author's personal copy - virginia tech · a major change in the intended mathematics...

This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution

and sharing with colleagues.

Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party

websites are prohibited.

In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information

regarding Elsevier’s archiving and manuscript policies areencouraged to visit:

http://www.elsevier.com/authorsrights

http://www.elsevier.com/authorsrights

Author's personal copy

Journal of Mathematical Behavior 32 (2013) 266– 279

Contents lists available at SciVerse ScienceDirect

The Journal of Mathematical Behavior

j ourna l ho me pag e: w ww.elsev ier .com/ locate / jmathb

A cognitive core for common state standards

Anderson Norton ∗, Steven BoyceVirginia Tech, United States

a r t i c l e i n f o

Keywords:Common Core State StandardsFractionsSchemesTeaching experiment

a b s t r a c t

The purpose of this paper is to illustrate cognitive challenges introduced by Common CoreState Standards for Mathematics (2010) with regard to conceptualizing fractions. We focuson a strand of standards that appear across grades three through five, which is best repre-sented in grade four, by standard 4.NF.4a: “[Students should] understand a fraction a/b asa multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).” We argue that suppor-ting such conceptualizations is a critical challenge for teachers and researchers to addressin US education, but one fraught with conceptual hurdles for students. We adopt a schemetheoretic perspective and share the case study of a sixth-grade student, to illustrate the waysof operating students need to develop for meaningful attainment of the new standards.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The adoption of the Common Core State Standards for Mathematics (CCSSM) in an overwhelming majority of statesindicates a major change in the intended mathematics curriculum for American students (Cobb & Jackson, 2011; Porter,McMaken, Hwang, & Yang, 2011). Expressing mathematics reform in a manner that will lead to faithful implementation byschool district leaders and teachers has been a difficult and complex task for policy makers, partly because it requires reform-ers to consider the knowledge and perspective of those using the reform documents (Spillane, 2000). Although alignmentwith existing state and international standards was a consideration in the development of their mathematics standards,the authors of the CCSSM claim that the development of sequencing “began with research-based learning progressionsdetailing what is known today about how students’ knowledge, skill, and understanding develop over time” (CCSSI, 2010,p. 4). In this paper, we elucidate and analyze one of the learning progressions prescribed in the CCSSM, within the “Numberand Operations—Fractions” domain; specifically the progression from a part-whole conception of fractions, to a measureconception of fractions, to an understanding of fractions as operators. The demands this intended learning progression willplace on students introduce additional considerations for faithful implementation of the new standards.

We focus on the development of fraction understandings described in the CCSSM grade levels 3–5, specifically standards3.NF.1, 4NF.4a, and 5.NF.5b (see Table 1). We chose these standards because they represent expectations for students’conceptualizations of fractions themselves (rather than computations with fractions), especially as they relate to the part-whole, measure, and operator sub-constructs (cf., Kieren, 1979). We investigate these demands through the case of a sixth-grade student, named Isaac.

The purpose of this paper is to illustrate the cognitive challenges students must overcome to meet the standards outlinedin Table 1. We begin with an overview of current research-based knowledge of how students come to understand fractions.We then apply a scheme theoretic perspective in the case of Isaac, to examine his available ways of operating and his

∗ Corresponding author. Tel.: +1 540 231 6942; fax: +1 540 231 5960.E-mail address: [email protected] (A. Norton).

0732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmathb.2013.01.001


A. Norton, S. Boyce / Journal of Mathematical Behavior 32 (2013) 266– 279 267

Table 1CCSSM relating to an intended fractions progression.

Standard Statement

Grade 3: Number and Operations—Fractions 1 (3.NF.1) “Understand a fraction a/b as the quantity formed by a parts when a whole ispartitioned into b equal parts.”

Grade 4: Number and Operations—Fractions 4a (4.NF.4a) “Understand a fraction a/b as a multiple of 1/b. For example, use a visual fractionmodel to represent 5/4 as the product 5 × (1/4), recording the conclusion by theequation 5/4 = 5 × (1/4).”

Grade 5: Number and Operations—Fractions 5b (5.NF.5b) “Interpret multiplication as scaling (resizing), by explaining why multiplying a numberby a fraction greater than 1 results in a product greater than the given number.”

constructions of new ways of operating that might undergird the new fractions standards. Isaac’s case holds implicationsfor the instructional challenges teachers will face while attempting faithful implementation of the fractions strand.

