Teachers' Knowledge of Children's Mathematics after Implementing a Student-Centered Curriculum

SUSAN B. EMPSON and DEBRA L. JUNK

TEACHERS’ KNOWLEDGE OF CHILDREN’S MATHEMATICSAFTER IMPLEMENTING A STUDENT-CENTERED CURRICULUM

ABSTRACT. Our study investigated the knowledge 13 elementary teachers gained imple-menting a student-centered curriculum in the context of district-wide reform. Participantscomprised all the teachers in grades three, four and five at a single elementary school. Webelieved that investigating teachers’ responses to fictional pedagogical scenarios involvingnonstandard algorithms would yield insights into critical components of their knowledgebase. We looked in particular at teachers’ knowledge of children’s mathematics. We foundthat teachers were in the midst of creating a knowledge base focused on children’s math-ematics and grounded in knowledge about alternative conceptual trajectories through theelementary curriculum. Teachers’ knowledge of nonstandard strategies supported by thecurriculum materials was stronger and more coherent than their knowledge of students’novel nonstandard strategies. Strong mathematical knowledge was not necessarily asso-ciated with strong knowledge of children’s mathematics. Teachers’ thinking varied by atopic’s treatment in the written curriculum materials used, suggesting implementation ofthe curriculum as a source of learning.

KEY WORDS: children’s mathematics, elementary, teacher knowledge, teacher learning

INTRODUCTION

As educational reform in the U.S. grows, greater numbers of teachersthan ever before have become involved in teaching curriculum programsthat require new kinds of mathematics knowledge (Ball, Lubienski &Mewborn, 2002; Floden, 1997; Richardson & Placier, 2001; Sherin, 2002).Some research has found that implementing these programs providessignificant opportunities for teacher learning, especially when accom-panied by professional development (Featherstone, Smith, Beasley, Corbin& Shank, 1995; Hull, 2000; Remillard, 2000; Sherin, 1996). But, witha few exceptions, teachers’ knowledge in these contexts remains largelyunexplored. Our study investigated the knowledge teachers gained throughimplementing an innovative, student-centered curriculum, and examinedthe implications of this knowledge for characterizing a knowledge base forteaching mathematics.

Because we start with the premise that teachers’ knowledge is situatedin practice (Ball et al., 2002), we examine teachers’ knowledge in the

Journal of Mathematics Teacher Education 7: 121–144, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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context of teachers’ thinking about children’s mathematical thinking. Wefocused on teachers’ knowledge of children’s mathematics because, as abody of information, it epitomizes some of the major characteristics anddilemmas of the new knowledge policy-setting documents (e.g., NationalCouncil of Teachers [NCTM], 2000). For example, teachers are requiredto know and be able to form relationships among a variety of commonlyused student-generated strategies for multidigit problems in addition to theusual standard algorithms. Teachers also need to be prepared to make senseof both common and novel strategies during lessons, as well as before andafter lessons (Even & Tirosh, 2002).

The research took place in the context of district-wide1 reform in math-ematics instruction in Austin, Texas (Batchelder, 2001). This reform wasa result of a grant from the National Science Foundation to implementa student-centered, Standards-based (NCTM, 1989, 2000) mathematicscurriculum, Investigations in Number, Data, and Space (TERC, 1995–98), at the elementary level. By “student-centered”, we mean a curriculumdesigned to elicit and build on students’ ways of understanding math-ematics. By “Standards-based”, we mean a curriculum designed to beconsistent with the vision for mathematics instruction put forth by NCTM(1989, 2000).

Our questions included: (1) What knowledge do teachers who haveimplemented a student-centered curriculum use to make sense of students’non-standard strategies? (2) How might teachers’ acquisition of thisknowledge be linked to the use of the new curriculum materials?

CONCEPTUAL FRAMEWORK

In this study, we looked specifically at teachers’ knowledge of students’non-standard strategies for multidigit operations. We focused on teachers’knowledge of children’s mathematics, but also considered teacher beliefs,as they were expressed in teachers’ responses about children’s mathe-matics. We defined non-standard strategies as student-generated strategiesinvolving number relationships and informal as well as formal models ofoperations (e.g., Carpenter, Franke, Jacobs, Fennema & Empson, 1998;Fuson, Wearne, Hiebert, Murray, Human, Olivier, Carpenter & Fennema,1997). For example, a non-standard strategy for adding 47 and 26 couldinvolve adding 40 and 20 to get 60, 7 and 6 to get 13, then combining thoseintermediate sums to get 73. Research has amply documented that childrenreadily generate strategies to solve problems involving number operations(Kilpatrick, Swafford & Findell, 2001). Further, compared to standardalgorithms, students’ non-standard strategies for multidigit operations are

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for the most part conceptually, rather than procedurally, driven (Fusonet al., 1997; Kamii, 1994). Thus, we reasoned, teachers’ knowledge ofstudents’ nonstandard strategies could provide information about teachers’conceptual knowledge of the mathematics of multidigit operations forteaching.

In this section, we review briefly research on teacher knowledge andbeliefs, then describe the framework we use to conceptualize teachers’knowledge for teaching mathematics.

