Preparing to Teach in the New Millennium: Algebra Through the

Eyes of Preservice Elementary and Middle School Teachers

Joyce Bishop	Sheryl Stump
eastern Illinois University	Ball State University
cfjdb1@eiu.edu	sstump@gw.bsu.edu

This study examined preservice elementary and middle school teachers’ conceptions of algebra using a framework with categories of procedural and conceptual perspectives. Throughout the semester course, the majority of preservice teachers provided either a non-algebraic or a procedural definition of algebra. At the beginning of the semester, more preservice teachers with a conceptual perspective took a problem-solving position, but at the end of the semester, more held a generalization view. The preservice teachers in this study had a limited understanding and appreciation of generalization.

One theme of mathematics education reform is that algebraic reasoning should be integrated throughout elementary and middle school mathematics (NCTM, 1998, 2000). To implement this objective, teachers in the new millennium will need to identify concepts of algebra that are appropriate for elementary and middle school grades and choose appropriate pedagogy for promoting algebraic reasoning. Effective teacher education must insure that preservice teachers are prepared to address the questions: “What is algebra?” and “How can it be taught effectively?” This study begins to explore these issues in the context of an algebra course for preservice K-8 teachers.

Objectives

The purpose of this investigation is to gather insights into preservice elementary and middle school teachers’ conceptions of algebra. The research addresses the following questions: 1) What are preservice elementary and middle school teachers’ conceptions of algebra? 2) How do their conceptions change while taking an algebra course for preservice elementary and middle school teachers? and 3) What is their understanding and appreciation of generalization?

Theoretical Framework

The nature and role of algebra in school mathematics has been the focus of extended discussion in recent years. Edwards (1990) recommended that we guarantee all students the opportunity to study algebra, and Silver (1997) emphasized that we should guarantee access to algebraic ideas, not just algebraic courses. A consensus has emerged that to ensure such broad access, algebra should be incorporated into elementary and middle school mathematics (NCTM, 1989, 1993, 1998). But what kind of algebra should we incorporate? Moses (1997) observed that the content of algebra is being transformed from a discipline involving the manipulation of symbols to a way of seeing and expressing relationships, “a way of generalizing the kinds of patterns that are part of everyday activities” (p.246). As this view is interpreted in elementary and middle school, the crucial issue appears to be the development of algebraic reasoning, not just the introduction of algebraic concepts (Yackel, 1997; Driscoll, 1999). Mason (1996, p. 65) describes school algebra as the “language for expression and manipulation of generalities,” and states that the essence of teaching mathematics is the “awakening of pupil sensitivity to the nature of mathematical generalization, and dually, to specialization.” To appreciate the potential of mathematical generalization is to grasp the power of mathematics. As teacher educators, our goal is to help preservice teachers develop an appreciation for these subtle notions.

Discussions of mathematical understanding often distinguish between procedural and conceptual knowledge where procedural knowledge encompasses the formal language and the algorithms used to complete mathematical tasks, and conceptual understanding is characterized as a network of rich relationships (Hiebert & Lefevre, 1986). Particularly where conceptual knowledge is concerned, we have little knowledge about the specifics of teachers’ understanding (Ma, 1999). It seems likely, however, that a close relationship exists between the richness of a teacher’s mathematical understanding and the quality of mathematical thinking that is promoted in his or her classroom (Ma, 1999; Mason, 1996). A teacher with a rich understanding of the connections between mathematical ideas is more likely to reveal and represent them (Ma, 1999) at the same time that a teacher who lacks this awareness is unlikely to promote deep insight in his or her students (Mason, 1996). We cannot assume that preservice teachers bring appropriate connected knowledge with them.

Although the Professional Standards for Teaching Mathematics (NCTM, 1991) calls for the mathematical preparation of all preservice elementary and middle school teachers to include the study of algebra, experiences focusing on algebra as symbol manipulation may do very little to help preservice teachers develop profound understanding of algebraic concepts. To address this goal, a framework is needed for organizing algebra instruction for preservice teachers that promotes understanding of fundamental concepts of algebra beyond the traditional rules and procedures. Bednarz, Kieran, and Lee (1996) present four alternative approaches to the development of algebraic ideas: generalization, problem solving, modeling, and functions. These perspectives suggest a framework for examining preservice teachers’ conceptions of algebra, and for providing beneficial experiences for preservice teachers.

Research Methods and Data Sources

The participants in this investigation were students in one of two different algebra classes for preservice elementary and middle school teachers. Two of the common goals for the two courses were: 1) to develop preservice teachers’ understanding of algebraic concepts, and 2) to explore ideas for teaching algebra to elementary and middle school students. Both classes addressed the first goal by having preservice teachers engage in college-level algebraic experiences involving generalization, problem solving, modeling and functions. They addressed the second goal by exploring algebraic activities for children involving variables, functions, and pattern generalization.

On the first day of class, each preservice teacher responded to two questions designed to probe their definitions of algebra and their beliefs about teaching algebra. Question 1: “How would you explain what algebra is to someone who has never heard of it?” Question 2: “Imagine that you are observing a sixth-grade class where the teacher, Ms. Jones, is trying to promote algebraic thinking. Describe what is happening in the classroom that supports the achievement of this goal.” The preservice teachers responded to Question 1 and Question 2 again on the last day of the class.