2. Research about the learning of fractions

Fractions have long been among the topics most difficult to teach, most cognitively challenging, and most essentialto higher mathematics, yet the amount of research on the teaching and learning of fractions is disproportionate to itsimportance (Lamon, 2007). This is not to say that there has been no progress. The Rational Number Project (e.g., Cramer,Post, & delMas, 2002) developed a curriculum for upper-elementary students emphasizing quantitative representations forfractions, rather than the development of procedures for fractions operations. Students taught using the Rational Num-ber Project (RNP) curriculum performed as well or better than students taught with commercial curricular (CC) materials.In their quasi-experimental longitudinal assessment, no differences were found between RNP and CC students on frac-tion equivalence items or items dealing with symbolic addition and subtraction. However, RNP students were more likelyto solve tasks correctly using a conceptual strategy and more likely to provide explanations for their procedures. Thefundamental difference in the curricula that led to this performance gap was greater emphasis in the RNP curriculumon translations between different modes of representation for fractions: verbal, real-world, enactive, iconic, as well assymbolic.

RNP also elaborated on a perspective with which to examine fractions teaching and learning, through the treatment of itsvarious sub-constructs. Initially introduced by Kieren (1979), Behr, Lesh, Post, and Silver (1983) described five sub-constructsfor interpreting fractions: part-whole, measure, ratio, quotient, and operator (see Table 2). The part-whole sub-constructhas traditionally been the focus of teaching fractions in the US (Streefland, 1993). Using structural equation modeling,Charalambous and Pitta-Pantazi (2007) found evidence supporting the existence of the distinct sub-constructs and thenecessity of the part-whole sub-construct for the development of the others. The dependency on the part-whole sub-construct was non-uniform, however, indicating that students might benefit from focus on other sub-constructs. The benefitof a curricular inclusion of multiple fraction sub-constructs to student achievement is supported by international curricularand textbook comparisons (e.g., Brenner, Herman, Ho, & Zimmer, 1999; Charalambous, Delaney, Hsu, & Mesa, 2010) andinternational mathematics assessments such as TIMSS 2007 (Mullis, Martin, & Foy, 2008).

In the United States, Lamon (2007) conducted a four-year longitudinal classroom teaching experiment to test the effec-tiveness of instructional focus on particular sub-constructs. She investigated whether instruction based on a particularsub-construct would result in a different type of understanding than a control group whose instruction was based on part-whole comparisons including rote computation. Similar to the RNP findings, experimental groups strongly outperformed thecontrol group on conceptual understanding and eventually caught up to the control group on computation. Instruction witha focus on measure was the most effective for inducing student understanding of other sub-constructs, but Lamon (2007)cautioned that anchoring in that sub-construct might not be the most valuable for all children. “In the end, the measuresub-construct seemed to be the strongest because it connected most naturally with the other sub-constructs. . . [but] a moresignificant factor in overall success [no matter the sub-construct focus placement] was the development of the central mul-tiplicative structures.” (Lamon, 2007, p. 660). In Section 3, we introduce a theoretical perspective to investigate the mentalactions that support students’ multiplicative structures.

Table 2Fractions sub-constructs.

Sub-construct Sample representation for two-thirds

Part-whole A whole is partitioned into three equal parts, and two of those parts are shadedMeasure Two iterations of a length that when iterated three times would give a length of one unitRatio Two dollars per three candy barsQuotient The amount each person would get when two cookies were shared amongst three peopleOperator A rule for making two-thirds of some other amount


268 A. Norton, S. Boyce / Journal of Mathematical Behavior 32 (2013) 266– 279

3. Conceptual framework

In considering students’ development of fractions concepts, we need to consider potential influences of whole numberconcepts. A primary and (still) widespread view is that whole number concepts interfere with fraction operations, that“presuppositions of the explanatory framework of natural number will inhibit the acquisition of fractions, causing system-wide misconceptions” (Stafylidou & Vosniadou, 2004, p. 505). An example of such interference is that “well-known intuitiverules children form about multiplication and division (for example, multiplication makes bigger and division makes smaller)apparently result from a curricular overemphasis on multiplication and division of whole numbers” (Behr, Harel, Post, &Lesh, 1992, p. 300). Researchers embracing the interference hypothesis focus on identifying (e.g., Hartnett & Gelman, 1998)or rectifying (e.g., Cramer, Wyberg, & Leavitt, 2009) the interpretations that children form of fractional representations oroperations that are inconsistent with the correct mathematical notions.

