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Mathematics Teachers' Reasoning AboutFractions and Decimals Using DrawnRepresentationsSoo Jin Lee a , Rachael Eriksen Brown b & Chandra Hawley Orrill ca Montclair State Universityb Knowles Science Teaching Foundationc University of Massachusetts DartmouthPublished online: 13 Jul 2011.

To cite this article: Soo Jin Lee , Rachael Eriksen Brown & Chandra Hawley Orrill (2011) MathematicsTeachers' Reasoning About Fractions and Decimals Using Drawn Representations, MathematicalThinking and Learning, 13:3, 198-220

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Mathematical Thinking and Learning, 13: 198–220, 2011Copyright © Taylor & Francis Group, LLCISSN: 1098-6065 print / 1532-7833 onlineDOI: 10.1080/10986065.2011.564993

Mathematics Teachers’ Reasoning About Fractionsand Decimals Using Drawn Representations

Soo Jin LeeMontclair State University

Rachael Eriksen Brown

Knowles Science Teaching Foundation

Chandra Hawley Orrill

University of Massachusetts Dartmouth

This qualitative study considers middle grades mathematics teachers’ reasoning about drawn rep-resentations of fractions and decimals. We analyzed teachers’ strategies based on their responseto multiple-choice tasks that required analysis of drawn representations. We found that teachers’flexibility with referent units played a significant role in understanding drawn representations withfractions and decimals. Teachers who could correctly identify or flexibly use the referent unit couldbetter adapt their mathematical knowledge of fractions validating their choice, whereas teachers whodid not attend to the referent unit demonstrated four problem-solving strategies for making sense ofthe tasks. These four approaches all proved to be limited in their generalizability, leading teachers tomake incorrect assumptions about and choices on the tasks.

In traditional mathematics classrooms, an emphasis on the use of abstract and symbolic rep-resentations (e.g., the long division algorithm, invert-and-multiply) has dominated. In contrast,standards-based classrooms provide students with opportunities to create and use a variety of rep-resentations (National Council of Teachers of Mathematics [NCTM], 2000). In fact, the NCTM(2000) and others (e.g., Kilpatrick, Swafford, & Findell, 2001) have suggested that engaging stu-dents in mathematics through multiple representations—such as diagrams, graphical displays,and symbolic expressions—is powerful, and flexibility in utilizing representations is considereda primary characteristic of a competent problem solver (e.g., Dreyfus & Eisenberg, 1996). Forteachers, having this flexibility is critical for opening up conversation with students about theirlearning (NCTM, 2000; Goldin, 2002), and provides gateways to abstraction and generalizationas their students develop the ability to mathematize situations (Smith, 2003, p. 264). Despitewidespread agreement that using multiple representations to support student learning is critical,teachers frequently still rely on one type of representation in their classrooms: symbolic notation.Too often, when teachers use other representations, they only use them to illustrate solutions as

Correspondence should be sent to Soo Jin Lee, PhD, Department of Mathematical Sciences, Montclair StateUniversity, NJ, 07043, USA. E-mail: [email protected]

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TEACHERS’ REASONING ABOUT DRAWN REPRESENTATIONS 199

opposed to adapting the representation to support the development of student understanding ofrational numbers (Izsák, 2008). Few teachers use representations to model concepts or activelyproblem solve despite the research that shows that drawn representations can be used to facilitatemoving from concrete to abstract understandings of mathematical concepts (Post, Wachsmuth,Lesh, & Behr, 1985).

The limited use of drawings may be linked to teachers’ beliefs that models might confusetheir students. This argument is somewhat substantiated by research indicating that students havelower success rates on assessment items with drawn representations than those without (Lesh,Behr, & Post, 1987). However, we assert that the underutilization of drawn representations sug-gests a gap in teachers’ mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008).That is, teachers either lack their own understanding of these representations, thereby limitingtheir ability to interact with them, or they lack the pedagogical knowledge necessary to sup-port students in learning with them. Teachers must understand their content, have strategies forworking with students and content, and understand the connections among ideas as well as “therepresentations for and the common student difficulties with particular ideas” (Ball, Lubienski,& Mewborn, 2001, p. 448). To better understand teacher knowledge and to explore our asser-tion about the lack of use of representations, in this study we examined teachers’ strategies forinterpreting drawn representations of fraction and decimal operations by addressing the follow-ing question: How does attention to referent units shape teachers’ problem-solving approachesin multiple choice tasks that require analysis of representations?

The data reported in this study were gathered as part of a larger study of teacher profes-sional development for which an assessment of middle grades teachers’ abilities to reasonwith rational numbers was developed. Items on the assessment were all multiple choice andincluded items taken, with permission, from the University of Michigan’s Learning Mathematicsfor Teaching measures of mathematical knowledge for teaching (Study of InstructionalImprovement/Learning Mathematics for Teaching [SII/LMT], 2004) as well as several itemscreated by the project team (see Izsák, Orrill, Cohen, & Brown, 2010 for a description of theinstrument). As part of the instrument validation effort, the assessment was administered to anationwide convenience sample of 201 teachers. Additionally, 25 interviews were conducted onselect items from the assessment as a means for understanding how teachers were determiningtheir answers on the assessment.

In this study, we analyzed a subset of 12 middle school teachers’ interview responses toassessment items that required them to make sense of area models and number lines for vari-ous operations with fractions and decimals. Our goal was to address two gaps in the literaturearound representations and fraction operations. First, most research done on representations hasfocused on graphs and verbal representations (e.g., Gagatsis & Shiakalli, 2004) rather than ondrawn representations such as area and number line models. Second, while researchers have pro-vided considerable insight into student conceptual understanding of rational numbers (e.g., Behr,Harel, Post, & Lesh, 1994; Behr, Wachsmuth, Post, & Lesh, 1984; D’Ambrosio & Mewborn,1994; Freudenthal, 1983; Moss & Case, 1999; Olive, 1999, 2001; Olive & Steffe, 2002; Steffe,2002; Steffe & Olive, 2010), the literature has yet to provide an equivalent insight into teachers’knowledge of these concepts. As assessment developers and professional developers, we foundthe lack of research in this area to be problematic for guiding our development of both the assess-ment and the professional development. The lack of prior research became particularly noticeablein the intersection of teachers’ conceptual understandings and their abilities to interpret a rangeof drawn representations modeling fraction operations.

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200 LEE, BROWN, AND ORRILL

LITERATURE REVIEW

As a way of understanding what is known about teachers’ abilities to interpret drawn representa-tions, we provide an overview of two areas of the literature: mathematical knowledge for teachingrational numbers and teacher knowledge of representations and problem-solving strategies.

