K-8 teacher candidates' use of mathematical representations and the development of their pedagogical content knowledge as exhibited in their lesson planning.

Abstract

A study was carried out involving thirty-one K-8 teacher candidates enrolled in an elementary mathematics methods course to investigate and document their thinking as they plan for mathematics instruction. The teacher candidates submitted lesson plans at three intervals during a semester-long methods course which were coded based on the planned uses of mathematical representations. Analysis of the data revealed trends in the choices of representations. Recommendations are presented highlighting the potential benefits of incorporating the knowledge base on mathematical representations into a mathematics methods course and a discussion ensues on the development of these teacher candidates' pedagogical content knowledge through their choices of mathematical representations in their lesson planning.

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Introduction

A primary goal of a methods course is to prepare future teachers to teach the subject matter in effective and engaging ways such that student understanding is maximized. In other words, the authors posit that methods courses provide opportunities for teacher candidates to develop and further their pedagogical content knowledge. In an attempt to document and explore how K-8 teacher candidates represent mathematical ideas in ways that are understandable to students, a study was carried out in which K-8 teacher candidates submitted lesson plans at three intervals during a semester. Serving as a lens during the coding and analyses of these lesson plans were the five representations defined by Lesh, Post, & Behr (1987). Of interest to the authors were the representations these teacher candidates might choose to use as they plan for teaching a mathematics topic in the most effective way as possible; that is, so that student understanding is maximized. Also of interest was the potential emergence of trends in these K-8 teacher candidates' choices of representations in their lesson planning as they progressed through their semester-long elementary mathematics methods course. After analyzing the lesson plans of these teacher candidates at three intervals, the authors were able to document the planned choices of and trends in both teacher and student use of various representations within their lesson plans. These choices and trends will be described and a discussion will follow presenting the potential benefits of incorporating the knowledge base on mathematical representations into a mathematics methods course to contribute to the development of K-8 teacher candidates' pedagogical content knowledge.

Research on pedagogical content knowledge

Many researchers have described the various types of knowledge that are needed by teachers. In particular, Shulman (1986) defines pedagogical content knowledge as the ability to represent ideas in ways that are understandable to students. Shulman argues that teachers need to know things such as what topics children find interesting or difficult and what the representations are that are most useful for teaching a specific content idea. Shulman (1987) further elaborates on pedagogical content knowledge as the capacity "to transform the content knowledge he or she possesses into forms that are pedagogically powerful and yet adaptive to the variations in ability and background presented by the students" (p. 15).

For Carter (1990), pedagogical content knowledge represents an attempt to determine what teachers know about their subject matter and how they translate that knowledge into classroom curricular events. Doyle (1992) contents that this ability distinguishes a teacher from a non-teaching specialist; for example, "Knowing biology is necessary, but certainly not sufficient, to know how to represent biological content to students in a teaching situation" (p. 498).

Kennedy (1998) offers that recitational knowledge; that is, the traditional mode of defining facts and terminology is not sufficient for teaching. Instead, other types of knowledge are needed such as the conceptual understanding of subject matter as well as pedagogical content knowledge. "Teachers need to be able to respond to questions and hypotheses that they might not have anticipated, provided students with guidance when they get in over their heads, clarify confusions, and ensure that misconceptions are not perpetuated" (Kennedy, 1998, p. 252). Additionally, Ball & Bass (2000) posit that in order to make mathematical knowledge usable, teachers must know content sufficiently and flexibly such that it can be used within a wide variety of contexts.

Research on representations

Lesh, Post, & Behr (1987) offer five different representations of mathematical ideas, namely, concrete (manipulatives), language, symbolism (notation), semi-concrete (pictorial), and contextual (real-world situations). As an example to the reader, the researchers present their interpretation of these five representations by considering how a teacher might represent the concept of 65 (see Table 1). A teacher could use counters or money to represent 65 concretely. A teacher could use language such as. "Sixty-five is five less than seventy" to represent this concept or use the notation, "100-35=65" to represent this concept symbolically. Similarly, a teacher might use a set model to represent 65 pictorially or a teacher might offer the real-life context of the cost of an item or the age at which one retires to assist a student in understanding the concept of 65. According to Lesh, Post, & Behr (1987), strengthening the ability to move between and among representations improves the growth of students' understanding of mathematical concepts.

