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Supporting PreserviceTeachers' Understanding ofPlace Value and MultidigitArithmeticKay McClainPublished online: 18 Nov 2009.
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
To link to this article: http://dx.doi.org/10.1207/S15327833MTL0504_03
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Supporting Preservice Teachers’Understanding of Place Value
and Multidigit Arithmetic
Kay McClainDepartment of Teaching and Learning
Vanderbilt University
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 students’ways of reasoning.
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, & McClain, 1999; Carpenter, Blume, Hiebert,
MATHEMATICAL THINKING AND LEARNING, 5(4), 281–306Copyright © 2003, Lawrence Erlbaum Associates, Inc.
Requests for reprints should be sent to Kay McClain, Box 330 GPC, Department of Teaching andLearning, Vanderbilt University, Nashville, TN 37203. E-mail: [email protected]
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Anick, & Pimm, 1982; Carpenter & Moser, 1984; Cauley, 1988; Cobb &Wheatley, 1988; Cobb, Yackel, & Wood, 1992; Fuson, 1986, 1990; Fuson &Briars, 1990; Hiebert & Wearne, 1992, 1996; Kamii, Lewis, & Livingston, 1993;McClain, Cobb, & Bowers, 1998; Sowder & Schappelle, 1994; Steffe, Cobb, &von Glasersfeld, 1988).
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 & Wearne, 1996; Steffe et al., 1988).
Prior research on students’understandings 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(Dörfler,1993;Kaput,1994;Meira,1998;Pea,1993).
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.
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 & Cobb, 2001; Yackel & Cobb, 1996). The role of the
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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 theteacher’s 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 teacher’s 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.
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 teachers’developing 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.
METHODOLOGY
The methodology employed during this research falls under the general heading ofa teaching experiment. Similar to Simon’s (2000) teacher development experiment
PRESERVICE TEACHERS’ UNDERSTANDING 283
FIGURE 1 Simon’s mathematics teaching cycle.
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(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.
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.
SETTING
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-
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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 & Thompson(2000) characterized as an explanatory framework. This is appropriate for guiding action that is adap-tive from particular cases.
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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.
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, & Cocking, 2000; CBMS, 2001;Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996; Grossman, 1990;Grossman, Wilson, & Schulman, 1989; Lampert, 1990; Ma, 1999; Morse, 2000;National Research Council, 2001; Schifter, 1990, 1995; Shulman, 1986; Simon,1995, 2000; Stein, Baxter, & Leinhardt, 1990; Stigler & 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-quently, it is placed in the background as the preservice teachers are asked to“change hats” and reflect on their own activity in light of how they might teachsuch concepts.
Concurrent with the mathematical focus is an ongoing focus on the negotia-tion of classroom norms to support the preservice teachers’ mathematical explo-rations. This includes the need to justify all mathematical processes and explainthe reasoning behind solution methods. The goal is to shift the emphasis from“answers” to “solution processes” where the mathematics takes a more promi-nent position in class discussions. This process supports the development of aclassroom microculture where argumentation and justification are the corner-stones of discussions.
INITIAL HYPOTHETICAL LEARNING TRAJECTORY
The unit on place value and multidigit addition and subtraction, which is the focusof this article, was conducted in five class sessions spanning a 3-week period dur-ing late January and early February. The instructional activities utilized during the
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teaching experiment were part of the Candy Factory instructional sequence2 thatwas developed for use with elementary students (for a detailed analysis of theCandy Factory instructional sequence see Bowers, 1996; Bowers et al., 1999;Cobb et al., 1992). The intent of the instructional sequence is to support elemen-tary students’ construction of increasingly sophisticated conceptions of placevalue numeration and increasingly efficient algorithms for adding and subtractingthree-digit numbers. An overarching goal is to support the development of under-standing and computational facility in an integrated manner.
The Candy Factory instructional sequence is built on the scenario of a candyfactory that initially involves Unifix cubes as substitutes for candies, and later in-volves the development of ways of recording transactions in the factory. Duringinitial whole-class discussions, the preservice teachers were introduced to the sce-nario of Ms. Wright and her Candy Factory. In the third-grade version of this activ-ity, the students are told that 10 candies are packed in each roll and 10 rolls arepacked in each box. However, the preservice teachers were told that in Ms.Wright’s Candy Factory, single pieces of candy are packed into rolls of 8, and 8rolls are packed into a box. The instructional sequence was modified for use withthe preservice teachers to problematize the mathematics. Despite these changes,the mathematical goal for the preservice teachers was the same as that for the ele-mentary students: (a) to develop an understanding of the multiplicative relationswithin place value, and (b) to develop ways of symbolizing transactions in theCandy Factory dealing with buying or adding and selling or subtracting candies.This would lead to invented algorithms for three-digit addition and subtraction. Inboth cases, the intent of the activities was that the students and preservice teacherswould come to understand the mathematics in the context of their own prob-lem-solving efforts. The reason for modifying the context was that if the mathe-matics had been trivial for the preservice teachers, the need to create ways to sym-bolize their transactions would not have emerged naturally. The goal was then tobuild from the preservice teachers’evolving notational schemes to support shifts intheir understandings of place value and multidigit addition and subtraction so thatthey might develop conceptual understanding instead of mere proficiency withmeaningless algorithms.