Mathematical Knowledge for Teaching Rational Numbers

Teacher knowledge related to student learning was conceptualized by Shulman (1986) as beingcomprised of a series of interconnected knowledge types. Important for the current study,Shulman described pedagogical content knowledge as knowing about students’ thinking aboutparticular topics, anticipating typical difficulties that students have, and having available a reper-toire of representations that make mathematical ideas accessible to students. Since its proposal,pedagogical content knowledge has been the basis for a plethora of research in mathematicsteaching (cf., Borko et al., 1992; Borko & Putnam, 1996). For instance, Borko and colleaguesexamined prospective teachers’ knowledge in a case study of a middle school teacher, Ms.Daniels. When she was asked by a child to explain why the invert-and-multiply algorithm worksfor dividing fractions, Ms. Daniels could not clearly represent fraction division. Instead, shemodeled fraction multiplication despite having taken several undergraduate mathematics courseswhere she demonstrated the ability to represent fraction division. The case of Ms. Daniels sug-gests that simply developing enough mathematical knowledge to solve problems is insufficientfor supporting student learning.

In a similar line of research, Ball and colleagues (2008) have conceptualized the mathemati-cal knowledge teachers need to support student learning. Their work has identified this body ofknowledge as mathematical knowledge for teaching. While differentiating between pedagogicalcontent knowledge and mathematical knowledge for teaching is outside the scope of the currentstudy, we draw from both frameworks in our thinking about what teachers need to know andbe able to do to support their students in learning mathematics. For ease of reading we refer tothis construct as “specialized knowledge for teaching” throughout this article. A teacher withstronger specialized knowledge is not only able to introduce content but also interpret studentwork and support students in moving from their current mathematical understandings to newunderstandings. From our perspective, specialized knowledge for teaching is critical for inter-preting students’ non-standard approaches to tasks to determine their viability and to supportstudents in developing richer understanding.

For our work, the specialized knowledge for teaching construct frames the assessment devel-opment efforts underway for measuring teacher knowledge. Because we are interested in both thedevelopment of instruments and the use of those instruments to drive professional development,the mathematical knowledge for teaching work (e.g., Ball et al., 2008) is particularly relevant.We build from the belief that there is knowledge critical for teachers that is unique to their pro-fession. Because of this belief, the assessment used in the present study focused on specializedknowledge for teaching. We began building from the Learning Mathematics for Teaching assess-ments of mathematical knowledge for teaching (SII/LMT, 2004) because they are currently themost widely used instruments of their kind. Further, studies with the instruments indicate math-ematical knowledge for teaching is both measurable and matters for student achievement. For

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example, one recent study showed third-grade teacher performance on the Learning Mathematicsfor Teaching assessment of mathematical knowledge for teaching was as strongly correlated tostudent achievement as was student socioeconomic status level (Hill, Rowan, & Ball, 2005).These findings suggest that understanding the nature of mathematical knowledge for teachingand better defining its components could impact student learning in important ways.

In both the Learning Mathematics for Teaching assessment of mathematical knowledge forteaching and our instrument, all the items attempted to address the specialized knowledge forteaching operations with fractions and decimals rather than just capturing whether teachers had away of solving the items. To this end, many assessment items asked teachers to analyze hypothet-ical student approaches to particular items or to determine appropriate approaches for classroominstruction rather than simply asking for solutions to mathematics problems.

Referent Units and Drawn Representations

Exploring the role of the mathematics of quantity has been one strand of research on the develop-ment of rational numbers concepts for children (Kaput, 1985; Schwartz, 1988). This perspectiveemphasizes the strong relationship between numbers and their referent units—that is the link-age of units of measurements and the magnitude of quantities—for understanding relations andoperations. For example, knowing that 2/3∗1/5 can be conceived of as having 1/5 of a unit andwanting to identify 2/3 of that 1/5. The resulting product, 2/15, refers back to the same wholeto which the 1/5 referred. As children encounter the domain of rational numbers, changes inthe nature of the unit largely account for the cognitive complexity entailed in linking meaning,symbols, and operations (Behr, Harel, Post, & Lesh, 1992; Harel & Confrey, 1994; Hiebert &Behr, 1988). In short, reasoning with referent units is both difficult and necessary for children’smathematical development.

Despite the literature indicating the importance of referent units for students’ operations withfractions, few studies have investigated the role of referent units in teachers’ understandingof rational numbers. One study examined two middle school teachers’ knowledge of fractionmultiplication as they interpreted students’ work with drawn representations and used drawnrepresentations to teach fraction operations in the classroom (Izsák, 2008). Izsák found that ateacher’s attention to units was a necessary component of effective instruction for incorporat-ing linear or area representations into teaching. Even though one teacher could attend to unitsin working with fraction multiplication tasks, she did not apply that reasoning as she interpretedstudents’ representations, thereby missing an opportunity to make adequate sense of the students’thinking. Izsák concluded that the teacher’s perspective about students’ learning and drawn repre-sentations all played significant roles in forming her mathematical knowledge for teaching. Basedon this study, it is clear that while insufficient for teaching fraction concepts, having a meaning-ful understanding of referent units is one requisite component of teachers’ specialized knowledgefor teaching. Teachers need to have such knowledge both to engage students in developing theirown understandings and to interpret their students’ reasoning about fractions.

In another study, Izsák, Tillema, and Tunç-Pekkan (2008) explored fraction addition usingnumber lines in one classroom. Based on analysis of interactions between the teacher and onestudent, the researchers asserted that drawings could lead to miscommunication between teachersand students. Differences between the teacher’s and students’ perspectives on fractions led to

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them confounding each other’s understanding of fractions. Together Izsák’s studies (Izsák, 2008;Izsák et al., 2008) suggest teaching fraction addition and multiplication with representationsrequires teachers to have the ability to attend to the appropriate referent unit and flexibly usedrawn representations for solving problems. The studies also suggest that teachers need to knowwhen and why representations should be used as well as how students might use them.

Building from the premise that referent-unit knowledge is a necessary component of special-ized knowledge for teaching and noting that teachers struggle with representations, we soughtto understand how our sample of teachers used referent-unit reasoning in items that use drawnrepresentations. We also identified strategies used in place of referent-unit reasoning across avariety of multiple-choice items. Our study contributes to the field by providing an initial presen-tation of teacher knowledge related to referent-unit reasoning and, more specifically, in providingan analysis of how teachers make sense of fraction concepts using drawn representations. Theunderlying motivation behind this research is to inform the professional development of teach-ers. By understanding how teachers used their knowledge to interpret a series of representations,mathematics educators are better able to develop quality professional learning.