3.1. Reorganization hypothesis

In Steffe’s (2001) reorganization hypothesis, rather than viewing whole number operations as obstacles to fractionsunderstanding, the mental operations underlying whole number conceptions are viewed as a foundation for children’sunderstanding of fractions. From this perspective the conceptual development of fractions results from a fundamentalreorganization of whole number schemes, where a scheme consists of three parts: a recognition template for assimilatingsituations (perceptions or conceptions) that activate the scheme, a mental action (operation) or sequence of mental opera-tions, and an expected result of this activity (von Glasersfeld, 1995). When an expected result of operating differs from anobserved result, students experience a perturbation, which often induces a modification, or accommodation, to the scheme(Piaget, 1970; von Glasersfeld, 1995). Such accommodations include modifying the operations of the scheme, modifying therecognition template of the scheme to include new situations, or constructing a new scheme. In the context of fractions, reor-ganization refers to the accommodations students make to their whole number schemes to resolve problematic situationsinvolving a larger class of numbers—fractions.

Via teaching experiments with several pairs of students (cf., Steffe & Thompson, 2000), Steffe and colleagues have inves-tigated the reorganization of students’ whole-number operations into new ways of operating with fractions, leading to theidentification of a hierarchy of fraction schemes (e.g., Hackenberg, 2007; Olive, 1999; Saenz-Ludlow, 1994; Steffe, 2001; Tzur,1999). Among others, they have identified a hierarchy of three schemes central to students’ construction of fractions: part-whole scheme (PWS), partitive unit fraction scheme (PUFS), and iterative fraction scheme (IFS). In line with Lamon’s findings(2007), they have also identified mental operations that support the multiplicative structures critical to the development offractions schemes (e.g., Hackenberg, 2007; Steffe & Olive, 2010).

3.2. Units coordinating

Units coordinating operations provide the basis for the multiplicative structures that Lamon (2007) advocated, becauseas Hackenberg (2007) noted, “students’ multiplicative structures are the units coordinations they can take as given priorto activity [emphasis added]” (p. 27). For example, with three levels of units coordination, a student might conceive of thequantity 35 as a unit, containing seven units of 5, which is itself a unit of five 1 s. Students with only two levels of unitscoordination can compute the value of 5 iterated seven times, but the multiplicative structure for 35 as seven 5 s doesnot exist prior to this activity. Here, we describe how various levels of units coordinating and other operations—typicallyconstructed in whole number contexts (Steffe & Olive, 2010)—play a vital role in the construction of fractions schemes.

3.3. Part-whole scheme

The PWS is characterized by the ability to separate a whole into n equally sized pieces (partitioning) and mentally dis-tinguish m of those pieces from the whole without destroying the whole (disembedding) (Steffe & Olive, 2010). This way ofoperating aligns well with the part-whole subconstruct (Norton & Wilkins, 2010), except that neither Kieren (1979) nor theRNP team (Behr et al., 1983) recognized disembedding as a psychological operation necessary for considering the part out ofthe whole. Because it involves a coordination of the m pieces, both within and without the n pieces in the whole, disembed-ding itself requires that students coordinate two levels of whole-number units (Steffe & Olive, 2010). Using a PWS, studentswill assimilate even a unit fraction, say 1/n, as one part out of n equal parts in the whole, rather than a 1-to-n size relationbetween part and whole. With the construction of a PUFS, unit fractions become quantities that do have multiplicative sizerelationships with the whole.

3.4. Partitive unit fraction scheme

As with the PWS, the PUFS includes mental actions of partitioning a whole into n equal parts and disembedding asingle part. However, with the PUFS, the two units being coordinated (the 1/n piece and the whole) now have a 1-to-nmultiplicative relationship because subsequent mental action involves making connected copies (iterating) of the 1/n piece,n times, to reproduce the whole (Steffe, 2001). Thus, a student who has constructed a PUFS can produce a specified unit



Table 3Fractions schemes and operations.

