The mathematics content knowledge role in developing preservice teachers' pedagogical content knowledge.

Abstract. This paper outlines the nexus between mathematics content knowledge and pedagogical knowledge in developing pedagogical content knowledge. Increasing expectations about what students should know and be able to do, breakthroughs in research on how children learn, and the increasing diversity of the student population have put significant pressure on the knowledge and skills teachers must have to meet education goals for the 21st century. Specifically, in undergraduate mathematics education, how pedagogical awareness is taught should relate to deeper and broader understandings of mathematical concepts for preservice teachers. The participants (n = 193) were enrolled in their senior integrated methods block in the semester prior to beginning their student teaching. Among the data analyzed were previous mathematics course performance, a pre- and post-assessment instrument, success on the state level teacher certification examination, and portfolios and journals. The results indicated that previous mathematics ability and posttest performance were valuable predictors to student success on all portions of the state-mandated teacher certification exam, ExCET The qualitative data indicated that mathematically competent preservice teachers exhibited progressively more pedagogical content knowledge as they were exposed to mathematics pedagogy during their mathematics methods course.

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Various reform initiatives have produced documents calling for a new vision for the teaching and learning of mathematics (National Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000; National Research Council, 2001). These documents describe a dynamic, invigorated role for the preparation of mathematics teachers as compared to the more traditional one described previously by other authors (cf., Romberg & Carpenter, 1986). This change of role has led to the need for those responsible for the preparation of prospective mathematics teachers to examine their own roles and how these new teachers are being prepared. The responsibility of teacher preparation institutions is to provide access to high-quality mathematical preparation and to create a supportive learning environment. These opportunities maximize the chances that prospective teachers will have the solid mathematical preparation needed to teach mathematics to students successfully (Texas Statewide Systemic Initiative, 1998).

Impact of Mathematics Content Knowledge on Preparing Mathematics Teachers

The Teaching Principle from the Principles and Standards for School Mathematics (PSSM) states that, "Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well" (NCTM, 2000, p. 16). Effective teachers must have a profound understanding of mathematics (Ma, 1999). A profound understanding, in Ma's description, has three related meanings: deep, vast, and thorough. A deep understanding is one that connects mathematics with ideas of greater conceptual power. Vast refers to connecting topics of similar conceptual power. Thoroughness is the capacity to weave all parts of the subject into a coherent whole. "Effective teachers are able to guide their students from their current understandings to further learning and prepare them for future travel" (National Research Council, 2001, p. 12).

Impact of Pedagogical Knowledge on Preparing Mathematics Teachers

Teaching and learning mathematics with understanding involves some fundamental forms of mental activity: 1) constructing relationships, 2) extending and applying knowledge, 3) reflecting about experiences, 4) articulating what one knows, and 5) making knowledge one's own (Carpenter & Lehrer, 1999). Some specific classroom activities and teaching strategies that support these mental activities include appropriate tasks, representational tools, and normative practices that engage students in structuring and applying their knowledge. There may be differential effects of this type of instruction for some students (Secada & Berman, 1999).

Interaction Between Mathematics and Pedagogical Knowledges

Classrooms that promote learning mathematics with understanding for all students involve a necessarily complex set of interactions and engagement of teacher and students with richly situated mathematical content (Cobb, 1988). Within that richly situated learning environment, teachers must be able to build on students' prior ideas and promote student thinking and reasoning about mathematics concepts in order to build understanding (Kulm, Capraro, Capraro, Burghardt, & Ford, 2001).

Teaching mathematics effectively is a complex task. The National Commission on Teaching and America's Future (1996) stated that in order to teach mathematics effectively, one must combine a profound understanding of mathematics with knowledge of students as learners, and to skillfully pick from and use a variety of pedagogical strategies. To complement this, The Texas Statewide Systemic Initiative (TSSI), in their document Guidelines for the Mathematical Preparation of Prospective Elementary Teachers (1998), states that the teaching of mathematics not only requires knowledge of content and pedagogy, but also requires an understanding of the "relationship and interdependence between the two" (p. 6). Shulman (1986) referred to this knowledge as "pedagogical content knowledge," one of the seven domains of teachers' professional knowledge. Shulman defined this as "a knowledge of subject matter for teaching which consists of an understanding of how to represent specific subject matter topics and issues appropriate to the diverse abilities and interest of learners" (p. 9). By developing this knowledge, preservice teachers become capable of making instructional decisions that lead to meaningful activities and real-world experiences for the students in their future classrooms (TSSI, 1998).

Lloyd and Frykholm (2000) found that future teachers need to develop both extensive subject matter background and pedagogical concepts and skills. In using middle-school reform-oriented teacher guides and student texts to work on activities, preservice teachers were able to recognize that "teaching demands extensive subject matter knowledge" (p. 578). These preservice teachers found that even 6th-grade activities posed significant mathematical difficulties for them.