Teachers’ Problem-Solving Strategies with Rational Numbers

Our search of the rational numbers literature yielded three studies of the strategies teachersemployed in solving rational number problems. The first two focused on teachers interpretingfractional quantities from drawn representations (Behr, Helen, Harel, Post, & Lesh, 1997; Harel& Behr, 1995) whereas the third study considered teachers’ responses to word problems (Seaman& Szydlik, 2007).

Of the two studies from the Rational Numbers Project (Behr et al., 1997; Harel & Behr,1995), one focused on practicing teachers, while the other considered preservice teachers who,by definition, have not yet developed high levels of specialized knowledge for teaching becausethey lack the experience needed for developing and refining that knowledge. Behr and associates(1997) provided an analysis of preservice teachers’ problem-solving strategies in answering thequestion “How many piles of sticks are in three-fourths of eight bundles of four sticks?” Behrand colleagues found that the successful preservice teachers commonly used two strategies tosolve the problem. However, in one strategy, teachers operated on the number of units in aunit of units, whereas the other strategy seemed more focused on the sizes of the units in aunit of units (see Behr et al., 1997 for more detail). This emphasizes a strong knowledge ofunits as a fundamental factor in successfully using representations of fraction multiplicationsituations. Like our research, this research focused on representations (e.g., bundles of sticks)and used conceptual analysis to make sense of the strategies used. However, we investigatedinservice teachers’ problem-solving strategies in responding to multiple-choice items and alsolooked across teachers’ strategies in the various drawn representations and operations. Moreover,the Rational Numbers Project study focused only on strategies that provided correct answers,whereas we accounted for strategies that were unsuccessful.

In the second study, Harel and Behr (1995) interviewed 32 inservice (grade 4–6) teachers toidentify and classify their strategies in solving rational number multiplication and division prob-lems, looking for violations of basic intuitive models (Fischbein, Deri, Nello, & Marino, 1985).For example, in multiplication, one basic intuitive model is that multiplication always makesnumbers bigger. The researchers found that teachers’ intuitive models influenced their problem

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solving just as they influenced children’s and preservice teachers’ problem solving. Interestingly,Harel and Behr found that only teachers who incorporated the concepts of ratio and proportionsolved the problems relationally and correctly without having constraints from intuitive models.Like our study, Harel and Behr (1995) identified teachers’ unsophisticated strategies for solvingproblems. In their study, these strategies included trying the four symbolic operations to find acorrect operation or using the keywords (e.g., give away for subtraction, of for multiplication,share for division, etc.) to identify the operation.

More recently, Seaman and Szydlik (2007) suggested that the problem-solving strategies ormathematical reasoning displayed by the participant elementary teachers differ from that of prac-ticing mathematicians in that the teachers’ reasoning was mathematically unsophisticated andimpoverished. In the study, 11 preservice elementary teachers were asked to interpret a frac-tion multiplication problem. Six participants misapplied keyword strategies to interpret the itemas being a subtraction problem. Finding keywords was not sophisticated enough to use in thissituation. In contrast, the researchers found that the three teachers who successfully identifiedthe problem as fraction multiplication used approaches similar to practicing mathematicians:they attended to the language, drew a model of the situation, and determined the answer wasreasonable given the context.

To summarize, existing research highlights teachers’ failures to invoke necessary knowledgewhen interpreting problem situations—whether from drawn representations, from word prob-lems, or from student’s questions. Further, as shown in Izsák’s work (Izsák, 2008; Izsák et al,2008), teachers struggle to use representations in clear ways for classroom instruction. We buildon these earlier studies by considering teachers’ approaches to interpreting a variety of fractionoperation items that use drawn representations. We consider teachers’ approaches to each itemto provide an analysis of the problem-solving strategies used. Our goal is to provide insight intothe knowledge the teachers rely on for interpreting these problems and representations.

THEORETICAL FRAMEWORK

Our work is grounded in the emerging line of thinking about specialized knowledge for teach-ing as well as in the work on knowledge in pieces (diSessa, 1988; Izsák, 2005, 2008; Smith,1995; Smith, diSessa, & Roschelle, 1993). Specifically, we draw on key ideas that inform ourthinking about how teachers understand the mathematics and what specialized knowledge forteaching should be in this domain. We agree with the position that for each person, knowledgeis distributed across a series of “interrelated general and context specific components” (Smithet al., 1993, p. 145) that manifest themselves in different ways in different situations. The impli-cation of this is that a teacher may draw on a variety of small components of understandingsin any given situation and that failure to invoke expected components of knowledge does notnecessarily imply their absence.

Building from knowledge in pieces, we propose that expertise may be about having greaterflexibility in approaches to solving problems rather than in having more elegant ways of usinga single approach. This is consistent with Smith and colleagues (1993) who showed while therewere three algorithms that might be applied to an entire problem set to produce correct responsesefficiently, experts in their study relied on a range of approaches that did not include the threealgorithms. Instead, the expert participants relied on perceptions of ease of use and elegance

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(e.g., working with nicer numbers) rather than on quick calculation in choosing their strategies.In other words, people with more expertise could essentially flip through a mental catalog ofoptions for working each problem, analyze the fit of each option, and apply an effective toolto the situation. For the experts in the study, this analysis and fitting of approaches was doneproblem-by-problem. In contrast, novices did not exhibit such flexibility.

We assert that specialized knowledge for teaching, as defined previously, necessarily requiresthis kind of ability. Teachers need to know and understand components of knowledge suchas identifying referent units, applying area conceptions to drawings, and applying keywordstrategies. In fact, we assert that one important component of knowledge for our study is theability to demonstrate flexibility with units. By flexibility, we refer to a teacher’s ability to keeptrack of the unit to which a fraction refers (e.g., in a problem such as 1/5∗1/4 = 1/20, the 1/4and 1/20 refer to a whole, but the 1/5 refers to a portion of the 1/4) and to shift their relativeunderstanding of the quantities as the referent unit changes. This ability, combined with theability to norm (i.e., identify the standard unit from which to measure; Lamon, 1994), seems tobe a critical component for teachers to use to interpret a variety of student solutions. However,we assert that knowing how to do this and knowing when to do it need to be coordinated insome deeper understanding that connects these individual units of knowledge together (diSessa,1988). This study rises out of the recognition that the research community currently lacks thedepth of understanding of how teachers think about mathematics that would allow professionaldevelopment and teacher preparation programs to support the development of not only theindividual components of knowledge but, more importantly, the theoretical foundation toconnect them into the flexible library that experts exhibit.

