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Developing preservice teachers' pedagogical content knowledge of slope
Journal of Mathematical Behavior 20 (2001) 207 – 227
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 realworld 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
* Tel.: +1-765-285-8662; fax: +1-765-285-1721. E-mail address: [email protected] (S.L. Stump). 0732-3123/01/$ – see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 2 - 3 1 2 3 ( 0 1 ) 0 0 0 7 1 - 2
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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 mathematics in order to broaden their perspectives in relation to mathematics and mathematics 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 pedagogical 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
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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. 197–198). Researchers have documented limitations in teachers’ knowledge of instructional representations. 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 teacher’s insufficient understanding of functions and the adverse affects on his teaching practices. In contrast, Lloyd and Wilson (1998) illustrated how another teacher’s 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 y2 y1 ratio rise run , a measure of the steepness of a line; algebraically, as the ratio x2 x1 or as the m in the equation y = mx + b; trigonometrically, as the tangent of a line’s angle of inclination, m = tan q; ðxÞ . and in calculus, as a limit, limh!0 f ðxþhÞf h It is believed that the use of real-world representations helps students develop understanding 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.
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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 Dy Dx, 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 representations—physical 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 one’s own practice in
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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 representations for slope, distinctions between algebraic, geometric, physical, and functional representations 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 of the methods course,
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Fig. 1. The development of preservice teachers’ pedagogical content knowledge of slope.
and again after they taught their slope lessons in Math 107. The interviews were audiotaped and the tapes were transcribed. Fig. 1 illustrates the framework for the methods course and for this investigation. It shows the process through which the knowledge that preservice teachers brought with them to the methods class was developed, with input from various activities. As illustrated, the two important products of the process are the preservice teachers’ slope lesson plans and the Math 107 lessons, the actual lessons they conducted with students of their own. The data sources for this investigation included an initial survey, the interview and analysis assignment, the textbook analysis, the slope lesson plans, the transcripts of the audiotaped and videotaped sessions, and the handouts prepared by the preservice teachers for their Math 107 lessons. The data analysis employed axial coding; the data were organized into categories and examined to identify relationships among categories (Strauss & Corbin, 1990). Tables served to organize information and illustrate relationships (Miles & Huberman, 1994).
4. Results The knowledge of three preservice teachers, Joe, Natalie, and Tracie, developed along three very different paths. (Pseudonyms replace the actual names of the preservice teachers.)
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Their progress is described here in relation to the two components of pedagogical content knowledge of slope: knowledge of students’ difficulties with slope and knowledge of representations for teaching slope. 4.1. Knowledge of students’ difficulties with slope The data sources used to examine the development of the preservice teachers’ knowledge of students’ difficulties with slope are the initial survey, the interview and analysis assignment, the slope lesson plans, transcripts of the Math 107 lessons, and interview transcripts. The preservice teachers came to the methods course with specific beliefs about students’ knowledge of slope, as evidenced by their responses to the question, ‘‘What difficulties do you think students might have with slope?’’ on the initial survey. These specific beliefs then appeared as themes throughout each preservice teachers’ work in the methods class and later in their teaching. For the interview and analysis assignment, the preservice teachers designed or selected their own interview tasks, many of which emerged from discussions in the methods class. The preservice teachers were encouraged to assess both conceptual and procedural knowledge and
Table 1 Interview tasks designed and/or selected by preservice teachers Preservice teachers Interview tasks
Joe
Natalie
Tracie
1. 2. 3. 4.
X X X X
X X
X X
X
X
X X X
X
X
X
In your own words, describe what slope is. Give three real-world examples of slope. Draw a line with slope 1 through (1,1). Find the equation of the line. Compare and arrange these slopes (triangles) from greatest to least. Explain. 5. Write a linear equation with positive slope and one with negative slope. 6. What does it mean for a line to have positive or negative slope? 7. What does it mean for a line to have zero or no slope? 8. Would a line with slope of 1/2 intersect the x-axis at an angle greater than, less than, or equal to 45? Why? 9. Determine which of three graphs appropriately fits the likely relation between price and sales. 10. Observe a graph of temperature versus time. What does the slope of the line represent? 11. Observe graph of distance versus time. Find slope. Explain what the slope means in terms of the situation. 12. Draw a graph of distance versus time. What does the slope of the line represent? 13. Air temperature decreases by 11 F for each mile you rise above ground. If starting temperature is 68 F, what is the temperature 3 miles up? How does this relate to slope? Graph the relationship. What is the slope of the line? 14. Determine the rate of growth of a child. Draw a graph. 15. Find the steepness that rises 1 foot for every 3.17 feet of horizontal distance.
X X
X X X
X X
X X
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Table 2 Preservice teachers’ comments about students’ difficulties with slope in various data sources Data sources Preservice teachers Initial surveya Joe
Procedures: Do not follow the formula.
Natalie
Tracie
Interview and analysis assignment Concepts: Very good on the conceptual questions.
Procedures: Not so good with algebraic representations and formulas. Concepts: Understanding Concepts: Do not realize what slope refers to. Do not that slope can be a associate visual diagram numerical value. with equation. Procedures: Did not know how to find slope. Concepts: Assigning a Concepts: Confused it with length. number to a slant.
Interview transcriptsb Concepts: Connecting slope as a fraction to rise over run.
Concepts: Connecting the notion of steepness to rise over run.
Concepts: They think it is a measurement you can measure.
Procedures: x and y values interchanged. a
Question: ‘‘What difficulties do you think students might have with slope?’’ Question: ‘‘What difficulties do you think students might have with slope?’’ and ‘‘What did you learn about students’ knowledge of slope from conducting the interviews?’’ b
to include various representations of slope in their interview tasks. A summary of their tasks appears in Table 1. Table 2 shows the progression of preservice teachers’ knowledge of students’ difficulties with slope by revealing the preservice teachers’ comments at various stages in the methods course. Comments that focused on students forming relationships among ideas were identified as ‘‘concepts,’’ and comments that focused on symbols and rules were called ‘‘procedures.’’ This knowledge of students’ difficulties with slope is then reflected in Table 3, which catalogues the various concepts and procedures contained in the preservice teachers’ slope lesson plans and Math 107 lessons. We may now examine a more detailed description of the development of each of the three preservice teachers’ knowledge of students’ difficulties with slope. 4.1.1. Joe: Concepts and procedures On the initial survey, Joe, a returning graduate seeking a teaching license, speculated that students ‘‘do not follow the formula Dy Dx: (a) they don’t understand it, (b) they switch the order, or (c) they make computation errors.’’ His response reveals a focus on procedures. In the methods class, however, Joe participated in discussions about the differences between conceptual and procedural knowledge (Hiebert & Lefevre, 1986), and later, when he completed his interview and analysis assignment, his responses reflected those discussions.
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Table 3 Concepts and procedures in preservice teachers’ slope lesson plans and Math 107 lessons Preservice teachers
Slope lesson plans
Math 107 lesson
Joe
Concepts: Graph equations, look for patterns. Interpret slope as a rate of change. Procedures: Compute rise over run. Equation: y = mx + b. Concepts: Develop slope as a measure of steepness. Procedures: Compute rise over run. Positive and negative slope. Horizontal and vertical lines. Parallel and perpendicular lines. Concepts: Steepness. Graph equations, look for patterns.
Concepts: Connects slope as a fraction to rise over run. Graph equations, look for patterns. Procedures: Compute rise over run. Equation: y = mx + b. Concepts: Interpret slope as a measure of rate of change. Procedures: Equation: y = mx + b. Compute rise over run. Graph equations. Horizontal and vertical lines. Parallel and perpendicular lines. Concepts: Steepness. Graph equations, look for patterns. Interpret slope as a rate of change. Procedures: Equation: y = mx + b. Compute rise over run.
Natalie
Tracie
Procedures: Compute rise over run. Horizontal and vertical lines. Equation: y = mx + b.