It is important to note that activities were not conducted using base eight nota-tion—the packing rule for the Candy Factory was merely that 8 candies compriseda roll and 8 rolls comprised a box. This was a conscious decision based on previousattempts to use the Candy Factory with base eight notation with preservice teach-ers. In prior instances, preservice teachers focused their time on attempting to cor-rectly notate transactions in base eight rather than exploring the underlying con-cepts of place value. This misdirected focus resulted in their developing tricks and
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2The Candy Factory instructional sequence used in the methods course was refined and developedin collaboration with Cobb and Bowers. It was originally developed by Cobb and Yackel.
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short cuts to facilitate their use of the notation. As a result, the preservice teachersdid not develop invented algorithms or notational schemes. Furthermore, theywere merely manipulating symbols instead of acting on quantities. As an example,in combining pieces and rolls, most of the preservice teachers developed the fol-lowing scheme: If the sum does not exceed seven or there is no need to regroup insubtraction, the notation is “the same as if there were ten” in a roll (see Figure 2).
The preservice teachers were not reasoning about the quantity represented bythe notation, but were instead determining ways to solve the problem that built ontheir knowledge of base 10 computation. Their reasoning was not based on rollscomposed of 8 candies or boxes composed of 8 rolls, but a modification of the base10 algorithm that yielded the correct answer.
When the problems became more complex, their methods involved findingways to modify their procedure such that their algorithm still yielded a correct re-sponse. As an example, for the problem 13 + 26, they reasoned that in base 10 theanswer is 39 so in base 8 it would be 2 more because they were adding, or 41. Theycame to this conclusion as a result of empirical testing, not because of mathemati-cal understanding. They found the procedure that resulted in the correct answerand created a justification, post hoc, grounded in the contrast between base 8 andbase 10. Their reasoning was not a result of conjectures about the mathematics. Forthe problem 43 – 14, they reasoned that in base 10, the answer is 29, and base 8 is 2less (because they were subtracting) so the answer is 27. Although their numeralswere correct, 27 was not two rolls of 8 and 7 pieces, but two 10s and 7 ones.
All of the preservice teachers’ methods produced a correct solution. However,there was no understanding to underlie their procedures; merely results of lookingfor patterns in solutions. In addition, their focus in class discussion was on the trickthat converted their base 10 answer into a correct base 8 answer. They also did nothave a need to create drawings or inscriptions to support their notation, becausetheir activity was not conceptual. When pushed to explore the mathematical reasonbehind their strategies, they were resistant because they did not have a need to con-ceptually understand what was happening mathematically. They were excited bytheir discoveries and felt as though they were really exploring important mathe-matical terrain. This created a conundrum for me as the teacher. Not only did I feelthat the preservice teachers were not furthering their own understanding of placevalue, I also judged that the nature of classroom discussions was supporting acalculational orientation (cf. Thompson, Philip, Thompson, & Boyd, 1994). As a
PRESERVICE TEACHERS’ UNDERSTANDING 287
FIGURE 2 Problems where solu-tion is notationally the same in base 8and base 10.
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result, at the end of the exploration of place value, I judged that my interactionswith the preservice teachers had not enhanced their understanding of place value ina fundamental way. It is for this reason that I decided to try a different approach inmy instruction. The change entailed no longer using base 8 notation, but focusingthe attention of the preservice teachers on the composition of units that built from8. My conjecture was that the preservice teachers needed to build the system bycomposing and decomposing quantities and creating drawings and notation sys-tems to symbolize their activity instead of deciphering the base 8 notation. Thiswould provide experiences similar to those of the elementary students andstrengthen my ability to support the preservice teachers’ understanding of themultiplicative structure of the place value system.
Against this background, and framed by Bowers’ (1996) analysis of the enact-ment of the Candy Factory instructional sequence in a third-grade classroom3, thehypothetical learning trajectory for the preservice teachers therefore included aninitial starting point of a packing rule of 8 with base 10 notation. Building on thisstarting point, the first phase of the hypothetical learning trajectory was intended toprovide opportunities for the preservice teachers to count collections of candies by64’s, 8’s and 1’s. The conjecture was that the preservice teachers had to be able toquantify collections by counting by 64’s, 8’s and 1’s before they could perform op-erations on collections. Following this first phase of the learning trajectory (seeFigure 3 for phases of the hypothetical learning trajectory), the preservice teachersthen would need to understand how to create different arrangements for the samequantity. In other words, 4 rolls and 3 pieces might also be arranged as 3 rolls and11 pieces or 2 rolls and 19 pieces.
This second phase of the hypothetical learning trajectory would involve instruc-tional tasks where the preservice teachers were asked to show different arrange-ments for a given amount of candies.
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3The Candy Factory instructional sequence was the focus of a developmental research project con-ducted with a group of third-grade students in the fall of 1994. The research team included Cobb,Gravemeijer, Yackel, Bowers, and McClain.
FIGURE 3 Chart showing phases of hypothetical learning trajectory.