Our interest in the use of drawings as a means of reasoning about fractional quantities createsan interesting testing ground for considering teacher knowledge because interpreting representa-tions requires teachers to make connections between and among various knowledge componentsthat may not be coordinated in algorithmic approaches. After all, a single drawing requires theteacher to be able to interpret both the mathematics that is supposed to be represented and themathematics that is being represented and to consider whether those are the same. We suggestthat this is difficult for teachers. In fact, in our own observations, consistent with Izsák (2008), werarely see teachers using drawn representations for purposes other than illustrating answers. Froma knowledge in pieces perspective, the use of representations provides a particular challenge forteachers, who were typically taught using memorized algorithms, to apply those algorithms andother approaches to drawings that rely on a coherent understanding of mathematics. After all,for teachers to support students in using representations, they need ways of coordinating theirown mathematical understandings that allow them to interpret various representations of mathe-matical ideas that may be idiosyncratic. The teacher’s role is to help students build bridges fromidiosyncratic to conventional representations (Smith, 2003).

We posit that knowledge in pieces provides insight into the specialized knowledge for teachingbecause it considers not just the knowledge that teachers have but also how the components ofunderstanding are connected. Our interest, ultimately, is in understanding what teachers needto know to support student learning rather than simply assessing what teachers do or do notknow. Indeed, all our participants were capable of generating correct answers to the underlyingmathematical situation in each task. However, they were challenged by having to reason withdifferent components of knowledge to make sense out of others’ reasoning. We assert that this isimportant because it is the task of teaching.

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METHODS

The dataset for this study includes interviews with 12 middle grades teachers who had completeda written multiple-choice assessment of rational numbers. The assessment was comprised ofitems from the Learning Mathematics for Teaching measures of mathematical knowledge forteaching (SII/LMT, 2004) as well as items developed by our research team. Once the teachershad completed the written assessment, they were invited to participate in a one-hour validationinterview that was intended to inform the development of the assessment items. In this interview,we asked the teachers to discuss how they understood each item and why they did or did notselect each of the given answer options.

For this study, we relied on a subset of eight assessment items (see Table 1 for the list of oper-ations and representation type of each item), with the eighth item, the fraction division numberline item, only being discussed by four participants because of time constraints in the interviews.Each item provided teachers with one or more drawings of a fraction or decimal operation andasked the teacher to interpret different aspects of the number line or area model representations.For example, some of the questions simply asked the teachers about what the drawing was show-ing them while others provided sample student drawings and asked the teacher to determine theircorrectness. The assessment was administered to a national convenience sample of 201 teachersand the teachers in this study were from a convenience sample of those teachers who had takenthe assessment and were geographically near the researchers. The 12 interviewees for this studyrepresented three school districts (one rural and two urban) and comprised the entire populationof the first round of interviews for the instrument. The sample of teachers included one spe-cial education teacher. The participants ranged in experience from second-year teachers to thosewith 20 or more years of experience and included teachers with traditional certificates as wellas those who had completed alternative certification programs. Several of the teachers had mas-ter’s degrees and some had completed educational specialist degrees. The sample included sevenurban and five rural teachers. Some of the teachers had experience teaching with standards-basedcurriculum such as Connected Mathematics Program (Lappan, Fey, Fitzgerald, Friel, & Phillips,2002) that promotes the use of representations (as laid out by NCTM, 2000).

The interviews were videotaped using two cameras—one focused on the participants’ writtenwork and hand gestures, the other on the interviewee’s face. These videos were mixed into onefile to create a restored view (Hall, 2000) and transcribed verbatim. Our initial analysis focusedon identifying emergent themes in the data. Each transcript was analyzed by at least two mem-bers of the research team using emergent codes to find pools of meaning in the participants’discussions (Coffey & Atkinson, 1996). Each researcher noted the trends that were emergingusing memoing techniques in concert with the coding scheme (Strauss & Corbin, 1998). As thefindings began to emerge, further analysis was used to make sense of the emerging trends andstrengthen our definitions of codes. To this end, each researcher focused on a subset of itemswithin each video.

Once most of the key strategies had been identified, two researchers analyzed all the videosagain to explain each teacher’s responses for all the items included in this analysis. In additionto coding for key strategies, we also made notes whenever the teachers expressed discomfort inusing the drawn representations or described his or her experience with drawn representations.The researchers compared their notes and reanalyzed any responses until there was 100% inter-rater agreement. In analyzing the data, we concentrated not only on their verbal interpretations

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of the work but also on the nonverbal cues such as tick marks and hand gestures. The data weretriangulated both between researchers and across participants (Denzin, 1989) to ensure that atrustworthy picture of teachers’ problem-solving strategies emerged.

FINDINGS

In our analysis of middle school teachers’ sense-making of drawn representations for fractionoperations, we found that teachers who demonstrated flexibility in their understanding of the ref-erent unit could adapt their mathematical knowledge of fractions to select reasonable responses.However, most of the teachers demonstrated one or more limitations in their ability to renormor work with a changing referent unit within the items. In these cases, teachers either appliedan inflexible approach to referent units or one (or more) of four strategies that did not rely onreferent units at all: identifying requisite features, looking for a diagram that matches a solution,using the process of solving to select solutions, and measuring to find a solution.

As shown in Table 1, of the 88 tasks analyzed (seven questions each for eight teachers andeight questions each for four teachers), 18 explanations (20%) could not be classified into ourcategories because teachers used anomalous approaches that were not seen across teachers oracross tasks. Of these 18, three were cases in which the participant either guessed or was unableto make sense of the item. The remaining 15 were either instances of the participant not remem-bering how they reasoned through the item (3 out of 15) or the strategy used did not fit into ourcategories and was deemed to be an outlier. Such responses included disliking the representationsprovided to the point of distraction or misreading the item in such a way that the solution did notfit with the stem (e.g., solving for length rather than area).

The presentation of our data focuses on the 80% of responses that did fit into the classificationcategories. In this section, we share data demonstrating the various strategies teachers used inorder to explain their choice of solutions (see Table 2 for strategy use by participant).

Attending to Referent Units

Teachers applied reasoning with referent units—correctly or incorrectly—in 30 of the 88 expla-nations analyzed (34%). Broadly, we classified these teachers into two groups: identifyingreferent units and inflexibility with referent units. Each item in which participants focused onthe referent unit as part of their strategy for making sense of the model was coded as identifyingreferent units. For situations where teachers focused on the referent unit as a fixed and inflexi-ble value rather than considering various possible wholes, we used the inflexibility with referentunits code. Both of these categories are described further next.