Scheme Operations involved Canonical tasks

Part-whole scheme (PWS) Partitioning a whole into n equal parts and disembedding a m ofthose parts to form a part to whole comparison, m out of n

Make 3/5 of the stick shown below

Partitive unit fraction scheme (PUFS) Partitioning a whole into n equal parts, disembedding one of thoseparts, and iterating that part n times to produce a 1-to-nrelationship between the unit fractional part and the whole

What fraction is the smaller stick of thelarger stick?

Iterative fraction scheme (IFS) Splitting the whole into n equal parts, disembedding one of thoseparts and iterating it r times to produce any fraction r/n, orreversing that way of operating to produce the whole from anygiven fraction

If the stick shown below is 7/5 of thewhole, draw the whole stick

fraction from an unpartitioned whole, or determine the unit fractional size of a given piece by iterating it within a givenwhole. It is important to note that, when iterating a unit fractional piece, students using a PUFS treat that piece as a unit of1, rather than a unit fraction, which has an invariant size relation to the whole (Steffe, 2001; Tzur, 1999). In other words,students use the PUFS to iterate pieces and, thus, establish a multiplicative size relation between the piece and the whole,but the iterated piece is not treated as a unit fraction that maintains this relationship when the whole is no longer present.In particular, students limited to a PUFS do not understand an improper fraction, say 7/5, as seven iterations of 1/5 becausethe whole is lost within the improper fraction. Such a conception requires additional ways of operating, such as splitting,coordinating three levels of units, and an iterative fraction scheme.

3.5. Splitting operation

The splitting operation is the simultaneous composition of partitioning and iterating operations (Steffe, 2001). Studentswho split understand, before carrying out either action, that partitioning a whole into n equal parts and iterating one of thoseparts n times are inverse actions, which nullify one another when composed in either order. This understanding manifestsitself in tasks that require partitioning and iterating operations be considered simultaneously, rather than just sequentially(as in the PUFS). For example, consider the following task: “This candy bar is five times as big as your candy bar; draw yourcandy bar.” The task is iterative in nature, as it refers to something that is five times as big; but to solve the task, studentsmust partition. Students who do not split will typically iterate the given bar five times, instead (Wilkins & Norton, 2011).Construction of the splitting operation marks a critical point in students’ development because splitting enables them toreverse their ways of operating so that they can reproduce the whole from a given fraction (Hackenberg, 2010). It also playsa key role in conceptualizing improper fractions.

3.6. Iterative fraction scheme

The iterative fraction scheme (IFS) supersedes PWS and PUFS because it enables students to treat all fractions—properand improper—as multiplicative size relations with the whole, and as “numbers in their own right” (Hackenberg, 2007, p.27), rather than a comparison of two whole numbers. Students can use this way of operating to produce the whole from anyfraction and produce any fraction from the whole. Operating with fractions in this way requires the coordination of threelevels of units in order to keep track of the improper fraction, the unit fraction being iterated, and the whole with whichthe unit fraction has an invariant multiplicative size relation (Tzur, 1999). For example, 7/5 is understood as a unit of sevenunits of 1/5, five of which comprise the whole.

3.7. Summary of schemes

Table 3 summarizes each of the fractions schemes described here, along with the operations involved with each. Couchedin these terms, our purpose is to investigate the schemes and operations available to a sixth-grade student, named Isaac, andto consider what accommodations would be necessary for him to meaningfully achieve the thread of fractions standardsincluded in Table 1. Whereas we might classify the three standards (3.NF.1, 4.NF.4a, 5.NF.5b) as involving part-whole,measure, and operator sub-constructs, respectively, our approach will be to consider the ways of operating that undergirdthose sub-constructs and related concepts.

4. Methods

The data analyzed for this article was obtained from a constructivist teaching experiment (Steffe & Thompson, 2000)with two sixth-grade students at a rural middle school in the Southeastern United States. Isaac and Kadyeisha were selectedbecause the classroom teacher indicated both students were struggling in her mathematics class, despite good attendanceand no identified special needs. Like most students in the United States, the students had been learning about fractions sincethird grade. In sixth grade, their teacher was using the Prentice Hall Math 6 (Course 1) textbook, which, like many textbooks



used in the United States (see Li, Chen, & An, 2009 for a characterization of US textbooks), concentrates on part-wholeconceptions. The teacher–researcher (first author) and witness met with both students (excepting student absences) forapproximately thirty minutes after school, once or twice per week, for seven sessions, each of which was video recorded.We focus on Isaac because he was the more expressive of the two students.