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The third phase of the hypothetical learning trajectory included conjecturesabout the importance of the preservice teachers being able to transform quantitiesby packing and unpacking. The instructional tasks developed in support of thisphase of the trajectory were important in helping the preservice teachers under-stand how to preserve quantities, which is necessary in renaming and regroupingwhen adding and subtracting. The conjectured end point of the trajectory was thatthe preservice teachers add and subtract structured quantities. Instructional tasks insupport of this last phase of the hypothetical learning trajectory included scenariosfrom the Candy Factory where additional candies were being produced or candieswere being sold from the factory. An overarching goal was that the preserviceteachers’ activity in the Candy Factory would have quantitative significance, pro-viding a meaningful basis for their computational activity.
The hypothetical learning trajectory was formulated around the premise that thepreservice teachers’ solutions would be critical in supporting the emergence of themathematical agenda. For this reason, the preservice teachers’diverse ways of rea-soning on tasks became the focus of class discussions where the mathematical sig-nificance of the varied solutions was highlighted. In addition, norms for mathemat-ical argumentation that supported the exploration of the solution methods wereconstantly being negotiated. As a result, the instructional tasks, the preserviceteachers’ varied solutions and notations, and the classroom norms were critical as-pects of the classroom in supporting the emergence of the mathematical agenda.
ANALYSIS OF CLASSROOM EPISODES
At the beginning of the semester, when asked about place value, the preserviceteachers defined it as being able to identify the correct digit in a given “place,” butdid not carry this idea onto its larger implications such as the relations betweenplace value and the primary operations. Furthermore, their understandings of bothplace value and multidigit addition and subtraction were very superficial andgrounded in rules for manipulating algorithms. While they openly embraced a“student-centered” approach to mathematics, they also held strong beliefs aboutthe importance of students being able to manipulate algorithms quickly and effi-ciently, especially on “timed” tests. Although they wanted their students to under-stand mathematics, they referred to the importance of performing well on stan-dardized tests where all that was judged was the correctness of an answer. As webegan to engage in tasks from the Candy Factory instructional sequence, my goalwas to support shifts in the preservice teachers’ ways of reasoning so that theydeveloped what Ma (1999) called a profound understanding of fundamentalmathematics.
The initial tasks in the Candy Factory instructional sequence involved estimat-ing how many rolls of “candy” could be formed from transparent bags of candies
PRESERVICE TEACHERS’ UNDERSTANDING 289
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(i.e., Unifix cubes). As the preservice teachers made their estimates, they were re-corded on the board. Afterward, I took eight “candies” from the bag and formed aroll and then asked if anyone wanted to change their estimate. We then engaged inan iterative process of refining estimates based on producing additional rolls fromthe collection. This whole-class activity was followed by the preservice teachersworking in pairs with bags of candies to complete a similar activity. The goal ofthese tasks was to support their estimating by creating images for them to use asthey examined the bag. The activity also helped build imagery for a roll being com-posed of eight pieces of candy and hence their conception of rolls as compositeunits.
The next set of instructional tasks involved estimating how many rolls ofcandy could be formed from rectangular arrays of candies such as the one shownin Figure 4. The arrays were shown on the overhead projector for 3 to 4 sec. Thepreservice teachers were then asked to estimate the number of rolls by envision-ing how the array might be “packed.” These tasks were intended to support thedevelopment of imagery for the activity of packing and unpacking and also tobuild on the preservice teachers’ imagery of rolls as composite units. These ini-tial estimation activities therefore provided a basis for subsequent activity in theCandy Factory.
At the end of the first class, the preservice teachers were given an assignmentin which they had to determine the total number of candies in collections ofboxes, rolls, and pieces. These quantifying tasks were intended to build from theestimating tasks toward counting collections by 64’s, 8’s and 1’s. Concurrentwith this activity, the preservice teachers were asked to make a record of theiractivity so that they could share their solution in the course of whole-class dis-cussion. The second day of the instructional sequence therefore began with dis-cussions of the preservice teachers’ solutions. The first task to be discussed in-volved determining the quantity of candies in one box, three rolls, and twopieces of candy. In solving the task, the preservice teachers developed a variety
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FIGURE4 Rectangulararrayofcan-dies for estimation tasks.
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of solutions. The first solution that I selected to be shared in the whole-class set-ting involved drawing pictures of all the pieces of candy including the 64 piecesin the box and the 8 pieces in each of the 3 rolls. My goal in selecting Brenda toshare her solution first was to make explicit the use of pictures and symbols tobuild imagery for the activity in the Candy Factory. I therefore first chosepreservice teachers whose solutions involved drawings of the candies and thenmoved to more symbolic ways of reasoning in hopes of initiating shifts in thepreservice teachers’ ways of reasoning. Had I started with the symbolic forms, Iconjectured that those preservice teachers who needed the drawings would at-tempt to decipher the symbols devoid of any real understanding. By ordering thesolutions in the manner I chose, I hoped to support the construction of imageryto underlie the symbols. My goal was to support the preservice teachers’ abilityto count collections by 64’s, 8’s and 1’s where the result of their activity ofcounting yielded a quantity, not a meaningless numeral. This goal was in keep-ing with the first phase of the hypothetical learning trajectory that focused onquantifying collections by counting by 64’s, 8’s and 1’s.