Identifying Referent Units

Identifying the referent unit was observed in 22 of the 30 cases in which referent unit wasexplicitly discussed (73%). The label was given only to those items in which teachers clearlydiscussed the whole (referent unit) and its relationship to the parts of the item. As shown inTable 2, only two of the participants failed to exhibit referent-unit reasoning at some point in the

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TABLE 2Strategies Used by Participants

Attending to Referent Units Not Attending to Referent Units

Name

IdentifyingReferent

Units

Inflexibilitywith Referent

UnitsRequisiteFeatures

Diagram thatMatches aSolution

Process ofSolving to

SelectSolutions

Measuring toFind a

Solution

Alicia 2 0 1 1 0 0Casey 3 0 0 0 0 0Edison 2 1 1 2 0 2Ellen 0 0 2 4 1 1Jaelyn 0 0 5 0 1 0Kendall 2 3 0 1 2 0Lydia 3 0 0 1 1 0Malcolm 1 1 1 1 0 1Nina 2 0 3 2 0 0Pamela 3 0 2 2 0 0Rick 2 1 2 0 2 0Sarah 2 2 1 2 0 0

interviews. Each of the other 10 participants used referent-unit reasoning in at least two items.Nine chose the correct answer for at least one item on which they attended to referent units. Thetenth participant selected incorrect responses when taking the test, but during the interview hebegan to reason with referent units to correct his responses and explain how his original responseswere incorrect. None of the participants relied on the referent-unit reasoning strategy more thanhalf of the time across the seven or eight items on which they were interviewed; however 15 ofthe 30 instances were with the area multiplication and area addition problems (see Table 1). Infact, participants were able to reason with referent units over 25% of the time with the fractionarea problems and less than 20% of the time with the fraction number line problems.

In the number line fraction multiplication problem1 (see Figure 1), we asked teachers to iden-tify the number line that correctly modeled 1/5∗1/4. Even though the portion of the bolded linesfor each choice represented one-twentieth, drawing 1a shows 1/5−1/4 = 1/20 while drawing 1bshows 1/5∗1/4 = 1/20. Four of the 12 teachers attended to the referent unit correctly and iden-tified that number line 1a in Figure 1 was an incorrect representation because the unit to whichone-fifth referred in the drawing was the whole rather than the one-fourth. For instance, Edison,2

who commented throughout the interview about his lack of familiarity with any drawn represen-tations, rejected choice 1a with the rationale, “I don’t know if I am wrong or right. But I did notchoose [1a] because it [pointing to the one-fifth line] wasn’t one-fifth of this one-fourth piece.It was one-fifth to the entire line, and I thought that I wanted one-fifth of just the one-fourth.”3

1Because the assessment items are secure, the items reported here have been modified with the intent of maintainingthe mathematical ideas and the complexity of the original items.

2All names are pseudonyms.3In all the transcripts reported in this manuscript, italics have been used to emphasize the statement or phrase that

aided us in classifying the strategy used.

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FIGURE 1 a. Number-line model of 1/4 – 1/5 = 1/20; b. Number-linemodel of 1/4 × 1/5 = 1/20.

FIGURE 2 Two area models of 1/3 of 3/4.

Unlike teachers who made their selection based on locating the correct amount (length) of theline shaded,4 Edison applied his understanding of referent units and his knowledge that fractionmultiplication acts to segment a part of a part of a whole. His correct attention to the referent unitof one-fifth in Figure 1a guided him toward correctly eliminating this answer choice.

Similarly in the area multiplication item (see Figures 2a and 2b), teachers attending to referentunits referred to the relationship of the parts to each other and to the whole. For example, Sarahnoted on the drawing like 2a:

The first thing I saw, well the first thing I thought was are they all accurate? And then I said, okaywell this [indicating the whole square divided] is four fourths. Here’s three fourths [indicating thethree lightly shaded segments]. And he took one of the three. So in my head he was thinking, here’smy three fourths, there’s three of them, I need one third of the three fourths so I take this one.

In this response, Sarah indicated that she was thinking about taking a 1/3 part from the 3/4of the whole, which is consistent with our definition of reasoning with referent units. In thiscase, the referent unit for the 3/4 was one whole and the referent unit for the 1/3 was the 3/4 ofone-whole.

Attending to the referent unit emerged as important in our efforts to understand how the teach-ers were interpreting the drawn representations. Reasoning flexibly with referent units required

4This is discussed later under teachers’ inefficient problem solving strategy, Looking for a diagram that matches asolution.

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teachers to apply knowledge about fractions and operations to a representational situation inmathematically sound ways.

Inflexibility with Referent Units

We defined inflexibility with referent units as focusing on the referent unit as a fixed andinflexible value throughout the task rather than considering there may be a number of wholesembodied in the task. In the data set, this was most commonly evidenced when teachers attemptedto reason with referent units, but arrived at incorrect conclusions because they were unable torenorm, that is, shift their understanding of what the whole in the situation was, or they failed toidentify the correct referent unit as part of their reasoning. Because our assessment tasks focusedon fractions and decimals, teachers who lacked flexibility with referent units struggled to makesense of the representations. Inflexibility with referent units was identified eight times (9% of the88 total tasks).

Five of the eight instances of limited flexibility occurred with the two area division problems(see Table 1). In one division task using an area model, the teachers were asked whether a draw-ing such as that in Figure 3 could be interpreted as modeling 3/2 as the quotient for 1÷2/3. Inthe interview, we specifically asked the teachers to discuss whether they interpreted the diagramas showing 2/3 of one whole shaded, 1/2 of 2/3 shaded, and/or the quotient 3/2 shaded becausewe wanted to understand whether these teachers recognized that the quotient refers to a differentwhole than the divisor and dividend. Four of the twelve teachers demonstrated inflexibility in

FIGURE 3 Area model that can be interpreted as showing 2/3 shaded;1/2 of 2/3 unshaded; or 3/2 in all.

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their interpretation of this drawing. All of those teachers explained that only the first interpre-tation, 2/3 of the whole is shaded, was correct. Sarah explained her reasoning for selecting thisinterpretation:

To show three halves, I would think, one, you need more than one picture because you would needto show the unit of two halves or one whole. So that you have a comparison so that you understandthat if this is one whole [she draws a unit square] or two halves [she partitions the unit square intotwo pieces]. You need to be able to see that that’s three halves. So that you can see that it’s biggerthan one unit [she draws a representation of the answer with a drawing bigger than the unit square].So, for me this picture wasn’t explaining what I needed it to explain cause I don’t see one whole.

Sarah knew that looking for the whole was important but determined that the whole squarecould only represent one whole. She wanted there to be two wholes, each divided in halves, tomodel 3/2. She either did not know or did not rely on knowledge of the referent whole changingin a division situation in her interpretation of this item. For this teacher, the answer was 3/2without flexibly realizing the whole could be the shaded parts.