When he interviewed each of the two students, he began by asking, ‘‘In your own words, describe what slope is.’’ One student, Bill, answered, ‘‘Relation in the change between x and y coordinates,’’ while the other student, Brad, answered, ‘‘Given two points on a graph, slope is x1 x2 y1 y2 .’’ In his analysis, Joe wrote, ‘‘It was interesting to see Bill’s memory of slope to be more conceptual and Brad’s memory of slope to be related to the formula (even if it was an incorrect answer).’’ It was encouraging to see evidence of Joe incorporating a new idea into his knowledge for teaching. The tasks he chose for his interview and analysis assignment included equations, graphs, and functional situations. However, the following comments suggested that he was mistakenly developing the notion that conceptual knowledge was good and procedural knowledge was bad: The area in which both of the students were weak was working with equations concerning an algebraic representation of slope. They both had trouble coming up with an equation for a line that was already graphed, and vice versa. This is something that I found very surprising, as well as encouraging [emphasis added]. What we have discussed so far in this course is the need to focus more on conceptual understanding, and not so much on memorizing formulas. Well, these students exhibited these characteristics almost to a fault. They were very good on the conceptual questions, and not so good with algebraic representations and formulas.
Fortunately, Joe did not completely dismiss the importance of algebraic representations and formulas. He developed these representations in both his slope lesson plans and later in his Math 107 lessons. However, despite his initial propensity toward procedural aspects of slope, Joe displayed a willingness to try teaching with a focus on the development of conceptual understanding. When he introduced the concept of slope to his Math 107 class, he
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spent the first few minutes ‘‘philosophizing’’ about slope, rambling a bit, setting the stage for a discussion about the meaning of slope. One of his students became impatient with his verbal treatise, devoid of symbols, and interrupted Joe by saying, ‘‘I have a question. What exactly is slope? Can you define it?’’ vertical change Joe eventually defined slope as ‘‘horizontal change,’’ and presented a graph of the line passing through the points (0,0) and (3,2). He emphasized that the slope was a fraction, 23, up 2, over 3. Later, while students were working at their seats, one student was having difficulty 5 and 5 understanding how the two fractions 6 6 could both represent the same slope. Although at the time Joe struggled in vain to help her understand, he later described her difficulty with the following insight: ‘‘They think you are describing a movement as opposed to you describing a number, a measurement.’’ Joe’s encounters with students revealed him carefully pondering various aspects of students’ difficulties with slope. The methods class provided ideas about conceptual and procedural knowledge to frame his thinking. He attempted to address conceptual knowledge in his teaching, but these attempts were awkward and both he and his students were more comfortable discussing procedures. Joe, however, continued to be thoughtful, and would most likely have benefited from some more discussion on the topic. 4.1.2. Natalie: What is slope? On the initial survey, Natalie conjectured, ‘‘Students will have difficulty first actually understanding what slope even refers to since it tends to be a difficult concept to grasp. Students then might not associate the visual diagram of the line with the equation.’’ However, Natalie’s findings on the interview and analysis assignment surprised her. Her interview tasks focused on graphs and functional situations, and she found that the students were unable to answer most of her questions with appropriate answers. ‘‘I had not thought much about how little prior knowledge a student may possess. Both of the students I interviewed had taken first year algebra and geometry. . . . I would assume they would at least realize . . . that slope can be a numerical value and how to find it (or at least its inverse, ‘run over rise’).’’ Later, Natalie said, ‘‘I don’t think they get a chance to go from what they see in real life and what they believe is steepness into a numerical sense that they are given this number. And even if they understand rise over run, they see it on a piece of paper on a graph or just two points and plugging in a formula.’’ She had assumed that students would remember how to calculate slope, but perhaps not understand what it means. She discovered, however, that students did not remember how to calculate slope. Her interview and analysis assignment included several tasks addressing slope as a measure of rate of change. Although both students she interviewed were unsuccessful with these tasks, Natalie chose to ignore this idea in her slope lesson plans. Instead, she pursued the concept of slope as a measure of steepness. Her slope lesson plans included an activity from the Mathematics Teacher (Anderson & Nelson, 1994) that focused on measuring the rise and run of various objects such as ramps and staircases. I wanted to see them develop it so that they could remember ‘‘rise over run.’’ . . . When I first began, I thought my goal would be y = mx + b, but that to me, after I started thinking about it,
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wasn’t what the concept of slope is at all. That is one place where it is applied, . . . but that, after the interview and what we looked at in class, was not the beginning of slope. Maybe that is where it needed to end up, but that wasn’t what developed the concept of slope. . . . The concept started some place way before that.
The following semester, for her Math 107 lesson, Natalie’s goal was ‘‘y = mx + b. . .’’ She took her class to the computer lab and directed them to an interactive website where they worked on finding the slope of a line given two points of the line, graphing the line given its slope and a point on the line, finding the slope of a line given its equation, finding the slopes of horizontal and vertical lines, and comparing the slopes of parallel and perpendicular lines. On the initial survey, when Natalie was asked the question, ‘‘What is slope?,’’ she answered, ‘‘Slope is a term used to associate the incline of a line with a numerical value.’’ Later, when she discovered that this association may not be automatic for students, she designed her slope lesson plans to help middle school or high school students develop this connection, focusing on the concept of slope as a measure of steepness. When she actually worked with college students, however, she skipped over the concept of slope as a measure of steepness, and focused on developing a series of procedures related to graphs and equations of lines. 4.1.3. Tracie: Measurement of slant On the initial survey, Tracie hypothesized that ‘‘assigning a number to a ‘slant’ is something that students just learning about slope are not accustomed to.’’ Later, when interviewing two students for the interview and analysis assignment, Tracie assessed this particular aspect of slope by posing the following problem: Portions of Filbert Street and 22nd Street in San Francisco are the world’s steepest streets, rising 1 foot for every 3.17 feet of horizontal distance. How steep do you think this is? How could you figure out the steepness of the streets?
According to Tracie, one student merely responded with, ‘‘Pretty steep, I guess,’’ and shrugged her shoulders. The other student drew a triangle and used the Pythagorean theorem to find the length of the hypotenuse. Later in the semester, I again asked Tracie, ‘‘What difficulties do you think students have with slope?’’ She maintained her initial belief, but now she added more details: Because you can represent it with a number, they think it is a measurement that you can measure, like you can measure 5 inches or 6 inches. They want to measure it in that way as a length instead of an idea of the steepness.
In her slope lesson plans, Tracie was careful to develop the notion of slope as a measurement of steepness. Her lesson plans began with an activity on the playground, with students using a tape measure to measure the rise and run of a slide. She defined slope as rise run and then asked, ‘‘So, what is the slope of the slide? What does this mean?’’ Tracie also created a worksheet with pictures of a slide, a hill, a ramp, a roller coaster, a roof, and a ski slope. Two numbers appeared on each picture, and the students were instructed to find the rise, the run, and the slope of each object.
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The following semester, when she taught a lesson involving slope to Math 107 students, Tracie again emphasized measurement of slant, using examples similar to those from her slope lesson plans. None of the other preservice teachers in the investigation included this aspect of slope in their Math 107 lessons. Tracie initially expressed concern about students understanding the notion of slant, or steepness. She maintained this concern throughout the course, and she created learning opportunities for students to discover relationships among various representations of slope. Her lesson plans and actual lessons reflected a balance between concepts and procedures. 4.2. Knowledge of representations for teaching slope The other aspect of pedagogical content knowledge addressed in this investigation was preservice teachers’ knowledge of representations for teaching slope. The data sources used to examine the preservice teachers’ knowledge of representations for teaching slope were the initial survey, the textbook analysis, the slope lesson plans, the Math 107 lessons, and interview transcripts. When the preservice teachers designed their slope lesson plans, they were encouraged to include various representations of slope. They made their own choices of which representations to include and which resources to use for designing their lessons. When teaching Math 107, however, they were constrained by the textbook they were using. The textbook used in Math 107 was a traditional basic algebra textbook. It emphasized algebraic representations of slope (the slope ratio and linear equations), with less accentuation on graphs and minimal attention to physical or functional situations. Furthermore, the textbook did not emphasize
Table 4 Representations of slope included by preservice teachers in various data sources Data sources Preservice teachers
Initial surveya
Slope lesson plan
Math 107 lesson
Joe
graphs
Natalie
graphs physical situations (I)
graphs and ratios graphs and equations functional situations (P) graphs and ratios physical situations (P)
Tracie
physical situations (I)
graphs and ratios graphs and equations functional situations (I) graphs and ratios graphs and equations functional situations (I) physical situations (I) graphs and ratios graphs and equations physical situations (P) functional situations (P)
graphs and ratios physical situations (P) functional situations (P)
I = used as an illustration; P = used as a problem. a Question: ‘‘What analogies, illustrations, examples, or explanations do you think are most useful of helpful for teaching the concept of the slope?’’
