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

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<ul><li><p>This article was downloaded by: [Gebze Yuksek Teknoloji Enstitsu ]On: 21 December 2014, At: 19:46Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK</p><p>Journal of Research in Childhood EducationPublication details, including instructions for authors and subscription information:</p><p>The Mathematics Content Knowledge Role inDeveloping Preservice Teachers' PedagogicalContent KnowledgeRobert M. Capraro a , Mary Margaret Capraro a , Dawn Parker a , Gerald Kulm a &amp;Tammy Raulerson aa Texas A&amp;M UniversityPublished online: 03 Nov 2009.</p><p>To cite this article: Robert M. Capraro , Mary Margaret Capraro , Dawn Parker , Gerald Kulm &amp; 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</p><p>To link to this article:</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content)contained in the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy, completeness, or suitabilityfor any purpose of the Content. Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed by Taylor &amp; Francis. The accuracy ofthe Content should not be relied upon and should be independently verified with primary sources ofinformation. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of the Content.</p><p>This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution inany form to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found at</p><p></p></li><li><p>Journal of Research in Childhood Education2005, Vol. 20, No. 2</p><p>Copyright 2005 by the Association forChildhood Education International</p><p>0256-8543/05</p><p>102</p><p>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 &amp; Carpenter, 1986) . This change of role has led to the need for those responsible for the preparation of prospective mathematics </p><p>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. </p><p>Robert M. CapraroMary Margaret CapraroDawn ParkerGerald KulmTammy RaulersonTexas A&amp;M University</p><p>The Mathematics Content Knowledge Role in Developing Preservice Teachers Pedagogical Content Knowledge</p><p>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) .</p><p>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 </p><p>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 .</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Geb</p><p>ze Y</p><p>ukse</p><p>k T</p><p>ekno</p><p>loji </p><p>Ens</p><p>tits</p><p>u ] </p><p>at 1</p><p>9:46</p><p> 21 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>PREsERVICE TEACHERs PEdAGoGICAl KNoWlEdGE</p><p>103</p><p>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 Mas 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) .</p><p>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 ones own (Carpenter &amp; 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 &amp; Berman, 1999) . </p><p>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, &amp; Ford, 2001) . </p><p> Teaching mathematics effectively is a complex task . The National Commission on Teaching and Americas 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 </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Geb</p><p>ze Y</p><p>ukse</p><p>k T</p><p>ekno</p><p>loji </p><p>Ens</p><p>tits</p><p>u ] </p><p>at 1</p><p>9:46</p><p> 21 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>CAPRARo, CAPRARo, PARKER, KUlM, ANd RAUlERsoN </p><p>104</p><p>Lehrers (1999) findings regarding reflection and Mas 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><p>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 &amp; 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 .</p><p>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 ones 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 </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Geb</p><p>ze Y</p><p>ukse</p><p>k T</p><p>ekno</p><p>loji </p><p>Ens</p><p>tits</p><p>u ] </p><p>at 1</p><p>9:46</p><p> 21 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>PREsERVICE TEACHERs PEdAGoGICAl KNoWlEdGE</p><p>105</p><p>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 contribut...</p></li></ul>


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