Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

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<ul><li><p>This article was downloaded by: [University of Glasgow]On: 19 December 2014, At: 08:53Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK</p><p>Mathematical Thinking andLearningPublication details, including instructions forauthors and subscription information:</p><p>Supporting PreserviceTeachers' Understanding ofPlace Value and MultidigitArithmeticKay McClainPublished online: 18 Nov 2009.</p><p>To cite this article: Kay McClain (2003) Supporting Preservice Teachers' Understandingof Place Value and Multidigit Arithmetic, Mathematical Thinking and Learning, 5:4,281-306, DOI: 10.1207/S15327833MTL0504_03</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 theinformation (the Content) contained in the publications on our platform.However, Taylor &amp; Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor &amp; Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, 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 theContent.</p><p></p></li><li><p>This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms &amp; Conditions of access and use can be found at</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f G</p><p>lasg</p><p>ow] </p><p>at 0</p><p>8:53</p><p> 19 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p><p></p></li><li><p>Supporting Preservice TeachersUnderstanding of Place Value</p><p>and Multidigit Arithmetic</p><p>Kay McClainDepartment of Teaching and Learning</p><p>Vanderbilt University</p><p>This article provides an analysis of a teaching experiment conducted in the context ofteacher education designed to support preservice teachers understandings of placevalue and multidigit addition and subtraction. The experiment addresses the follow-ing research question: Can the results from research conducted in elementary mathe-matics classrooms guide preservice elementary teachers development of conceptualunderstanding of the same concepts? In both cases, the students (e.g., elementary stu-dents and preservice teachers) participated in activities from an instructional se-quence designed to support conceptual understanding of both place value andmultidigit addition and subtraction. Analyses of the episodes from the teaching ex-periment document the learning of the preservice teachers and how that learning wassupported by initial conjectures grounded in the research on elementary studentsways of reasoning.</p><p>The Conference Board of the Mathematical Sciences (2001) characterized placevalue and multidigit arithmetic as a substantial mathematical concept where adeep understanding is needed in order for preservice teachers to help their stu-dents use it as a foundation for the successful learning of integer arithmetic, andlater decimal arithmetic and symbolic calculations in algebra (p. 5). Unfortu-nately, there is a dearth of research to document preservice teachers learning inthis area. However, current research in mathematics education has provided nu-merous accounts of elementary students understandings and misunderstandingsof place value and their developing algorithms for multidigit addition and subtrac-tion (Bowers, 1996; Bowers, Cobb, &amp; McClain, 1999; Carpenter, Blume, Hiebert,</p><p>MATHEMATICAL THINKING AND LEARNING, 5(4), 281306Copyright 2003, Lawrence Erlbaum Associates, Inc.</p><p>Requests for reprints should be sent to Kay McClain, Box 330 GPC, Department of Teaching andLearning, Vanderbilt University, Nashville, TN 37203. E-mail:</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f G</p><p>lasg</p><p>ow] </p><p>at 0</p><p>8:53</p><p> 19 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>Anick, &amp; Pimm, 1982; Carpenter &amp; Moser, 1984; Cauley, 1988; Cobb &amp;Wheatley, 1988; Cobb, Yackel, &amp; Wood, 1992; Fuson, 1986, 1990; Fuson &amp;Briars, 1990; Hiebert &amp; Wearne, 1992, 1996; Kamii, Lewis, &amp; Livingston, 1993;McClain, Cobb, &amp; Bowers, 1998; Sowder &amp; Schappelle, 1994; Steffe, Cobb, &amp;von Glasersfeld, 1988).</p><p>These studies highlight the importance of supporting students development ofconceptual understanding of both place value and multidigit addition and subtrac-tion so that they develop strategies for carrying out what Fuson (1990) called in-creasingly abstract and efficient addition and subtraction solution procedures (p.273). This involves developing an understanding of and investigating the relationsbetween such ideas as quantifying sets of objects in groups of 10 and treating thegroups as composite units, understanding the preservation of quantity when onedecomposes multidigit numbers into different groupings, and the composition ofquantities (Fuson, 1988; Hiebert &amp; Wearne, 1996; Steffe et al., 1988).