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problem solving. Throughout the course, though, the preservice teachers were encouraged to supplement the textbook with ideas of their own. Table 4 shows the progression of evidence of the preservice teachers’ knowledge of representations for teaching slope. Once again, we look at the patterns of development for Joe, Natalie, and Tracie.
4.2.1. Joe: Graphs and functional situations—What does the slope of the line represent? On the initial survey, Joe wrote, ‘‘(1) Draw as many graphs as you have time for, so they can see it. (2) Make x, y chart. (3) Try to make it as simple as possible. Don’t overemphasize the why until they know the how [emphasis in the original].’’ In the first lesson in his series of slope lesson plans, Joe wrote, The objective of this lesson is to provide an opportunity for students to experience the process of making and testing conjectures. Students will develop, on their own, an understanding of slope and y-intercept. Through observation of how a multiplicative coefficient or an additive constant changes the orientation of the graph of y = x, students will gain insight into these concepts. To conclude, students will be introduced to the slope –intercept form as the equation of a line.
He planned to develop these objectives by having students graph sets of equations on graph paper. He would then describe slope as m ¼ Dy Dx with a graph of a line, carefully showing Dy and Dx. Finally, he would connect the formula y = mx + b to the graphical work that the students had completed. In his second lesson, Joe planned to involve the students in collecting data that involved a linear relationship. The students would enter the data into a graphing calculator and produce a scatterplot. They would then use their knowledge of slope and y-intercept to produce an equation of a line to fit the data. One of the questions Joe planned to ask the students is, ‘‘What does the slope of this line represent?’’ Joe’s third lesson stressed the notion of slope as rate. It included three different functional situations involving slope: drawing the graph of time versus distance for a traveling bicycle, interpreting a graph of time versus temperature for a substance being heated, and interpreting a graph of time versus velocity for a traveling car. For the first two situations, Joe again asked, ‘‘What does the slope of the line represent?’’ The framework that Joe created for his textbook analysis included the following: How does the textbook address the concept of slope? (a) conceptual focus, (b) computational focus, (c) answers the question: ‘‘What does the slope of the line represent?’’, (c) introduces slope as a rate, (e) connects slope to real-world situations. in the analysis, he wrote: This book looks at slope as a computational tool, and wants students to be able to find slope under several different conditions. It is formula based, and memorization will play a large role in the successful completion of the material. It is not to say that these are not important parts of understanding slope. It is just that the students are never given an opportunity to investigate what slope represents, and that is what our discussions have determined is the most beneficial part of learning slope.
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When interviewed, Joe said: I think that maybe you can gain the most from a graph. If you look at slope on a graph that is where you gain the most insight. If that graph is a model of something they have done before in their life, if there is something outside in their lives that they can connect it to, then that is even more important, I think.
The following semester, Joe taught two lessons involving slope to his Math 107 class. He introduced the concept of slope by stating, ‘‘We are going to discuss slope on two different levels. (1) Mathematically: How do you compute slope? (a) from a graph (b) from an equation; and (2) What does the slope of a line represent?’’ Joe’s first lesson only addressed the first part of the first level. His partner addressed the second level the following day, and Joe discussed equations later in the semester. vertical change In his first lesson, Joe defined slope as horizontal change, and presented a graph of the line passing through the points (0,0) and (3,2). He emphasized that the slope was a fraction 23, up 2, over 3. After discussing a second example, he introduced a paper-and-pencil game called ‘‘Rapid Fire’’ that he had created for the students to practice finding the slope of a line connecting two points. The students spent the remainder of the period playing in pairs. The following class day, Joe’s teaching partner introduced some functional situations involving slope. A couple of weeks later, Joe took his Math 107 students to the computer lab for a graphing activity. The students used a computer graphing program and a worksheet that Joe had prepared for students to discover answers to such questions as the following: What effect does a coefficient multiplier have on the graph of y = x? In mathematical terms, we would call this a change in the _____ of the line. What does it mean for a graph to have a negative slope? What does it mean for a graph to have a positive slope? At the end of this lesson, seemingly as an afterthought, Joe drew a graph of distance versus time on the board to illustrate a real-world application of slope. However, he did not ask the question, ‘‘What does the slope of the line represent?’’ Based on the initial survey, Joe’s repertoire of representations for teaching slope did not include real-world situations. Later, though, for his slope lesson plans, he created a variety of rich problems involving functional situations. These problems emphasized the notion of slope as rate of change. These real-world problems were ignored, however, when he actually taught the concept of slope in Math 107. Instead, Joe focused on graphs, using only one functional situation as an illustration. 4.2.2. Natalie: Physical situations and ‘‘rise over run’’ On the initial survey, Natalie said, ‘‘I think the example of an inclined plan is very important. I also think the saying ‘rise over run’ helps students remember the order.’’ Her slope lesson plans focused on physical situations and developing the ratio ‘‘rise over run’’ as a measure of steepness.
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They need to be able to understand algebraically what it means and be able to graph, eventually, linear equations and for parallel and perpendicular lines. . . . They should be able to distinguish between, if they don’t get anything else out of it, they should be able to see the difference in steepness—of a roof’s pitch or grade.
In her textbook analysis, Natalie used the framework suggested by her instructor to discuss the various representations of slope: The text begins the lesson with physical representations of slope by using a short paragraph of the grade of a hill. Through the geometric representation, the text develops ‘‘rise over run.’’ The functional representation appears, but the text only eludes to it. The ratio representation is used most frequently. The majority of practice problems involve it. Finally, the next lesson involves the algebraic representation.
Natalie used these results from her textbook analysis to justify not including functional situations in her slope lesson plans. In an ideal world, at some place and time, I would include the way that slope can compare two different things like distance and time. Given an infinite amount of time in my class, I certainly do think that would be important. But judging from where this context is placed in algebra books that I have looked at, by the end of the book, there is not going to be enough time to cover the concepts that are going to be on the exam, in order for them to go onto the next class.
Perhaps to please her instructor, Natalie’s Math 107 lesson in the computer lab began with a functional situation, ‘‘To create a custom-made dress, one seamstress charges a set price plus a certain price per yard of fabric. The following graph shows her prices.’’ Natalie then posed the following questions: (1) ‘‘How much would it cost if I wanted a dress that uses 10 yards of fabric?’’ and (2) ‘‘What is her starting price without any fabric and how much does she charge per yard of fabric?’’ She told the class that their answers had ‘‘something to do with what we call the slope of a line,’’ but she did not ask the question, ‘‘What does the slope of the line represent?’’ She then asked where else they might use slope. One student mentioned a roof and another mentioned a ramp. Then Natalie told the students how to locate the website they would be using, and she handed out an activity sheet to accompany the lesson on the website, a lesson devoid of real-world context. She spent the remainder of the lesson circulating through the computer lab, answering the individual students’ questions about graphs and equations. Natalie’s maintained her initial appreciation for physical situations involving slope throughout her experiences in the methods course and her actual teaching. Of the three preservice teachers, Natalie was the most resistant to including the notion of slope as a measure of rate of change in her work. 4.2.3. Tracie: Physical and functional situations In response to this question, Tracie replied, ‘‘Something tangible. Examples of common uses are the slope or pitch of a roof and the slope or grade of a mountain road.’’ Later, these physical representations for slope appeared in her textbook analysis, her slope lesson plans and her Math 107 lesson.
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Tracie did, however, began to develop a repertoire of other representations for slope. Her slope lesson plans included an activity using a geometry computer program to construct lines and calculate slopes. She asked the students to make conjectures about the appearance of the graphed lines and the slope measurements. She also presented five different problems involving real-world representations of slope. One problem, entitled ‘‘Carpentry,’’ asked students to determine how to fit a staircase into a given space. The other four problems involved linear functions. ‘‘Class Fund Raiser’’ was about selling pizzas. ‘‘Population Growth’’ compared the rate of growth of two high schools. ‘‘Communications’’ asked for a comparison of the rates of three major long-distance companies. ‘‘Financing the Prom’’ involved magazine sales. For each of these linear functions, Tracie asked for a graph of the data, but she did not ask students to determine the slope of the line or interpret its meaning within the context of the situation. In her textbook analysis, Tracie wrote, Slope is defined as rise over run, initially in Chapter 11. [The textbook] does not consider the many different interpretations and applications of slope. There is one application for slope on page 423. It deals with temperature and altitude and provides only one of many good concrete applications for the concept of slope. Unfortunately, slope is developed in this text as a means to graph equations and determine equations of graphs. The importance of slope is not stressed.