Capraro, Capraro, and Lamb (2001) found that having preservice teachers critically evaluate videotapes of an experienced classroom teacher, while using a lesson-planning document as a basis for discussion, helped them to critically evaluate their mathematics teaching, which supports Carpenter and Lehrer's (1999) findings regarding reflection and Ma's ideas of profundity. The review of videos resulted in preservice teachers' improved ability to critically examine their mathematics knowledge and educational practices. As preservice teachers become aware of the intricacies of teaching, they begin to exhibit a greater awareness of guiding students from current understanding to deeper conceptualization.

Ball and Wilson (1990) found that teachers are tied, in general, to procedural knowledge and are not "equipped to represent mathematical ideas to students in ways that will connect their prior knowledge with the mathematics they are expected to learn, a critical dimension of pedagogical content knowledge" (Fuller, 1997, p. 10). This researcher found that experienced classroom teachers had a better conceptual understanding of numbers and operations than did preservice teachers; however, both groups relied mainly on procedural knowledge of fractions. Both practicing teachers and preservice teachers believed that a good teacher was one who demonstrated to students exactly how to solve problems, again supporting evidence that they relied on procedural knowledge (Fuller, 1997).

Realizing the importance of conceptual understanding, Ginsburg et al. (1992) suggested that mathematics should be taught as a thinking activity. Doing this requires that assessment methods provide ways of obtaining information concerning students' thinking, efforts at understanding, and procedural and conceptual difficulties. These assessments can provide those involved in preparing teachers with a richer level of understanding of what knowledge preservice teachers have as they move into their first years of teaching.

Preservice teachers must handle many different problems during their field experiences and ultimate future careers. "Because teaching and learning in increasingly diverse contexts are complex, prospective teachers cannot come to understand the dilemmas of teaching only through the presentation of techniques and methods" (Harrington, 1995, p. 203). To be effective, preservice teachers must comprehend the responsibilities and situations that lie ahead. Field-based assignments and clinical internships have provided students with limited opportunities due to their unique placements (Feinman-Nemser & Buchmann, 1986). Therefore, to determine whether pedagogical content knowledge can be gained through experiences in a methods class or in a field-based classroom demands further study.

Statement of the Problem

Increasing expectations about what children should know and be able to do, breakthroughs in research on how children learn, and the increasing diversity of the student population have put significant pressure on the knowledge and skills teachers must have to meet education goals set for the 21st century. Specifically, in undergraduate mathematics teacher preparation, instruction in pedagogy must develop deeper and broader understandings of mathematical concepts for preservice teachers. Teacher preparation programs are often measured by teacher certification examinations, but these examinations may not be aligned well with specific grade levels or require content-specific subtests for preservice teachers. Teacher preparation programs may choose to focus mainly on presenting mathematics content, with little consideration of preparing teachers to actively inquire about mathematics teaching and learning, or focus on presenting pedagogical issues with little regard for depth of mathematical content. This study investigates the belief that the nexus for developing pedagogical content knowledge lies in the interaction between constructing relationships, extending and applying knowledge, reflecting about experiences, articulating what one knows, and making knowledge one's own and situated within sound mathematical preparation (see Figure 1). It is this interaction between profound mathematical knowledge and pedagogy that forms pedagogical content knowledge. This investigation assesses the effectiveness of an elementary teacher preparation program to develop pedagogical content knowledge. Do mathematically proficient preservice teachers differ in their acquisition of pedagogical content knowledge? Do preservice teachers who differ on mathematical ability, score differentially on a standardized measure of elementary pedagogical content knowledge? How do the components of prospective teacher knowledge relate to early performances as mathematics teachers? How do mathematically proficient preservice teachers perform in teaching situations? The results of this study will contribute three things: 1) the development of two instruments intended to help mathematics teacher educators assess content and pedagogical content knowledge of prospective elementary teachers, 2) the triangulation of results through qualitative methods to offer rich descriptions ofpreservice teacher performance, and 3) an exploration of a process often reserved for states in the assessment of teachers at the end of a teacher preparation program. These findings can provide an opportunity for communities of stakeholders to respond and discuss these and related issues related to preparing mathematics teachers who can meet the needs of today's and future students.

Methodology

Participants

The study was conducted at a large southwestern U.S. state public university during the spring semester of 2001 through the spring semester of 2002 longitudinally, for three semesters, with different students each semester enrolled in the senior methods block (1). The participants (n = 193) were 20- to 22-year-old females in their senior year of undergraduate education. Ethnicity was predominantly white, with a few Hispanic and African American students. Participants completed between 28 and 56 full days of elementary classrooms field experience. During the senior methods block, the participants developed a week-long integrated thematic unit, wrote lesson plans, and taught a minimum of four lessons. The students were also involved in inquiry-type, hands-on, cooperative group activities involving the five process and five content strands from the Principles and Standards for School Mathematics (NCTM, 2000) during their mathematics methods instruction. In addition, each participant maintained a reflective journal of classroom activities and field experiences.

Study Design

This study employed a mixed method within stage design, where the quantitative analysis was used to inform the qualitative analysis of the variables indicative of success in mathematics teaching.