During the course, the preservice teachers worked in groups to solve two problems selected from Driscoll (1999). The Sums of Consecutive Numbers problem required preservice teachers to identify and extend number patterns and focused on the notion of generalization. The Painted Cubes problem asked preservice teachers to use variables to describe patterns in the number of cubes painted on three faces, two faces, one face, and zero faces of different sized cubes. For each of the two problems, the preservice teachers responded to a set of reflection questions, including the following: “Describe how you used algebra in this problem.”

Toward the end of the course, each preservice teachers wrote a lesson plan for students in grade four, five, or six. Choosing from one of three topics (variable, generalizing patterns, or functions), they were to find or create an activity to promote the development of algebraic reasoning. This assignment provided preservice teachers an opportunity to demonstrate pedagogical content knowledge as framed by their conception of algebra.

Data analysis employed axial coding during which data were organized into categories and subcategories and examined to identify relationships among categories (Strauss & Corbin, 1990). Tables were then prepared to organize information and illustrate relationships (Miles & Huberman, 1994).

Table 1. Conceptions of Algebra

Algebraic	Describes algebraic notions such as variables, formulas, unknowns, relationships, patterns, functions, or generalizations.
Non-algebraic	Describes arithmetic or general mathematics without reference to algebraic notions.
Algebraic Subcategories:
Procedural	Describes rules, procedures, or symbolic manipulation, without relevance to underlying concepts.
Conceptual (one of the following):
Generalization	Describes the construction of formulas that account for a general procedure or relationship between quantities. The goal is to find an expression or a formula.
Problem Solving	Describes the forming and solving of equations, using letters as unknowns. The goal is to find an answer, to find a numerical value.
Functions	Describes dependence relationships among real-world quantities. The focus is on how a change in one variable produces a variation in the value of the function.

The researchers worked together to code each piece of data. In order to examine preservice teachers’ conceptions of algebra, they analyzed both sets of responses to Question 1 and Question 2, reflections for the Sums of Consecutive Numbers problem and the Painted Cubes problem, and

the Lesson Plan. Table 1 explains the categories used to code data addressing the preservice teachers’ conceptions of algebra. The responses to various questions were first coded as Algebraic or Non-algebraic. Algebraic responses were then coded as Procedural or as one of three Conceptual subcategories: Generalization, Problem Solving, or Functions. Although Bednarz et al. (1996) included a modeling approach to algebra, that perspective did not appear in this study.

To examine preservice teachers’ understanding and appreciation of generalization, the researchers looked for patterns in the categories of preservice teachers’ conceptions of algebra across the course. In addition, they analyzed preservice teachers’ responses to the following reflection question: “Consider the discoveries you described above (on the Sums of Consecutive Numbers problem). Would you describe them as specific facts or generalizations?”

Results and Implications

At the beginning and the end of the semester, the majority of preservice teachers in the two classes provided either a Non-algebraic or a Procedural definition of algebra. Similarly, when asked to describe a classroom in which the teacher is promoting algebraic reasoning, the scenes they described often contained no algebra or merely procedural aspects of algebra. Table 2 shows the number of preservice teachers in each category of each component of this study.

The Sums of Consecutive Numbers problem and the Painted Cubes problem were chosen as class activities because they provided preservice teachers with opportunities to use algebra as a tool for pattern generalization. However, when asked to describe how they used algebra in the Sums of Consecutive Numbers problem, the majority of preservice teachers focused on procedural aspects. When reflecting on the Painted Cubes problem, a greater number of preservice teachers described using algebra to generalize patterns, but an even greater number focused on the problem-solving aspects of the problem, describing how they used algebra to write and solve equations. Despite having had these experiences with pattern generalization (and problem solving), more than half of preservice teachers in the two classes later described algebra in procedural terms.

The preservice teachers in this investigation had a limited understanding and appreciation of generalization. In their reflections, only nine students demonstrated on the Sums of Consecutive Numbers problem that they understood the meaning of the term “generalization.” Sixteen of the students, 50%, indicated that they thought that a specific fact was more powerful than a generalization about the relationships. Of 15 people who chose to write a lesson plan about generalizing patterns, six worked with patterns but did not actually incorporate generalization. At the beginning of the semester, the majority of preservice teachers with a conceptual view of algebra expressed a Problem Solving conception. On the other hand, at the end of the semester, the distribution of conceptual views shifted toward a Generalization perspective.

Table 2. Preservice Teachers’ Conceptions of Algebra Throughout the Course (N = 32)___

	Non-algebraic	Algebraic: Procedural	Algebraic: Conceptual
			Problem Solving	Generaliza-tion	Functions
Question 1 (first day)	8	16	7	1
Question 2 (first day)	17	9	5	1
Consecutive Numbers Problem	6	15	2	6	1
Painted Cubes Problem	1	4	14	12
Lesson Plan	12	3	2	9	6
Question 1 (last day)	3	19	2	5	3
Question 2 (last day)	18	6	1	6	1

Although it is desirable for preservice teachers to recognize a variety of perspectives to algebra and prepare to teach in a manner that incorporates these varied perspectives, it appears that many do not understand what distinguishes arithmetic from algebra, and of those who do, a majority perceive algebra mainly from a procedural perspective. This continues to be true after they participate in either of two courses which emphasize conceptual approaches to algebra.

References

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Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10. Portsmouth, NH: Heinemann.

Edwards, E. L. (Ed.). (1990). Algebra for everyone. Reston, VA: NCTM.

Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-23). Hillsdale, NJ: Lawrence Erlbaum.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahway, NJ: Lawrence, Erlbaum.

Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65-86). Boston: Kluwer.

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Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.

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