When I interviewed Tracie later, I asked her, ‘‘What representations are most important when teaching the concept of slope?’’ She said, In the beginning, I think it would have to be very concrete and the idea of just what slope is, as in steepness and stuff like that. And the further along you get, things like rate and talking about speed and acceleration and distance and finding what slope means when you graph those things.
The following semester, Tracie’s aspirations for her Math 107 class were evidently ‘‘further along’’ on her continuum, because she did address the notion of rate of change. She designed a worksheet that included the following set of questions designed to help students discover the connections among a functional situation, equations, and graphs. Here are some of the questions: Anissa is reading a book for her English class. She decides to read two chapters each day in order to avoid having to read it all at the last minute. Graph Anissa’s reading progress below. Which equation represents the graph of Anissa’s reading? When an equation is in the form y = mx + b, the numbers m and b tell us something about the graph. What part of the equation describes the slant (or slope) of the graph? Look back at the graph and linear equation of Anissa’s reading. What is the slope of this line? What does this slope represent?
Tracie’s worksheet for her Math 107 lesson also included illustrations of several physical situations — a slide, a ski slope, a skateboard ramp — for which students were supposed to identify the rise and the run and then determine the slope. Tracie was predisposed to using physical representations of slope. In fact, she mentioned that she had first learned about slope from her father, a carpenter. She did expand her
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repertoire to include functional representations, and she demonstrated her ability to develop connections between slope as the property of a line and slope as a measure of rate of change.
5. Discussion The methods course in this investigation provided a space in which the preservice teachers’ initial ideas about slope could be developed and even challenged. Reading assignments and class discussions exposed the preservice teachers to theoretical notions related to teaching and learning the concept of slope. Class activities and assignments provided opportunities for them to connect the theory to practical situations. Later, their evolving ideas about slope appeared in their lessons, suggesting a transformation in their pedagogical content knowledge about slope and how they would apply this knowledge. Because the methods course in this investigation assumed a constructivist perspective, there was an expectation that the preservice teachers would develop their pedagogical content knowledge in perhaps similar yet different ways. This research provides insight into the consistencies and the variations in their knowledge of students’ difficulties with slope and their knowledge of representations for teaching slope. 5.1. Knowledge of students’ difficulties with slope 5.1.1. Focus on concepts and procedures In the methods course, the interview and analysis assignment served as an important instrument for developing preservice teachers’ pedagogical content knowledge. By requiring the preservice teachers to focus on both conceptual and procedural knowledge, the assignment provided an opportunity for the preservice teachers to think about and examine students’ knowledge in new ways, thus expanding their own views about students’ knowledge of slope. It gave them a chance to discover for themselves the difficulties that students have with slope, and also to explore why students have those difficulties. When the preservice teachers entered the methods course, their initial ideas about students’ potential difficulties with slope were limited to vague assumptions. After interviewing students to assess their knowledge of slope, the preservice teachers did not abandon their initial assumptions, but they added more dimensions to their thoughts about students’ difficulties. Unlike the teachers in a previous investigation (Stump, 1999) who focused only students’ computational difficulties with slope, the preservice teachers in this study took advantage of the opportunity to examine both conceptual and procedural knowledge of slope. Not only did they learn more about students’ knowledge and skills with graphs and equations, they also discovered the limitations in students’ understanding of slope as a measure of steepness and as measure of rate of change. 5.1.2. Knowledge of students’ cognitions is valuable The interview and analysis assignment confirmed some of what the preservice teachers already believed, but it also provided new insights. For example, Joe was initially concerned
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with students understanding and using a formula, and he indeed witnessed such difficulties when he conducted his interviews. However, he also discovered students had conceptual understanding that he might have otherwise ignored. Natalie, on the other hand, was surprised to find that the students she interviewed lacked a basic understanding of the numerical value of slope. Tracie maintained her initial concern with students’ understanding of slope as a measure of slant, and the interview and analysis assignment provided her with more evidence of their lack of understanding. The information these preservice teachers gained from the interviews was reflected in specific ways in both their lesson plans and their actual teaching. To illustrate, Joe highlighted the meaning of slope, Natalie developed the notion of steepness, and Tracie emphasized slope as a measure of slant. Because they had evidence of students’ cognitions, the preservice teachers created lessons to carefully develop the concepts they knew were difficult for students to understand. Furthermore, they created ways to engage students in meaningful activities involving these concepts. 5.2. Knowledge of representations for teaching slope 5.2.1. Real-world situations When the three preservice teachers entered the methods course, their knowledge of representations for teaching slope was dominated by graphs and physical situations. Throughout the methods course, they continued to value slope as a measure of steepness, but they also became conscious of the notion of slope as a measure of rate of change. When they wrote lesson plans for middle or high school students, all three preservice teachers developed the concept of slope through problems involving real-world situations. Joe emphasized functional situations, Natalie emphasized physical situations, and Tracie included both. All of their lessons included questions designed to focus students’ attention on the meaning of slope in these real-world situations. The actual lessons they taught to college students, however, focused more on graphs and equations. All three preservice teachers presented lessons that were developed through problem solving and discovery, and all three did include real-world situations somewhere in their lessons. Joe and Natalie, however, used physical and functional situations merely as illustrations, not as problems for students to solve. They did not, for example, pose questions for students to interpret the meaning of slope in real-world situations. Only Tracie used realworld situations as problems in her Math 107 lesson. She asked students to interpret slope both as the measure of steepness in physical situations and measure of rate of change in functional situations. We should not assume that students will automatically develop an understanding of slope as measure when real-world situations are mentioned only as illustrations for the concept of slope (Stump, 2001). Instead, students must be purposefully challenged to construct connections among geometric, algebraic, and real-world representations of slope. In this investigation, the preservice teachers’ Math 107 lessons reflected limited appreciation for the importance of developing the concept of slope as a measure of steepness or rate of change through problem solving.
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5.2.2. The role of the textbook This investigation provides some insight into the representations for slope that preservice teachers choose when attempting to adapt a traditional textbook in meaningful ways. As is the case with many teachers, the preservice teachers’ lessons were constrained by the textbook they were using. The textbook used in Math 107 did not emphasize real-world applications of slope, and did not focus on problem solving. All three of the preservice teachers in this investigation, though, found creative ways to supplement the textbook. Joe created a paper-and-pencil game and devised a discovery lesson using a computer graphing program, Natalie prepared a packet for use with an interactive website, and Tracie designed a written discovery lesson using illustrations and word problems. Although most of these supplementary activities did not focus on real-world applications of slope, they did the engage students in problem solving.
6. Conclusion Throughout their experiences in the methods course and their actual teaching, the three preservice teachers in this investigation developed sensitivity to both conceptual and procedural aspects of students’ knowledge of slope. They also expanded their repertoires of representations for teaching the concept of slope and they created meaningful problemsolving activities for their students. The greatest difference, however, among the three preservice teachers was in their display of appreciation for slope as a measure of steepness and as a measure of rate of change. While one preservice teacher used real-world situations to engage students in problem solving, another used them merely as illustrations of slope. As one preservice teacher focused on physical situations, another emphasized functional situations. Perhaps the tenuous knowledge of these preservice teachers for developing the concept of slope in real-world situations can be explained by their limited exposure to nontraditional curriculum materials. The circumstances and available resources for these preservice teachers required them to supplement a traditional algebra textbook. The result was a demonstration of their ability to create worthwhile, engaging lessons involving slope, but also a hesitation to develop the concept of slope as a measure of steepness and as a measure of rate of change. Although they worked to include physical and functional representations of slope somewhere in their lessons, the framework in which these preservice teachers worked did not require them to use these representations as the primary focus of their lessons. Perhaps an alternative framework, one that requires preservice teachers to develop lessons using nontraditional curriculum materials, would help preservice teachers further develop their pedagogical content knowledge for teaching the concept of slope.