Quantitative. A regression analysis was used to identify variables useful in predicting preservice teacher success on the ExCET test, as well as the mathematics-teaching subtest. These variables include methods block section, success in previous mathematics courses, pedagogical content, mathematics methods course grade, and the open-ended posttest. Methods block section variable was used to identify differences between various enactments of the mathematics methods curriculum. Previous mathematics courses refer te university-required core mathematics courses for teachers. Success was determined by averaging their mathematics grades for these courses. Students whose average was 3.0 or greater were considered successful. Pedagogical content and the mathematics content tests were scale scores with a larger score demonstrating greater proficiency. Mathematics methods course grade was the final grade received at the completion of the course on a scale of 1 to 4.

Qualitative. A qualitative case study design was undertaken in an attempt to describe the performances and dispositions present in a purposeful sample of one high, one medium, and one low (n = 3) mathematics-achieving undergraduate education major, as evidenced by the open-ended instrument. The three cases were studied for insights into the understanding of how preservice teachers with mixed mathematics backgrounds develop pedagogical skills, plan for conceptual development, promote student thinking and reflection, and build on student ideas in the development of mathematics conceptualization. The mentor teachers of the case study participants were similar, based on their years of teaching, teaching practices, and educational philosophy. The mentors ranged in their practice from traditional to reform.

These three preservice teachers were observed during their individual teaching sessions with one 4th-grader from each of their field-based classrooms. Their reflection journals and their comments were noted during methods course discussions.

To explore the research question regarding how preservice teachers of varying mathematics content ability develop pedagogical content knowledge, individual cases were explored to describe the mathematics teaching performance of three senior preservice teachers in a classroom setting. These were not case studies in the sense of in-depth ethnographies; rather, each was a snapshot of mathematics teaching experiences and performance as part of a senior-level mathematics methods course that included an intensive field component in an elementary school setting.

Each snapshot represented opportunities for senior preservice teachers to teach mathematics concepts to 4th-grade students in a public school setting. These experiences occurred as one assignment for the methods course and were scheduled as one-hour sessions outside of their regularly scheduled field experiences for a period of five weeks. A theme was provided for each week (i.e., computational fluency, problem solving, communication, and estimation). Based on pre-assessment data from 4th-grade students, preservice teachers designed mathematics lessons based on objectives related to the theme standard for the week. After each session, preservice teachers reflected on their teaching performance with their peers and in an individual, written reflection submitted for evaluation.

Each of the three preservice teacher cases took their mathematics courses through the university and received grades of either A or B. Scores on the mathematics portion of the Scholastic Aptitude Test (SAT) were 540 or better. Table 1 further summarizes the mathematics background and performance on the two instruments administered as part of this study of the three preservice teachers presented in the snapshots.

Instrumentation 

The state teacher certification instrument, ExCET, was used as a benchmark for assessing preservice teacher competence in mathematics content and pedagogical content knowledge. In an attempt to determine the effectiveness of the mathematics teacher preparation program, the participants were administered two assessment measures (pre and post) during the first week and again during the last week of mathematics methods class, which coincided with the culmination of the field placement. The first instrument was a 15-item, multiple-choice mathematics pedagogical content knowledge instrument. Appendix A contains two sample items. This instrument was designed to mirror the pedagogical content questions contained on the ExCET test, which is the state test required for teacher certification. Participants also completed a four-item, open-ended, rubric-scored content and application instrument. Appendix B contains two sample items. This instrument was adapted from the Program for International Student Assessment (PISA, 2000) mathematics items, which covered the domains tested on the mathematics portion of the ExCET test (see Appendix C for a listing of these domains.).

In an attempt to achieve uniformity in administration, a test administration document was written and provided to all administrators of the instruments. Content and construct validity was achieved by having four classroom teachers and two mathematics teacher educators (not involved in the teacher preparation program) review the questions. After review, the original multiple-choice instrument was reduced from 20 items to the current 15 items. Based on responses from the reviewers, it was believed that the multiple-choice items sufficiently assessed the understanding of pedagogical content knowledge specific to mathematics; the Pearson r between the two versions was .95. The reviewers of the second instrument believed that it adequately assessed the narrow band of conceptual mathematics understanding that was the target of the investigation. An analysis was done to determine the alignment between the items on each test and the ExCET content mathematics domains. Following this alignment, a content analysis procedure was used to determine the content knowledge required for success on each item. An item analysis was conducted to ensure that proper alignment was achieved between the NCTM standards and State Board for Educator Certification (SBEC) standards, as tested on the ExCET content mathematics domains.

The test was administered across all sections of elementary school methods blocks (n=193) for three consecutive semesters beginning in spring 2001. Each mathematics methods instructor was responsible for administration of the instrument. Multiple choice answers were scored 1 correct and 0 incorrect. The rubric-scoring guide, included in Appendix D for each item, ranged from 0-4. Cronbach's alpha reliability estimates for the multiple choice mathematics pedagogical content (.74) and the open-ended mathematics content (.81) were obtained.