The National Council of Teachers of Mathematics (NCTM, 2000) states that:

    Representations should be treated as essential elements in
    supporting students' understanding of mathematical concepts and
    relationships; in communicating mathematical approaches, arguments
    and understandings to one's self and to others; in recognizing
    connections among related mathematical concepts; and in applying
    mathematics to realistic problem situations through modeling
    (p. 67).

The NCTM (2000) continues on to say that, "When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically" (p. 67). Thus, representations can then be thought of as thinking tools to communicate mathematical ideas. The model that Lesh, Post, and Behr (1987) have developed of external representations allows for teachers and students to have a common language for communicating internal mathematical ideas.

Methodology

The study involved thirty-one elementary education majors enrolled in a field-based mathematics methods course at a large southwestern university. All of these individuals had successfully completed a mathematics content course, a prerequisite to the methods course. Two of the participants were male and 29 were female. Throughout the semester, the teacher candidates were taught in a constructivist manner (using manipulatives, technology; engaging in problem solving, hands-on exploration, writing, discourse; making real-world connections, etc.) consistent with the reform standards (NCTM, 1991, 2000). During the methods course, the teacher candidates simultaneously participated in the course's field component, where they observed and assisted in an elementary classroom daily.

At the beginning of the semester, the authors compiled a list of eight mathematical topics that would be explored throughout the methods course and which represented typical K-8 mathematics topics these teacher candidates would be expected to teach. These topics included: multiplication of fractions, division of fractions, area of a circle, area of a trapezoid, area of a parallelogram, perimeter of polygons, addition and subtraction of integers, and mean. The topics, along with a corresponding and appropriate grade level, were written individually on index cards prior to class and one index card was distributed to each teacher candidate during the beginning portion of the semester. The teacher candidates were then instructed to develop and submit a lesson plan one week later reflecting what they considered to be an effective way to teach that particular topic; that is, such that student understanding would be maximized. The teacher candidates were encouraged, but not required, to seek out resources such as books, the Internet, speak to inservice teachers, consult with their field component's mentor teacher, etc., to assist them in completing this assignment. The teacher candidates were asked that when designing their lesson plans, they include along with the topic and grade level, any materials, the procedure, a closure, and the source (if any) of their ideas. For coding purposes, the lesson plans submitted at this initial juncture were labeled as "initial" lesson plans.

After completing and submitting their individual lesson plans, the teacher candidates were placed in groups with those sharing the same topic and, with their classmates, were asked to reach a group consensus on how to best teach their shared topic. During this time, the teacher candidates were encouraged to share their individual ideas described in their lesson plans and to justify to their classmates why they believed their method and approach to teaching this topic was effective. The teacher candidates were given one hour of class time to engage in discussion. Additionally, the groups were asked to identify a note-taker, who would capture the finalized ideas on paper and a typist, who would electronically submit the group's finalized lesson plan using the aforementioned format. For coding purposes, the lesson plans submitted at this middle juncture were labeled as "group" lesson plans.

While the teacher candidates were sharing their ideas and deliberating over how to best teach their given topic, each group was videotaped and voice recorded in an attempt to capture their conversational journey as they each first presented their individual arguments and justifications for how to best teach their topic and then concluded with how the group came to a consensus on how to teach the topic.

Near the end of the semester, the teacher candidates were asked to individually submit one final lesson plan, again using the previously described format, but this time detailing how to best teach a K-8 mathematical topic of their choice. Topics chosen by the teacher candidates included a wide range of grade levels and concepts, such as addition, area of a circle, counting, fractions, graphs, least common multiple, money, multiplication facts, probability, rate of acceleration, rounding, shapes, solids, subtraction, tessellations, and time. For coding purposes, the lesson plans submitted at this final juncture were labeled as "final" lesson plans.

Upon receiving the teacher candidates' lesson plans at each of the three intervals, the authors carefully coded the lessons plans noting each time one of the five aforementioned representations (Lesh, Post, Behr, 1987) was used. However, after coding several of the "initial" lesson plans and recognizing the apparent emerging gap between the number of times the teacher was using a representation versus the students, the authors refined their coding process by delineating between each time a representation was used by the teacher in the lesson plan versus students.

Additionally, during the coding process, the authors sub-divided the language representation into two sub-categories hereto referred to as, [L.sub.1] and [L.sub.2]. [L.sub.1] referred to language that is used to talk about mathematical procedures and defining mathematical terms whereas [L.sub.2] referred to mathematics discourse; that is, rich and thoughtful discussion about the mathematics and questions that encourage higher-level thinking as well as explaining, justifying, questioning, and challenging (Wood, 1999). This subcategorization of the language representation demonstrated the extent to which these teacher candidates adhered to the NCTM Standards (2000), which strongly advocate discourse in the mathematics classroom.