Acknowledgments This research was completed with funds from the Office of Academic Research and Sponsored Programs at Ball State University.
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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 realworld 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
* Tel.: +1-765-285-8662; fax: +1-765-285-1721. E-mail address: [email protected] (S.L. Stump). 0732-3123/01/$ – see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 2 - 3 1 2 3 ( 0 1 ) 0 0 0 7 1 - 2
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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 mathematics in order to broaden their perspectives in relation to mathematics and mathematics 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 pedagogical 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
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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. 197–198). Researchers have documented limitations in teachers’ knowledge of instructional representations. 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 teacher’s insufficient understanding of functions and the adverse affects on his teaching practices. In contrast, Lloyd and Wilson (1998) illustrated how another teacher’s 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 y2 y1 ratio rise run , a measure of the steepness of a line; algebraically, as the ratio x2 x1 or as the m in the equation y = mx + b; trigonometrically, as the tangent of a line’s angle of inclination, m = tan q; ðxÞ . and in calculus, as a limit, limh!0 f ðxþhÞf h It is believed that the use of real-world representations helps students develop understanding 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.
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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 Dy Dx, 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 representations—physical 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 one’s own practice in
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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 representations for slope, distinctions between algebraic, geometric, physical, and functional representations 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 of the methods course,
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Fig. 1. The development of preservice teachers’ pedagogical content knowledge of slope.
and again after they taught their slope lessons in Math 107. The interviews were audiotaped and the tapes were transcribed. Fig. 1 illustrates the framework for the methods course and for this investigation. It shows the process through which the knowledge that preservice teachers brought with them to the methods class was developed, with input from various activities. As illustrated, the two important products of the process are the preservice teachers’ slope lesson plans and the Math 107 lessons, the actual lessons they conducted with students of their own. The data sources for this investigation included an initial survey, the interview and analysis assignment, the textbook analysis, the slope lesson plans, the transcripts of the audiotaped and videotaped sessions, and the handouts prepared by the preservice teachers for their Math 107 lessons. The data analysis employed axial coding; the data were organized into categories and examined to identify relationships among categories (Strauss & Corbin, 1990). Tables served to organize information and illustrate relationships (Miles & Huberman, 1994).
4. Results The knowledge of three preservice teachers, Joe, Natalie, and Tracie, developed along three very different paths. (Pseudonyms replace the actual names of the preservice teachers.)
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Their progress is described here in relation to the two components of pedagogical content knowledge of slope: knowledge of students’ difficulties with slope and knowledge of representations for teaching slope. 4.1. Knowledge of students’ difficulties with slope The data sources used to examine the development of the preservice teachers’ knowledge of students’ difficulties with slope are the initial survey, the interview and analysis assignment, the slope lesson plans, transcripts of the Math 107 lessons, and interview transcripts. The preservice teachers came to the methods course with specific beliefs about students’ knowledge of slope, as evidenced by their responses to the question, ‘‘What difficulties do you think students might have with slope?’’ on the initial survey. These specific beliefs then appeared as themes throughout each preservice teachers’ work in the methods class and later in their teaching. For the interview and analysis assignment, the preservice teachers designed or selected their own interview tasks, many of which emerged from discussions in the methods class. The preservice teachers were encouraged to assess both conceptual and procedural knowledge and
Table 1 Interview tasks designed and/or selected by preservice teachers Preservice teachers Interview tasks
Joe
Natalie
Tracie
1. 2. 3. 4.
X X X X
X X
X X
X
X
X X X
X
X
X
In your own words, describe what slope is. Give three real-world examples of slope. Draw a line with slope 1 through (1,1). Find the equation of the line. Compare and arrange these slopes (triangles) from greatest to least. Explain. 5. Write a linear equation with positive slope and one with negative slope. 6. What does it mean for a line to have positive or negative slope? 7. What does it mean for a line to have zero or no slope? 8. Would a line with slope of 1/2 intersect the x-axis at an angle greater than, less than, or equal to 45? Why? 9. Determine which of three graphs appropriately fits the likely relation between price and sales. 10. Observe a graph of temperature versus time. What does the slope of the line represent? 11. Observe graph of distance versus time. Find slope. Explain what the slope means in terms of the situation. 12. Draw a graph of distance versus time. What does the slope of the line represent? 13. Air temperature decreases by 11 F for each mile you rise above ground. If starting temperature is 68 F, what is the temperature 3 miles up? How does this relate to slope? Graph the relationship. What is the slope of the line? 14. Determine the rate of growth of a child. Draw a graph. 15. Find the steepness that rises 1 foot for every 3.17 feet of horizontal distance.
X X
X X X
X X
X X
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Table 2 Preservice teachers’ comments about students’ difficulties with slope in various data sources Data sources Preservice teachers Initial surveya Joe
Procedures: Do not follow the formula.
Natalie
Tracie
Interview and analysis assignment Concepts: Very good on the conceptual questions.
Procedures: Not so good with algebraic representations and formulas. Concepts: Understanding Concepts: Do not realize what slope refers to. Do not that slope can be a associate visual diagram numerical value. with equation. Procedures: Did not know how to find slope. Concepts: Assigning a Concepts: Confused it with length. number to a slant.
Interview transcriptsb Concepts: Connecting slope as a fraction to rise over run.
Concepts: Connecting the notion of steepness to rise over run.
Concepts: They think it is a measurement you can measure.
Procedures: x and y values interchanged. a
Question: ‘‘What difficulties do you think students might have with slope?’’ Question: ‘‘What difficulties do you think students might have with slope?’’ and ‘‘What did you learn about students’ knowledge of slope from conducting the interviews?’’ b
to include various representations of slope in their interview tasks. A summary of their tasks appears in Table 1. Table 2 shows the progression of preservice teachers’ knowledge of students’ difficulties with slope by revealing the preservice teachers’ comments at various stages in the methods course. Comments that focused on students forming relationships among ideas were identified as ‘‘concepts,’’ and comments that focused on symbols and rules were called ‘‘procedures.’’ This knowledge of students’ difficulties with slope is then reflected in Table 3, which catalogues the various concepts and procedures contained in the preservice teachers’ slope lesson plans and Math 107 lessons. We may now examine a more detailed description of the development of each of the three preservice teachers’ knowledge of students’ difficulties with slope. 4.1.1. Joe: Concepts and procedures On the initial survey, Joe, a returning graduate seeking a teaching license, speculated that students ‘‘do not follow the formula Dy Dx: (a) they don’t understand it, (b) they switch the order, or (c) they make computation errors.’’ His response reveals a focus on procedures. In the methods class, however, Joe participated in discussions about the differences between conceptual and procedural knowledge (Hiebert & Lefevre, 1986), and later, when he completed his interview and analysis assignment, his responses reflected those discussions.
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Table 3 Concepts and procedures in preservice teachers’ slope lesson plans and Math 107 lessons Preservice teachers
Slope lesson plans
Math 107 lesson
Joe
Concepts: Graph equations, look for patterns. Interpret slope as a rate of change. Procedures: Compute rise over run. Equation: y = mx + b. Concepts: Develop slope as a measure of steepness. Procedures: Compute rise over run. Positive and negative slope. Horizontal and vertical lines. Parallel and perpendicular lines. Concepts: Steepness. Graph equations, look for patterns.
Concepts: Connects slope as a fraction to rise over run. Graph equations, look for patterns. Procedures: Compute rise over run. Equation: y = mx + b. Concepts: Interpret slope as a measure of rate of change. Procedures: Equation: y = mx + b. Compute rise over run. Graph equations. Horizontal and vertical lines. Parallel and perpendicular lines. Concepts: Steepness. Graph equations, look for patterns. Interpret slope as a rate of change. Procedures: Equation: y = mx + b. Compute rise over run.
Natalie
Tracie
Procedures: Compute rise over run. Horizontal and vertical lines. Equation: y = mx + b.