Research on Teachers’ Knowledge

Prior research on U.S. elementary teachers’ subject-matter knowledge, inparticular, has found that many teachers hold narrow, procedural under-standings of algorithms and have difficulty generating multiple, flexiblerepresentations of operations beyond addition and subtraction (Ball, 1990;Kennedy, 1990; Ma, 1999; Post, Harel, Behr & Lesh, 1991; Simon, 1993).For instance, teachers in a National Center for Research on Teacher Educa-tion study (NCRTE, 1992) did not know why zeros should be used as“place holders” in the standard algorithm for multidigit multiplication, andU.S. teachers in Ma’s (1999) study generated few if any models for frac-tion division. Mewborn (2000) reported similar results across a range ofresearch on pre-service and in-service teachers’ subject-matter knowledgeat all levels. These findings suggest that many teachers do not understanddeeply the mathematics they teach. Much of this research, however, hasfocused on teachers who are learning to teach, or those who teach with afocus on computational fluency, rather than those who have been involvedin instructional reform.

In contrast to studies documenting deficits in teachers’ knowledge,some research suggests that teachers’ knowledge can become deeperand more flexible as teachers implement programs designed to developstudents’ conceptual understanding of mathematics, as opposed to compu-tational fluency alone (Featherstone et al., 1995; Hull, 2000; Sherin, 1996;Schifter, 1998; Sowder, Philipp, Armstrong & Schappelle, 1998). Forexample, Hull (2000) found that after two years of implementing a student-centered, standards-based middle-school curriculum at seven sites acrossTexas, teachers’ knowledge of proportional reasoning increased, in somecases dramatically. Written pre- and post-assessments documented growthin accuracy, number of strategies, and variety of problem representationsin teachers’ knowledge of proportional reasoning. Schifter (1998) andSowder and colleagues (1998) have argued that teachers who concentratedon understanding children’s thinking may also develop broad, deep mathe-matical understanding. It is not clear, however, how much or what kind

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of knowledge for teaching mathematics teachers may learn under thesecircumstances.

Research on Teachers’ Beliefs

Teachers’ beliefs about the nature of mathematics and children’s learningplay a significant role in teaching (Thompson, 1992). Several researchershave argued that knowledge and beliefs must both change if teachers are tochange how they teach (Fennema & Nelson, 1997), although it is not clearif changes in beliefs precede changes in knowledge, or vice versa (Franke,Carpenter, Levi & Fennema, 1998; Philipp, Clement, Thanheiser, Schap-pelle & Sowder, 2003). In any case, these researchers have emphasized,changes in beliefs accompany changes in teacher knowledge.

Framework for Teachers’ Knowledge

Researchers have proposed that knowledge for teaching mathematics hasseveral important characteristics. First, knowledge of mathematics is inte-grated with knowledge of teaching and learning mathematics (Ball et al.,2002; Philipp et al., 2003; Schifter, 2001). We suggest, in particular, thatteachers’ knowledge of concepts, procedures, and mathematical practicesneeds to be integrated with knowledge of children’s thinking. We refer tothis integrated knowledge as knowledge of children’s mathematics.

Consider the topic of division as an example of how knowledge ofmathematics can be integrated with knowledge of children’s thinking.Mathematicians define the division of a by b in terms of the multiplicationof a by the multiplicative inverse of b, 1/b, if it exists.2 This definitionmakes no distinction between dividing into b groups and groups of b.However, informally, division can be defined in several ways, includingsituations in which a total needs to be partitioned into a given numberof equally-sized groups, or into groups of a given size (Greer, 1992).Although this distinction does not exist in formal mathematics, childreninitially interpret these different situations as two distinct processes, andlearn only later that they are united by a single mathematical operation.Knowledge about how these two informal models of division developin children’s early problem solving can be used by teachers to integratechildren’s informal mathematics with more formal mathematics.

Second, Ma (1999) argued that highly developed teacher knowledge isbroad, deep, and thorough. In her framework, breadth of knowledge refersto connections across several topics of similar conceptual complexity;depth of knowledge refers to connections across the longitudinal devel-opment of a single topic; and thoroughness refers to connections acrossseveral key topics and their longitudinal development. Depending on the

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content of the knowledge base, however, we suggest there are differencesin the nature of these connections within and across topics. Whereas Ma’s(1999) content framework emphasizes the structure of fundamental math-ematics, ours emphasizes what we refer to as children’s mathematics.3

For example, building on Bruner’s (1960) notion of structure, Ma (1999)posited that mathematics knowledge for teaching consists of proceduraltopics, conceptual topics, and basic principles. Basic principles, such as“inverse operations”, and “rate of composing a higher value unit”, exem-plify mathematical structure, because they afford mathematical connec-tions to a multitude of related topics, and they represent elementary butpowerful ideas that will be revisited in advanced mathematics.

Thus, we suggest that teachers’ knowledge of children’s mathematicsshould also be broad, deep, and thorough. Building on the notion thatthe integration of children’s thinking and mathematics provides a criticalfoundation for teachers’ knowledge, we use breadth to refer also to knowl-edge of a variety of child-generated strategies and their interrelationshipsfor a given type problem; depth to refer to mathematical justification fora given child-generated strategy; and thoroughness to refer to the integra-tion of knowledge of children’s thinking for a given set of problems in atopic area with knowledge of the longitudinal development of children’sthinking about that topic.

METHOD

Participants

We interviewed 13 teachers about what they had learned while imple-menting Investigations in Number, Data, and Space (TERC, 1995–1998), focusing in particular on how teachers made sense of students’nonstandard strategies for multidigit operations. Participants comprised allthe teachers in grades three, four and five at a single elementary school,located in an urban district in Texas that had adopted Investigations as itselementary mathematics program. The implementation of Investigationswas part of a larger district initiative, described above (Batchelder, 2001).All participants had been teaching Investigations for one or two years atthe time of the study. They represented a range of experience: four of theteachers were first-year teachers, and five had 15 or more years’ experi-ence. One teacher had begun her career before the advent of New Math,and so had experienced in one way or another every major curriculummovement conceived in the past 40 years. The student body was demo-

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graphically typical of the district; over half of the students came fromlow-income families.