In discussing her solution, Brenda acknowledged that drawing all the candieswas a tedious approach and stated that on subsequent tasks she would use thefact that a box contained 64 candies and a roll contained 8 candies. She noted,however, that she would still need to draw “empty” boxes and rolls so she could“keep track.” There were in fact several preservice teachers who drew circles todenote all of the candies, even those inside the box. However, in the course ofdiscussing the solution, they also agreed that the process of drawing pictureswith all the candies shown was too labor intensive but stated that they wouldneed some type of picture to help them solve the tasks. As a result, a symbolsystem for use on these tasks emerged; an example of which is shown in Figure5. In subsequent tasks, the preservice teachers who needed to create the candiesbefore they could act on them (i.e., pack or unpack them) found these drawingsuseful. They could manipulate them to solve the tasks and began to developsome efficiency with the process. I found this significant because a similar sys-tem had emerged in the third-grade classroom. In particular, most of thethird-grade students relied on the use of these symbols and the symbol-based no-tation system was an integral aspect of their problem-solving activity for an ex-tended period of time. Those preservice teachers who needed the drawings were,in fact, able to reflect on their activity of creating the drawings as a basis forshifting to more efficient notation schemes.
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FIGURE5 Picture showing onebox,three rolls, and two pieces.
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In contrast to Brenda’s solution, the next three solutions shared in the whole-class discussion each involved the use of a numerical notational scheme (see Fig-ure 6). In each of these solutions, we see evidence that the preservice teacher wasable to determine the total number of candies in the collection by building on therelations between the number of candies in the boxes, rolls, and pieces (e.g., count-ing collections by 64’s, 8’s and 1’s). They used the fact that 8 pieces comprised aroll, 8 rolls comprised a box, and 64 pieces comprised a box to determine the totalnumber of candies. It is important to clarify that the preservice teachers developedthese schemes to make a record of their own activity as they solved the tasks. Theywere working to devise ways to communicate their thinking. In doing so, they weredeveloping schemes that built on an understanding of the multiplicative structurethat underlies the place value system. In this way, their inscriptions became a toolthat supported the emergence of their understandings.
Discussions of these solutions initially focused on ways a record could bekept to indicate the process as the preservice teachers unpacked the boxes androlls to determine the total number of candies. As the preservice teachers contin-ued the discussion, they began to tease out the similarities and differences in thesolution methods. In doing so, they pointed to aspects of the schemes that madethe task easier to solve and the record easier to understand. The focus of thesediscussions was on making sure that the notational scheme clearly explained themathematical process. In addition, the preservice teachers also highlighted as-pects of the solutions that they judged to be efficient. As a result, as they solvedadditional tasks and then engaged in discussions of the various solution meth-ods, preservice teachers who had earlier used only pictures began to developmore cryptic versions of their drawings in tandem with a numerical record thatreflected aspects of the notational schemes shown in Figure 6. Through this pro-cess, a shift was occurring in that the preservice teachers were taking their con-versations as a basis for reflecting on their own activity and thinking about howthey might solve the tasks in a more efficient manner while still understandingthe process. As a result, there was an evolution in the schemes that paralleled thepreservice teachers’ developing understandings. This was evident in the shift thatoccurred from focusing on solutions that entailed drawing the boxes and rolls tofocusing on aspects of numerical schemes that represented these quantities andthe relations across them. Throughout this process, the activity of counting col-
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FIGURE 6 Three solutions to the total candies in one box, three rolls, two pieces.
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lections by 64’s, 8’s and 1’s was emerging as a normative way of solving instruc-tional tasks in the classroom.
At the close of the second class session in the sequence, I assigned instructionaltasks that involved finding different ways that given amounts of candies could be inthe storeroom (e.g., 13 candies as 1 roll and 5 pieces). The intent of these tasks wasto focus on the composition and decomposition of arithmetical units with attentionbeing given to the preservice teachers’ ways of symbolizing the process. The tasksbuilt on counting collections by 64’s, 8’s and 1’s and led to the next phase in the hy-pothetical learning trajectory, that of creating different arrangements for the samequantity.
The homework tasks were discussed during the third class session as thepreservice teachers shared their solution methods. Two distinct ways of solving thetask emerged in the course of the preservice teachers’ discussions. Although bothinvolved developing a process of generating a pattern by either continuing to packor unpack the candies, the distinction lay in the preservice teachers either perform-ing the computations mentally (see Figure 7a) or actually showing the subtractionthat yielded the rolls (see Figure 7b).
For the task involving 4 rolls and 16 pieces, some of the preservice teacherspacked the candies into 6 rolls and then began unpacking. Others unpacked all therolls then went back and found the solutions from packing them up again. In gen-eral, these tasks proved to be relatively unproblematic for the preservice teachers.Furthermore, when asked how they could be sure they had listed all the ways, theywere able to point to the pattern they had created in their tables as justification.Similar types of arguments were given by the third-grade students who alsopointed to the patterns in the result of their packing as justification for finding allthe ways. Here again, the preservice teachers’prior activity in creating the table of-fered means of initiating shifts in ways of reasoning. This was supported by the
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FIGURE 7 Table showing solutions to different ways Ms. Wright could have 43 candies.
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normative ways of justifying solutions that had been negotiated in the course ofprior problem-solving activities.
At this point in the sequence, I judged that the preservice teachers were not onlyable to create different arrangements for the same quantity (e.g., phase two of thehypothetical learning trajectory), but could also check their solutions by generat-ing patterns of all possible solutions. For this reason, I presented a new set of tasksin which the preservice teachers were given a quantity of candies and asked to packit up completely. The goal of these tasks was to introduce the inventory form thatMs. Wright used to keep track of the candies in the storeroom (see Figure 8).
