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The Mathematics Content Knowledge Role inDeveloping Preservice Teachers' PedagogicalContent KnowledgeRobert M. Capraro a , Mary Margaret Capraro a , Dawn Parker a , Gerald Kulm a &Tammy Raulerson aa Texas A&M UniversityPublished online: 03 Nov 2009.

To cite this article: Robert M. Capraro , Mary Margaret Capraro , Dawn Parker , Gerald Kulm & Tammy Raulerson(2005) The Mathematics Content Knowledge Role in Developing Preservice Teachers' Pedagogical Content Knowledge,Journal of Research in Childhood Education, 20:2, 102-118, DOI: 10.1080/02568540509594555

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Journal of Research in Childhood Education2005, Vol. 20, No. 2

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Various reform initiatives have produced documents calling for a new vision for the teaching and learning of mathematics (Na-tional Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000; National Re-search Council, 2001) . These documents describe a dynamic, invigorated role for the preparation of mathematics teachers as com-pared 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

Abstract. This paper outlines the nexus between mathematics content knowl-edge 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 teach-ing. 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.

Robert M. CapraroMary Margaret CapraroDawn ParkerGerald KulmTammy RaulersonTexas A&M University

The Mathematics Content Knowledge Role in Developing Preservice Teachers’ Pedagogical Content Knowledge

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 Knowl-edge on Preparing Mathematics TeachersThe Teaching Principle from the Principles and Standards for School Mathematics (PSSM) states that, “Effective mathemat-ics teaching requires understanding what

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

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students know and need to learn and then challenging and supporting them to learn it well” (NCTM, 2000, p . 16) . Effective teach-ers must have a profound understanding of mathematics (Ma, 1999) . A profound un-derstanding, in Ma’s description, has three related meanings: deep, vast, and thorough . A deep understanding is one that connects mathematics with ideas of greater conceptu-al 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” (Na-tional Research Council, 2001, p . 12) .

Impact of Pedagogical Knowledgeon Preparing Mathematics TeachersTeaching 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, repre-sentational tools, and normative practices that engage students in structuring and applying their knowledge . There may be dif-ferential effects of this type of instruction for some students (Secada & Berman, 1999) .

Interaction Between Mathematicsand Pedagogical KnowledgesClassrooms that promote learning math-ematics with understanding for all stu-dents 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, Cap-raro, 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 mathemat-ics effectively, one must combine a pro-found 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 Initia-tive (TSSI), in their document Guidelines for the Mathematical Preparation of Pro-spective Elementary Teachers (1998), states that the teaching of mathematics not only requires knowledge of content and peda-gogy, 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 do-mains 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 fu-ture 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 teach-ers were able to recognize that “teaching de-mands extensive subject matter knowledge” (p . 578) . These preservice teachers found that even 6th-grade activities posed signifi-cant mathematical difficulties for them. Capraro, Capraro, and Lamb (2001) found that having preservice teachers critically evaluate videotapes of an experienced class-room teacher, while using a lesson-planning document as a basis for discussion, helped them to critically evaluate their mathemat-ics teaching, which supports Carpenter and

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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 guid-ing 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 math-ematical ideas to students in ways that will connect their prior knowledge with the math-ematics they are expected to learn, a critical dimension of pedagogical content knowl-edge” (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) sug-gested 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 pro-cedural and conceptual difficulties. These assessments can provide those involved in preparing teachers with a richer level of un-derstanding 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 experi-ences 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 teach-ers 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 experi-ences in a methods class or in a field-based classroom demands further study .

Statement of the ProblemIncreasing expectations about what chil-dren should know and be able to do, break-throughs 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 cen-tury. Specifically, in undergraduate math-ematics 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 exami-nations may not be aligned well with spe-cific 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 teach-ers to actively inquire about mathematics teaching and learning, or focus on present-ing 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 construct-ing 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

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develop pedagogical content knowledge . Do mathematically proficient preservice teach-ers 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 teach-er knowledge relate to early performances as mathematics teachers? How do mathemati-cally 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 of preservice teacher performance, and 3) an exploration of a process often reserved for states in the as-sessment 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 .