Teachers had participated in up to four district-led workshops eachschool year on implementing Investigations in Number, Data, and Space.These workshops were designed to help the teachers become familiar withthe organization of the text materials and learn the mathematics by doingand discussing activities (Batchelder, 2001). There was little examinationof student work beyond what was presented in the curriculum materialsdialogue boxes.

Instrument

All teachers were interviewed once. The interview consisted of five ques-tions, beginning with an open-ended question about what and how teachershad learned implementing Investigations (Table I, Question 1). The inter-view ended with an open-ended question about which Investigations unitsteachers thought had contributed the most to student and teacher learning(Table I, Question 5). These two questions were designed to elicit teachers’reports of knowledge gained and, in particular, their beliefs of the role ofimplementing Investigations curriculum materials in their learning.

TABLE I

Interview Questions

Type of question Question (without probes)

1. Open-ended question aboutlearning

(Given prior to interview) I would like you to describea time when you learned something as a result ofteaching Investigations. I’m interested in hearingabout something you learned that stands out in yourmind as being important to teaching mathematics. Iwould like to hear what you learned, and how youthink you learned it.

2. Multidigit multiplicationscenario: Commonnostandard strategies

One goal of Investigations is to get students to solveproblems in many different ways. Suppose that youwere teaching multidigit multiplication. What are atleast three different strategies that children might useto solve 18×25?

3. Multidigit division scenario:Novel nonstandard strategywith mistake

A student was solving 144÷8 (show card). She said,“I know, I can just split it in half. So I will keepdividing by 2. I need to do that 4 times, since2+2+2+2 is 8.” As she talked, she wrote:

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TABLE I

Continued

Type of question Question (without probes)

How would you respond to this student?

4. Multidigit subtractionscenario: Novelnonstandard strategy

Your class is working on subtraction with regrouping.A student says she has come up with a simple method.She explains that 6–9=–3 (“negative 3”), and 30–10=20, and –3+20=17. Does this strategy makesense? Why or why not?

5. Stimulated recall of learning (Provide list of appropriate grade units.)

a) Which units do you think helped your studentsreally learn important mathematics?

b) Which units, if any, do you think helped YOU learnmore about mathematics?

To focus the inquiry, we posed three scenarios in which teachers wereasked to speculate about how they would respond to students’ nonstandardstrategies during instruction (Table I, Questions 2–4). We modeled ourscenarios on items from A Study Package for Examining and TrackingChanges in Teachers’ Knowledge, published by NCRTL (Kennedy, Ball& McDiarmid, 1993). The first scenario (Table I, Question 2) askedteachers to generate at least three strategies for multidigit multiplica-tion that students might use. This question was designed to measure thebreadth of teachers’ knowledge of nonstandard strategies, their conceptualunderstanding of those strategies, and their knowledge of their relativedevelopmental sophistication. If one of the strategies listed included the

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standard U.S. algorithm for multidigit multiplication, the teacher wasasked to describe an additional strategy.

The next two scenarios presented nonstandard strategies for teachersto evaluate. They represented the kinds of scenarios a teacher in a student-centered instructional setting could encounter, and assessed the knowledgeteachers would to bring to bear to make sense of such strategies. Specific-ally, the second scenario (Table I, Question 3) was designed to investigatethe depth of knowledge teachers used to interpret a student’s novel divi-sion strategy, including whether teachers distinguished between additiveand multiplicative composition in division, and the types of connectionsteachers made between informal and formal models of division. Thestrategy was based on a an intuitive model of division based on repeatedhalving, which requires keeping track of the number of partitions madeeach time. In the scenario given to the teachers the student repeatedlyhalves 4 times which results in 16 partitions rather than 8. This strategywas consistent with a partitive model of division, because it involved split-ting 144 into 8 equal groups – in contrast to a measurement model, whichwould have involved finding how many groups of 8 in 144.

The third scenario (Table I, Question 4), which we adapted froma questionnaire item in the NCRTE (1992) study package, involved astudent’s nonstandard strategy for subtraction with regrouping that incor-porated negative numbers. The item was intended to assess how teachersdetermined the validity of a novel nonstandard strategy for a specificnumber combination, and, more generally, for any number combina-tion. In designing all the items, we included strategies that we knew,through experience or empirical research, that elementary mathematicsstudents could generate independently of direct instruction. All three scen-arios were designed to assess the connections teachers made betweenstudent-generated strategies and mathematics.

Data Collection and Analysis

We interviewed all 13 teachers in March of the school year. Six of theteachers were just finishing their first year of implementing Investigations,and the rest were finishing their second year. All the interviews were audiorecorded and supplemented by notes. Most took place during teachers’planning periods and lasted about 45 minutes.

We transcribed all interviews and analyzed them by first looking forthemes, then refining those themes into codes. We revisited and revisedour codes several times to establish consistency within and across teachers.Codes were based on well-established findings in the literature concerningchildren’s thinking, Investigations-based strategies, and our own inter-

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pretations. For example, codes addressed the kinds of strategies teachersdescribed, models of operations, and nature of justification. Half of thedata was coded by both of us to ensure reliability. There were veryfew disagreements about our coding of problem-solving strategies thatwere well documented in the literature, such as using partial products tocompute a multidigit product. Disagreements tended to concern details ofteachers’ thinking, such as whether the use of paired addends was a kindof doubling strategy, and were easily resolved through discussion.

