Developing preservice teachers' pedagogical content knowledge of slope

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  • Developing preservice teachers pedagogical content

    knowledge of slope

    Sheryl L. Stump*

    Department of Mathematical Sciences, Ball State University, Muncie, IN 47306-0490, USA

    Received 8 January 2001; received in revised form 15 August 2001; accepted 15 August 2001


    Three preservice teachers participated in a secondary mathematics methods course and then taught a

    basic algebra course. The study examined the development of their knowledge of students difficulties

    with slope and their knowledge of representations for teaching slope. Data sources included written

    assignments, interview transcripts, and transcripts of the basic algebra lessons. The preservice teachers

    focused on conceptual and procedural aspects of students knowledge and developed a variety of

    representations for teaching slope. However, they inconsistently developed the concept of slope in real-

    world situations. The development of pedagogical content knowledge of slope may require the use of

    nontraditional curriculum materials. D 2001 Elsevier Science Inc. All rights reserved.

    Keywords: Slope; Pedagogical content knowledge; Teacher knowledge; Mathematics education; Preservice

    teacher education; Secondary mathematics; Methods course; Research; Algebra

    1. Introduction

    With the aim of teaching mathematics for understanding, the curriculum and pedagogy

    of school mathematics have come under much scrutiny in recent years. For example, the

    National Council of Teachers of Mathematics (NCTM, 1989, 2000) suggests that the

    emphasis of mathematics curricula should move away from rote memorization of facts and

    procedures to the development of mathematical concepts, and that connections among

    0732-3123/01/$ see front matter D 2001 Elsevier Science Inc. All rights reserved.

    PII: S0732 -3123 (01 )00071 -2

    * Tel.: +1-765-285-8662; fax: +1-765-285-1721.

    E-mail address: (S.L. Stump).

    Journal of Mathematical Behavior

    20 (2001) 207227

  • various representations of those concepts be investigated by students through problem

    solving. In order to facilitate these reforms in mathematics education, teachers must have a

    strong knowledge base including knowledge of mathematics, knowledge of student

    learning, and knowledge of mathematics pedagogy (NCTM, 1991). It is the task of

    teacher educators to help preservice and inservice teachers develop these types of

    knowledge, yet research has indicated that the task is often difficult (Brown, Cooney, &

    Jones, 1990).

    Ball (1993) suggests two reasons why learning to teach mathematics for understanding is

    not easy:

    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 (p. 162).

    In order to address these concerns, it seems reasonable that a mathematics teacher

    education program should provide opportunities for preservice teachers to construct and

    orchestrate various representational contexts. Furthermore, teacher educators should try

    to expose preservice teachers to nontraditional experiences and orientations to math-

    ematics in order to broaden their perspectives in relation to mathematics and math-

    ematics teaching.

    This project analyzed an attempt to provide such opportunities to preservice teachers in a

    secondary mathematics methods course in order to investigate how a teacher education

    program can help preservice teachers develop the knowledge they need for teaching. Building

    on a previous investigation of teachers knowledge of slope (Stump, 1999), this study focused

    on the development of preservice teachers pedagogical content knowledge of slope.

    2. Theoretical framework

    In his framework for analyzing teachers knowledge, Shulman (1986) described pedago-

    gical content knowledge as the ways of representing and formulating the subject that make it

    comprehensible to others (p. 9). Two important components of pedagogical content

    knowledge are insight into students potential misconceptions of particular mathematical

    topics, and understanding of representations for these topics.

    2.1. Knowledge of students understanding

    Fennema and Franke (1992) suggest that knowledge of students cognitions is more

    valuable to teachers than knowledge of learning theories. The authors described a set of

    studies conducted as part of a National Science Foundation-sponsored project called

    Cognitively Guided Instruction. The researchers found that elementary teachers were able

    S.L. Stump / Journal of Mathematical Behavior 20 (2001) 207227208

  • to gain knowledge about their students thinking about mathematics and this knowledge

    favorably influenced their teaching and the students learning.

    Clinical interviews provide important opportunities for preservice teachers to gain

    knowledge about students mathematical understanding (Cooney, 1994). In order to obtain

    a deeper, fuller perception of students mathematical thinking, assessment should focus on

    both mathematical concepts and mathematical procedures (NCTM, 1989, 2000).

    2.2. Knowledge of representations

    McDiarmid, Ball, and Anderson (1989) suggest that mathematics pedagogy may be

    viewed as a repertoire of instructional representations. By shifting the emphasis from methods

    or strategies of teaching to instructional representations, the focus of teaching mathematics

    moves from the teacher to the mathematics, and the connection between what the teacher

    knows and what the teacher does is tightened. In order to develop appropriate instructional

    representations, teachers must understand the content they are representing, the ways of

    thinking associated with the content, and the students they are teaching (pp. 197198).Researchers have documented limitations in teachers knowledge of instructional repre-

    sentations. For example, Ball (1993) described several studies in which elementary teachers

    were unable to use instructional representations effectively because of the limitations in their

    own mathematical understanding. Even (1993), Norman (1992), and Wilson (1994) observed

    that preservice secondary teachers had limited repertoires of instructional representations for

    the concept of function. Stein, Baxter, and Leinhardt (1990) described one teachers

    insufficient understanding of functions and the adverse affects on his teaching practices. In

    contrast, Lloyd and Wilson (1998) illustrated how another teachers strong understanding of

    functions led to skillful implementation of a reform curriculum.

    2.3. The concept of slope

    Representations of slope exist in both school mathematics and the real world. Within the

    secondary mathematics curriculum, slope emerges in various forms: geometrically, as the

    ratio riserun, a measure of the steepness of a line; algebraically, as the ratio y2y1

    x2x1 or as the m in theequation y =mx + b; trigonometrically, as the tangent of a lines angle of inclination, m = tan q;and in calculus, as a limit, limh!0

    f xhf xh


    It is believed that the use of real-world representations helps students develop understand-

    ing of abstract mathematics (Fennema & Franke, 1992). In the real world, slope appears in

    two different types of situations: physical situations such as mountain roads, ski slopes, and

    wheelchair ramps, involving slope as a measure of steepness and functional situations such as

    time versus distance or quantity versus cost, involving slope as measure of rate of change.

    Research has documented students difficulties with understanding slope in both functional

    and physical situations (Bell & Janvier, 1981; Janvier, 1981; McDermott, Rosenquist, & van

    Zee, 1987; Orton, 1984; Simon & Blume, 1994; Stump, 2001). With recent recommendations

    emphasizing the study of functions in high school (NCTM, 1989, 2000), functional situations

    involving slope are especially important.

    S.L. Stump / Journal of Mathematical Behavior 20 (2001) 207227 209

  • According to Hiebert and Lefevre (1986), meaningful understanding of mathematics

    includes relationships between conceptual and procedural knowledge. Conceptual knowledge

    is knowledge that is rich in relationships, linking new ideas to ideas that are already

    understood, and procedural knowledge consists of formal language and symbol systems, as

    well as algorithms and rules. Thus, conceptual knowledge of slope includes understanding the

    relationships among the various representations of slope that typically appear in school

    (algebraic, geometric, trigonometric, and calculus), as well as understanding slope as a

    measure of steepness and rate of change in real-world situations. Procedural knowledge of

    slope includes familiarity with the symbols typically used in relation to slope, for example, m

    and DyDx, and the rules used to calculate slope.

    A previous investigation of teachers knowledge of slope revealed that both preservice and

    inservice teachers were more likely to include physical situations than functional situations in

    their descriptions of classroom instruction, but some teachers failed to mention ei