Key Words

Introduction

A central and enduring problem in undergraduate mathematics education is how to structure and design learning environments so that students feel the need to develop for themselves (and in conjunction with their instructor), formal and abstract mathematical ideas. Too often powerful and beautiful mathematical ideas are learned (and taught) in a procedural manner, thus depriving students of an experience in which they create and refine ideas for themselves. A basic premise of our collective work is that theory and practice are mutually informative. Thus, to make progress on the actual teaching and learning of mathematics, one needs to deepen and further one’s understanding of theoretical ideas that frame teaching and learning. In this chapter we examine different theoretical perspectives on how models and modeling can inform undergraduate mathematics education. Currently, the field is developing several such approaches, each of which is informed by somewhat different orientations. Our conjecture is that there are significant points of compatibility between these different perspectives, but also important differences.

This chapter takes a first step towards articulating and synthesizing some of these comparisons. In the sections that follow, each of the five different perspectives on modeling address the following three questions, with illustrative examples as space permits: (1) What is a model? (2) What are the research goals for students in classrooms? (3) How does the modeling perspective relate to the broad learning goals for students in classrooms? …Answers to these questions depend on the modeling perspective adopted.

Spesier and Walter on Models

For us, mathematics is something that one does. Specifically, one solves a problem. A model, in this context, can function as a tool to help make sense of something that we want to understand. We begin with the world, and in particular the world of the student’s experience, where mathematics has the power to put order into that world and help us understand its behavior (Taylor, 1992, p. 7). The game gets interesting when learners feel impelled to build or reinvent the mathematics that might help. Our collective research focuses on how groups of learners build key mathematical ideas and understanding as they work on problems that demand new insights. To begin such work, we concentrate initially on task design in order to offer problems that give opportunities for key ideas or understandings to be built collectively. Given such tasks, our research concentrates on how groups of learners explore, reason, and communicate.

For concreteness, here is an example (Speiser and Walter, 2004) from undergraduate first-semester calculus in which students are provided enlarged copies of the photograph in Fig. 5.1. Students also were given metric rulers and a suggestion that they measure distances in millimeters and angles in radians. In this way, they had to choose an origin and polar axis, then decide for themselves how to meaningfully organize and interpret the data to be investigated. In their text (a preliminary version of Hughes-Hallett et al., 2002), the students already had seen exponential functions, and they knew how to look for constant ratios to make sense of them. Working in groups, they built three models in succession.

Fig. 5.1
figure 1

Placenticeras. A fossil ammonite, 170 million years old, from Glendive, Montana. About half actual size. (Photo: Two Samurai Graphics)

Full size image

Task (Speiser and Walter, 2004): It is natural to represent the spiral of this shell in polar coordinates (r, θ). What can you say about the relation between r and θ?

A consensus rapidly emerged that r should vary exponentially with θ. Hence they proposed r = r 0 a θ as a first model, with reasonable values for the two constants, r 0 and a. Inspecting graphs (software at hand) most students thought they found reasonable fits. One student, however, thought this fit, though reasonable, might still deserve examination. “Have you noticed,” he asked, “that the data points kind of snake back and forth across the exponential?” The students graphed the difference between the data points and what their model had predicted. They recognized an oscillation, whose period might well have been 2π. Hence it made sense to modify the initial model by adding a term of the form \( b\sin (\theta + c) \), for suitable constants b and c to be determined from their data. For this second model, each group reported a somewhat closer fit. But then came a surprise: each group had found significantly different constants b and c.

Let’s pause a moment to take stock. If we simply take the measured r-values as given, then each group obtained a closer fit with a model of the form \( r = r_0 a^\theta + b\sin (\theta + c) \). Hence, perhaps, we should have stopped right there. (Rationale: the model fits the data.) To stop, however, would have left unresolved an interesting question: Why did different groups find different b and c? Could each group’s pair of constants b and c depend on where one chose to place the origin? With this possibility in mind, each group used the specific b and c they had determined to relocate the origin and on this basis reconstruct their data. This relocation led to a third model, of the same form r = r 0 a θ as before, but in each case with new r 0 and a. These final models gave the closest fits so far obtained (Speiser and Walter, 2004; Speiser et al., 2007; Speiser and Walter, in press)…. We explore important psychological and epistemological implications of our particular approach to models in a further chapter in this volume.

Harel on Models

Rather than asking, “What is a model?”, the DNRFootnote 1 theory that I have developed asks, “What is Mathematics?” The answer to this question determines in large part my research goals, both locally (Question 1) and globally (Question 2). DNR-based instruction in mathematics is a theoretical framework that can be thought of as a system consisting of three categories of constructs: premises – explicit assumptions underlying the DNR concepts and claims; concepts – referred to as DNR determinants; and instructional principles – claims about the potential effect of teaching actions on student learning.