Limitations

Clearly there are limitations to using a single interview to assess teachers’knowledge. Although we made use of teachers’ self reports of theirlearning activities, for instance, we were unable to assess with confidencehow teachers’ knowledge changed. Because we collected no classroomdata, we had no information about how teachers would actually respondto the kinds of strategies emulated in the interview scenarios. Nonethe-less, there is precedent in research on teacher knowledge for using itemsformatted as were our interview items (Ball, 1990; Kennedy et al., 1993;Ma, 1999). Although the interview represents a single point in time, ourinterview protocol provided multiple opportunities for teachers to thinkabout the fictional strategies, and consequently we were able to docu-ment detailed, plausible responses. We do not claim that these responsescharacterize adequately each teacher’s knowledge, but we do claim that,collectively, they yield a valid analytic depiction of a knowledge base indevelopment for teaching mathematics.

FINDINGS

In this section we report and present evidence for our main findings.These findings are organized by broad claims about teachers’ knowledge,including the extent to which it was integrated, and its relationship tocurriculum use. We discuss areas in which teachers’ knowledge of mathe-matics was more and less integrated with knowledge of children’s thinking,the role we believe the curriculum played in the development of thisknowledge, and teachers’ beliefs about children’s mathematics.

Analysis of Multidigit Multiplication Elicited Integrated Knowledge ofChildren’s Mathematics

Teachers’ knowledge of children’s mathematics was broadest and deepestin the topic multidigit multiplication. The evidence for this claim includes

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the variety of nonstandard strategies teachers generated for multidigitmultiplication, their explanations of these strategies, and, for someteachers, the developmental progression among the variety of strategiesthey identified. All the strategies that teachers described were consistentwith children’s invented strategies documented elsewhere (e.g., Baek,1998), and several teachers framed their comments in terms of what theyhad observed children doing. Together, these two observations suggest weelicited teachers’ knowledge of children’s mathematical thinking, ratherthan only knowledge of mathematics.

When teachers were asked to generate three different strategies studentswould use to solve 18×25, all but one of the 13 teachers described at leastthree nonstandard strategies. As a group, teachers described six distinctstrategies beyond direct modeling and repeated addition (not includingthe standard algorithm for multiplication), suggesting breadth in teachers’knowledge of children’s mathematical thinking. These strategies also drewon conceptual features of the number system and the operation of multi-plication, indicating an integration of teachers’ knowledge of children’sthinking with knowledge of mathematics. We describe these strategies inorder of increasing mathematical sophistication.

The most basic strategy, after direct modeling and repeated addition,involved chunking the addends or successively doubling. Eight teachersdescribed how children would chunk 25s into 50s or 100s, then add,and another teacher said children would know that 10 25s is 250, thenrepeatedly add onto that. These teachers all mentioned the use of 25 as a“landmark” number, that is, an important “number we can use . . . to helpus tell where we are when we are counting or calculating with numbers”(Russell & Rubin, 1997, p. 27). Several teachers pointed out that childrenmight relate these numbers to money, facilitating chunking strategies.

Four teachers reported that children might round one of the factors toa multiple of 10 to make the multiplication easier, then compensate bysubtracting the extra factors. For example, Ms. Barill4 said “they wouldround up 18 to 20 and say, ‘20 times 25 is 500. Now I know that 20 is 2more than 18, so I need to take away 2 times 25 which is 50’ ”.

Finally, a majority of the teachers described breaking the multidigitmultiplication down into two, three, or four partial products, making useof the base-ten structure of the numbers and the distributive property.Using the terminology found in Investigations, teachers referred to thesestrategies as “cluster strategies”. For example, Ms. Rojas computed fourpartial products:

Okay, so I know that we do the 10 times 20, the 8 times 20 and then they would do the10 times 5 and the 8 times 5 (see Figure 1) . . . . And even in some of the children who

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may mathematically be even at a lower stage, if 20 still seems even bigger for them, they’llbreak that down even smaller and do 10 times 10, 10 times 10 twice as opposed to the 10times 20 . . . . They would go ahead and multiply 10 times 20 and get the 200. And they’ddo the 8, 2 times, ah, the 20 times 8, excuse me, would be 160. And then they would do10 times 5, would get the 50, and the 8 times 5 would get 40. And then they would turnaround and add ‘em all up. Which is what I am going to attempt to do. And get 450.

Figure 1. Ms. Rojas’s nonstandard algorithm for multiplication.

Though Ms. Rojas’s strategy is isomorphic to the steps in the standardalgorithm for multiplication, her explanation of the strategy is couched interms that relate to how children might actually work through this problem,and demonstrates a flexible understanding of multiplication as an oper-ation. Specifically, the conceptual organization of the strategy suggeststhat she, along with the eight other teachers who described this familyof strategies, understood the implicit use of the distributive property tocompute partial products.5 This finding is in stark contrast to prior researchdocumenting teachers’ limited understanding of “why the numbers ‘moveover’ [and] what the number in the partial product on the second rowmean[s]” (NCTRE, 1992, p. 30).

Several teachers went beyond identifying a variety of strategies todescribe longitudinal connections among children’s strategies. Theseteachers appeared to have in mind a model for the genesis of multidigitmultiplication in children, from repeated addition, to chunking, to theuse of partial products in cluster strategies. Their observations were inaccord what has been documented about the development of children’smultidigit strategies elsewhere (Ambrose, Baek & Carpenter, in press;Baek, 1998). This literature describes strategies involving chunking asthe children’s first move away from repeated addition toward efficiencyin solving multidigit multiplication.