Mathematics Methods Course Structure

The mathematics methods course sections (6) were all structured by the PSSM within the conceptual framework for the development of pedagogical knowledge articulated in Figure 1. Two sections used a four-day-a-week model of field placement for preservice teachers and four sections used a two-day-a-week model of field placement. The two-day model included two full days of field experiences and the four-day model consisted of four full days of field experience. Both models had methods courses following field experiences. These preservice teachers were placed randomly in extant elementary (K-4) classes in one of two school districts near the university and the classroom teacher functioned as a school-site mentor who had three or more years of teaching experience. Mentor teachers varied from those teaching traditionally to those teaching standards-based mathematics. One consideration for this study was to investigate whether one model or another provided greater impact on the development of pedagogical content knowledge.

Results

The results of the multiple linear regression analyses with ExCET scores as the dependent variable are contained in Table 2. The independent variables included: 1) the section in which the student was enrolled (section), 2) previous mathematics courses (math courses), 3) score on the posttest pedagogical content knowledge test (Ped. Cont.), 4) final (grade) in the mathematics methods course, and 5) the Open-Ended Content Knowledge posttest (O-E Post Test). In the regression model, of the 10.7 percent multiple R squared effect, the Beta weight of success in previous mathematics courses appeared to be the most important predictor at p = .024. In examining the squared structure coefficients, both the pedagogical content knowledge test and the open-ended content test were practically important predictors. The value of the predictors was not evidenced in the regression B weights because the variance accounted for was allocated arbitrarily by formula and another variable may be equally important. For this reason, it was important to compute and review squared structure coefficients to determine the practical importance of each variable in predicting the dependent variable. As Thompson and Borrello (1985) noted, "Logically, coefficients which are important in the canonical case may also be important in the case of multiple regression" (p. 208). These results seem to indicate that success in previous mathematics courses was strongly correlated to success on the ExCET content portion of the ExCET exam; therefore, without profound mathematical understandings it is difficult to develop adequate pedagogical content knowledge. However, having profound mathematical understandings does not ensure preservice teachers develop pedagogical content knowledge. When considering the variable section, there was no statistically significant effect. A review of the B weight for section revealed that the .256 weight was near last and its square structure coefficient accounted for only 3.9 percent of the variance accounted for in the model. These findings indicated that the impact of a four-day versus a two-day field experience variable was neither statistically nor practically important.

Several Pearson correlations indicated some interesting findings. First, as seen in Table 3, the correlation between performance in prerequisite mathematics courses and performance on subtests of the ExCET exam was statistically significant. In addition, the correlation between the performance in mathematics courses was strongly correlated with performance on the professional portion and more moderately correlated with the ExCET content. This result suggested that students who do better in mathematics courses also do better on the yardstick by which the mathematics teacher preparation program was measured. The correlation was also strong between previous mathematics performance and the pretest administered in the mathematics methods courses. This seemed to match the results of the earlier finding that students who demonstrated lower performance levels in mathematics courses entered the mathematics methods course exhibiting many of the same deficiencies. However, when considering the correlation between previous mathematics courses and the mathematics content posttest administered in the mathematics methods course, the correlation was almost zero. Given the comparison of the pre- and post-content means of 16.2 (SD = 7.1) and 22.6 (SD = 3.6), respectively, this finding seemed to indicate that the previous student performance in mathematics was no longer important to performance on the posttest and that the focus on pedagogical knowledge of the construct (Figure 1) improved mathematics content knowledge. The grade earned in mathematics methods courses was negatively correlated with grades earned in mathematics courses and with the mathematics portion of the ExCET content test.

When considering a participant's section on student performance, the only correlation was between previous mathematics courses and the professional development portion of the ExCET test. The correlation was weak but indicated that students who had preformed better in previous mathematics classes had opted for sections offering four-day field placements. Because of the strong correlation between previous mathematics courses and performance on the ExCET in general, this same effect was seen in the correlation with section as well. Although the sections were originally dissimilar on student mathematics ability, that difference did not appreciably affect measured outcomes. Given that students with higher mathematics ability also participated in the four-day field experience, the results seem to indicate that the quantity of field experiences did not influence measured outcomes.

Case Study Results

Sally. Sally claimed to enjoy mathematics and showed excitement about having the opportunity to design activities around a mathematics concept to teach to her 4th-grade student. Sally was confident in her ability to teach mathematics, designed meaningful learning opportunities each week, and her reflections identified strengths and weaknesses of her teaching. She was able to identify and discuss the strategies used by her student as she approached a learning task or solved a problem. Sally also explained that she realized students often select different strategies than those she would use, based on their current level of mathematics understanding. Sally's mathematical ability allowed her to look beyond the traditional algorithm and see the possibilities of divergent but equivalent alternatives to find the same solution. She also was comfortable allowing her student to use a representational model that helped her to see the solution, rather than prompting for another representation or questioning her model. From a discussion during her mathematics methods course, Sally explained that she learned that students use different representations to solve similar problems.