In clarifying the intent of the instructional task, I asked the preservice teachersto devise a way of keeping track of their packing process so that others in the classcould understand how they thought about the problems. In doing so, the preserviceteachers developed inscriptions and other means of symbolizing as models of theirmathematical reasoning. However, introduction of the inventory form shifted thefocus from drawings to more formalized notational schemes that could be re-corded on the form. Therefore, the preservice teachers found ways to symbolizetheir activity of composing and decomposing arithmetical units.
The first preservice teacher I selected to share her solution to the task of packing1 box, 23 rolls, and 19 pieces was Anita. She had used a nonconventional approachthat involved first unpacking all the rolls so she would know how many candies shehad. She then packed up the candies into boxes of 64. Afterwards, she packed theremaining candies into a roll (see Figure 9).
I selected Anita because I wanted to juxtapose her process with others thatmight appear more efficient to provide a contrast in ways of approaching the task.As Anita explained her process, other members of the class began to talk quietlyamong themselves, expressing some concern with Anita’s process. They ques-tioned whether it was necessary or even reasonable to unpack all the rolls. Giventhat the other preservice teachers reasoned that packing the rolls into boxes was a
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FIGURE 8 Ms. Wright’s inventoryform.
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more logical approach, there was a tension in the discussions. The norms for theclass were that everyone was to respect each other while they shared their mathe-matical thinking. The preservice teachers’ initial obligation was to ensure that theyunderstood the solution method; afterwards they could engage in discussions of ef-ficiency. This was an important aspect of the classroom microculture, because Iwas trying to ensure that all of the preservice teachers felt comfortable talkingabout their mathematics. In addition, I was attempting to model the type of class-room norms I hoped the preservice teachers might negotiate in their own class-rooms. However, in this instance, the other members of the class were attemptingto deal with their lack of agreement with Anita’s process and their commitment torespect each other’s thinking. This was the first instance where a significant num-ber of the members of the class strongly disagreed with a way of reasoning. Theywanted to ask Anita why she solved the task in the manner that she had and thenshow her what they deemed was an easier way. However, instead of discussing theefficiency of Anita’s solution, I directed the class to focus on making sure they un-derstood Anita’s method so they could subsequently compare and contrast it withother solutions.
I then asked if anyone in the class had solved the task a different way. My goalwas to elicit a variety of solutions to the task so that the preservice teachers couldbegin to tease out significant mathematical differences in the solutions. This wouldshift the focus to understanding the diversity instead of judging it. I hoped to en-courage shifts in the preservice teachers’ ways of reasoning toward the next phaseof the hypothetical learning trajectory, that of transforming quantities by packingand unpacking.
Martin was the first to respond and explained that he had seen that 23 rolls wasonly one short of “a multiple of eight,” so he began by packing up 1 roll from thepieces, then packing the resulting 24 rolls into 3 boxes. He finished by returning tothe pieces and packing a roll (see Figure 10).
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FIGURE 9 Anita’s solution to pack-ing 1 box, 23 rolls, and 19 pieces.
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After clarifying that everyone understood Martin’s solution method, I askedJeanne and then Regan to each share their procedure. Both Jeanne and Regan hadsolved the task in a similar way. They had started with the pieces and worked theirway “across” (see Figure 10). However, although Jeanne had to pack up the boxesand rolls one at a time, Regan was able to reason about the total number of rolls thatcould be made from the collection of pieces and then the total number of boxes thatcould be made from the collection of rolls. The mathematical goal in comparingand contrasting these two solutions was to highlight the efficiency in reasoningabout collections of rolls and pieces instead of treating them individually. This fo-cus provided means of supporting a shift toward efficient ways of transformingquantities.
After the four solutions were on the board, I asked if anyone had solved thetask in a different way. Louise then offered that she had drawn pictures of theboxes, rolls, and pieces and used them to actually show the packing. Afterwards,she took the result of her packing activity and recorded it in the inventory formas shown in Figure 11. When finished, Louise commented that she had com-pleted only the first task this way. She stated that, “it took too long.” She ex-plained that she did the remaining ones “in my head by taking away eights.”However, she needed to complete one task by creating the candies to build imag-ery for her subsequent activity.
At this point, I clarified my intent for the lesson by stating that I had planned todiscuss four different problems in the whole-class setting. My goal was thatthrough the sequence of tasks, the preservice teachers might begin to shift towardmore efficient ways of solving the task (as was the case in the third-grade teachingexperiment). Based on the lack of diversity in the preservice teachers’ solutions tothe “different ways” tasks, I did not anticipate the variety of solutions that emergedfrom this task. As a result, the whole-class discussion shifted to a focus on the dif-ferences in the offered solutions instead of contrasting emerging solutions to dif-
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FIGURE 10 Martin’s, Jeanne’s, and Regan’s solution to the packing task.
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ferent problems. Because my goal was to support the emergence of transformingquantities by packing and unpacking, I judged that it was important to highlight thediversity in methods. Preservice teachers who were still drawing candies needed tobe aware of more efficient strategies for their problem-solving activity. Likewise,the efficiency of Martin’s, Jeanne’s and Regan’s solutions offered a contrast forways of thinking about how to transform the quantities. The schemes that thepreservice teachers devised built off the earlier practice of creating different ar-rangements while offering an opportunity to explore the composition and decom-position of boxes and rolls.