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Essentially, these early invented strategies involve decomposing themultiplicand additively, and using that decomposition to chunk the multi-plier. As teachers here reported, children do not necessarily choose effi-cient chunks, although it is the teachers’ goal for students that they movetoward using cluster strategies based on multiplicative decomposition offactors. Several of these observations extended the information providedby Investigations, and so suggest that some of the teachers were engagedin extending their models of children’s thinking based on practical inquiry(Franke, Carpenter, Levi & Fennema, 2001; Richardson, 1990).

Analysis of Novel Division and Subtraction Strategies Elicited LessIntegrated Knowledge of Children’s Mathematics

When asked to interpret students’ novel nonstandard strategies, teachersdrew on knowledge that was, in comparison to knowledge of multi-digit multiplication, less integrated. In particular, teachers who wereable to make sense of students’ novel strategies within the constraintsof the interview tended to call on knowledge of mathematics discon-nected from knowledge of children’s thinking. Individual exceptions to thisfinding, however, provide instructive examples of teachers’ knowledge ofchildren’s mathematics. Evidence for these claims comes from teachers’responses to students’ strategies for multidigit division and subtraction.

Teachers found the division item more difficult than the multiplicationitem. When presented with a student’s novel strategy for dividing 144 by 8(see Table 1 above), which contained a mistake, only four out of the thir-teen teachers were able to make sense of the strategy within the constraintsof the interview. Three of these four teachers did so using explanations thatdrew on formal mathematics concepts, but did not connect with knowl-edge of children’s thinking. In particular, they recognized the student haddecomposed the divisor additively, when it should have been decomposedmultiplicatively, a significant insight (Simon, 1993). For example, Ms. Lee,who described herself as “a very mathematical person”, concluded that

[Decomposing the divisor multiplicatively] is a valid way to do it . . . if you’re at the pointwhere you can break down the number . . . where you can say [if dividing by 12], 12 is 2times 2 times 3. Yeah. Then it will work. (Solves 24 divided by 12 by dividing successivelyby 2, 2 then 3.) So if they understand factoring, then it’s a valid way to do it, but if they’renot at that point, then I would stop with, ‘It didn’t work.’

Ms. Lee further noted that if breaking the divisor down additively – as thefictional student had – worked, then dividing 144 by 7 then 1 (i.e., 7+1instead of 2+2+2+2) should also yield the correct answer. Other teachersarrived at a conclusion similar to Ms. Lee’s by referring to inverse opera-tions. Ms. Nichol worked her way back up the chain of repeated division by

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two by repeatedly doubling the divisors, and discovered that the net resultwas that the student had divided by 16 instead of 8. Ms. Devine noted thatthe student

kept cutting in half, in half again, in half again, until you got two plus two plus two plustwo, when in essence that’s not really what you’re doing. This is really two times twotimes two . . . They have to know that division and multiplication are, you know, together.They’re opposites.

These teachers’ explanations demonstrated critical understanding of math-ematics, but little integration of mathematics and children’s thinking. Theyused the definition of division as the inverse of multiplication correctly topoint out why the strategy does not work. However, this type of knowledgedoes not provide a way to understand the possible origin of division in thischild’s thinking.

Ms. Tilden was the only teacher to use a partitive model of division –in which the goal is to partition the total into a given number of groups– consistently throughout her response. Her response, more than the otherteachers, preserved the integrity of the child’s strategy, because it referredto the steps the child had already taken to solve the problem, and reinter-preted them in light of what could have been the child’s goal – to partition144 into 8 groups. Talking about the first division by two, Ms. Tilden said,“Okay, there are two 72s in 144. And I would just ask her, ‘Well doesthis mean there are two 36s in 144? Or four 36s in 144?’ And I meanat this point I would probably be asking her to draw pictures, you know,have some kind of visual to show me what that would look like” (drawsFigure 2). Then later: “Maybe she’s saying there are two 18s in 36, sohow many 18s are there in 144? I’m just trying to figure out how theconversation would work . . .”.

Figure 2. Ms. Tilden’s model of student’s division strategy.

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This explanation built on the child’s mathematics. Ms. Tilden’sapproach to helping the student map out all of the groups that were createdwith each successive halving of the partial quotient provided a way toconnect repeated halving and partitive division, an informal model that isreadily accessible to students. To make this connection, the student neededa way to keep track of the number of groups created by halving, since theprocess caused the number of groups to double each time, not increase bytwo, and should have stopped when eight groups have been created (thepartitive meaning of 144 ÷ 8).

Ms Tilden’s response suggests the power of children’s mathematicsas a knowledge base for teaching. Rather than treating the mistake as amisconception to be suppressed, Ms. Tilden found a way to build on it.Interestingly, unlike some of the other teachers, she did not discuss themistake in the child’s partitioning of the divisor. It is not clear, however,whether the distinction between multiplicative and additive partitions issalient in her interpretation of the strategy, since the mistake could beattributed to not keeping track of the number of groups, instead of whetherto add or multiply the twos. In other words, a child who is using repeatedhalving to divide may not be thinking ahead about when to stop.

