# 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

Abstract

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: sstump@bsu.edu (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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