As the preservice teachers continued to solve similar tasks, my assessment oftheir activity indicated that they were building an understanding of place value andits multiplicative structure. This was based on my observations that they had devel-oped a variety of strategies for solving tasks situated in the Candy Factory andcould reason about others’ solutions. In addition, they understood the relation be-tween boxes, rolls, and pieces and could act on these quantities in a reasonableway. Most of the preservice teachers also began showing evidence of being able toexpress their ways of reasoning more easily through numerical schemes, therebyrelying less on the use of pictures and drawings (cf. Fuson, 1986).
Against this background, in the fourth session I introduced the final series oftasks in the instructional sequence. These involved transactions such as selling orsubtracting and buying or adding candies that were written in paragraph form asopposed to being presented with pictures. The tasks were introduced to support thepreservice teachers’ development of informal algorithms for adding and subtract-
PRESERVICE TEACHERS’ UNDERSTANDING 297
FIGURE 11 Louise’s solution to the packing task.
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ing. As the preservice teachers worked on these tasks, most of them developednontraditional yet personally meaningful algorithms for addition and subtractionto symbolize their activity. As an example, consider an episode that occurred onthe last day of the instructional sequence. The preservice teachers had worked sev-eral problems outside of class that involved buying and selling candies. The firsttask discussed in the whole-class setting asked: There are 2 rolls and 5 pieces ofcandy in the storeroom. The workers make 4 rolls and 6 pieces. How many candiesare in the storeroom? As preservice teachers shared their ways of solving the task,it became evident that they found it useful to first find the total number of piecesand record that amount before considering if the pieces could be packed into a roll.This can be seen in several of the preservice teachers’ solutions to 2 rolls and 5pieces plus 4 rolls and 6 pieces as shown in Figure 12.
After the three solutions in Figure 12 were on the board, I posed the followingquestion: “I want you to notice that in each case you put a two-digit number in thecolumn for pieces. Is that okay?” The subsequent discussion then focused on thepreservice teachers’ ways of solving the task and the relation between their nota-tion and the conventional algorithm. Several of the preservice teachers stated that itwas important for them to keep a record of all the pieces before they thought about“packing up.” They related this to their understanding of the algorithm for additionand noted that you could do the same thing as long as you could devise a way to“keep track.” Other preservice teachers then offered that they really did not under-stand what the “carrying notation” in the traditional algorithm signified, but ac-knowledged that it “gave you the right answer.”
At this point, I felt that there was a disconnect between the preservice teachers’activity in the Candy Factory and their understanding of conventional algorithmsfor addition and subtraction. I decided to use the problem being discussed as a wayto investigate the preservice teachers’ understandings of the relation between theirways of solving tasks in the Candy Factory and the more formal algorithmic ap-proach. I began by pointing to their activity in the Candy Factory of first recordingthe total candies and then devising a method that noted the packing of eight candiesinto a roll. Similar to the process of what the preservice teachers called “carrying,”they were able to fully explain the process by comparing their notational schemes
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FIGURE 12 Three solutions to the task of combining 2 rolls and 5 pieces with 4 rolls and 6pieces.
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to the conventional algorithms. They were able to explain that the one that was“carried” in the conventional algorithm really signified one 10. However, theypointed to the importance of it indicating 10 of something. The imagery of theCandy Factory had supported their activity of acting on quantities and they wereasking for a similar image. Their schemes then became a tool to support the link tothe traditional algorithm.
Following this discussion, I selected a subtraction task to be discussed in thewhole-class setting: “There are 2 boxes, 5 rolls, and 3 pieces of candy in the store-room. If you send out 4 rolls and 5 pieces, how many candies are left in the store-room.” I wanted to investigate the preservice teachers’ use of notation for subtrac-tion and their understanding of the links to conventional notation there also. Thefirst preservice teacher to share her solution noted that she had started workingfrom the left, but realized she did not have enough pieces. This process of startingfrom the left was very common among the preservice teachers and supports whatMadell (1985) and Kamii et al. (1993) found when elementary students are al-lowed to invent their own algorithms. As a result, she began again, this time work-ing from the right. She stated, “I realized I had to get more pieces.” She thenworked the problem on the board as shown in Figure 13.
At this point, I built from her initial attempt by asking the preservice teachers ifthe task could be solved by starting with the boxes. This teaching move wasprompted by a very significant event that had occurred in the third-grade teachingexperiment where the same issue arose among the students (for a detailed analysis,see McClain et al., 1998). In the discussion that followed, several of the preserviceteachers argued that you could still unpack a roll to get more pieces, even if youstarted with the boxes. They then discussed how that could be notated to clarifywhat had happened in the storeroom. In doing so, they devised the notation shownin Figure 14.
Discussion then focused on the mathematical accuracy of each way, the similar-ities and differences of each approach, and what the notation represented. The sig-nificant aspect of the discussion was that the preservice teachers’ explanations in-
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FIGURE 13 Solution to transactiontask.
FIGURE 14 Notation resulting fromstarting with boxes.
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volved composing and decomposing quantities. In this way, they were building thebasis for a strong understanding of the algorithm for multidigit subtraction.