The rest of the teachers used neither mathematical principles norchildren’s thinking to make sense of the strategy, and consequently couldnot explain why dividing by two four times in succession was not equiva-lent to dividing by eight. For example, Ms. Puma said she did not considerrepeated halving to be division – “she’s splitting it in half each time, butit’s not really divided” – suggesting she held an isolated understandingof division. Four teachers interpreted division primarily in measurementterms, that is, in terms of the number of eights that would fit into 144.Measurement division is a useful, accessible model of division for youngchildren (Carpenter, Fennema, Franke, Levi & Empson, 1999); however,its use in this context is not consistent with the fictional student’s strategy,and so limits the opportunity for a teacher to engage the child’s emergentthinking about the relationship between repeated halving and division.

The subtraction strategy (see Table I above) was easier than the divisionstrategy for teachers to interpret, yet, as with the division strategy, teachers’knowledge of mathematics and the extent to which it was integrated withknowledge of children’s thinking influenced the quality of the explanationsteachers offered.

Although most of the teachers were convinced of the strategy’s validityafter working a few well-chosen examples, one, Ms. Lee, constructed ajustification for the strategy. In reference to 36–19, Ms. Lee said, “It’sbasically the inverse of regrouping. Instead of regrouping and taking a 10

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from here [the tens’ place], you are subtracting this [the ones’ column] andgetting negative three, which means you’re going to have to eventually . . .

take three away from [the tens’]”. In other words, Ms. Lee argued that thenegative difference in the ones’ place represented a deficit to be accountedfor in the tens’ place.

Additionally, Ms. Tilden and Mr. Garcia also argued based on principlesnot dependent on specific number combinations that the strategy general-ized. For instance, Mr. Garcia said it should work for combinations otherthan the ones he tried because

what you are dealing with is negative and positive numbers in essence and you are takingit one place at a time. You are dealing with the ones place and then coming up with ananswer and then dealing with the tens’ place and coming up with an answer . . . . Becausewhen you can deal with owing or negativeness, you realize that numbers can be owed aswell as had.

In essence, Mr. Garcia argued that, in problems that involve column addi-tion or subtraction, one can compute each column independently of theothers, and combine the partial sums or differences at the end. The strategyof treating ones and tens separately, then combining the results, is anapproach that has been noted in children’s strategies for addition as wellas subtraction (Carpenter et al., 1999; Fuson, 1992).

Curriculum Materials Influenced the Development of Teachers’Knowledge of Children’s Mathematics

Teachers’ knowledge of mathematics and children’s thinking was rela-tively well integrated for multiplication, but much less so for division, arelated topic, and subtraction. We speculate that differences in the generalquality of teachers’ responses to the multiplication item, compared to thedivision item in particular, may be explained in terms of the opportuni-ties for teacher learning created by teaching the curriculum. Teachers’self-reports of their learning and analysis of the curriculum materialscorroborate this claim.

First, most teachers reported that they had learned nonstandardstrategies for multidigit multiplication as a direct result of teaching Investi-gations – from interacting with students, preparing for lessons, or partici-pating in professional development workshops. Seven teachers specificallydescribed how they had increased their knowledge of multidigit multiplic-ation beyond knowing how to execute the standard algorithm. Each onereported that, before teaching Investigations, he or she knew only one wayto multiply multidigit numbers. Mr. Garcia, for example, said:

I was always very quick mathematically, mentally. And I would do it just in the way that Iwas taught . . . But through clustering what you can do is break a number down into more

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familiar parts and numbers that are easier to work with . . . . By watching kids and doingwhat they do I’ve become better at breaking down numbers and understanding how to dealwith large numbers.

These teachers’ self reports were corroborated by their ability to generatetwo or more examples of students’ nonstandard multidigit multiplicationalgorithms later in the interview. Further, their descriptions of nonstandardstrategies for multidigit multiplication tended to be the most mathematic-ally detailed of any the teachers gave.

Second, in our analysis of Investigations units, we noted differencesin the treatment of multidigit multiplication and division in grades fourand five, where these topics are central to the curriculum. The activities incurriculum units such as “Arrays and Shares”, “Landmarks in the Thou-sands”, “Packages and Groups”, “Mathematical Thinking at Grade Five”,and “Building on Numbers You Know” develop multidigit multiplicationthrough skip counting, breaking apart numbers into more familiar parts,solving clusters of related problems,6 building towers of multiples, andusing landmark numbers, such as multiples of ten, to estimate products.These activities appear repeatedly and with increasing sophisticationthroughout these curriculum units, yielding a coherent, developmentaltreatment of multidigit multiplication. In contrast, there is less overallemphasis on the development of multidigit division. Further, althoughboth partitive and measurement interpretations of division are describedand illustrated, the kinds of strategies students learn for multidigit multi-plication, such as the use of skip counting and multiples towers, favora measurement interpretation when translated to multidigit division. Thecurriculum also encouraged teachers to reformulate numeral-only divisionsentences such x ÷ y as “how many ys are in x”, a further reinforce-ment of the measurement interpretation. The net result of this differentialtreatment of multiplication and division appeared to be the creation oflimited opportunities to learn about children’s informal partitive strategiesfor division.

Teachers Gained New Beliefs About Children’s Mathematics

Overall, 10 out of the 13 teachers expressed what they described as newbeliefs about children’s mathematical thinking that were a direct resultof teaching Investigations. The majority of these beliefs – expressed byseven teachers – had to do with students’ ability to solve problems ontheir own and to generate new mathematics. Mr. Jaimez noted that “kidsdo have an intuitive sense of math” and “can really be in charge of theirmathematical learning”. Ms. Capa said, “We’ve always thought, well,we’re the ones that give information. But they [the children] give me

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information . . . They bring things to my mind”. These beliefs were alsoevidenced in teachers’ responses to the instructional scenarios, especiallyin how teachers valued the nonstandard strategies. Two teachers describedspecifically how meaning needed to be built for operations and proceduresbefore children learned to compute efficiently.