In an activity designed to encourage critical thinking and the use of problem solving skills, Sally had her 4th-grade student use pattern blocks to cover different animal shapes. Each pattern block was given a cost value. The student then determined the least expensive and most expensive way to construct the animals. Sally explained in her reflection:

I observed Christie talking herself through each step. As she selected blocks for her animal, she would comment about it fitting or not fitting or wanting to try another block. After constructing her animal, Christie decided to make a chart. She labeled each of her columns without being prompted. Once she counted the number of different blocks used, she recorded them to her chart. Then she decided that she could either add all the values, or she could multiply the cost value of the block by the number of times it was used. She chose to use multiplication. I asked her why she chose that method, and she told me she thought it would be easier and quicker. While watching her multiply, I noticed that instead of carrying her numbers and placing them above the appropriate number, she wrote the number that needed to be carried to the right of the problem. This was a new strategy for me, but it definitely worked for her. ... After Christie found the total of her animal, she began looking at the turtle and began to notice that she could trade in two triangles for a parallelogram or two trapezoids for a hexagon depending on whether she wanted to make her animal cost more or less. ... Another preservice teacher did this same activity with her 4th-grade student and he used a bar graph to organize how many pattern blocks he used. He ended up adding the cost values together to get the total cost of the animal.

In a computational fluency activity, Sally worked with a 4th-grade student on the conversion of fractions to decimals. Sally asks the student to change the fraction 2/5 to its decimal equivalent. Sally displayed evidence of vastness and depth while reflecting on her own ideas and constructing relationships. She discussed the teaching experience in her reflection in the following way:

I asked Christie how she would change the fraction 2/5 into its decimal equivalent. Christie said we should try to get a 100. I wondered why Christie made this comment because I was thinking she would just tell me to divide. I asked Christie why she thought we needed to get a 100, and she said so we can write the fraction as a decimal. I'm thinking she is confused, so I prompt her and say why don't we just divide. I divided 2 by 5 and showed her how to write the decimal and get .4. Christie then said, "What about this?," and proceeded to multiply the 2 by 20 and the 5 by 20 to get 40 over 100; then she wrote it equaled .40. I quickly responded to Christie by saying: "Oh yeah, you can do it that way" and realized at that point why she wanted to get a 100.

Sally had in mind that her 4th-grade student would use the method of dividing to get the decimal equivalent of the fraction 2/5 since she herself would select that strategy, and to her it was the simplest way to solve the problem. However, her student made a connection to making an equivalent fraction for 2/5 with a denominator of 100. Sally learned that a different perspective often results in an alternative solution, which may demonstrate a new relationship and burgeoning depth and thoroughness of mathematical profundity.

Jane. Jane was a conscientious preservice teacher with good planning skills, a strong academic background, and experience with the use of technology. However, she indicated that her confidence in teaching mathematics effectively to elementary students was weak at the beginning of the semester and, indeed, she did have only the minimum amount of university-required mathematics courses. She explained that she decided on a social studies emphasis because she felt she could not pass the additional required mathematics classes. Jane was capable of designing meaningful mathematics activities for her 4th-grade student, but was always very critical of her teaching ability. During the five weeks of the teaching sessions and throughout the methods semester, Jane gained confidence in her teaching ability. The journal entry below showed how she was growing more confident in her ability to teach mathematics due to positive student responses to activities she planned. Jane's confidence is bolstered by her growing ability to articulate what she knows and does not know and by her making the knowledge her own through the eyes of her student as her mathematical profundity grew during this polygon lesson.

After two sessions with my 4th-grader, planning has gotten fun and it has become a challenge to find activities that will be at her level and keep her interest. ... I found out last week that Alexus really liked to work with geoboards. She was good at it. She was better than I was at it. The geoboard activity I planned focused on learning about polygons. We each made shapes on our geoboards and then Alexus copied them onto geoboard paper. I would ask her if the shape was a polygon or not. I did not give her the definition of what a polygon was at this point, but I would tell her if it was or wasn't so we could group them. We continued to make shapes and copy them to the paper. Once we had enough shapes that were polygons and some that were not polygons, I asked Alexus to look for patterns to try to make a definition of polygon in her own words based on the examples we created. She looked for what the polygons had in common and what the non-polygons had in common. She then wrote and explained to me that a polygon is closed and the lines do not cross. I couldn't believe she was able to explain this by looking at the examples. I had to look up the definition in the book when I was planning this teaching session. I don't think I knew what a polygon was when I was in 4th grade.

Jane is not unlike other senior preservice teachers whose perception of their mathematics teaching ability has been framed by their own feelings of inadequacies in mathematics courses taken in preparation for certification. Jane's perception of inadequacy arose from not having achieved A's in all mathematics courses; however, she never received lower than a B. This preservice teacher made a special effort to improve her mathematics skills in preparation for this lesson and, as a result, felt more efficacious. This suggests that some preservice teachers need the motivation of lesson preparation as the impetus to improve specific mathematics skills.