Likewise, Kendall demonstrated either successful or limited flexibility with referent units forall five of the area model items (see Table 2). For example, on the same division item describedpreviously (Figure 3), Kendall indicated that the drawing could show 2/3 or it could show 2. Sheexplained, “It’s not 3/2 because it’s either wholes—three separate wholes or it’s 2/3. That’s whatwe have.” While she showed more flexibility than Sarah, she was unable to reason flexibly abouta variety of interpretations for the whole. Thus, this item was classified as limited flexibility.

Teachers Who Did Not Attend to Referent Units

The four key strategies in which teachers did not attend to referent units at all were usedto categorize 40 of the 88 responses analyzed (45%). Five of these include instances inwhich teachers used two strategies in a single item. In these cases, the item was classifiedfor each. This resulted in 45 strategies being used across the 40 instances. While the fournon-referent unit strategies were used in nearly half of the instances, each proved to have signif-icant limitations in its applicability. However, the teachers never indicated recognition of theselimitations.

Identifying Requisite Features

This was the most common of the four non-referent unit approaches (45%, 18 of 40 instances)and was used by 9 of the 12 teachers at least once. In the requisite features approach, teachersinvoked knowledge about perceptual features (e.g., symbols or numbers on a given numericalexpression) in choosing the correct drawing rather than using conceptual operations or quan-titative reasoning. To be more specific, when applying this strategy, the teachers focused onconnecting parts of the diagram to their corresponding parts in the numerical problem (e.g.,locating both factors and the product in a multiplication item).

The area model for fraction multiplication task (see Figure 2) led to this solution strat-egy the most with 4 of the 12 participants relying on requisite features in their explanation

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(see Table 1). We asked teachers to determine whether area models drawn by four hypothet-ical students accurately represented a quantity such as 1/3 of 3/4. Despite all four choicesmodeling 1/3 of 3/4 using a part-of-part model, distributive reasoning, or equivalent frac-tions, teachers using the requisite features strategy accepted only the drawing like Figure 2b.The typical explanation for this preference was that the factors, one-third and three-fourths,were clearly presented and the overlapping part was the answer. This explanation impliedthat the representations somehow contained specific stepwise elements that determined theircorrectness. Teachers who attended to requisite features were less likely to accept represen-tations with slight modifications as being correct. We posit that this conception may be aby-product of the ways textbooks introduce these models and that the knowledge teachersrelied on may have been knowledge of their curricular materials rather than mathematicalunderstanding.

Jaelyn was one teacher who relied on this approach. She used the strategy in five of eightitems and was the only teacher to apply the strategy to both area model fraction division items.In responding to the multiplication item described previously, Jaelyn explained her reasoning:

I looked at this [she indicates the choice like Figure 2b], the reason why I only picked [2b] was causethat was the only one that looked right to me. [She laughs.] Because this was saying you had 1/3going this way once. [She runs her index finger through the shaded third in the first row.] And thenone three-fourths. The fourths are going this way [she runs the same index finger along the threevertically shaded regions] . . . so the answer of two fractions multiplication is the double shaded part,which is three . . . um [pauses for about 5 seconds] twelfths. You see what I’m saying?

Jaelyn’s explanation highlights that she located each factor and noted that the double-shadedregion was the product. She made her choice based on looking for requisite features in thediagrams because she saw the one-third, the three-fourths, and the double-shaded product. Itseemed in the interview that she already had a numerical answer, one-fourth, in her mind as sheconsidered the choices.

Nina also rejected the representation like 2a but favored the drawing like 2b by thinkingonly of requisite features. She explained, “I was clear on [this task]. 1/3 of 3/4. [2a] was obvi-ously wrong because there was nothing broken into thirds. 2b . . . this one was, if you hadto pick one, I would have said I like that one the best because it most obviously shows thirdsand fourths.” It is clear in Nina’s explanation that, for her, the representation needed to havesomething divided into thirds and not just have one of the three-fourths shaded in order to becorrect.

This approach seemed favored by teachers who were either trying to link the algorithm tothe drawn representation or who were drawing on their past textbook experience with areamodels. Teachers who used the requisite features approach did not include more meaningfulconceptual discussion of multiplication and how it works in their discussion of these items.We note that while the requisite features approach was problematic because it led teachersto reject drawings that were correct when aspects of the formula were not readily apparent.However, this approach did consistently lead teachers to work in the problem context by attend-ing to all parts of the situation including the operation. As will be discussed elsewhere in thisarticle and as discussed elsewhere (e.g., Orrill, Sexton, Lee, & Gerde, 2008), teachers solvingthe assessment items often lost track of the operation involved in an item as they discussed it.Therefore, using a requisite features approach did have benefits for supporting mathematicalsense making.

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Looking for a Diagram that Matches a Solution

This was the second most popular strategy (40%, 16 of 40 tasks) and was used on at leastone item by 9 of the 12 teachers in our analysis. Responses coded as looking for a diagram thatmatches a solution were those in which the teacher not only calculated the solution before select-ing but also relied on that solution to select the representation that best modeled the solution.This was evidenced both in the teachers’ scratch work and in their discussion during the inter-views. Because we anticipated this strategy, we designed the responses on the assessment in waysthat uncovered the approach. In fact, the assessment provided the solution as part of the stem inmost cases because we were interested in understanding how teachers interpreted the situationrather than their ability to solve the task. Despite this, many teachers calculated the solution forthemselves.

When teachers used the matching a solution strategy, their criteria for selection was whetherthe correct quantity was shown in the drawing, regardless of the operation being modeled withinthe diagram. The implication of this was that they did not invoke knowledge other than calcula-tional ability. For example, the drawing like Figure 1a was selected as correct by teachers usingthe matching a solution strategy because the correct quantity was shaded even though the repre-sentation did not model multiplication. Ellen’s explanation of why she accepted both drawingson the item like Figure 1 (1/5∗1/4) highlights this approach.

Ellen: I just said all of them [referring to the number line drawings given in the item]. Why?Cause I was looking at these. [She points at bolded portions of a and b.] I was thinking20 parts. I don’t know. I missed; I didn’t understand it to tell you the truth.

I: You didn’t understand it?Ellen: You didn’t understand it?

I: I’m not sure about multiplying with number lines, but then . . .I: Ok. Were there some of these diagrams that you felt more comfortable with than others?

Ellen: Sure.I: Which one did you feel the most comfortable with?

Ellen: I felt comfortable with [1a] with the 1/5 times 1/4 equals 1/20. One-fifth times 1/4 equalsthis. I don’t know, I guess this cause it’s got the numbers there. I don’t know.

I: Okay. What about the [other choices]? There’s no [labels] on those.Ellen: I know there are no numbers, but I was looking at the portion. That’s 1/20, that’s 1/20,

that’s 1/20. [She points to the bold line segments in each of the number line drawings.]