As an example to the reader, consider the following portion of a "group" lesson plan submitted by a group of teacher candidates on the topic of perimeter of polygons. The lesson, designed for third graders, begins:

    "Discuss the characteristics of a polygon with the class. Explain
    that a polygon has straight lines with no arcs.

    Pass out a geoboard to each student or pair of students.

    Have the students construct various polygons on the geoboards
    emphasizing the attributes of a polygon.

    As the students are working, tape large shapes onto the floor, one
    shape per group of students. Distribute colored construction paper
    and instruct students to trace their foot and cut it out.

    Divide the class into groups explaining that they will use their
    paper foot to determine the distance around the shape. They are to
    find out how many of their paper feet will go around the shape.

The first two sentences above were coded as the teacher using language ([L.sub.1]) as a representation of the idea of a polygon. The third sentence was not coded since it is a logistical aspect of the lesson plan. The fourth sentence was coded as students using a concrete representation of a polygon along with them using [L.sub.1]. The fifth and sixth sentences were coded as pictorial representations of polygons that the students would be potentially using to find perimeters. The seventh and eighth sentences were coded as the teacher's use of [L.sub.1] to represent the notion of perimeter.

Listed in Table 2 are the percentages of times the various representations were used within the lesson plans submitted at each of the three intervals. Table 3 presents these same percentages, but separated out by teacher versus student use.

Discussion of the findings

The authors caution that the reader keep in mind that these lesson plans represent hypothetical teaching acts, as these lesson plans were not enacted in an actual classroom setting. Thus, the teaching approaches and ideas presented in the lesson plans provide us with insight into these teacher candidates' thinking about how to plan for and teach mathematics.

After analyzing the data, language was found to be the most used representation in the lesson plans submitted by the teacher candidates at all three intervals. Moreover, the use of [L.sub.1] (definitions and procedures) heavily outweighed the use of [L.sub.2] (discourse) and it was one of the most used representations overall. Also, in comparing the initial lesson plans to the final lesson plans, the use of [L.sub.1] increased from 33% to 46%. A disappointing trend was that [L.sub.2] was least used representation overall. One possible explanation for this might be the reluctance, apprehension, or lack of recognition of its importance, on the part of these teacher candidates, to plan for and encourage rich discussion and discourse in their lesson planning, despite this type of language being modeled and encouraged regularly in their mathematics methods course. Also noted was the very infrequent use of and decreasing trend in the contextual (real-world) representation, again, despite its frequent modeling by the methods instructor and the discussion of its importance during class.

Although its use sharply declined (32% to 14%) when comparing the initially submitted lesson plans to those submitted at the end of the semester, symbolism, like language, was also a heavily used representation. Little change was noted in the semi-concrete (pictorial) representation, decreasing ever so slightly over the course of the semester from 16% to 15%.

Although its use did increase from 10% to 18%, prior to coding the lesson plans, the authors anticipated a higher occurrence of the concrete representation in all three of the submitted lesson plans, primarily due to the fact that manipulatives were used in every class meeting of both the methods course and the prerequisite mathematics content course. This assumption was also supported by the fact that during the videotaping of the teacher candidates' group discussions, they repeatedly voiced the importance of and predilection for using manipulatives to provide a "hands-on approach." For example, an individual in the group whose topic was mean suggested to her classmates that the students should all have "some sort of manipulative for them to actually see it [the mean] with their hands." In another group, one teacher candidate reiterated her group members' sentiments stating, "Yeah, we all like hands-on" while another individual in that same group stated, "These [geo]-boards help out a lot." Consequently, the group members unanimously agreed to use geoboards to teach their given topic of perimeter of polygons.

Upon comparing the percentages of planned uses of the various representations by the teacher as opposed to the students, (see Table 3), the authors noted that although student use of language increased over time (from 7% to 19%) it was the teacher who used the language representation far more often in the lesson plans. Additionally, teacher use of [L.sub.1] outweighed student use of [L.sub.1] and, although it was the least often used representation, [L.sub.2] was used more often by the students than by the teacher. The heavy use of [L.sub.1] by the teacher in the lesson plans could be attributed to the fact that the teacher candidates specified how the teacher would have to "introduce the formula" and "explain" and "tell" students how a formula was derived or "how it worked." One student whose topic was the area of a trapezoid told his group. "The actual formula will come later in the lesson. You're not gonna blast them with the formula right away." When the teacher candidates were able to collaborate and negotiate with their classmates in their respective groups, the student use of [L.sub.2] increased as the teacher candidates began to allow for the students, as opposed to the teacher, to explain, apply, justify, and "figure out for themselves" the formulas necessary to understand their topic.