When he interviewed each of the two students, he began by asking, ‘‘In your own words, describe what slope is.’’ One student, Bill, answered, ‘‘Relation in the change between x and y coordinates,’’ while the other student, Brad, answered, ‘‘Given two points on a graph, slope is x1 x2 y1 y2 .’’ In his analysis, Joe wrote, ‘‘It was interesting to see Bill’s memory of slope to be more conceptual and Brad’s memory of slope to be related to the formula (even if it was an incorrect answer).’’ It was encouraging to see evidence of Joe incorporating a new idea into his knowledge for teaching. The tasks he chose for his interview and analysis assignment included equations, graphs, and functional situations. However, the following comments suggested that he was mistakenly developing the notion that conceptual knowledge was good and procedural knowledge was bad: The area in which both of the students were weak was working with equations concerning an algebraic representation of slope. They both had trouble coming up with an equation for a line that was already graphed, and vice versa. This is something that I found very surprising, as well as encouraging [emphasis added]. What we have discussed so far in this course is the need to focus more on conceptual understanding, and not so much on memorizing formulas. Well, these students exhibited these characteristics almost to a fault. They were very good on the conceptual questions, and not so good with algebraic representations and formulas.
Fortunately, Joe did not completely dismiss the importance of algebraic representations and formulas. He developed these representations in both his slope lesson plans and later in his Math 107 lessons. However, despite his initial propensity toward procedural aspects of slope, Joe displayed a willingness to try teaching with a focus on the development of conceptual understanding. When he introduced the concept of slope to his Math 107 class, he
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spent the first few minutes ‘‘philosophizing’’ about slope, rambling a bit, setting the stage for a discussion about the meaning of slope. One of his students became impatient with his verbal treatise, devoid of symbols, and interrupted Joe by saying, ‘‘I have a question. What exactly is slope? Can you define it?’’ vertical change Joe eventually defined slope as ‘‘horizontal change,’’ and presented a graph of the line passing through the points (0,0) and (3,2). He emphasized that the slope was a fraction, 23, up 2, over 3. Later, while students were working at their seats, one student was having difficulty 5 and 5 understanding how the two fractions 6 6 could both represent the same slope. Although at the time Joe struggled in vain to help her understand, he later described her difficulty with the following insight: ‘‘They think you are describing a movement as opposed to you describing a number, a measurement.’’ Joe’s encounters with students revealed him carefully pondering various aspects of students’ difficulties with slope. The methods class provided ideas about conceptual and procedural knowledge to frame his thinking. He attempted to address conceptual knowledge in his teaching, but these attempts were awkward and both he and his students were more comfortable discussing procedures. Joe, however, continued to be thoughtful, and would most likely have benefited from some more discussion on the topic. 4.1.2. Natalie: What is slope? On the initial survey, Natalie conjectured, ‘‘Students will have difficulty first actually understanding what slope even refers to since it tends to be a difficult concept to grasp. Students then might not associate the visual diagram of the line with the equation.’’ However, Natalie’s findings on the interview and analysis assignment surprised her. Her interview tasks focused on graphs and functional situations, and she found that the students were unable to answer most of her questions with appropriate answers. ‘‘I had not thought much about how little prior knowledge a student may possess. Both of the students I interviewed had taken first year algebra and geometry. . . . I would assume they would at least realize . . . that slope can be a numerical value and how to find it (or at least its inverse, ‘run over rise’).’’ Later, Natalie said, ‘‘I don’t think they get a chance to go from what they see in real life and what they believe is steepness into a numerical sense that they are given this number. And even if they understand rise over run, they see it on a piece of paper on a graph or just two points and plugging in a formula.’’ She had assumed that students would remember how to calculate slope, but perhaps not understand what it means. She discovered, however, that students did not remember how to calculate slope. Her interview and analysis assignment included several tasks addressing slope as a measure of rate of change. Although both students she interviewed were unsuccessful with these tasks, Natalie chose to ignore this idea in her slope lesson plans. Instead, she pursued the concept of slope as a measure of steepness. Her slope lesson plans included an activity from the Mathematics Teacher (Anderson & Nelson, 1994) that focused on measuring the rise and run of various objects such as ramps and staircases. I wanted to see them develop it so that they could remember ‘‘rise over run.’’ . . . When I first began, I thought my goal would be y = mx + b, but that to me, after I started thinking about it,
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wasn’t what the concept of slope is at all. That is one place where it is applied, . . . but that, after the interview and what we looked at in class, was not the beginning of slope. Maybe that is where it needed to end up, but that wasn’t what developed the concept of slope. . . . The concept started some place way before that.
The following semester, for her Math 107 lesson, Natalie’s goal was ‘‘y = mx + b. . .’’ She took her class to the computer lab and directed them to an interactive website where they worked on finding the slope of a line given two points of the line, graphing the line given its slope and a point on the line, finding the slope of a line given its equation, finding the slopes of horizontal and vertical lines, and comparing the slopes of parallel and perpendicular lines. On the initial survey, when Natalie was asked the question, ‘‘What is slope?,’’ she answered, ‘‘Slope is a term used to associate the incline of a line with a numerical value.’’ Later, when she discovered that this association may not be automatic for students, she designed her slope lesson plans to help middle school or high school students develop this connection, focusing on the concept of slope as a measure of steepness. When she actually worked with college students, however, she skipped over the concept of slope as a measure of steepness, and focused on developing a series of procedures related to graphs and equations of lines. 4.1.3. Tracie: Measurement of slant On the initial survey, Tracie hypothesized that ‘‘assigning a number to a ‘slant’ is something that students just learning about slope are not accustomed to.’’ Later, when interviewing two students for the interview and analysis assignment, Tracie assessed this particular aspect of slope by posing the following problem: Portions of Filbert Street and 22nd Street in San Francisco are the world’s steepest streets, rising 1 foot for every 3.17 feet of horizontal distance. How steep do you think this is? How could you figure out the steepness of the streets?
According to Tracie, one student merely responded with, ‘‘Pretty steep, I guess,’’ and shrugged her shoulders. The other student drew a triangle and used the Pythagorean theorem to find the length of the hypotenuse. Later in the semester, I again asked Tracie, ‘‘What difficulties do you think students have with slope?’’ She maintained her initial belief, but now she added more details: Because you can represent it with a number, they think it is a measurement that you can measure, like you can measure 5 inches or 6 inches. They want to measure it in that way as a length instead of an idea of the steepness.
In her slope lesson plans, Tracie was careful to develop the notion of slope as a measurement of steepness. Her lesson plans began with an activity on the playground, with students using a tape measure to measure the rise and run of a slide. She defined slope as rise run and then asked, ‘‘So, what is the slope of the slide? What does this mean?’’ Tracie also created a worksheet with pictures of a slide, a hill, a ramp, a roller coaster, a roof, and a ski slope. Two numbers appeared on each picture, and the students were instructed to find the rise, the run, and the slope of each object.
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The following semester, when she taught a lesson involving slope to Math 107 students, Tracie again emphasized measurement of slant, using examples similar to those from her slope lesson plans. None of the other preservice teachers in the investigation included this aspect of slope in their Math 107 lessons. Tracie initially expressed concern about students understanding the notion of slant, or steepness. She maintained this concern throughout the course, and she created learning opportunities for students to discover relationships among various representations of slope. Her lesson plans and actual lessons reflected a balance between concepts and procedures. 4.2. Knowledge of representations for teaching slope The other aspect of pedagogical content knowledge addressed in this investigation was preservice teachers’ knowledge of representations for teaching slope. The data sources used to examine the preservice teachers’ knowledge of representations for teaching slope were the initial survey, the textbook analysis, the slope lesson plans, the Math 107 lessons, and interview transcripts. When the preservice teachers designed their slope lesson plans, they were encouraged to include various representations of slope. They made their own choices of which representations to include and which resources to use for designing their lessons. When teaching Math 107, however, they were constrained by the textbook they were using. The textbook used in Math 107 was a traditional basic algebra textbook. It emphasized algebraic representations of slope (the slope ratio and linear equations), with less accentuation on graphs and minimal attention to physical or functional situations. Furthermore, the textbook did not emphasize
Table 4 Representations of slope included by preservice teachers in various data sources Data sources Preservice teachers
Initial surveya
Slope lesson plan
Math 107 lesson
Joe
graphs
Natalie
graphs physical situations (I)
graphs and ratios graphs and equations functional situations (P) graphs and ratios physical situations (P)
Tracie
physical situations (I)
graphs and ratios graphs and equations functional situations (I) graphs and ratios graphs and equations functional situations (I) physical situations (I) graphs and ratios graphs and equations physical situations (P) functional situations (P)
graphs and ratios physical situations (P) functional situations (P)
I = used as an illustration; P = used as a problem. a Question: ‘‘What analogies, illustrations, examples, or explanations do you think are most useful of helpful for teaching the concept of the slope?’’
