# «HALVES, PIECES, AND TWOTHS: CONSTRUCTING 1 REPRESENTATIONAL CONTEXTS IN TEACHING FRACTIONS 2 Deborah Loewenberg Ball Goals of Teaching and Learning ...»

## HALVES, PIECES, AND TWOTHS: CONSTRUCTING

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## REPRESENTATIONAL CONTEXTS IN TEACHING FRACTIONS

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Deborah Loewenberg Ball

Goals of Teaching and Learning Mathematics

Current discourse about the desirable ends of mathematics teaching and learning centers on the

development of mathematical understanding and mathematical power——the capacity to make sense

with and about mathematics (cf. California State Department of Education, 1985; National Council of Teachers of Mathematics, 1989a; National Research Council, 1989). Learning mathematics with understanding, according to this view, entails making connections between informal understandings— —about mathematical ideas, quantitative and spatial patterns, and relationships——and more formal mathematical ideas. Connections must be forged among mathematical ideas (Fennema, Carpenter, and Peterson, 1989). Students must develop the tools and dispositions to frame and solve problems, reason mathematically, and communicate about mathematics (National Council of Teachers of Mathematics, 1989a).

These goals go beyond understanding of particular ideas——place value, functions, triangles, area measurement. "Knowing mathematics" includes knowing how to do mathematics: "To know mathematics is to investigate and express relationships among patterns, to be able to discern patterns in complex and obscure contexts, to understand and transform relationships among patterns" (National Research Council, 1990, p. 12). Included in this view of understanding mathematics also are ways of seeing, interpreting, thinking, doing, and communicating that are special to the community of mathematicians. These specialized skills and ways of framing and solving problems can contribute to everyday confidence and competence; they are personally as well as intellectually empowering.

**Schoenfeld (1989) summarizes this dimension of mathematical knowledge:**

Learning to think mathematically means (a) developing a mathematical point of view— —valuing the process of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of understanding structure——mathematical sense-making. (p. 9) This sense-making is both individual and consensual, for mathematical knowledge is socially constructed and validated. Drawing mathematically reasonable conclusions involves the capacity to make 1 This will appear as a chapter in T. P. Carpenter and E. Fennema (Eds.), Learning, Teaching, and Assessing Rational Number Concepts.

2 Deborah Loewenberg Ball, assistant professor of teacher education at Michigan State University, is a senior researcher with the National Center for Research on Teacher Education.

Contemplating Content and Students Helping students develop this kind of mathematical power depends on insightful consideration of both content and learners, consideration that is both general and situated. Figuring out how to help students develop this kind of mathematical knowledge depends on a careful analysis of the specific content to be learned: the ideas, procedures, and ways of reasoning. Such analyses must examine the particular: Probability, for instance, is a domain that differs in some important ways from number theory, both in the nature of the ideas themselves and in their justification, as well as in the kinds of reasoning entailed. Similarly, an argument in geometry is distinctive from one in arithmetic. Differences in how a given topic evolved may also be useful in considering how students may encounter and develop its ideas: That it took the mathematical community centuries to accept negative numbers in a "felt way" (Kline, 1970) may help to explain students' struggles to make sense of quantities that are less than zero (Ball, 1990b).

But analyzing the content——concepts and ways of knowing——is insufficient. Helping students develop the kind of knowledge described above also depends crucially on understandings of students themselves and how they learn the particular content. Careful analyses of the content cannot suffice to map the terrain through the eyes of the prospective child-explorer. As Dewey (1902) puts it aptly, "The map does not take the place of the actual journey" (p. 20). The teacher must simultaneously maintain a complex and wide-angled view of the territory, all the while trying to see it through the eyes of the learner exploring it for the first time (Lampert, in preparation). How does the mathematics appear to a nine-year-old? Nine-year-olds' ideas and ways of thinking approach formal mathematical ideas and ways of thinking unpredictably and, at times, with breathtaking elegance. Teachers, argues Hawkins (1972), must be able to "sense when a child's interests and proposals... are taking him near to mathematically sacred ground" (p. 113). This bifocal perspective——perceiving the mathematics through the mind of the learner while perceiving the mind of the learner through the mathematics——is central to the teacher's role in helping students learn with understanding.

Representational Contexts for Learning Mathematics But this contemplation of content and students is not passive.

**The teacher is not, as Hawkins (1972), points out, simply an observer; the teacher's role is to participate in students' development:**

In order to help students develop mathematical understanding and power, the teacher must select and construct models, examples, stories, illustrations, and problems that can foster students' mathematical development. Lampert (1989) writes of the need to select a representational domain with which the children are familiar and in which they are competent to make sense——in other words, in which they can extend and develop their understandings of the ideas, as well as their capacity to reason with and about those ideas. For instance, because students are familiar with relationships among pennies, dimes, and dollars, and because they are comfortable with the notation, Lampert argues that money may provide one helpful terrain in which they can extend their understanding of decimal