In regard to the other representations, student use of the concrete representation outweighed teacher use primarily because, as noted during the coding of the lesson plans, the students were described as actively engaged in using the manipulatives, as opposed to simply watching the teacher demonstrate some concept using the manipulative solely. During the second and third intervals, student and teacher use of symbolism and the contextual representation were somewhat balanced while there existed a transition from teacher-use to a more balanced teacher/student-use for the semi-concrete (pictorial) representation.

Table 3 also illustrates the overall percentage of time the representations were planned to be used by the teacher (T) as opposed to the students (S). In the initial lesson plans, the authors noted that the teacher (66% vs. 34%) used the representations approximately two-thirds of the time. In the group lesson plans, the teacher and the students were using the representations almost equally (47% vs. 53%) whereas in the final lesson plans, the teacher and the students were using the representations the same percentage of time (50% vs. 50%). The authors posit that this more balanced approach to planning for their teaching, where both the teacher and the students are equally engaged in the mathematics, could be attributed to the modeling that these teacher candidates observed in their mathematics methods class, where they were equally responsible for and engaged in understanding the material. Another contributing factor could be the added observation of the use of manipulatives in the elementary classrooms that they observed and participated in during the field component of their methods course. At some point during this time, as these teacher candidates prepared to become teachers, they may have begun the process of valuing the use of concrete materials when exploring mathematics concepts.

Other findings

After reading the transcriptions of the videotaped group discussions, the authors noted that not only did the teacher candidates negotiate, question, challenge, and justify their choices for how and why they planned to teach the topic, but the discussions also included tending to the logistics of teaching. For example, within some groups, the conversation at times overly focused on such logistical issues as the use of worksheets, grouping, and time limitations. Their comments included:

    "Maybe we should just do a worksheet ... We could give them a few
    problems."

    "How many examples do we need to use on one paper? Do you think
    four? Is this sufficient?"

    "We should put a time limit; for fifth graders, maybe 45 minutes. Do
    we want to do groups or individuals?"

    "We need to decide what will happen on each day."

Some of these logistical issues demonstrated the teacher candidates' growth in other areas of pedagogical knowledge; in particular, practical knowledge as defined by Carter (1990).

After coding and analyzing all of the data, the mean number of representations used per lesson plan was computed (see Table 4). The mean number of representations used in the lesson plans increased over the course of the semester from seven to twelve. The authors attribute this increase to the teacher candidates having acquired more comfort, fluency, and flexibility with planning to use a variety of representations as a result of their engagement in their methods course and their participation in the field component of the course. By choosing to use more representations and moving between these representations flexibly in their lesson planning, these teacher candidates demonstrated their ability to make mathematical knowledge usable, which supports the work of Ball & Bass (2000). Thus, these learning opportunities presented in the methods course could have possibly increased the breadth of these teacher candidates' pedagogical content knowledge resulting in their articulation and inclusion of more approaches and ideas; that is, representations, in their lesson plans. Certainly, learning to teach with understanding is a process that takes time, experience, and continued growth of the knowledge base and how students learn.

Closing remarks

Serving as one aspect of the framework to this study was Shulman's (1986) definition of "pedagogical content knowledge" which he defines as the ability to represent ideas in ways that are understandable to students. Considering the fact that these teacher candidates were instructed to develop lesson plans that would describe effective ways to teach a particular K-8 mathematics topic, the authors argue that these individuals were applying their developing pedagogical content knowledge, illustrated by their choices of the various representations as defined by Lesh, Post, and Behr (1987). This supports the authors' aforementioned claim that methods courses provide opportunities for teacher candidates to develop their pedagogical content knowledge further and, one method of observing this pedagogical content knowledge is through the coding and analyses of their lesson plans via the lens of representation.