While her explanation for preferring 1a included aspects of the requisite features strategy,Ellen’s insecurity and her repeated highlighting of the shaded product for all the given numberlines led us to determine that she was more concerned with the solution quantity than with therequisite parts. On the other number lines provided, Ellen explicitly stated that she looked for thesolution, 1/20, without regard to the presence of labels. In fact, the third and fourth choices (notdrawn in Figure 1) simply showed a segment 1/20 long placed in different parts of the numberline with no labeling at all.

Combined, our teachers relied on identifying requisite features or matching a solution for 85%of the non-referent unit strategies (34 of 40 tasks). This suggests that teachers may approachdrawn representations by applying superficial or answer-oriented understandings rather thanrelying on conceptual knowledge of operations. This is significant because relying on these

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approaches led the teachers to frequently select incorrect responses as they were not focusedon particular aspects of the mathematics in the situation.

Using the Process of Solving to Select Solutions

Using the process of solving a problem was identified when the teacher used reasoningthrough the solution process to inform the choice. This strategy differed from the matching asolution strategy in that teachers using the process of solving looked for elements of the algo-rithm in the representation, not just the solution. Interestingly, this strategy was not invoked forany fraction multiplication items. We conjecture that the division models and the number linemodels may have been less familiar to the teachers, therefore they lacked preconceived ideasof how of interpret them. Thus, they drew on the mathematics they knew (e.g., algorithms) andtried to use that knowledge to make sense of the drawings. We observed 5 of the 12 teachers usethis strategy in 7 of the 40 instances in which referent unit reasoning was not used (17.5%). Inall seven instances of using the process of solving strategy, the teachers relied on algorithms tomake sense of the drawn models. For example, three teachers looked for a common denominatorbetween the two given fractions when identifying the correct number line model in the fractionsubtraction problem. Similarly, three of the four teachers interviewed about the number line frac-tion division problem (see Figure 4) changed the problem into a fraction multiplication situationby using the invert-and-multiply algorithm and chose the number line that showed the process ofthe algorithm. As shown in Figure 4, the number line division problem provided teachers with amodel that showed a partitive (sharing) division interpretation of 2/3÷1/4. The intention behindthis item was to see if teachers would use a partitive interpretation of division, which asks, “Howmuch is the share of one?”

Teachers relying on the process of solving knew that when they used the invert-and-multiplyalgorithm, they would have four groups of 2/3 because multiplication was interpreted as groupsof. The following two teachers’—Rick and Kendall—explanations highlight this strategy.

Rick: Okay [35 second silence]. One, two, three, four. [He points at each of the four arcs with hispencil.] I was happy with that part—the fact that there are four bumps. [He laughs.] Now,how did I get 2/3 here? [After a pause, he explains that he began with the 2/3 on the numberline. He then points to each of the arcs over the 2/3 segments in Figure 4.] . . . Four two-thirds is that it? Yeah four two-thirds that’s what that was. Because it matched the algorithmafter I yeah that is what it did it matched the algorithm that I depend on for—yeah that iswhat it did.

Kendall: If you want them to show that the answer as being it is two and two-thirds when you putit as a mixed number. That [she points at the number line in Figure 4] does show two andtwo-thirds. And if he did—let’s see when you do, when you’re doing division of fractions

FIGURE 4 One choice for the number-line fraction division problem.

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then you’re going to flip. It’s going to be 2/3 times 4/1. [She writes the numerical expression2/3 × 4/1.] So this [she points at the two-thirds] is 2/3 of a whole and he did it four, youknow, times four. One, two, three, four. Four. [She indicates the four arcs.] And that’s wherehe ended up. [She indicates the solution point.]

Rick explicitly stated that he applied his algorithmic knowledge in making his selection. WhileKendall also clearly considered the invert-and-multiply algorithm, she also attended to whetherthe diagram matched a solution. However, we identified this as process of solving because morethan one diagram showed the correct quantity (e.g., one had two groups of 4/3), but only thechosen diagram included four groups of 2/3 and the four groups seemed to be important to herstrategy. Therefore, we classified her approach as process of solving.

Among the four non-referent unit strategies, this one was arguably the most sophisticatedin that it required an application of understanding of the problem situations to make a correctselection. However, it allowed teachers to focus on aspects of the context that were perhapssecondary to the intended foci. For example, in the division task, the teachers using process ofsolving applied multiplication as their interpretation lens for a division situation. While this lensallowed the selection of a correct response, focusing on the inverted divisor maintains a connec-tion to the algorithm that does not require conceptual understanding of division and overlooks thefundamental question that the division problem is trying to model, “What number is two-thirdsone-fourth of?” In terms of specialized knowledge for teaching, this has significant implica-tions in that the use of this strategy suggests a challenge in supporting students in developing aconceptual understanding of partitive division.

Measuring to Find a Solution

This category was less prevalent than the others with only 4 of the 40 instances (10%) usingthis strategy. In fact, two teachers used this strategy one time in the interviews while one teacherimplemented it twice. The measuring to find a solution strategy was only used with the fractionsubtraction number line problem, fraction multiplication area problem, and the decimal multi-plication area problem (see Table 1). Measuring instances included all cases in which a teacherturned to linear measurement, typically with an improvised measuring instrument such as a linesegment drawn on a sheet of paper or with fingers. This strategy invoked teachers’ understand-ings of measurement and demonstrated some connection between teachers’ number sense andtheir geometric knowledge. While limited in its effectiveness, this strategy did place explicitfocus on the quantities of interest in the representations and not just numeric answers. For exam-ple, when Edison was asked to explain how the area of an entire figure such as that shown inFigure 5 represented the final answer of the computation 0.3∗1.2, he stated that he knew therepresentation needed to be divided into 36 equal pieces in order to show the answer and thateach of those pieces needed to be one-hundredth of a unit of area. Then he said, “I was goingto try splitting [the shaded part] up so that I can get [the 36 pieces] but I couldn’t so I drew. . .” Then he folded the bottom of the page toward the figure (see Figure 6) so he could lay thewidth side under the 1.2 units, then he continued to say, “I just knew that that [he refers to thesix-hundredths that is unshaded] was six pieces. So I said that’s 6 [he marks a line segment thewidth of the white columns on the back of the page and begins to count as he moves the line

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FIGURE 5 Decimal multiplication area model showing 0.3 × 1.2 (colorfigure available online).

FIGURE 6 Edison measuring to find the solution (color figure availableonline).

segment along the base of the rectangle], 12, 18, 24, 30, 36, so I figured that the statement wascorrect.”