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problem solving. Throughout the course, though, the preservice teachers were encouraged to supplement the textbook with ideas of their own. Table 4 shows the progression of evidence of the preservice teachers’ knowledge of representations for teaching slope. Once again, we look at the patterns of development for Joe, Natalie, and Tracie.
4.2.1. Joe: Graphs and functional situations—What does the slope of the line represent? On the initial survey, Joe wrote, ‘‘(1) Draw as many graphs as you have time for, so they can see it. (2) Make x, y chart. (3) Try to make it as simple as possible. Don’t overemphasize the why until they know the how [emphasis in the original].’’ In the first lesson in his series of slope lesson plans, Joe wrote, The objective of this lesson is to provide an opportunity for students to experience the process of making and testing conjectures. Students will develop, on their own, an understanding of slope and y-intercept. Through observation of how a multiplicative coefficient or an additive constant changes the orientation of the graph of y = x, students will gain insight into these concepts. To conclude, students will be introduced to the slope –intercept form as the equation of a line.
He planned to develop these objectives by having students graph sets of equations on graph paper. He would then describe slope as m ¼ Dy Dx with a graph of a line, carefully showing Dy and Dx. Finally, he would connect the formula y = mx + b to the graphical work that the students had completed. In his second lesson, Joe planned to involve the students in collecting data that involved a linear relationship. The students would enter the data into a graphing calculator and produce a scatterplot. They would then use their knowledge of slope and y-intercept to produce an equation of a line to fit the data. One of the questions Joe planned to ask the students is, ‘‘What does the slope of this line represent?’’ Joe’s third lesson stressed the notion of slope as rate. It included three different functional situations involving slope: drawing the graph of time versus distance for a traveling bicycle, interpreting a graph of time versus temperature for a substance being heated, and interpreting a graph of time versus velocity for a traveling car. For the first two situations, Joe again asked, ‘‘What does the slope of the line represent?’’ The framework that Joe created for his textbook analysis included the following: How does the textbook address the concept of slope? (a) conceptual focus, (b) computational focus, (c) answers the question: ‘‘What does the slope of the line represent?’’, (c) introduces slope as a rate, (e) connects slope to real-world situations. in the analysis, he wrote: This book looks at slope as a computational tool, and wants students to be able to find slope under several different conditions. It is formula based, and memorization will play a large role in the successful completion of the material. It is not to say that these are not important parts of understanding slope. It is just that the students are never given an opportunity to investigate what slope represents, and that is what our discussions have determined is the most beneficial part of learning slope.
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When interviewed, Joe said: I think that maybe you can gain the most from a graph. If you look at slope on a graph that is where you gain the most insight. If that graph is a model of something they have done before in their life, if there is something outside in their lives that they can connect it to, then that is even more important, I think.
The following semester, Joe taught two lessons involving slope to his Math 107 class. He introduced the concept of slope by stating, ‘‘We are going to discuss slope on two different levels. (1) Mathematically: How do you compute slope? (a) from a graph (b) from an equation; and (2) What does the slope of a line represent?’’ Joe’s first lesson only addressed the first part of the first level. His partner addressed the second level the following day, and Joe discussed equations later in the semester. vertical change In his first lesson, Joe defined slope as horizontal change, and presented a graph of the line passing through the points (0,0) and (3,2). He emphasized that the slope was a fraction 23, up 2, over 3. After discussing a second example, he introduced a paper-and-pencil game called ‘‘Rapid Fire’’ that he had created for the students to practice finding the slope of a line connecting two points. The students spent the remainder of the period playing in pairs. The following class day, Joe’s teaching partner introduced some functional situations involving slope. A couple of weeks later, Joe took his Math 107 students to the computer lab for a graphing activity. The students used a computer graphing program and a worksheet that Joe had prepared for students to discover answers to such questions as the following: What effect does a coefficient multiplier have on the graph of y = x? In mathematical terms, we would call this a change in the _____ of the line. What does it mean for a graph to have a negative slope? What does it mean for a graph to have a positive slope? At the end of this lesson, seemingly as an afterthought, Joe drew a graph of distance versus time on the board to illustrate a real-world application of slope. However, he did not ask the question, ‘‘What does the slope of the line represent?’’ Based on the initial survey, Joe’s repertoire of representations for teaching slope did not include real-world situations. Later, though, for his slope lesson plans, he created a variety of rich problems involving functional situations. These problems emphasized the notion of slope as rate of change. These real-world problems were ignored, however, when he actually taught the concept of slope in Math 107. Instead, Joe focused on graphs, using only one functional situation as an illustration. 4.2.2. Natalie: Physical situations and ‘‘rise over run’’ On the initial survey, Natalie said, ‘‘I think the example of an inclined plan is very important. I also think the saying ‘rise over run’ helps students remember the order.’’ Her slope lesson plans focused on physical situations and developing the ratio ‘‘rise over run’’ as a measure of steepness.
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They need to be able to understand algebraically what it means and be able to graph, eventually, linear equations and for parallel and perpendicular lines. . . . They should be able to distinguish between, if they don’t get anything else out of it, they should be able to see the difference in steepness—of a roof’s pitch or grade.
In her textbook analysis, Natalie used the framework suggested by her instructor to discuss the various representations of slope: The text begins the lesson with physical representations of slope by using a short paragraph of the grade of a hill. Through the geometric representation, the text develops ‘‘rise over run.’’ The functional representation appears, but the text only eludes to it. The ratio representation is used most frequently. The majority of practice problems involve it. Finally, the next lesson involves the algebraic representation.
Natalie used these results from her textbook analysis to justify not including functional situations in her slope lesson plans. In an ideal world, at some place and time, I would include the way that slope can compare two different things like distance and time. Given an infinite amount of time in my class, I certainly do think that would be important. But judging from where this context is placed in algebra books that I have looked at, by the end of the book, there is not going to be enough time to cover the concepts that are going to be on the exam, in order for them to go onto the next class.
Perhaps to please her instructor, Natalie’s Math 107 lesson in the computer lab began with a functional situation, ‘‘To create a custom-made dress, one seamstress charges a set price plus a certain price per yard of fabric. The following graph shows her prices.’’ Natalie then posed the following questions: (1) ‘‘How much would it cost if I wanted a dress that uses 10 yards of fabric?’’ and (2) ‘‘What is her starting price without any fabric and how much does she charge per yard of fabric?’’ She told the class that their answers had ‘‘something to do with what we call the slope of a line,’’ but she did not ask the question, ‘‘What does the slope of the line represent?’’ She then asked where else they might use slope. One student mentioned a roof and another mentioned a ramp. Then Natalie told the students how to locate the website they would be using, and she handed out an activity sheet to accompany the lesson on the website, a lesson devoid of real-world context. She spent the remainder of the lesson circulating through the computer lab, answering the individual students’ questions about graphs and equations. Natalie’s maintained her initial appreciation for physical situations involving slope throughout her experiences in the methods course and her actual teaching. Of the three preservice teachers, Natalie was the most resistant to including the notion of slope as a measure of rate of change in her work. 4.2.3. Tracie: Physical and functional situations In response to this question, Tracie replied, ‘‘Something tangible. Examples of common uses are the slope or pitch of a roof and the slope or grade of a mountain road.’’ Later, these physical representations for slope appeared in her textbook analysis, her slope lesson plans and her Math 107 lesson.
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Tracie did, however, began to develop a repertoire of other representations for slope. Her slope lesson plans included an activity using a geometry computer program to construct lines and calculate slopes. She asked the students to make conjectures about the appearance of the graphed lines and the slope measurements. She also presented five different problems involving real-world representations of slope. One problem, entitled ‘‘Carpentry,’’ asked students to determine how to fit a staircase into a given space. The other four problems involved linear functions. ‘‘Class Fund Raiser’’ was about selling pizzas. ‘‘Population Growth’’ compared the rate of growth of two high schools. ‘‘Communications’’ asked for a comparison of the rates of three major long-distance companies. ‘‘Financing the Prom’’ involved magazine sales. For each of these linear functions, Tracie asked for a graph of the data, but she did not ask students to determine the slope of the line or interpret its meaning within the context of the situation. In her textbook analysis, Tracie wrote, Slope is defined as rise over run, initially in Chapter 11. [The textbook] does not consider the many different interpretations and applications of slope. There is one application for slope on page 423. It deals with temperature and altitude and provides only one of many good concrete applications for the concept of slope. Unfortunately, slope is developed in this text as a means to graph equations and determine equations of graphs. The importance of slope is not stressed.