Central to DNR is the distinction between “ways of understanding” and “ways of thinking.” “Ways of understanding” refers to a cognitive product of a person’s mental actions, whereas “way of thinking” refers to its cognitive characteristic. Accordingly, mathematics is defined as the union of two sets: the set WoU, which consists of all the institutionalized ways of understanding in mathematics throughout history, and the set WoT, which consists of all the ways of thinking that characterize the mental acts whose products comprise the first set.

The members of WoT are largely unidentified in the literature, though some significant work was done on the problem-solving act (e.g., Schoenfeld, 1985; Silver, 1985) and the proving act (see an extensive literature review in Harel and Sowder, 2007). The members of WoU include all the statements appearing in mathematical publications, such as books and research chapters, but it is not listable because individuals (e.g., mathematicians) have their idiosyncratic ways of understanding. A pedagogical consequence of this fact is that a way of understanding should not be treated by teachers as an absolute universal entity shared by all students, for it is inevitable that each individual student is likely to possess an idiosyncratic way of understanding that depends on her or his experience and background. Together with helping students develop desirable ways of understanding, the goal of the teacher should be to promote interactions among students so that their necessarily different ways of understanding become compatible with each other and with that of the mathematical community.

Since mathematics, according to the above definition, includes historical ways of understanding and ways of thinking, it must include ones that might be judged as imperfect or even erroneous by contemporary mathematicians. The boundaries as to what is included in mathematics are in harmony with the nature of the process of learning, which necessarily involves the construction of imperfect and erroneous ways of understanding and deficient and faulty ways of thinking. These boundaries, however, are not to imply acceptance of the radical view that particular mathematical statements could be true for some people and false for others.

My definition of mathematics implies that an important goal of research is to identify desirable ways of understanding and ways of thinking, recognize their development in the history of mathematics, and, accordingly, develop and implement mathematics curricula that aim at helping students construct them. As an example, I mention the algebraic invariance way of thinking. With it, students learn to manipulate symbols with a goal in mind – that of changing the form of an entity without changing a certain property of the entity. Another example involves the role of Aristotelian causality, as a way of thinking, in the development of mathematics. This raises the question of whether the development of students’ proof schemes parallels those of the mathematicians of these periods. An answer to this question would likely have important curricular implications.

Pedagogically, the most critical question is how to achieve such a vital goal as helping students construct desirable ways of understanding and ways of thinking. DNR has been developed to achieve this very goal. As such, it is rooted in a perspective that positions the mathematical integrity of the content taught and the intellectual need of the student at the center of the instructional effort. The mathematical integrity of a curricular content is determined by the ways of understanding and ways of thinking that have evolved over many centuries of mathematical practice and continue to be the ground for scientific advances. To address the need of the student as a learner, a subjective approach to knowledge is necessary. For example, the definitions of the process of “proving” and “proof scheme” are deliberately student-centered (see Harel and Sowder, 1998). It is so because the construction of new knowledge does not take place in a vacuum but is shaped by one’s current knowledge. What a learner knows now constitutes a basis for what he or she will know in the future. This fundamental, well-documented fact has far-reaching instructional implications. When applied to the concept of proof, for example, this fact requires that instruction takes into account students’ current proof schemes, independent of their quality. Despite this subjective definition the goal of instruction, according to DNR, must be unambiguous – namely, to gradually refine current students’ proof schemes toward the proof scheme shared and practiced by contemporary mathematicians. This claim is based on the premise that such a shared scheme exists and is part of the ground for advances in mathematics.

Larson on Models

What is a Model? Lesh and Doerr (2003) characterize models as

…conceptual systems (consisting of elements, relations, operations, and rules governing interaction) that are expressed using external notation systems, and that are used to construct, describe, or explain the behaviors of other systems(s) – perhaps so that the other system can be manipulated or predicted intelligently. (p. 10)

Or, even more specifically, Lesh et al. (2000) provide the following definition of a model:

A model is a system that consists of (a) elements; (b) relationships among elements; (c) operations that describe how the elements interact; and (d) patterns or rules … that apply to the relationships and operations. However, not all systems function as models. To be a model, a system must be used to describe another system, or to think about it, or to make sense of it, or to explain it, or to make predictions about it. (p. 609)

According to this models and modeling perspective (MMP), the term model is used to refer to the conceptual system with which people make sense of their experiences (Lesh and Doerr, 2003). Models may be expressed externally using any combination of representational media ranging from physical actions (including gestures), to spoken words, to written words or symbols (including metaphors and equations), to images (including pictures, graphs, tables, and diagrams).

MMP Research Goals. Characterizing the research goals of an entire community of researchers who identify themselves with this theoretical perspective is a daunting and perhaps, in some sense, dangerous task. I do not claim to completely and exhaustively articulate the research goals of the broader community of researchers associated with and influenced by MMP. Rather, I offer a set of general impressions that are a reflection of my experiences as a graduate student and researcher who has spent several years working with Dick Lesh.