Additionally, after analyzing the results of this study, the authors do not claim that any one representation is better than another, nor do they suggest the existence of some optimal number of representations that would yield a highly effective lesson plan. The authors offer that it is not the representation that is used, but how it is used and who uses it, whether by the teacher or the students, that seemed to contribute to the effectiveness of a lesson plan. Lesh, Post, & Behr (1987) offer that strengthening the ability to move between and among representations improves the growth of students' understanding of mathematical concepts. The authors offer that teacher candidates need assistance in developing and strengthening this same ability, but on a metacognitive level, thereby enhancing their pedagogical content knowledge and their ability to make mathematics usable. Further analysis of representation within the context of planning for mathematics instruction coupled with teaching for understanding will yield additional and valuable knowledge in observing and describing teacher candidates' growth in their pedagogical content knowledge.

This study suggests potential benefits for incorporating the knowledge base on mathematical representations into a math methods course, as by using mathematical representations, teacher candidates can demonstrate their developing pedagogical content knowledge. After carrying out this study, Lesh, Post and Behr's (1987) research on the five mathematical representations was shared and discussed with the teacher candidates. In closing interviews, these teacher candidates shared their advocacy for including this body of research into a mathematics methods course. Some of their comments included:

    "Mathematical representations could help in planning for teaching
    lessons in mathematics because I can have a variety of ways to show
    a concept."

    "Being aware of different representations to teach would help you
    reach each of the students in your class in the most effective way.
    Not all kids learn the same."

    "Knowing about mathematical representations would help me think
    outside of the box and be more creative in my lesson plans."

As stated earlier, given that a primary goal of a methods course is to prepare future teachers to teach the subject matter in effective and engaging ways such that student understanding is maximized, the authors strongly recommend the inclusion of Lesh, Post and Behr's (1987) work on representation in mathematics methods courses.

[TABLE 1 OMITTED]

          Concrete  Language      Symbolism  Semi-        Context  Total
          (Manip.)  [L.sub.1]                concrete
                    [L.sub.2]                (pictorial)

Initial      10     33               32          16          9     100
  LP                [L.sub.1]=31
                    [L.sub.2]=2

Group        12     31               30          18          9     100
  LP                [L.sub.1]=28
                    [L.sub.2]=3

Final LP     18     46               14          15          7     100
                    [L.sub.1]=39
                    [L.sub.2]=7

Table 2. Overall summary of planned uses of mathematical representations
(by percent).

         Concrete           Lang.                Symb.  Semi-
         (Manip.)  [L.sub.1]      [L.sub.2]             concrete
                                                        (pictorial)

Initial    S=7     S=7            T=26           S=13       S=5
  LP       T=3                                   T=19       T=11
                   [S.sub.L1]=6   [T.sub.L1]=25
                   [S.sub.L2]=1   [T.sub.L.2]=1

Group      S=11    S=13           T=18           S=14       S=11
  LP       T=1                                   T=16       T=7
                   [S.sub.L1]=10  [T.sub.L1]=18
                   [S.sub.L2]=3   [T.sub.L2]=0
Final      S=13    S=19           T=27           S=8        S=7
  LP       T=5                                   T=6        T=8
                   [S.sub.L1]=15  [T.sub.L1]=24
                   [S.sub.L2]=4   [T.sub.L2]=3

         Context  S   T   Total

Initial    S=2    34  66   100
  LP       T=7
Group      S=4    53  47   100
  LP       T=5
Final      S=3    50  50   100
  LP       T=4

Table 3. Teacher (T) vs. students (S) planned use of representations
(by percent).

Initial LP   7
Group LP    13
Final LP    12

Table 4. Mean number of repretations used per lesson.

REFERENCES

Ball, D.L. and Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.) Multiple Perspectives Mathematics Teaching and Learning, 83-104. Westport, CT: ABLEX Publishing.

Carter, K. (1990). Teachers' knowledge and learning to teach. In W.R. Houston (Ed.), Handbook of research on teacher education (pp. 291-310). New York: Macmillan.

Doyle, W. (1992). Curriculum and pedagogy. In P. Jackson (Ed.), Handbook of research on curriculum (pp. 486-516). New York: Macmillan.

Kennedy, M. (1998). Education reform and subject matter knowledge. Journal of Research in Science Teaching, 35(3), 249-263.

Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.) Problems in representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston. VA: NCTM.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4-14.

Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22.

Wood, T. (1999). Creating a context for argument in mathematics class. Journal for Research in Mathematics Education, 30(2), 171-191.

Robin A. Ward, Cynthia O. Anhalt, and Kevin D. Vinson

University of Arizona