As shown in this analysis, measuring did not require the teachers to have any particular under-standing of the operations but did rely on their ability to correctly calculate the answer to use asthe basis for comparing their measurements. One of the most significant shortcomings of thismethod was the lack of precision used in measuring and the lack of recognition of this as alimitation by the teachers.

DISCUSSION AND FUTURE RESEARCH

In the introduction to this article we suggested that teachers may lack either the pedagogicalknowledge or the mathematical knowledge to make sense of drawn representations. Throughthis investigation of 12 teachers’ reasoning across eight items, we noted that there were certainlysome components of knowledge (e.g., diSessa, 1988) that these teachers were either unable toapply to the drawings or were missing. We note that there were no instances of teachers blindly

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guessing without some basis for their decision—even in the unclassifiable strategies, suggestingthat the teachers were relying on components of their knowledge in their analysis of the items.Based on these findings, we assert that the teachers were flexible in the strategies that they usedand that they invoked different criteria for solving different items. This was similar to the expertsin the knowledge in pieces work (Smith et al., 1993). However, unlike those experts, our par-ticipating teachers were unable to flexibly and effectively apply different strategies across theproblem types (operation and drawn representation).

In response to our primary research question for this study, we had hoped to see these teachersreason with the representations in place of algorithms. Reasoning with units was also highlightedas desirable in the work done by the Rational Numbers Project (Behr et al., 1997; Harel & Behr,1995). We view the 22 successful instances of reasoning with referent units as demonstrating adesirable and effective strategy. However, correct referent-unit reasoning accounted for only 25%of the 88 cases analyzed. The most common strategies used by teachers who did not attend to thereferent unit were requisite features and matching a solution. These strategies led to the correctanswer only 20.5% of the time (7 of the 34 instances) they were used. Thus, these strategies areineffective for reaching the correct solution.

The purpose of understanding teachers’ reasoning about these representations is to under-stand how teachers reason about mathematical situations and to provide guidance for effectiveprofessional development opportunities. Our analysis raises questions about the reasonableexpectations for using drawn representations in classrooms as suggested by NCTM (2000) andsuggests professional development is needed to support teachers in thinking about representa-tions in different ways—particularly in learning to reason with representations rather than tryingto interpret them as drawings of solutions (e.g., Izsák, 2008). We further assert that professionaldevelopment needs to support the broadening of teachers’ abilities to reason with representationsbeyond just area models in multiplication situations. In this sample, the teachers struggled morewith number lines than area models, but also struggled with area model representations that werenot focused on multiplication in ways consistent with Figure 2b. In short, to address the NCTMstandards, teachers need to be able to make sense of student work—which is neither predictablenor prescribed—in the classroom, and teachers who are not relying on conceptual understand-ings of mathematics (e.g., not relying on referent units) may incorrectly discourage a student’smathematical ideas. This is consistent with the limited literature base on teachers’ interactionswith students in fraction operation instruction (e.g., Borko et al., 1992; Izsák, 2008).

Our study points to particular needs for professional development. These teachers exhibitedcertain expert-like qualities (Seaman & Szydlik, 2007) in their approaches to these items, whichsuggests that professional development needs to capitalize on the knowledge and strategies theteachers have while problematizing a variety of situations to support teachers in making con-nections among their uncoordinated ideas about fraction operations. Specifically, professionaldevelopment needs to support the teachers in developing a theoretical foundation to invokeknowledge they have about the operations and move them from relying on applications ofalgorithms or searches for correct answers as criteria for accepting drawings as models of math-ematical problem solving. This is consistent with the idea of supporting teachers to reason asmathematicians by looking for reasonable answers and attending to language, as was seen inSeaman and Szydlik’s study.

Simply focusing professional development on practice with representations is insufficient. Allthe teachers in this sample had previous knowledge of at least one representation used on this

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assessment, yet the items with the most familiar representations (e.g., area model for fractionmultiplication) often led to the highest instances of rote applications supported by little more thanalgorithmic knowledge rather than conceptual understandings of the operations. For example, thedrawings most common in textbooks (e.g., Figure 2b) seemed to elicit matching a solution andrequisite features approaches. In contrast, those that were less common (e.g., distributing 1/3into each of three segments that was 1/4 of the whole) elicited more mathematically complexreasoning from some teachers. Specifically, some teachers elicited their understandings of thedistributive property as they interpreted the representation rather than simply identifying whetherthe correct pieces were shown. We propose, therefore, that teachers should be encouraged toengage with representations that are non-standard and that lend themselves to mathematicallyrich conversations. By making their thinking explicit, teachers can identify places in which theirown knowledge may be faulty and engage with other teachers around the mathematics uponwhich representations build. For example, an item such as that shown in Figure 4 opens upmultiple opportunities to explore division as being richer than repeated subtraction (a commonwhole-number conception) and more knowable than simply “invert-and-multiply,” which was theoverwhelming explanation for division we heard from these teachers (note that this is analogousto the findings of Harel & Behr, 1995). These kinds of opportunities can support the developmentof quantitative mathematical understandings as well as the development of a theoretical founda-tion to connect those ideas consistent with the development of expertise (e.g., Smith et al., 1993).

Clearly, supporting teachers in linking the mathematics they know to a wider array of repre-sentations (not just drawn) is a vital role for professional developers interested in the specializedknowledge teachers need. By shifting the focus to modeling the problem-solving process ratherthan representing the answer and by supporting teachers in developing ways of interpretingrepresentations, professional learning can lay the groundwork for teachers to engage their ownstudents in mathematically meaningful uses of drawings for reasoning about quantities. This willhelp realize the Standards (NCTM, 2000) and is something that cannot happen if teachers lackthe understandings of representations necessary to link symbolic representations of mathematicsto drawn representations of the same mathematics.

While this study presents a starting point for considering the knowledge teachers have andthe need to make sense of representations of fraction operations, it falls short of presenting acoherent theory of teacher knowledge or its development. Future research should focus on howto develop reasonable learning opportunities for teachers that simultaneously acknowledge theexpertise they have while also supporting them in making new connections among and adding totheir knowledge pieces.

ACKNOWLEDGMENTS

The work reported here is supported by the National Science Foundation under grant DRL-0633975. The results reported here are the opinions of the authors and may not reflect those ofNSF. The authors wish to thank the Does it Work team for their support and particularly DanieBrink, Andrew Izsák, and Susan Sexton for help in conducting the interviews analyzed for thisreport. The authors also wish to thank three anonymous reviewers for their thoughtful—andthought-provoking—comments. Earlier versions of this report were presented at the ResearchPresession of the 87th Annual Meeting of the National Council of Teachers of Mathematics, the

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2009 Annual Meeting of the American Educational Research Association, and the InternationalConference of the Learning Sciences 2008.

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