MMP is a broad perspective, and thus has the potential to inform a wide variety of research questions. Much of the work that has been done using MMP focuses on problem solving and local conceptual development (heavily influenced by Piaget), teacher development, and the nature of mathematical knowledge needed by students to succeed in an increasingly technologically sophisticated society. Central to much of this work has been the use and development of so-called model-eliciting activities (MEAs) which are “real-life” tasks that are designed to require students to invent, refine, and generalize powerful mathematical constructs (Lesh and Doerr, 2003). An MEA is a task that is carefully designed to act as a research tool to help document student thinking. In particular, these tasks are designed so that students are able to test the quality of their own solutions (and thus revise those solution strategies as they deem necessary), and so that students’ solutions provide descriptions of the ways of thinking that they used to solve the problem. Thus, solutions provide trails of documentation of students’ thinking (Lesh et al., 2000).

M&M Learning Goals. A central goal of MMP research is to facilitate students development and refinement of their own abilities to mathematize, “…by quantifying, dimensionalizing, coordinatizing, categorizing, algebratizing, and systematizing relevant objects, relationships, actions, patterns, and regularities” (Lesh and Doerr, 2003, p. 5). This perspective places a particular emphasis on the importance of presenting students with opportunities to cyclically improve the mathematically significant systems that they develop to reason with and about “real-life” situations such as the contexts described in particular instances of model-eliciting activities.

Oehrtman on Models

My research has drawn significantly from Max Black’s interactionist theory of metaphorical attribution to identify and characterize mental models constructed by students as they reason about new mathematical ideas. Employing a design research methodology, I assess which sets of student metaphors are amenable to instruction to form more mathematically rigorous and powerful conceptual foundations.

Black’s description of strong metaphors, those accounting for new ways of understanding, differs slightly from recent influential characterizations of metaphor by cognitive linguists. Lakoff and Johnson (1980), Lakoff (1987), and Lakoff and Núñez (2000), for example, describe conceptual metaphors as mappings in which abstract concepts are generated and made meaningful through the projection of preconceptual structures from domains of “embodied experience.” Black refers to this type of projection as a “theoretical model” (1962). This is the nature of reasoning employed, for example, by Clerk Maxwell in invoking the image of motion in an incompressible fluid to represent his ideas about an electrical field. Such a model provides a well-understood source domain (e.g., fluids) that is imagined to be isomorphic with respect to certain structures and properties of the new scientific domain (e.g., electrical fields) so that inferences may be transferred.

Black draws a distinction between theoretical models and metaphorical thinking in that the creator of a theoretical model “must have prior control of a well-knit scientific theory if he is to do more than hang an attractive picture on an algebraic formula. Systematic complexity of the source of the model and capacity for analogical development are of the essence” (Black, 1962, p. 239). This description of scientific systematicity is far stronger than what most introductory calculus students are likely to display with respect to building models for complex ideas such as limits. Furthermore, typical calculus instruction does not expect students to develop the solutions to (or be engaged by) the types of technical problems that gave rise to the formal structures that are the targets of such expert mappings.

Metaphorical attribution generating new ways of understanding does not simply involve antecedently formed concepts of the domains involved, but is achieved through an implicative dialectic between conceptual domains (Black, 1962, 1977, 1979). Thus, in identifying student metaphors, I focus on characterizing emerging concepts through changes in ways of viewing both the source and target domains. This perspective on metaphorical reasoning is consequently closer in form to characterizations of “conceptual blends” by Fauconnier and Turner (2002) than it is to the projections of conceptual metaphors.

Central properties of “strong” metaphors drive my methodological practices. Strong metaphors are “emphatic,” commanding commitment to their particular structure. As a result, I look for a convergence of repeated application of a metaphor to any given concept, applications across a variety of situations involving the concept, and use of similar metaphors by a large number of students. Strong metaphors are also “resonant,” supporting a substantial degree of implicative elaboration. Thus I also require the presence of conceptual consequences of the application of the metaphors. Finally, they are “ontologically creative,” establishing new perspectives that would not have otherwise existed for the student. To capture this aspect of students’ reasoning, I seek evidence that the claims being made within the application of a particular metaphor are unique to that way of reasoning.

The design research component of my work may best be described as choosing student metaphors that have potential to be restructured to guide students in more rigorous mathematical reasoning and developing, evaluating, and refining curricular activities that help students incorporate these new structures into their reasoning in a systematic way. In terms of Max Black’s categorizations, I aim to assist students in converting their “metaphorical thinking” into “theoretical models.”