MethodologyParticipantsThe study was conducted at a large south-western 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 meth-ods block1 . The participants (n = 193) were 20- to 22-year-old females in their senior year of undergraduate education . Ethnic-ity 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 les-sons . 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 DesignThis study employed a mixed method within stage design, where the quantitative analysis was used to inform the qualitative analysis of the variables indicative of suc-cess in mathematics teaching . Quantitative. A regression analysis was used to identify variables useful in pre-dicting 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 differ-ences between various enactments of the mathematics methods curriculum . Previous mathematics courses refer to university-required core mathematics courses for teach-ers . 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 dem-onstrating 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) mathemat-ics-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

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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 tra-ditional 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 reflec-tion journals and their comments were noted during methods course discussions . To explore the research question regard-ing how preservice teachers of varying mathematics content ability develop peda-gogical content knowledge, individual cases were explored to describe the mathemat-ics 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 math-ematics 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 ses-sions 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 stu-dents, preservice teachers designed math-ematics 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 reflec-tion 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 as-sessing preservice teacher competence in mathematics content and pedagogical con-tent knowledge . In an attempt to determine

Table 1 Summary of Mathematics Background and Scores on Instruments for Preservice Teacher Cases

Certification Math Courses

GPR SAT(Math)

ExCET(Math)

Open-Ended Pre

Open-EndedPost

Multiple ChoicePre

MultipleChoicePost

Sally 4-yr INSTMathematics

18 hrsA

3 .36 550 -- 22 27 13 14

Jane 4-yr INSTSocial Studies

9 hrsB

3 .43 540 85 19 20 11 12

Molly EarlyChildhood

15 hrsA-B

3 .84 550 -- 7 17 13 14

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 .

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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 knowl-edge 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 con-tent and application instrument . Appendix B contains two sample items . This instru-ment 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 docu-ment was written and provided to all admin-istrators of the instruments . Content and construct validity was achieved by having four classroom teachers and two mathemat-ics teacher educators (not involved in the teacher preparation program) review the questions . After review, the original mul-tiple-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 instru-ment believed that it adequately assessed the narrow band of conceptual mathemat-ics 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 math-ematics domains . Following this alignment, a content analysis procedure was used to de-termine 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 sec-tions of elementary school methods blocks (n=193) for three consecutive semesters beginning in spring 2001 . Each mathemat-ics methods instructor was responsible for administration of the instrument . Multiple choice answers were scored 1 correct and 0 incorrect . The rubric-scoring guide, includ-ed in Appendix D for each item, ranged from 0-4 . Cronbach’s alpha reliability estimates for the multiple choice mathematics peda-gogical content ( .74) and the open-ended mathematics content ( .81) were obtained .

Mathematics Methods Course StructureThe mathematics methods course sections (6) were all structured by the PSSM within the conceptual framework for the develop-ment 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 experi-ence . Both models had methods courses fol-lowing 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 mathemat-ics . One consideration for this study was to investigate whether one model or another provided greater impact on the development of pedagogical content knowledge .

ResultsThe results of the multiple linear regres-sion analyses with ExCET scores as the

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dependent variable are contained in Table 2 . The independent variables included: 1) the section in which the student was en-rolled (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 ef-fect, the Beta weight of success in previous mathematics courses appeared to be the most important predictor at p = .024 . In ex-amining 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 prac-tical importance of each variable in predict-ing the dependent variable . As Thompson and Borrello (1985) noted, “Logically, coeffi-cients 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 corre-lated to success on the ExCET content por-tion of the ExCET exam; therefore, without profound mathematical understandings it is difficult to develop adequate pedagogi-cal content knowledge . However, having profound mathematical understandings does not ensure preservice teachers develop pedagogical content knowledge . When con-sidering 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

