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 either type of

    representation. Both preservice and inservice teachers expressed concern with students

    understanding of the meaning of slope, but the specific student difficulties they identified

    focused on procedures rather than conceptual notions of slope (Stump, 1999).

    2.4. Developing teachers pedagogical content knowledge

    Research supports an emphasis on the development of pedagogical content knowledge in

    preservice teacher education programs (Brown & Borko, 1992). Swafford (1995) suggests

    that the content preparation of preservice teachers should include opportunities to revisit

    school mathematics from a deeper perspective. McDiarmid et al. (1989) list several

    suggestions for methods courses, including the following: (a) challenge preservice teachers

    conceptions of teaching and learning; (b) help preservice teachers develop their own

    understanding of specific subject matter; (c) provide opportunities for preservice teachers

    to learn more about students in relation to specific subject matter; (d) help preservice teachers

    develop skill in evaluating representations; and (e) concentrate on developing a wide range of

    representations for a limited number of topics.

    The purpose of this investigation was to see if experiences in a mathematics methods

    course can help preservice secondary mathematics teachers develop two specific components

    of their pedagogical content knowledge for the concept of slope. The research focused on the

    following questions: (1) What do preservice teachers learn about students difficulties with

    slope, and how is this knowledge reflected in their lesson plans and in their teaching? (2)

    What do preservice teachers learn about various representations for teaching slope, and how

    is this knowledge reflected in their lesson plans and in their teaching? Specifically, how do

    they use real-world representationsphysical and functional situations involving slope?

    3. Method

    This study employed the perspective of practitioner research, as defined by Liston and

    Zeichner (1991), who referred to inquiries that are conducted into ones own practice in

    S.L. Stump / Journal of Mathematical Behavior 20 (2001) 207227210

  • teaching or teacher education (p. 147). The participants in this investigation were six

    mathematics-teaching majors enrolled for one semester in a secondary mathematics methods

    course at a mid-sized midwestern university. After the course, five of the preservice teachers

    worked in pairs or individually to teach Math 107, a basic algebra course at the university, for

    the entire semester. This report focuses on three of the preservice teachers. The various roles

    of the practitioner researcher included instructor of the methods course, supervisor of Math

    107, and researcher for this study.

    The methods course adopted a constructivist perspective similar to that of a classroom

    devoted to the development of mathematics content knowledge. That is, the course was

    based on the assumption that the preservice teachers learning was contingent on their own

    activity and involvement in the various readings, discussions, activities, and assignments.

    Furthermore, the course was structured to acknowledge the important role of the preservice

    teachers prior knowledge (Maher & Alston, 1990; von Glasersfeld, 1990). This was

    accomplished through various writing assignments and class discussions in which the

    preservice teachers described their own experiences in relation to teaching and learning

    mathematics. The preservice teachers were encouraged to contemplate the differences

    between conceptual and procedural knowledge (Hiebert & Lefevre, 1986; NCTM, 1989),

    and to explore the notion of developing understanding in mathematics via problem solving

    (Schroeder & Lester, 1989).

    One goal of the methods course was to help preservice teachers develop their knowledge

    about students difficulties with slope. Thus, they completed an interview and analysis

    assignment, in which they interviewed one high school chemistry student and one college

    student enrolled in Math 107 to learn about their understanding of slope, and then compared

    the interviews through written analysis.

    Another goal of the methods course was to expand their repertoires of representations for

    teaching the concept of slope. In class discussion, as students began to generate representa-

    tions for slope, distinctions between algebraic, geometric, physical, and functional repre-

    sentations were offered by the instructor as a framework. An attempt was made to emphasize

    slope in functional situations. First, the Junior Class Dance problem from page 155 of the

    NCTM Curriculum and Evaluation Standards (1989) was presented to the class. Later, the

    class used graphing calculators in two activities designed to focus on the interpretation of

    slope as rate of change: Testing Paper Bridges, from Lappan, Fey, Fitzgerald, Friel, and

    Phillips (1998), and Walk the Line, from Brueningsen, Brueningsen, and Bower (1997). In

    class, minimal attention was paid to physical situations involving slope because the

    preservice teachers were more familiar with these representations. As a midterm assignment

    in the methods course, each preservice teacher selected an algebra textbook, created a

    framework for analyzing the textbook, and completed a textbook analysis, focusing on the

    concept of slope among other topics.

    Also in the methods course, each preservice teacher wrote a series of slope lesson plans for

    a hypothetical middle school or high school algebra class. The following semester, in Math

    107, each preservice teacher taught a lesson involving slope, not necessarily based on the

    lesson plans from the previous semester. These lessons were videotaped and the tapes were

    transcribed. The preservice teachers were interviewed toward the end...


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