As an example, consider the metaphor of limit as approximation. An expert version of such a metaphor applied to the limit of a sequence might include

  • the limit, L, as a quantity to be approximated,

  • the terms, a n , as approximations,

  • the magnitude of the difference, |L – a n |, as the error

  • epsilon as a bound on the size of the error, |L – a n | < ε,

  • the condition n > N for some N as a way of controlling the size of the error,

  • convergence as a statement about obtainable accuracy,

  • Cauchy convergence as a statement about obtainable precision, etc.

While students use many aspects of this structure in their own reasoning, their spontaneous approximation metaphors contain only fragments of these ideas plus idiosyncrasies such as approximations “always becoming more accurate” and errors being “mistakes” (Oehrtman, 2007). Research on the implementation of our activities indicates that using approximation as a thematic foundation for calculus instruction does help students develop more refined versions of these metaphors and systematic control over their use (Oehrtman, in press). Ongoing research also indicates that this approach strengthens students conceptual understanding of the Riemann integral (Sealey and Oehrtman, 2005, 2007), and subsequent studies are planned to investigate its impact on students’ development of other central concepts in calculus defined in terms of limits such as continuity, derivatives, and series.

Rasmussen and Zandieh on Models

What is a model? We define models to be student generated ways of organizing their mathematical activity with physical and mental tools. Thus, models are not simply the “things” that one uses (e.g., a graph, an equation, etc.) but rather the ways in which learners structure their activity with and conceptions of graphs, equations, and even definitions. Moreover, from a Realistic Mathematics Education (RME) point of view, there is not just one model, but a sequence of models that develop (or emerge) through student activity. This progressive characteristic is typically referred to as the Emergent Model heuristic. As described by Gravemeijer (1999), students first develop models-of their mathematical activity, which later become models-for more sophisticated mathematical reasoning. This model-of/model-for transition is commensurate with the creation of a new mathematical reality (for learners).

Zandieh and Rasmussen (2007), drawing on their work with undergraduate students in a proof-oriented geometry class, extend the construct of emergent models to the activity of defining. In a case study that involves student work with both the plane and sphere, they detail the evolution of models in terms of four layers of activity, referred to as Situational, Referential, General, and Formal (Gravemeijer, 1999). The table below defines each of these four layers and gives a brief example in the case of defining.

Layers of activity

Defining example

Situational activity involves students working toward mathematical goals in an experientially real setting

Students create a definition for triangle on the plane and in the process revisit some of their concept images for planar triangles. Planar triangles are, for these students, experientially real in the sense that they have rich concept images of planar triangles

Referential activity involves models-of that refer to physical and mental activity in the original task setting

Students use the planar triangle definition (slightly modified for the sphere), along with the properties that they associate with this definition, to create examples of spherical triangles and to notice some of their properties. Students’ organizing activity with the definition and associated concept images of planar triangle applied to the sphere functions as a model-of (or definition-of) the relevant physical and mental activity in the plane

General activity involves models-for that facilitate a focus on interpretations and solutions independent of the original task setting

Students’ organizing activity with refined definitions and concept images of spherical triangles functions as models-for (or definitions-for) enlarging the new mathematical reality of spherical triangles, and making generalizations about these objects in ways that do not refer to the plane

Formal activity involves students reasoning in ways that reflect the emergence of a new mathematical reality and consequently no longer require the support of prior models-for activity

Students reason about spherical triangles in ways that reflect new structural relationships between these objects and consequently use definitions as links in chains of reasoning without having to revisit or unpack the meaning of these definitions

What are our research and learning goals? We view our research as falling within the larger category of Design Research (Cobb and Gravemeijer, in press). Design Research has a dual focus in which research on teaching and learning guides the implementation of the instructional design, and the implementation of the instructional design provides the data for both an ongoing and a retrospective analysis. There are several different types of instructional design products that this process has the potential to generate. One overarching product is a local instructional theory (LIT) specific to the teaching and learning of a specific topic area, such as linear algebra or differential equations. A LIT is an essential support for teachers. This is not meant to be a step-by-step instruction for the teacher but rather a “description of, and rationale for, the envisioned learning route as it relates to a set of instructional activities for a specific topic (Gravemeijer, 2004, p. 107).” For us, a LIT includes four related and revisable aspects: (1) learning goals about student reasoning, (2) a storyline of how students’ mathematical learning experience will evolve, (3) the role of the teacher in the storyline, and (4) a sequence of instructional tasks that students will engage in.

These somewhat pragmatic products develop concurrently with our broader, theoretical goals. For example, in ongoing work we have three core research goals: (1) To create theoretical means for understanding and interpreting student learning (from both cognitive and sociocultural perspectives) as it relates to the model-of/model-for transition, (2) To develop methodological approaches that enable these kind of analyses, and (3) To apply these analyses to instructional design in the teaching of undergraduate mathematics in general, and in linear algebra in particular.

The strength of this work is the tight integration of basic research on student learning and instructional design work with students in their normal classroom settings to develop products that are both practical and theoretical.