Molly. Molly's emphasis was early childhood and she had experience in working with and teaching elementary students. She had been a HOSTS (Helping One Student To Succeed) volunteer and had been a substitute in a local school district to gain experience teaching children prior to the methods block semester. Molly was at the stage of "trying to put it all together," as she commented. Molly's reflection demonstrates her lack of mathematical sophistication and poor internalization of pedagogical knowledge. To begin this session, Molly provided learning experiences for her 4th-grader that focused on equivalent fractions. Fraction squares and circles were used during the session to model the fractions. Molly wrote:

After we had all the pieces out, I went back to try and assess his understanding of equivalent fractions. I asked Kerry how many fourths make one-half. He struggled with this and looked at me with a blank stare and then guessed four. I asked him why he thought four, and he couldn't give any explanation. So I had Kerry show me one-half of a circle, and then I had him cover it with fourths. Kerry then realized that it only took two fourths to make one half. So I said, are the fractions 1/2 and 2/4 equivalent? He said yes and explained because they take up the same amount of area. I continued this with him for fourths, eighths, and sixteenths. Kerry was able to do this by putting pieces on top of the others. ... After I felt he had a good grasp of equivalent fractions, I moved on to a game. The game required him to turn over two fraction cards and decide whether they were equivalent. When he turned over the first two cards, I realized that he did not have a good understanding of how to simplify fractions. So, I didn't get to play the game as planned. Instead, I decided to use the fraction cards to decide if two fractions were equal. The first two fractions he turned over were 1/8 and 3/24. He didn't know where to begin, so I asked him to show me 24 divided by 3.

During the session, Molly began with a concrete model to demonstrate the concept of equivalent fractions, demonstrating pedagogical content understanding. However, her intent was to have Kerry simplify fractions, which was a different concept from simply generating equivalency. Molly demonstrated that she lacked mathematical profundity and was also unable to identify an appropriate pedagogical strategy to help Kerry develop the necessary skills. After feeling like she had spent enough time with concrete models and making equivalent fractions, Molly tried to engage her student in a game about equivalent fractions. This game required her student to be able to look at two fractions and decide if they were equivalent. However, her intention was for Kerry to simplify the fractions into simplest terms in order to be successful. Molly failed to persist in the use of concrete models, which Kerry obviously still needed to use to complete the activity. This resulted in Kerry having difficulty in determining if the fractions were equivalent. Molly resorted to the use of division to determine equivalency and this complicated the lesson and moved her pedagogically away from her initial activity of concretely determining equivalency. Allowing additional time early in the session for more practice in representing equivalent fractions with concrete materials and connecting them to the symbolic forms may have allowed for greater understanding and eliminated the division strategy. Molly did not reflect on her own knowledge and was not able to construct relationships between her own mathematical knowledge and the concept she was attempting to teach.

Discussion

It is evident from this study that preservice teachers learn and develop as teachers throughout their education (Onslow, Beynon, & Geddis, 1992). There is no magic that takes place during methods courses or field experience that either makes or breaks a future teacher. However, indications from this study highlight some important factors that lead to success as measured by state accountability instruments. First, as one might expect, mathematics content measures were the best predictors of the mathematics portion of the state test. Since this test is required for teacher certification, it is important to continue to include strong content preparation. Our findings are contrary to the findings of Hadfield, Littleton, Steiner, and Woods (1998), who indicated that they were unable to identify any statistically significant differences in teacher effectiveness as measured by a course quiz and a pedagogical content knowledge (PCK) test. Our results indicated that a lack of mathematical content knowledge leads to ineffective mathematics instruction (cf., Leinhardt, Putnam, Stein, & Baxter, 1991; Schwartz & Riedsel, 1994).

The use of previous grades earned in a specific field of study has been found practically useless in substantive data analyses. Grades earned in the methods course were used in the regression equation to investigate whether similar findings were evident in our data. The findings from the regression analysis supported findings related to grade inflation that accounts for such loss of predictability.

The moderate correlation between mathematics course success and the open-ended mathematics items suggested that these courses are not effective in developing problem solving or mathematical reasoning ability. Similar to previous research (cf., Ball, 1996, 2000), the case studies provided some evidence that mathematics course-taking success does not guarantee that preservice teachers apply their knowledge correctly in the classroom. Additionally, preservice teachers need opportunities to understand how mathematics content is delivered to elementary students in a school setting. These experiences can build teaching confidence, allow preservice teachers to work on questioning strategies, and increase understanding of strategies used by students at various grade levels.

The grades in the mathematics methods course were poorly correlated with any of the measures. Most likely, grades are too crude a measure for this purpose because the focus is less centered on mastery than on developing teaching skills. The case studies provide some evidence that students who comprehend the ideas contained in a mathematics methods course did well in their work with students. Because the methods courses were so closely integrated with the field-based work, it is difficult to assess these experiences separately.