Teachers valued students’ novel strategies in particular, even when theydid not have the knowledge to assess thoroughly the mathematical orpedagogical viability of these strategies. Although these kinds of beliefsdo not require sophisticated domain knowledge, they also do not engagethe potential in students’ invented strategies to ground new mathema-tical understanding in children’s mathematics. For example, although twoteachers emphasized the importance of making productive use of students’mistakes in other parts of the interview, neither teacher did so for thedivision strategy, which contained a mistake; lack of specific knowledgeof children’s mathematics may have limited teachers’ ability to act on thisbelief is this situation.

The data suggest teachers had acquired a predisposition to elicitstrategies from children, and to expect a variety of responses. Theybelieved quite strongly that they, as well as their children, were bene-fiting from this kind of approach to mathematics. All these beliefs wereconsistent with instructional frameworks based on current mathematicslearning research (e.g., Hiebert, Carpenter, Fennema, Fuson, Wearne,Murray, Olivier & Human, 1997), and so suggest movement toward beliefsthat reinforced teachers’ use of children’s mathematics as a foundationfor instruction and as a context for their own learning. However, withoutsubstantial knowledge of children’s mathematics, it may be difficult, if notimpossible, fully to actualize these kinds of beliefs.

DISCUSSION

We began this study with two questions: (1) What knowledge do teacherswho have implemented a student-centered curriculum designed to supportteacher learning use to make sense of students’ nonstandard strategies?and, (2) in what ways might teachers’ acquisition of this knowledge belinked to the use of the new curriculum materials?

We found that, collectively, the teachers exhibited a fair amount ofknowledge of children’s mathematics in a single topic area, although thestory is not one of uniformly high knowledge across topic areas. Teachershad broad, and in some cases deep, knowledge of nonstandard strategiesfor multidigit multiplication represented in the curriculum materials, butnot of less common strategies. We also found some teachers’ knowledge of

138 SUSAN B. EMPSON AND DEBRA L. JUNK

mathematics was disconnected from knowledge of children’s thinking, andthis disconnection influenced their hypothesized responses to children’sthinking. These teachers treated a mistake in thinking about division as awrong idea to be corrected. One teacher, using knowledge of children’sthinking about division, saw, within the child’s mistake, productive, albeitinformal, thinking about division, and described an interaction that couldbuild on her thinking.

We also found that teachers’ knowledge of children’s mathematicsappeared to be, in large part, a result of implementing Investigations inNumber, Data, and Space. Knowledge of children’s mathematics was,for these teachers, most robust in multiplication, a topic that was treatedexplicitly in the curriculum materials in terms of its mathematical organi-zation and the development of children’s thinking. We speculate thatthis attention to children’s mathematical thinking allowed the curriculumwriters to design tasks that elicited fairly predictable patterns of reasoning,and to alert teachers to the variety and meanings of those strategies. Onceteachers realized their students could indeed generate their own strategiesto solve problems, it appears that they were motivated to understand andextend these strategies as much as possible. Several teachers, such asMs. Rojas, related personally to the mathematical empowerment they felttheir students were experiencing from implementing the curriculum andinteracting with students.

Significantly, there were limitations in teachers’ knowledge that were aconsequence, we believe, of limitations in teachers’ opportunities to learnfrom the curriculum, as described above, and from each other. Consider,for example, the fact that the Chinese teachers in Ma’s (1999) study whoexhibited Profound Understanding of Fundamental Mathematics (PUFM)had, on average, 18 years of experience teaching in ways that empha-sized conceptual understanding and gave rise to PUFM. During this time,teachers regularly planned together and discussed their teaching with eachother. The teachers in our study, who ranged in total teaching experience,had at most two years’ experience teaching in a way that could, we haveargued, give rise to knowledge children’s mathematics. Thus, experienceswith this type of teaching and opportunities to plan and discuss with peerswere limited for these U.S. teachers in comparison with Chinese teacherswith PUFM.

How might these limitations be addressed? A persistent theme,throughout our findings, was teachers’ reliance on what students said anddid to help them see new ways of thinking mathematically. When teacherswere not sure how to interpret a strategy, they often expressed a desire totalk further with the fictional student. More teachers singled out students

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over curriculum materials and workshops as a source of learning. If dataabout students’ thinking are marshaled so that teachers may debate andanalyze what their students understand, students represent a potentiallypowerful learning resource for teachers.

We suggest, in particular, that teachers’ knowledge of mathematicsneeds to be developed in tandem with knowledge of children’s thinking(Philipp et al., 2003; Schifter, 2001), and that this learning should be basedon teachers’ interactions with their own students when possible (Franke,Kazemi, Shih & Biagetti, 1998). In addition to using knowledge frame-works of common developmental patterns in children’s thinking as a basisfor learning mathematics (e.g., Carpenter et al., 1999), our findings suggestthat instances of novel or mistaken thinking can also serve as catalysts tolearning mathematics. Although teachers struggled, as individuals, withthe division item, for example, they had the collective mathematical andpedagogical knowledge to make sense of it. We can imagine that, had theteachers had opportunities to listen to and to discuss each other’s responsesin a formally organized school-based forum, teachers’ understanding couldhave been measurably enhanced. As teachers become more attuned tostudents’ invented strategies beyond those that are common, collectiveexamination of novel or mistaken strategies could facilitate the kind ofmathematics learning called for in the literature (Ball et al., 2002; Even &Tirosh, 2002).