**numeration. Dewey (1902) writes:**

What concerns [the teacher] is the ways in which that subject may become part of experience; what there is in the child's present that is usable in reference to it; how such elements are to be used; how his own knowledge of the subject-matter may assist in interpreting the child's need and doings, and determine the medium in which the child should be placed in order that his growth may be properly directed. [The teacher] is concerned, not with the subject-matter as such, but with the subject-matter as a related factor in a total and growing experience. (p. 23, emphasis added) The issue of selecting, developing, and shaping instructional representations has been the focus of a wide range of inquiry (e.g., Ball, 1988; Kaput, 1987, 1988; Lampert, 1986, 1989; Lesh, Behr, and Post, 1987; Lesh, Post, and Behr, 1987; McDiarmid, Ball, and Anderson, 1989; Wilson, 1988; Wilson, Shulman, and Richert, 1987). Shulman (1986) and his colleagues (Wilson, Shulman, and Richert, 1987) have developed a construct which they call pedagogical content knowledge: an "amalgam" of knowledge of subject matter and students, of knowledge and learning. Pedagogical content knowledge includes understandings about what students find interesting and difficult as well as a repertoire of representations, tasks, and ways of engaging students in the content. Nesher (1989) frames the problem for the teacher of mathematics in terms of two main needs: "(a) the need for a young child to construct his knowledge through interaction with the environment, and (b) the need to arrive at mathematical truths" (p. 188). The teacher must structure what Nesher calls a "learning system"——in which learners can explore and test mathematical ideas. Nesher's framework reminds us that the representation of ideas is more than just a catalog of ideas or a series of models——rather it is interactive and takes place within a larger context of ideas, individuals, and their discourse.

3 Dewey's (1902) problem of "determining the medium," or weaving what I will call a representational context in which children can do——explore, test, reason, and argue about——and consequently, learn, particular mathematical ideas and tools is at the heart of the difficult work of teaching for understanding in mathematics. Such representational contexts must balance respect for the integrity and spirit of mathematics with an equal and serious respect for learners, serving as an "anchor" for the development of learners' mathematical ideas, tools, and ways of reasoning. These contexts must provide rich opportunities for both individual and group discourse. All this sounds both sensible and elegant——pulling it off, however, is difficult.

Learning to Teach Mathematics for Understanding Learning to teach mathematics for understanding is not easy. This paper examines two reasons for this. First, practice itself is complex. Constructing and orchestrating fruitful representational contexts, for example, is inherently difficult and uncertain, requiring considerable knowledge and skill.

Second, many teachers' traditional experiences with and orientations to mathematics and its pedagogy hinder their ability to conceive and enact a kind of practice that centers on mathematical understanding and reasoning and that situates skill in context. Helping teachers develop their practice in the direction of teaching mathematics for understanding requires a deep respect for the complexity of such teaching and depends on taking teachers seriously as learners. In this paper I explore and provide evidence for this claim.

Considering the content. Substantively, a representation should make prominent conceptual dimensions of the content at hand, not just its surface or procedural characteristics. Important to bear in mind is that representations are metaphorical, borrowing meaning from one domain to clarify or illuminate something in another. As with metaphors——where objects are never isomorphic with their comparative referents——mathematical ideas are by definition broader than any specific representation.

**For example, area models——such as a circle model of 1/2:**

represent only one of several meanings of fractions (Ohlsson, 1988). Despite the fact that this is the most frequent representation that children will give if asked what one-half means, 1/2 also refers to the point halfway between 0 and 1 on a number line, the ratio of one day of sunshine to every two of clouds, or the probability of getting one true-false test item right.

No representational context is perfect. A particular representation may be skewed toward one meaning of a mathematical idea, obscuring other, equally important ones. For example, the number line as a context for exploring negative numbers highlights the positional or absolute value aspect of integers: that -5 and 5 are each five units away from 0. It does not necessarily help students come to grips with the idea that -5 is less than 5. Using bundling sticks to explore multidigit addition and subtraction directs attention to the centrality of grouping in place value, but may hide the importance of the positional nature of our decimal number system.

Beyond the substance of the topic itself, another layer of complexity rests with the fact that representation is fundamental to mathematics itself (Kaput, 1987; Putnam, Lampert, and Peterson, 1990). One power of mathematics lies in its capacity to represent important relationships and patterns in 3 My thinking about representations in teaching has been influenced by conversations over time with Suzanne Wilson and G.

Williamson McDiarmid. Wilson's (1988) work on representations in the teaching of U.S. history as well as my work with McDiarmid (cf. McDiarmid, Ball, and Anderson, [1989]) have also extended my ideas about this aspect of teaching for understanding. In addition, the conversations I have had with Sylvia Rundquist over the past two years——in particular the insightful questions she asks me about my teaching——have contributed significantly to my work on representation.

5 ways that enable the knower to generalize, abstract, analyze, understand. Learning to represent is therefore a goal of mathematics instruction, not just a means to an end. The teacher must figure out ways to help students learn to build their own models and representations——of real world phenomena as well as of mathematical ideas (Putnam, Lampert, and Peterson, 1990).

In teaching fractions, the teacher must weigh the relative advantages in providing students with structured representational materials (such as fraction bars that are already ruled into certain fixed partition sets) versus having students refine existing models and develop their own representational media (e.g., drawing circular regions and subdividing portions thereof). Take the idea of unit, which is central to fraction knowledge. If students are comparing 4/4 with 4/8, fraction bars will force them to

**the right answer that 4/4 is more than 4/8:**

They do not have to consider directly the role of the common unit, for it is implicit within the material.

Yet, if students construct their own models, they may confront and have to struggle with this essential

**concept, as one nine-year-old did when he drew, at first:**