There was evidence that the methods semester resulted in gains in the participants' scores on the two measures developed in the study. Although it was unclear which components of the experience contributed the most to these gains, the study showed clearly that participating in a field-based assignment for a longer period of time had no measurable short-term effects by the methods used in this study. In fact, the results seem to confirm the findings of Quinn (1997) that inadequate mathematical content often precedes enacted mathematical issues, which is evidence of the old adage noted in Adding It Up (Kilpatrick, Swafford, & Findell, 2001) that "You cannot teach what you don't know" (p. 373). Long-term effects are yet to be determined. The cases showed that one preservice teacher being allowed opportunities to teach mathematics to students as part of her field experience gained a better understanding of strategies used by students, while another gained confidence in her mathematics teaching ability. Results of the study also showed the importance that rigorous coursework in mathematics plays in building a knowledge base that prepares preservice teachers for certification testing.

Test scores, course grades, and performance on the administered instruments indicated the preservice teachers had a mathematics background sufficient to qualify for certification in the teaching of elementary school mathematics but clearly displayed diversity in mathematical profundity. Evidence from their written reflections of lessons to 4th-grade students indicated a wide degree of variance in confidence levels. Some preservice teachers struggled with appropriate questions to access student understanding of the concept or to facilitate a deeper understanding of the concept.

Quality mentors are critical. For instance, if a preservice teacher works in a prolonged field experience with a mentor who exhibits the qualities of a "master" mathematics teacher, the ideas, beliefs, and interpersonal abilities of the mentor would be aligned with that of the university instruction, leading to a reasonable opportunity for learning by the prospective teacher (Merseth, 1996). In contrast, if the preservice teacher is placed in the classroom where the teacher lacks "math power" and uses methods that do not promote student understanding, the methods instructor will find it difficult to convey the importance of teaching conceptually.

The measures developed in this study were moderately correlated with a state standardized test, and with other components of a mathematics teacher preparation program. Further measures are needed for other parts of the program, including the outcomes of methods courses and the field-based teaching experiences. Because a large investment is required for an intensive field-based program, it is important to continue development of these measures to assess the effectiveness of such programs. Ma (1999) articulates both the problems facing preservice teachers in becoming effective teachers as well as those facing researchers who seek solutions to complex questions,

One thing is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed ... it is very complicated, subtle, and takes a lot of time. It is easy to be an elementary school teacher, but it is difficult to be a good elementary school teacher. (p. 136)

It is evident that limited differences exist between the two- and four-day field experiences in relation to how well a preservice teacher develops mathematical ideas conceptually for students. The idealistic differences are that preservice teachers who engage in prolonged field experiences become better mathematics teachers and that more experience in classrooms encourages deeper understandings of the teaching and learning process. Results of this study indicate that the benefit of field experience is dependent on several factors, including the quality of the mentor, the rigor of the pedagogical expectations, and the willingness of the preservice teacher to fully engage the content and pedagogy. From this study, more time in the classroom alone is not useful. Programs need to make use of time appropriately and ensure that preservice teachers see best practices and have meaningful experiences, which cannot be measured by hours in a field experience.

Appendix A--Sample Items From the Pedagogical Content Knowledge Test

Use the student work sample below to answer the question that follows. Name: Juanita Problem: The sun is 785,354 miles away from the earth. If it takes a spaceship 4 days to go from the earth to the sun, how fast did the space ship travel? Use your calculator to solve the problem, and explain how you got your answer.

Answer: 81.8077 miles per hour

How did you get your answer? First I figured out how many hours there are in 4 days, which is 96 hours. Then I divided the distance to the sun by the time it took the spaceship to get there.

Juanita, a 6th-grade student, used a calculator to solve the word problem above. When going over Juanita's work with her, the teacher should place the greatest importance on which of the following?

A. reminding Juanita that she should always do each calculation several times whenever she is using a calculator

B. asking Juanita to estimate the answer to the problem in order to assess the reasonableness of the answer on the calculator

C. reviewing with Juanita the rules for the conversion of units within the same system of measurement

D. asking Juanita to try to think of another method to use to solve the problem

Students in a 4th-grade class are measuring the circumference and diameter of common objects to the nearest centimeter. Some of their data are displayed in the table below.

Object      Diameter    Circumference

Soup can    2 cm         6 cm
Frisbee     4 cm        12 cm
Dish        6 cm        18 cm

The teacher could best develop students' understanding of the concept of a function by posing which of the following questions about the data?

A. Do objects with larger diameters always have larger circumferences than objects with smaller diameters?

B. Do you think the data in your table would show a different trend if you were using more precise measurement tools?

C. How can you use the data in your table to calculate the area of the circles you have measured?

D. If you knew the diameter of a circle, how could you determine the circumference without measuring it?

Appendix B--Sample Items From the Open-Ended Content Knowledge Test

Question 1: Pizzas

A pizzeria serves two round pizzas of the same thickness in different sizes. The smaller one has a diameter of 30 cm and costs 30 zeds. The larger one has a diameter of 40 cm and costs 40 zeds. Which pizza is better value for money? Show your reasoning.