When I interviewed Tracie later, I asked her, ‘‘What representations are most important when teaching the concept of slope?’’ She said, In the beginning, I think it would have to be very concrete and the idea of just what slope is, as in steepness and stuff like that. And the further along you get, things like rate and talking about speed and acceleration and distance and finding what slope means when you graph those things.
The following semester, Tracie’s aspirations for her Math 107 class were evidently ‘‘further along’’ on her continuum, because she did address the notion of rate of change. She designed a worksheet that included the following set of questions designed to help students discover the connections among a functional situation, equations, and graphs. Here are some of the questions: Anissa is reading a book for her English class. She decides to read two chapters each day in order to avoid having to read it all at the last minute. Graph Anissa’s reading progress below. Which equation represents the graph of Anissa’s reading? When an equation is in the form y = mx + b, the numbers m and b tell us something about the graph. What part of the equation describes the slant (or slope) of the graph? Look back at the graph and linear equation of Anissa’s reading. What is the slope of this line? What does this slope represent?
Tracie’s worksheet for her Math 107 lesson also included illustrations of several physical situations — a slide, a ski slope, a skateboard ramp — for which students were supposed to identify the rise and the run and then determine the slope. Tracie was predisposed to using physical representations of slope. In fact, she mentioned that she had first learned about slope from her father, a carpenter. She did expand her
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repertoire to include functional representations, and she demonstrated her ability to develop connections between slope as the property of a line and slope as a measure of rate of change.
5. Discussion The methods course in this investigation provided a space in which the preservice teachers’ initial ideas about slope could be developed and even challenged. Reading assignments and class discussions exposed the preservice teachers to theoretical notions related to teaching and learning the concept of slope. Class activities and assignments provided opportunities for them to connect the theory to practical situations. Later, their evolving ideas about slope appeared in their lessons, suggesting a transformation in their pedagogical content knowledge about slope and how they would apply this knowledge. Because the methods course in this investigation assumed a constructivist perspective, there was an expectation that the preservice teachers would develop their pedagogical content knowledge in perhaps similar yet different ways. This research provides insight into the consistencies and the variations in their knowledge of students’ difficulties with slope and their knowledge of representations for teaching slope. 5.1. Knowledge of students’ difficulties with slope 5.1.1. Focus on concepts and procedures In the methods course, the interview and analysis assignment served as an important instrument for developing preservice teachers’ pedagogical content knowledge. By requiring the preservice teachers to focus on both conceptual and procedural knowledge, the assignment provided an opportunity for the preservice teachers to think about and examine students’ knowledge in new ways, thus expanding their own views about students’ knowledge of slope. It gave them a chance to discover for themselves the difficulties that students have with slope, and also to explore why students have those difficulties. When the preservice teachers entered the methods course, their initial ideas about students’ potential difficulties with slope were limited to vague assumptions. After interviewing students to assess their knowledge of slope, the preservice teachers did not abandon their initial assumptions, but they added more dimensions to their thoughts about students’ difficulties. Unlike the teachers in a previous investigation (Stump, 1999) who focused only students’ computational difficulties with slope, the preservice teachers in this study took advantage of the opportunity to examine both conceptual and procedural knowledge of slope. Not only did they learn more about students’ knowledge and skills with graphs and equations, they also discovered the limitations in students’ understanding of slope as a measure of steepness and as measure of rate of change. 5.1.2. Knowledge of students’ cognitions is valuable The interview and analysis assignment confirmed some of what the preservice teachers already believed, but it also provided new insights. For example, Joe was initially concerned
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with students understanding and using a formula, and he indeed witnessed such difficulties when he conducted his interviews. However, he also discovered students had conceptual understanding that he might have otherwise ignored. Natalie, on the other hand, was surprised to find that the students she interviewed lacked a basic understanding of the numerical value of slope. Tracie maintained her initial concern with students’ understanding of slope as a measure of slant, and the interview and analysis assignment provided her with more evidence of their lack of understanding. The information these preservice teachers gained from the interviews was reflected in specific ways in both their lesson plans and their actual teaching. To illustrate, Joe highlighted the meaning of slope, Natalie developed the notion of steepness, and Tracie emphasized slope as a measure of slant. Because they had evidence of students’ cognitions, the preservice teachers created lessons to carefully develop the concepts they knew were difficult for students to understand. Furthermore, they created ways to engage students in meaningful activities involving these concepts. 5.2. Knowledge of representations for teaching slope 5.2.1. Real-world situations When the three preservice teachers entered the methods course, their knowledge of representations for teaching slope was dominated by graphs and physical situations. Throughout the methods course, they continued to value slope as a measure of steepness, but they also became conscious of the notion of slope as a measure of rate of change. When they wrote lesson plans for middle or high school students, all three preservice teachers developed the concept of slope through problems involving real-world situations. Joe emphasized functional situations, Natalie emphasized physical situations, and Tracie included both. All of their lessons included questions designed to focus students’ attention on the meaning of slope in these real-world situations. The actual lessons they taught to college students, however, focused more on graphs and equations. All three preservice teachers presented lessons that were developed through problem solving and discovery, and all three did include real-world situations somewhere in their lessons. Joe and Natalie, however, used physical and functional situations merely as illustrations, not as problems for students to solve. They did not, for example, pose questions for students to interpret the meaning of slope in real-world situations. Only Tracie used realworld situations as problems in her Math 107 lesson. She asked students to interpret slope both as the measure of steepness in physical situations and measure of rate of change in functional situations. We should not assume that students will automatically develop an understanding of slope as measure when real-world situations are mentioned only as illustrations for the concept of slope (Stump, 2001). Instead, students must be purposefully challenged to construct connections among geometric, algebraic, and real-world representations of slope. In this investigation, the preservice teachers’ Math 107 lessons reflected limited appreciation for the importance of developing the concept of slope as a measure of steepness or rate of change through problem solving.
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5.2.2. The role of the textbook This investigation provides some insight into the representations for slope that preservice teachers choose when attempting to adapt a traditional textbook in meaningful ways. As is the case with many teachers, the preservice teachers’ lessons were constrained by the textbook they were using. The textbook used in Math 107 did not emphasize real-world applications of slope, and did not focus on problem solving. All three of the preservice teachers in this investigation, though, found creative ways to supplement the textbook. Joe created a paper-and-pencil game and devised a discovery lesson using a computer graphing program, Natalie prepared a packet for use with an interactive website, and Tracie designed a written discovery lesson using illustrations and word problems. Although most of these supplementary activities did not focus on real-world applications of slope, they did the engage students in problem solving.
6. Conclusion Throughout their experiences in the methods course and their actual teaching, the three preservice teachers in this investigation developed sensitivity to both conceptual and procedural aspects of students’ knowledge of slope. They also expanded their repertoires of representations for teaching the concept of slope and they created meaningful problemsolving activities for their students. The greatest difference, however, among the three preservice teachers was in their display of appreciation for slope as a measure of steepness and as a measure of rate of change. While one preservice teacher used real-world situations to engage students in problem solving, another used them merely as illustrations of slope. As one preservice teacher focused on physical situations, another emphasized functional situations. Perhaps the tenuous knowledge of these preservice teachers for developing the concept of slope in real-world situations can be explained by their limited exposure to nontraditional curriculum materials. The circumstances and available resources for these preservice teachers required them to supplement a traditional algebra textbook. The result was a demonstration of their ability to create worthwhile, engaging lessons involving slope, but also a hesitation to develop the concept of slope as a measure of steepness and as a measure of rate of change. Although they worked to include physical and functional representations of slope somewhere in their lessons, the framework in which these preservice teachers worked did not require them to use these representations as the primary focus of their lessons. Perhaps an alternative framework, one that requires preservice teachers to develop lessons using nontraditional curriculum materials, would help preservice teachers further develop their pedagogical content knowledge for teaching the concept of slope.
Acknowledgments This research was completed with funds from the Office of Academic Research and Sponsored Programs at Ball State University.
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