As a follow-up, I asked the preservice teachers to discuss how they would solvethe task: “5 boxes and 2 pieces take away 4 rolls and 6 pieces.” This decision wasalso motivated by experiences from the third-grade classroom teaching experimentwhere the students wrestled with what to unpack when there were no rolls (e.g., 1box as 100 candies or 1 box to 10 rolls, and then 1 roll). The first solution was of-fered by Anne who stated, “First I unpacked a box then I had to unpack a roll. So Isent out my pieces and that left four then three rolls and four boxes.” As Anne ex-plained, I notated as shown in Figure 15.
After she finished, Louise asked, “Why did she put eight then seven on the rolls?”This question prompted a discussion about how to notate unpacking across a zero.The preservice teachers engaged in a lively discussion about how to subtract when“you don’t have enough.” Although many of the preservice teachers understood theprocedure as explained by Anne, others commented that they had never understoodwhat to do “when you have a zero.” At this point, Caroline offered that she knew thatin the algorithm “when you have a zero, you put a ten and then a nine.” However, shecontinued by stating that she could never remember which came first. For her, thesewere meaningless rules in a procedure that were intended to produce a correct an-swer. They did not involve decomposing composite units to be able to perform a cal-culation. For that reason, the 10 and the 9 were just numerals that were used to com-plete the task. In the course of answering Louise’s questions, Caroline was able tomake sense of the procedure with respect to the place value system.
Discussion of the remaining tasks indicated that the preservice teachers weredeveloping proficiency in their ability to act on quantities in the Candy Factory. Asactivity on the tasks progressed, the preservice teachers began to curtail their nota-tional schemes so that they paralleled the notation inherent in the standard algo-rithms. In this way, they were able to construct links between their activity in theCandy Factory and the algorithms that earlier had limited meaning.
A RETURN TO THE HYPOTHETICALLEARNING TRAJECTORY
Throughout the analysis, I have documented the development that occurred as thepreservice teachers and I engaged together in a series of tasks from the modified
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FIGURE 15 Notating Anne’s solu-tion.
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Candy Factory instructional sequence. In doing so, I have pointed to the preserviceteachers’ developing ways of both symbolizing and understanding their own andothers’mathematical activity. In particular, I have shown a progression from a needto have visual representations of all the candies to be able to perform even the sim-plest transaction to developed notational schemes that parallel efficient algorithmsfor addition and subtraction of three-digit numbers. The hypothetical learning tra-jectory for supporting this mathematical development began with conjecturesabout the preservice teachers coming to view the boxes as both individual units andcomposite units composed of 8 smaller units. This allowed them to be able to quan-tify collections by counting by 64’s, 8’s and 1’s. This development is very similarto the developments in the third-grade classroom where students came to under-stand that the boxes were “both individual units that could be counted and compos-ites composed of 10 smaller units” (Bowers et al., 1999, p. 42).
The next phase in the hypothetical learning trajectory involved creating differ-ent arrangements of boxes, rolls, and pieces for the same quantity. Unlike the de-velopment of the third-grade students, this aspect of the instructional sequence ap-peared unproblematic to the preservice teachers and did not comprise a significantportion of instructional time. However, like the third-grade classroom, the nextphase of the hypothetical learning trajectory, transforming quantities by packingand unpacking (which led to the creation of self-developed algorithms for additionand subtraction), proved to be a pivotal aspect of the preservice teachers’ develop-ment. As was the case with the third-grade students, discussions in this phase of theinstructional sequence focused on ways to “manipulate and notate actions with theboxes, rolls, and pieces” (Bowers et al., 1999, p.52). The preservice teachers’waysof symbolizing their activity as they transformed a single collection of candies wascritical in providing a basis for their activity in the final phase of the learning tra-jectory, that of adding and subtracting structured quantities by packing and un-packing. This phase therefore served to support their understanding of the pres-ervation of quantity when one decomposes multidigit numbers into differentgroupings (Fuson, 1988; Hiebert & Wearne, 1996; Steffe & Cobb, 1988).
Although the instructional tasks in the teaching experiment reported in this arti-cle were modified from those in the third-grade classroom teaching experiment, itappears that the underlying learning trajectory that guided the third-grade students’development offers the means of supporting preservice teachers’ development ofunderstandings of place value and multidigit addition and subtraction. This is animportant finding in that the broad strokes of the significant mathematical issuesthat emerge from investigating ways to support preservice teachers’ understandingof place value and multidigit addition and subtraction can be painted with theCandy Factory and its underlying learning trajectory. This finding also has broaderimplications—that the broad base of research conducted in elementary classroomscan feed forward to inform efforts at supporting the development of preserviceteachers’ content knowledge.
