, 1988; Fennema, Carpenter, & Lamon, 1991). Attempting to integrate a number of different research perspectives is a complex task, and ways need to be found both for organizing a group of scholars and for constraining the task so that complexity will be reduced without sacrificing integration.

One strategy for organizing a group has been to hold a conference that brings to the same venue researchers with a common interest. This book is a product of such a conference— one at which scholars interested in the graphical representation of functions had a chance to share their work. Precedence for such a gathering came from the Wingspread Conference on addition and subtraction held in 1979 (Carpenter, Moser, & Romberg, 1982). At that conference, researchers interested in the learning of early number concepts and operations found that there was general agreement as to the basic questions of interest regarding what had been accomplished, and what had yet to be done. The participants were able to come to terms with a common vocabulary and a common agenda for future work. Thus, they were able to integrate their research endeavors into a common framework. As Sowder (1989) pointed out, it is difficult to find research in the learning of early number concepts that has not been positively affected by the Wingspread Conference.

The common method of structuring the work of such a research group has involved studying specific mathematical domains. The domain knowledge strategy (Romberg, 1987) is based on the philosophic premise that the power of mathematics lies in the fact that a small number of symbols and symbolic statements can be used to represent a vast array of different problem situations. The properties of a domain include a set of problem situations that make a set of related concepts meaningful (problems that give rise to the need to relate two variables), a set of invariants that constitute the concept (e.g., the concept of a variable), and a set of symbolic situations used to represent the concept, its properties, and the situations it refers to (tables, equations, and graphs). In the past scholars have started with a semantic analysis of the problem situations. This has been done for such domains as addition and subtraction of whole numbers, multiplication and division of whole numbers, rational numbers, algebra, and geometry. This strategy has helped researchers coalesce their work because:

1. They were able to agree on the mathematical and semantic structure underlying a particular domain.
2. They could relate student and teacher cognitions directly to this underlying structure.
3. They could use this knowledge to enhance learning in classrooms.

In this volume a different point of departure has been chosen—the focus is the impact of technology on the way functions are represented. This choice was made due to the fact that the number of situations in which even simple linear functions can apply is phenomenally large. Thus, nothing akin to the semantic analysis of problem types seemed realistic for functions. Instead, we chose to confine work in this domain to an examination of the impact of graphing technology on the domain. Whether this will prove to be a viable structure for organizing a program of research is open to question. The perspectives presented in this volume illustrate the potential for adopting this starting point. However, its full potential will not be realized until problems related to three aspects of the teaching and learning of functions are addressed and resolved. These are agreement about what constitutes the domain of mathematical functions, the impact of graphing technology on the domain, and research on the teaching and learning of functions.

The Domain: Mathematical Functions

The evolution of functions in mathematics clearly reflects the fact that mathematical objects are created in response to social problem situations (Romberg, 1992). Furthermore, their development involves invention and abstraction, and even today mathematicians’ views about the domain are changing (Kleiner, 1989). Several chapters in this volume (those by Moschkovich, Schoenfeld, & Arcavi [Chapter 4], Cooney & Wilson [Chapter 6], and Kieran [Chapter 8]) describe aspects of this evolution both conceptually and pedagogically. The initial situations that gave rise to functions were problems involving related variables. For example, the distance an object moves in relationship to its speed, a child’s weight in relationship to his or her age, and the amount of postage required to mail a letter in relationship to its weight are typical situations. How best to consider functions has not been agreed upon. However, most authors agree that a functional relationship involves some sort of rule that assigns to each element of a given set a unique element of some other, not necessarily distinct, set. Freudenthal (1983) claimed that there are two essential features of the function domain that have evolved with the structure of the sets and the rules relating them— arbitrariness and univalence.

The arbitrary nature of functions refers to both the relationship between the two sets on which the function is defined and the sets themselves. The first means that functions do not have to exhibit some regularity—that is, functions do not have to be described by any specific algebraic expression or by a graph of a particular shape. For example, the relationship between time and temperature is an illustration of this kind of function. The arbitrary nature of the two sets means that functions do not have to be defined on any specific sets of objects; in particular, the sets do not have to be sets of numbers. A rotation of the plane is an example of this type of function, because it is defined on sets of points.