</p><p>Prior research on studentsunderstandings of place value and multidigit arithme-tic also highlights the importance of building connections between students repre-sentations of their mathematical ideas and actions on quantities as opposed to ma-nipulating meaningless symbols. In this process, students representations aretreated as tools that provide a record of their activity that aid both in communicatingabout these activities and in providing connections between key ideas. These toolstherefore provide the means for students to reorganize their activity in the course oftheirongoing investigations(Drfler,1993;Kaput,1994;Meira,1998;Pea,1993).</p><p>The purpose of this article is to extend the research on students understandingsof place value and multidigit addition and subtraction to provide an analysis of thedevelopment of one group of preservice teachers understandings of the same con-cepts. In particular, the analysis will document how conjectures about the learningof preservice teachers were informed by research efforts conducted with elemen-tary students. These conjectures contributed to the formulation of a hypotheticallearning trajectory (cf. Simon, 1995) for the preservice teachers and the means ofsupporting their learning along that trajectory. The purpose of the hypotheticallearning trajectory is to provide a conjectured learning route through the mathe-matical terrain. As part of this process, the normative ways of reasoning thatemerged in research settings with elementary students (e.g., preserving quantity bycreating different arrangements for the same quantity or transforming quantities bycomposing and decomposing) are taken as a basis for the hypothetical learning tra-jectory for the preservice teachers. In essence, the results of prior research form thebasis of starting points and conjectured ways of proceeding.</p><p>Inherent in the hypothetical learning trajectory are conjectures about the ac-companying means of support. These support mechanisms include the sequence ofinstructional tasks, the use of the preservice teachers inscriptions and solutions asa focus of whole-class discussions (cf. McClain, 2000), and the norms for argu-mentation (cf. McClain &amp; Cobb, 2001; Yackel &amp; Cobb, 1996). The role of the</p><p>282 McCLAIN</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f G</p><p>lasg</p><p>ow] </p><p>at 0</p><p>8:53</p><p> 19 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>teacher during classroom interactions is then to test and refine the hypotheticallearning trajectory based on the contributions of the preservice teachers against thebackground of the overarching mathematical agenda. In this process, the means ofsupport provide tools for advancing the mathematical agenda. Simon (1997) de-scribed this process in his Mathematics Teaching Cycle as (a) building from theteachers knowledge to formulate a conjecture about a hypothetical learning trajec-tory, (b) using the trajectory as a basis for interacting with preservice teachers, (c)informing the teachers knowledge as a result of the interactions, and then (d) re-formulating the trajectory. And so the process continues as shown in Figure 1. Theimage that results is that of testing and revising conjectures on an ongoing basiswhile interacting with the preservice teachers in a classroom setting where mathe-matical understanding is the goal.</p><p>In the following sections of this article, I begin by describing the methodologyused in the analysis. I then describe the methods classroom that was the site of theresearch. Next, I outline the hypothetical learning trajectory that guided the mathe-matical investigations. Against this background, I provide an analysis of classroomepisodes intended to document the preservice teachersdeveloping understandingsof place value and algorithms for multidigit addition and subtraction. I then returnto the hypothetical learning trajectory and the similarities and differences betweenthe development of the preservice teachers and that of the third-grade students onwhich the trajectory was based. I conclude by addressing implications of this studyas they relate to the professional growth and development of preservice teachers.</p><p>METHODOLOGY</p><p>The methodology employed during this research falls under the general heading ofa teaching experiment. Similar to Simons (2000) teacher development experiment</p><p>PRESERVICE TEACHERS UNDERSTANDING 283</p><p>FIGURE 1 Simons mathematics teaching cycle.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f G</p><p>lasg</p><p>ow] </p><p>at 0</p><p>8:53</p><p> 19 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>(TDE), the study was focused on the learning of preservice teachers. Simon char-acterizes a TDE as a whole-class teaching experiment in the context of teacher de-velopment (p. 345). However, a TDE attempts to account for changes in thepreservice teachers professional development. In contrast, the study reported inthis article focuses on the preservice teachers mathematical development.