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CONCLUSION: IMPLICATIONSFOR PRESERVICE TEACHER CHANGE
Although the primary goal of the Candy Factory instructional sequence was to sup-port the preservice teachers’ development of understanding of place value andmultidigit addition and subtraction, a concurrent goal for the course was the devel-opment of preservice teachers’ ability to reason pedagogically about their ownlearning as they engaged in the Candy Factory instructional sequence. This in-volved the preservice teachers thinking about the mathematics from the perspec-tive of a student as they explored the problems, and then stepping back from theirown activity and reflecting on how these activities could be used to support themathematical development of their students. This required that the preserviceteachers assume a dual role in the methods classroom, that of student and of pro-spective teacher. Shifting between these roles was problematic for the preserviceteachers. They were working to understand the mathematics as a student and hadnot begun to reflect on how their activity could support their ability to better teachtheir own students. They were unable to deal with the complexity involved in con-currently teasing out the mathematical knowledge involved in understanding placevalue for themselves while also reflecting on how they might design a lesson se-quence aimed at supporting their students’ mathematical development or orches-trate a whole-class discussion aimed toward a particular mathematical endpoint. Itis only in retrospect that I have come to appreciate the difficulty in expectingpreservice teachers to be able to operate in this dual mode. This difficulty was fur-ther complicated for me by the fact that in making decisions about how to proceedmathematically, I was able to draw on my prior experience with the Candy Factoryinstructional sequence and the hypothetical learning trajectory. This guided deci-sions about what tasks to select, how to anticipate what might occur, and how todeliberately facilitate discussions around significant issues. However, in makingdecisions about how to proceed pedagogically, there was no similar grounding.
One of the pedagogical issues that had been discussed in the class focused onthe importance of the teacher’s role in the classroom. In particular, discussions hadfocused on defining several key aspects of the teacher’s role which included:(a) choosing worthwhile tasks, (b) negotiating classroom norms, (c) deliberatelyfacilitating whole-class discussions, (d) continually monitoring and assessing stu-dents’ activity to make informed decisions about mathematical next steps, and(e) having a deep understanding of the mathematics to be taught. However, be-cause these preservice teachers were not currently involved in the process of teach-ing, these discussions had been framed as issues to consider as they began to inter-act with students. As a result, asking them to place this layer on top of their effortsto understand the mathematics was problematic. Although they may not have ap-preciated the intricacies of the pedagogy, they did learn a larger lesson: They hadcome to realize the importance of teaching mathematics in a conceptual way, and
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that they would not be able to teach for conceptual understanding if they, them-selves, did not have such an understanding of the content.
As a result of what transpired, I took additional class sessions at the end of theCandy Factory instructional sequence to help the preservice teachers reflect ontheir own activity and think through the process again, but from the viewpoint of aprospective teacher. In doing so, the preservice teachers were able to point to issuesthat had emerged, which they deemed problematic such as how to handle Anita’ssolution of unpacking the rolls and then packing boxes. They also discussed atlength the importance of getting a variety of solution methods on the board to bediscussed and began to shift from a need to correct all the tasks, to focusing moreclosely on a few, well-chosen ones. Throughout these discussions, I constantly re-minded the preservice teachers of the need to situate their decisions against thebackground of what they wanted to achieve mathematically. In other words,whole-class discussion should be used to advance the mathematical agenda.Therefore, problems and solutions should be carefully selected for discussion.
An additional goal of the instructional unit was to create perturbations in thepreservice teachers’ beliefs about what it means to “assess.” Therefore, at the endof the unit, the preservice teachers were given a timed test on three-digit problemsthat represented inventory sheets from the Candy Factory where eight pieces com-prised a roll and eight rolls comprised a box. Afterwards, they exchanged papersand graded each other’s work. The preservice teachers’ reactions to the test and themethod of grading revealed their frustration with a timed test as a true assessmentof what they understood. They also noted that as a result of how the tests weregraded, I had no way of determining whether or not they understood. They pointedto the fact that they could have produced correct answers by incorrect means ormade careless errors. In either case, their true understanding would go unchecked.They argued that a much better form of assessment would be for me to select threeor four different types of problems and allow them to explain how they would solvethe problems, showing all their work. This was a radical shift from their earliercomments where they had focused on the importance of students performing wellon timed tests. When questioned about this, several of the preservice teachers com-mented, “I will never do this to my students.” They also noted that if students un-derstood, they could probably get all the questions that they answered correct,which might be just as effective as missing several due to careless mistakes andfrom not understanding. They also pointed to the fact that if students are just per-forming meaningless algorithms, they have no basis for self-checking their work.Any answer would seem reasonable. However, when the problems have meaning,the students are more likely to realize their mistakes, thereby improving theiraccuracy.
As the discussion continued, it was apparent that many of the preservice teach-ers were beginning to reconceptualize what it means to teach mathematics for un-derstanding. Their goals for their classrooms were shifting from a focus on correct
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procedures to an emphasis on students’ understanding in the context of their math-ematical activity. This shift was based on their experience at having developedconceptual understandings related to place value and multidigit addition and sub-traction. It is important to note, however, that this process was supported bynumerous tools including the instructional tasks, the preservice teachers’ in-scriptions, and the classroom norms for argumentation. In addition, their prior ex-perience provided a window into what an exploration of mathematics could entail.Their reflection on this experience then provided an opportunity for me to makemy pedagogical decisions explicit. As a result of this extensive process, thepreservice teachers were able to begin to reconstruct their image of what it meansto teach mathematics.
ACKNOWLEDGMENTS
The analysis reported in this article was supported by the National Science Foun-dation under Grant REC-0135062 and REC-9814898 and by the Office of Educa-tional Research and Improvement under Grant R305A60007. The opinions ex-pressed do not necessarily reflect the views of the Foundation or OERI. I amgrateful to Janet Bowers and Marty Simon for their constructive critiques of a pre-vious draft of this article.
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