Whereas the arbitrary nature of functions is implicit in the common definition of a function, the univalence requirement, that for each element of the first set the rule assigns a unique element of the second set, is explicit. Freudenthal (1983) attributed this requirement to the desire of mathematicians to keep things manageable. Keeping track of meanings of multivalued symbols (such as √), and taking care that they have the same meaning in the same context, became too messy in advanced analysis. Thus, the univalent restriction on functions became an accepted part of all definitions of functions.

Pedagogically, the notion that the relationship does not have to have any regularity causes both teachers and students problems. It is clear from the chapters in this volume by Yerushalmy and Schwartz (Chapter 3), Moschkovich, Schoenfeld, and Arcavi (Chapter 4), Cooney and Wilson (Chapter 6), and Kieran (Chapter 8) that the way one considers the “rule or recipe” relating elements from one set to elements of another is critical. Although indeed logically a relationship may be arbitrary, in almost all practical situations there is an implicit dependency relationship. For example, the area of a square can be expressed as the length of a side squared (A = s²). Such a functional relationship between the length of a side of a square and its area is not arbitrary. The current pedagogic view as strongly endorsed by Thorpe (1989) is to emphasize such practical relationships in school mathematics.

In order to develop an integrated research program for school mathematics in this domain, agreement needs to be reached on how best to characterize the domain. The semantic analyses used as a starting point for many other mathematical domains would be unwieldy for functions, but a conceptual and pedagogic framework to tie scholarly work together is still needed.

Graphing Technology

There are three primary ways of representing a functional relationship —by presenting ordered pairs of values in a table, by creating algebraic expressions to represent the relationship, and by constructing graphs to picture the relationship. For purely arbitrary functions, tables of related values (e.g., time and temperature) are commonly used. For most mathematical applications, such sets of ordered pairs are of interest if one suspects the existence of a pattern or regularity. The identification of such a pattern is then described. At this point the pattern itself may become the object of study. As Moschkovich, Schoenfeld, and Arcavi (Chapter 4) point out, this is a critical step in the mathematical modeling process, but it shifts one’s attention from the relationship to the model.

The identification of patterns, while based on the observed ordered pairs, involves generalization to unobserved data. The primary means of representing such patterns in the past has been to use algebraic equations. In fact, it is the creation of such equations that many view as the most important pedagogic feature of functions. Unfortunately, as several authors in this volume (e.g., Kieran, Chapter 8; Philipp, Martin, & Richgels, Chapter 9) point out, too often the study of functions is limited to the study of a small number of given algebraic functions. In fact, students are not expected to create an expression for a set of data; they are only expected to learn the properties of certain recipes. Even and Ball (1989) showed that many mathematics teachers and students think that functions must be expressible as algebraic formulas and, indeed, must express a regularity.

The third method of representing functional relationships, via graphs, is the theme of this book. Graphs were initially created from the ordered pairs in a table to facilitate discovery of some regularity. Also, they were sometimes created to picture common algebraic functions (e.g., y = 3x − 4 as a straight line; or y = sin x as a wave with a particular shape). This is still usually done pedagogically for motivational purposes.

However, until recently the creation of a graph for most functions was difficult. Many points needed to be plotted and sketched. The ready availability of graphing technology has changed that. This fact informs all of the chapters in this volume. If one has an algebraic representation and enters it on a graphing utility, its graph is immediately created. Similarly, it is now easy to graph data sets. The easy creation of such graphs should lead to the introduction of such important topics as curve fitting, and why some classes of functions (e.g., polynomial, exponen...

Table of contents

• Cover Page
• Half Title page
• Series page
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Toward a Common Research Perspective
• I Content
• II Student Thinking
• III Teacher Thinking
• IV Teacher Knowledge
• V Classroom Instruction
• VI Curricular Implications
• VII Reactions
• Author Index
• Subject Index