Figure 1

Constructing Relationships

Extending &Applying

Knowledge

ReflectingAbout

Experiences

Articulating What One

Knows

MakingKnowledgeOne’s Own

PedagogicalContent

Knowledge

MathematicalProfundity

Deep Vast Thorough

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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 statisti-cally nor practically important . Several Pearson correlations indicated some interesting findings. First, as seen in Table 3, the correlation between perfor-mance in prerequisite mathematics courses and performance on subtests of the ExCET exam was statistically significant. In ad-dition, the correlation between the perfor-mance in mathematics courses was strongly correlated with performance on the profes-sional portion and more moderately corre-lated 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 pre-

vious 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 math-ematics methods course exhibiting many of the same deficiencies. However, when con-sidering the correlation between previous mathematics courses and the mathematics content posttest administered in the math-ematics 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 pedagogi-cal knowledge of the construct (Figure 1) improved mathematics content knowledge .

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 Rs2 t Sig .

Constant -57 .101 - .374 .709Section .256 .003 .039 .840 .402Math Courses 3 .704 .191 .489 2 .281 .024Pedagogical Content .611 .069 .387 1 .644 .102Methods Grade .037 .139 .024 .032 .975Open-Ended Posttest -1 .022 .156 .415 1 .809 .043

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

Table 3 Correlation Matrix for Participant Variables

MathCourses

PDExCET

ExCET Ped . Cont . O-E Content Section Grade

Math Courses 1 .000PD ExCET ** .466 1 .000Content ExCET ** .442 ** .733 1 .000Ped . Cont . .097 .139 ** .225 1 .000O-E Content .140 .118 ** .227 ** .210 1 .000Section * .119 * .144 .109 .035 .025 1 .000Methods Grade **- .154 * .225 ** .214 .041 .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 .

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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 differ-ence 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 mathemat-ics 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, de-signed meaningful learning opportunities each week, and her reflections identified strengths and weaknesses of her teach-ing . She was able to identify and discuss the strategies used by her student as she approached a learning task or solved a prob-lem . Sally also explained that she realized students often select different strategies than those she would use, based on their current level of mathematics understand-ing . 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 discus-sion 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 criti-cal 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 differ-ent 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 add-ing 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 con-

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version 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 frac-tion 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 stu-dent 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 indi-cated that her confidence in teaching math-ematics 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 ac-tivities for her 4th-grade student, but was always very critical of her teaching ability . During the five weeks of the teaching ses-sions 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 grow-ing 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 math-

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ematics teaching ability has been framed by their own feelings of inadequacies in mathematics courses taken in prepara-tion 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 prepa-ration 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 im-prove specific mathematics skills. Molly. Molly’s emphasis was early child-hood and she had experience in working with and teaching elementary students . She had been a HOSTS (Helping One Student To Suc-ceed) 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 in-ternalization 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 . How-ever, 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 re-sorted to the use of division to determine equivalency and this complicated the lesson and moved her pedagogically away from her initial activity of concretely determin-ing 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 .

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DiscussionIt is evident from this study that preservice teachers learn and develop as teachers throughout their education (Onslow, Bey-non, & 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 con-trary to the findings of Hadfield, Littleton, Steiner, and Woods (1998), who indicated that they were unable to identify any sta-tistically 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 practi-cally useless in substantive data analyses . Grades earned in the methods course were used in the regression equation to investi-gate whether similar findings were evident in our data. The findings from the regres-sion analysis supported findings related to grade inflation that accounts for such loss of predictability . The moderate correlation between math-ematics 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 teach-ers 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 stu-dents 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 se-mester 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 re-sults seem to confirm the findings of Quinn (1997) that inadequate mathematical con-tent 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 perfor-

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mance 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 profun-dity. 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 in-stance, 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 univer-sity instruction, leading to a reasonable opportunity for learning by the prospec-tive 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 pro-mote 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 compo-nents 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 preser-vice 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 teach-ing . 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 pre-service 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 encour-ages 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 willing-ness of the preservice teacher to fully en-gage 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 preser-vice teachers see best practices and have meaningful experiences, which cannot be measured by hours in a field experience.