For many teachers, implementing Investigations fostered a desire tocontinue learning because, we maintain, the curriculum helped themformulate and address problems at the heart of their mathematical workwith children. At the end of the interview, Mr. Jaimez said, “What Iwould like to do is take some time, if I can, to spend some time lookingat these problems and, I mean, I’d love to figure out some more aboutit”. Perhaps it is not surprising that the learning most salient to teachersdid not take place in district-wide workshops. The most pressing instruc-tional dilemmas teachers face with reform have more to do with decidingwhether or not a student’s nonstandard algorithm for multidigit subtractionis legitimate in the classroom context of discussing and extending manyother students’ strategies, than with following a curriculum script (Junk, inprogress).7 Solutions to instructional dilemmas cannot be provided in theform of curriculum materials or knowledge frameworks, although thesetypes of resources play a facilitative role, as the current study suggests.

One could argue that our study shows, once again, how weak U.S.elementary teachers’ knowledge of mathematics is. After all, the teachershad participated in district-sponsored professional development sessionson the implementation of Investigations in Number, Data, and Space four

140 SUSAN B. EMPSON AND DEBRA L. JUNK

– eight times each, yet had difficulty making sense of alternative strategiesfor subtracting and dividing. We reject this interpretation, however,because we identified critical differences between teachers in our studyand teachers in prior studies (e.g., Ma, 1999; Mewborn, 2000). Further, weinterpret teachers’ difficulties as an indication of the complexity of the kindof knowledge teachers are expected to have and to use in order to imple-ment reforms in mathematics education. We have argued, in particular,that teachers’ integrated knowledge of children’s thinking and mathe-matics, in combination with productive beliefs, differs from knowledgeof mathematics in isolation from knowledge of children’s thinking.

CONCLUSION

The most astonishing result of this study is that every teacher noted someaspect of their newfound respect for children’s mathematics capabilities.If nothing else, the teachers had acquired an epistemological dispositionthat acknowledged the existence of a multiplicity of strategies, and wasfounded on a high regard for children’s thinking.

However, one thing that Investigations did not do – and, as a set ofcurriculum materials, cannot be expected to do – was to prepare teachersto detect and build on the mathematical potential in students’ nonstandardstrategies, in addition to what was presented in the curriculum. Thereis a great deal of intellectual work involved in learning to hear and tounderstand the mathematical significance of children’s thinking that goesbeyond acquiring the specifics of knowledge of mathematics. It is anongoing process that requires formal, school-based structures for profes-sional learning be put in place, such as those reported by Ma (1999)for Chinese teachers and by Lewis (2000) for Japanese teachers. Neitherformal preparation in mathematics nor mathematics teaching methodscourses can fully prepare teachers to engage with student thinking in thisway.

There has been a great deal of attention to research on teacher knowl-edge that is focused on teachers’ insufficient understanding of the mathe-matics. In fact, several teachers in this study commented ruefully on theirown mathematics preparation. Noting that she used to hate mathematics asa student, Ms. Rojas said, “But now I’m like, ‘I could’ve been an awesomemath student . . . had somebody taught it this way!’ ” Our study suggeststhat the implementation of an innovative curriculum fostered knowledge,beliefs, and values that were to a notable extent shared by the teachers inone school. The challenge for U.S. schools is to recognize the learning thatteachers like Mr. Jaimez and Ms. Rojas are eager to take on, and to organize

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school-based learning structures that build on teachers’ daily work withchildren.

ACKNOWLEDGEMENTS

We gratefully acknowledge Linda Levi and Corey Drake for their feedbackon earlier drafts of this manuscript. We also thank three anonymous JMTEreviewers and Editor Peter Sullivan for their numerous helpful suggestionsfor improvement. The first author thanks the Department of Curriculumand Instruction, University of Wisconsin-Madison, for support as a visitingscholar during the time the majority of this report was written.

NOTES

1 In the U.S. schools are grouped into administrative and policy-setting units called“districts”. Districts can correspond to cities (so, for example, Austin, Texas has one schooldistrict, the Austin Independent School District), though – especially in the case of largercities – not necessarily (for example schools in San Antonio, Texas are grouped into severaldistinct districts). The decision to use Investigations for elementary mathematics instruc-tion was made at the district, rather than school, level.2 For example, in the set of integers, only two elements have a multiplicative inverse (1and –1). In the set of rationals (all a/b with b not equal to 0), every element but 0 has amultiplicative inverse. One way to think about rational numbers is as an extension of theintegers so that the operation of division is closed (i.e., for all a and b in the set of rationals,with b not equal to 0, a ÷ b yields a unique rational).3 We do not consider these two content frameworks to be mutually exclusive; in fact webelieve they can inform each other. However, we conjecture they orient teachers to funda-mentally different kinds of teacher-student interactions, and perhaps ultimately, differentkinds of student outcomes.4 All teachers’ names are pseudonyms.5 The other 4 teachers did not use strategies that demonstrated understanding of thedistributed property.6 E.g., 3×3, 6×3, 12×3; i.e., problems “arranged in such a way that if you figure out oneof the facts, it may help you find a clever strategy to solve some of the others in the cluster”(“Arrays and Shares”, p. 22).7 Junk, D.L. (in progress). Teaching mathematics and the dilemmas of practice. Unpub-lished doctoral dissertation, University of Texas at Austin.

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Science and Mathematics Education1 University Station, D5705University of Texas at AustinAustin, TX 78712-0382USAE-mail: [email protected]