</p><p>Data for the study consist of videorecordings of each class session. Copies ofthe preservice teachers work, my daily reflective journal (as the instructor of theclass) and a set of field notes taken by a research assistant comprise the written arti-facts. The approach taken when conducting retrospective analyses of the data in-volves a method described by Cobb and Whitenack (1996) for analyzing sets ofclassroom data. This method is an adaptation of Glaser and Strauss (1967) con-stant comparative method. The initial orientation for a retrospective analysis isprovided by the tentative and eminently revisable conjectures that were developedboth prior to and while actually conducting the teaching experiment. Once theteaching experiment was in progress, initial conjectures were tested, refined, andrevised through ongoing discussions between the instructor and the research assis-tant where each used their data collection method (daily journal and detailed fieldnotes, respectively) as a basis for claims. The next phase of the analysis involvedtesting and revising these conjectures and working through the videotapes chrono-logically. This part of the analytic process entails carefully analyzing the videowhile corroborating events against the field notes. The conjectures that emergedfocused on such issues as the evolution of the preservice teachers understandingsof the mathematics and the means by which preservice teachers learning was sup-ported and organized. The constant comparison of conjectures with data resultedin the formulation of claims or assertions that span the data set but yet remain em-pirically grounded in the details of specific episodes.</p><p>SETTING</p><p>The setting for the methods course in which the preservice teachers were enrolledis a private university in the southeast United States. The teaching experiment thatis the focus of this article was conducted in the spring semester of 1999. The coursemet twice weekly for 50 min over a period of 14 weeks. There were 24 students en-rolled in the course, 22 women and two men. Twenty-one of the students were intheir third year and 3 in their fourth year.1 The course is the second of a two-coursesequence focusing on teaching mathematics in the elementary school and is typi-</p><p>284 McCLAIN</p><p>1The fact that this study was conducted in a single, private university has the potential to limit theapplicability of the findings. However, the claims are generalizeable through what Steffe &amp; Thompson(2000) characterized as an explanatory framework. This is appropriate for guiding action that is adap-tive from particular cases.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f G</p><p>lasg</p><p>ow] </p><p>at 0</p><p>8:53</p><p> 19 </p><p>Dec</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>cally taken in the junior (third) year. The first course in the sequence is taken in thesophomore (second) year and has no associated field experience. As a result, theinstructor of the first course uses a series of three CD-ROM based cases to createclassroom contexts that can be explored by the preservice teachers. A primary goalof the first course is to support the preservice teachers as they begin to tease out thecomplexities involved in teaching mathematics at the elementary-school level. Assuch, the National Council of Teachers of Mathematics [NCTM] Standards docu-ments, numerous articles from the NCTM practitioner journals, and CD-ROMbased cases comprise the basis of the resources.</p><p>The second course involves a concurrent practicum placement that requires ob-serving elementary classrooms for 6 hours per week. In addition, as part of a dualassignment for both the methods course and the field placement, the preserviceteachers are expected, by the end of the semester, to teach a 3-day lesson sequencein the classroom in which they observe. The focus of the methods course is themathematics that comprises the elementary curriculum and how that can be taughtfor conceptual understanding. An underlying premise is that the preservice teach-ers must possess conceptual understanding themselves to teach in a conceptualmanner (cf. Ball, 1989, 1993; Bransford, Brown, &amp; Cocking, 2000; CBMS, 2001;Fennema, Carpenter, Franke, Levi, Jacobs, &amp; Empson, 1996; Grossman, 1990;Grossman, Wilson, &amp; Schulman, 1989; Lampert, 1990; Ma, 1999; Morse, 2000;National Research Council, 2001; Schifter, 1990, 1995; Shulman, 1986; Simon,1995, 2000; Stein, Baxter, &amp; Leinhardt, 1990; Stigler &amp; Hiebert, 1999). To thisend, the mathematical focus is placed in the foreground as the preservice teachersengage in tasks intended to challenge them to expand their understandings. Subse-que...</p></li></ul>