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Carpenter, T . P ., & Lehrer, R . (1999) . Teach-ing and learning with understanding . In E . Fennema & T . Romberg (Eds .), Mathematics classrooms that promote understanding (pp .

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33-58) . Mahwah, NJ: Lawrence Erlbaum .Cobb, P . (1988) . The tension between theories of

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Hadfield, O. D., Littleton, C. E., Steiner, R. L., & Woods, E . S . (1998) . Predictors of preservice elementary teacher effectiveness in the micro-teaching of mathematics lessons . Journal of Instructional Psychology, 25, 34-46 .

Harrington, H . L . (1995) . Fostering reasoned decisions: Case-based pedagogy and the pro-fessional development of teachers . Teaching and Teacher Education, 11(3), 203-214 .

Kilpatrick, J ., Swafford, J ., & Findell, B . (Eds .) . (2001) . Adding it up: Helping children learn mathematics . Washington, DC: National Academy Press .

Kulm, G ., Capraro, R . M ., Capraro, M . M ., Burghardt, R ., & Ford, K . (2001, April) . Teach-ing and learning mathematics with understand-ing in an era of accountability and high-stakes testing . Paper presented at the research presession of the 79th annual meeting of the National Council of Teachers of Mathematics, Orlando, FL

Leinhardt, G ., Putnam, R . T ., Stein, M . T ., & Baxter, J . (1991) . Where subject knowledge matters . In J . E . Brophy (Ed .), Advances in research on teaching: Teachers’ subject matter knowledge and classroom instruction (Vol . 2, pp . 87-113) . Greenwich, CT: JAI Press .

Lloyd, G ., & Frykholm, J . (2000) . How innova-tive middle school mathematics can change prospective elementary teachers’ conceptions. Education, 120, 575-580 .

Ma, L . (1999) . Knowing and teaching elementary mathematics: Teachers’ understanding of fun-damental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum .

Merseth, K . K . (1996) . Cases and case methods in teacher education . In T . J . Buttery & E . Guyton (Eds .), Handbook of research on teacher educa-tion (pp . 722-744) . New York: Macmillan .

National Commission on Teaching and America’s Future . (1996) . What matters most: Teaching for America’s future. New York: National Com-mission on Teaching and America’s Future .

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National Research Council . (2001) . Knowing and learning mathematics for teaching. Washing-ton, DC: National Academy Press .

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Quinn, R . (1997) . Effects of mathematics meth-ods courses on the mathematical attitudes and content knowledge of preservice teachers . The Journal of Educational Research, 91, 108-113 .

Romberg, T ., & Carpenter, T . (1986) . Research on teaching and learning mathematics: Two dis-ciplines of scientific inquiry. In M. C. Wittrock (Ed .), The third handbook of research on teach-ing (pp . 850-873) . New York: Macmillan .

Schwartz, J . E ., & Riedsel, C . A . (1994, February) . The relationship between teachers’ knowledge and beliefs and the teaching of elementary math-ematics. Paper presented at the annual meet-ing of the American Association of College for Teacher Education, Chicago, IL . (ERIC Docu-ment Reproduction Service No . ED366585)

Secada, W . G ., & Berman, P . W . (1999) . Equity as a value-added dimension in teaching for under-standing in school mathematics . In E . Fennema & T . Romberg (Eds .), Mathematics classrooms that promote understanding (pp . 76-94) . Mah-wah, NJ: Erlbaum .

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

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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 calculatorB . asking Juanita to estimate the answer to the problem in order to assess the reasonableness

of the answer on the calculatorC . reviewing with Juanita the rules for the conversion of units within the same system of measurementD . 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?

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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: diam-eters 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 machin-ery 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 From 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

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Appendix D—Rubrics for Open-Ended Content Knowledge Test

Question 1 (Pizza)Student A: Incomplete or no process without any demonstration of mathematical solution (intui-tive 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 arith-metic 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 .

Footnote1 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 .

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