Question 2: Coins

You are asked to design a new set of coins. All coins will be circular and colored silver, but of different diameters.

Researchers have found out that an ideal coin system meets the following requirements: diameters of coins should not be smaller than 15 mm and not larger than 45 mm.

Given a coin, the diameter of the next coin must be at least 30% larger. The minting machinery can only produce coins with diameters of a whole number of mm (e.g., 17 mm is allowed, 17.3 mm is not).

You are asked to design a set of coins that satisfy the above requirements. You should start with a 15 mm coin and your set should contain as many coins as possible.

Appendix C--Mathematical Domains Front the ExCET Mathematics Subtests

ExCET Content    Domain Descriptions

020              Higher-order Thinking and Questioning
021              Problem-Solving Strategies
022              Mathematical Communication
023              Mathematics in Various Contexts
024              Number and Numeration Concepts
025              Patterns and Relationships
026              Mathematical Operations
027              Geometry and Spatial Sense
028              Measurement
029              Statistics and Probability
030              Recent Developments and Issues in Mathematics

Appendix D--Rubrics for Open-Ended Content Knowledge Test

Question 1 (Pizza)

Student A: Incomplete or no process without any demonstration of mathematical solution (intuitive solution).

Student B: Incomplete process. Demonstrates some mathematical understanding of the concept. No or partial incorrect solution. No or partial process or explanation.

Student C: Complete process. Proper application of mathematical relationships. Incorrect arithmetic or incorrect interpretation of numerical results.

Student D: Complete process. Proper application of mathematics relationships. Correct solution and interpretation. Evidence of understanding that the comparison is based on cost per unit.

Question 2 (Coins)

Student A: Incomplete or no process shown whether answer is correct or not.

Student B: Incorrect process shown, such as 30% uniformly added to first coin and that amount added to all succeeding coins.

Student C: Correct process shown, minor miscalculations; started correctly but did not complete all five coins.

Student D: Logical correct process carried out, all steps shown, correct coins created.

Student N: No response.

Footnote 

(1) The methods block of course work (17 semester hours) occurs during the first semester of the senior year. Preservice teachers take three semester hours each in mathematics, science, social studies, reading, and special education, and two semester hours in behavior management. The course work is situated in local schools where the preservice teachers are assigned a school-site mentor teacher for between two and fours days per week.

References

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Authors' Note. Special thanks to Dr. John Helfeldt, Department Head, and Dr. James Kracht, Associate Dean for Undergraduate Affairs, for their support of this research.

Robert M. Capraro

Mary Margaret Capraro

Dawn Parker

Gerald Kulm

Tammy Raulerson

Texas A&M University

Table 1

Summary of Mathematics Background and Scores on Instruments for
Preservice Teacher Cases

        Certification    Math      GPR     SAT        ExCET
                         Courses           (Math)     (Math)

Sally   4-yr INST        18 hrs    3.36    550        --
        Mathematics      A
Jane    4-yr INST        9 hrs     3.43    540        85
        Social Studies   B
Molly   Early            15 hrs    3.84    550        --
        Childhood        A-B

        Certification    Open-     Open-   Multiple   Multiple
                         Ended     Ended   Choice     Choice
                         Pre       Post    Pre        Post

Sally   4-yr INST         22        27      13          14
        Mathematics
Jane    4-yr INST         19        20      11          12
        Social Studies
Molly   Early              7        17      13          14
        Childhood

Note. Mathematics courses consistent among all preservice teachers:
Maximum score on open-ended instrument was 30. Maximum score on the
multiple-choice instrument was 15.

Table 2

Summary of Regression Analysis for Variables Predicting a Passing Score
on the Elementary Comprehensive (ExCET Content) Portion of ExCET Exam
(n = 193)

  Variable         B      Beta   [R.sub.s.sup.2]     t       Sig.

Constant        -57.101                             -.374    .709
Section          .256     .003        .039          .840     .402
Math Courses     3.704    .191        .489          2.281    .024
Pedagogical
  Content        .611     .069        .387          1.644    .102
Methods Grade    .037     .139        .024          .032     .975
Open-Ended
  Posttest       -1.022   .156        .415          1.809    .043

Note. R Square=.107; p = .008

Table 3

Correlation Matrix for Participant Variables

                      Math       PD        ExCET     Ped. Cont.
                      Courses    ExCET

Math Courses         1.000
PD ExCET             ** .466     1.000
Content ExCET        ** .442     ** .733   1.000
Ped. Cont.           .097        .139      ** .225   1.000
0-E Content          .140        .118      ** .227   ** .210
Section              * .119      * .144    .109      .035
Methods Grade        ** -.154    * .225    ** .214   .041

                     O-E Content   Section    Grade

Math Courses
PD ExCET
Content ExCET
Ped. Cont.
0-E Content            1.000
Section                .025         1.000
Methods Grade          .115          .085      1.000

** Correlation is significant at the 0.01 level (2-tailed).

* Correlation is significant at the 0.05 level (2-tailed).

a Cannot be computed because at least one of the variables is constant.