Teaching sessions began with hypothetical models of student thinking, which were adjusted in attempts to explain stu-dents’ responses to tasks that were specifically designed and posed to test the viability of the models (von Glasersfeld, 1995).An important characteristic of the nature of tasks was to encourage students to create objects of their own understanding,instead of an answer that the teacher–researcher or another adult might call correct. Manipulatives included constructionpaper, markers, scissors, tape, index cards, and a ruler. Fraction rods, which are rectangular shapes of multiple colors withuniform width and lengths between one and ten times the width, were also used extensively in latter sessions.

Developing models relied on careful observation of student–student, student–teacher, and student–manipulative inter-actions, as well as consideration of the researchers’ perspective. The witness assisted in videotaping and acted as an observer,with particular attention to student–teacher interaction. The teacher–researcher and the second author conducted retro-spective analysis of the videotapes in order to refine models of student thinking and to design new tasks. These tasks servedtwo purposes: to test existing models of students’ thinking and related hypotheses about how students would respond; andto provoke new ways of acting in fractions situations that might promote construction of new schemes and operations.

5. Analysis of Isaac’s ways of operating

In this section, we chart Isaac’s ways of operating through seven sessions in the teaching experiment, in which weattempted to support his progress from part-whole conceptions toward measure conceptions for fractions, as called for bystandard 4.NF.4a. The first four sessions illustrate Isaac’s ways of operating with fractions. Session 1 implicates Isaac as anemerging splitter (i.e., the splitting operation was available to Isaac but not well-established for him yet); Sessions 2 and 3indicate that, as an emerging splitter, Isaac was beginning to coordinate partitioning and iterating as inverse operations, toproduce the whole from a unit fraction and vice versa. The beginning of Session 4 indicates that Isaac had a second way ofoperating with fractions, based on his part-whole scheme. In line with CCSSM 3.NF.1, Isaac could use this way of operatingto conceive of proper fractions, such as 3/7, as three parts out of seven equal parts in the whole. When working with unitfractions, Isaac used the former way of operating, to conceive of unit fractions, 1/n, as one part that can be iterated n timesto produce the whole—consistent with the operations of a PUFS. However, when working with non-unit fractions, conflictsemerge between this way of operating and Isaac’s part-whole scheme. These conflicts first become apparent later in Session4 and again in Sessions 6 and 7. Once we have established a model for Isaac’s initial ways of operating, analysis will focus onsuch conflicts, because they have strong implications for successful modifications to curriculum and instruction that addressCCSSM.

5.1. Session 1: April 21, 2010

During this first meeting, the teacher–researcher posed tasks to Isaac and his partner Kadyeisha to test whether theycould split. The teacher–researcher handed the students a blank piece of construction paper and told them that it representeda candy bar four times as big as theirs, then asked the students to show him how big their candy bar would be. Kadyeishabegan by measuring the length of the candy bar (eleven inches) and attempted to use a division algorithm to divide 11 by4. Kadyeisha struggled to complete the algorithm and, after assistance from the teacher–researcher, both students demon-strated difficulty in interpreting the decimal values that resulted from it. The students did not identify a candy bar based onKadyeisha’s strategy. Protocol 1 begins twelve minutes into the session, when Isaac attempted his own solution.

Protocol 1:

I: You could put 2 lines [draws a line about half-way across length of paper and then draws a line about half-way from thereto the left end. Pauses, smiles, and looks up]. I don’t know.T-R: Well, you did that for a reason. . .T-R: Do you remember the original question? What was the question?I: This candy bar is four times as big as hersT-R: For some reason you drew two lines. . .I: [Draws third line and traces his finger across the left-most part.] I drew three lines and that divides up into four, so. . .you said that is four times as big as her candy, so yeah.A few minutes later in the session, the teacher–researcher asked the students to pose similar problems to one another.Kadyeisha began by showing Isaac a new blank sheet of construction paper and asking him to “make one five times as big.”I: So it’s bigger than this. . . Five times as big as this [with right hand on whole sheet. Puts together five sheets of constructionpaper.].K: [indicates she wanted Isaac to solve it differently]I: She said make one that’s five times as big as this, so I put five out. . . they’re all the same.



Fig. 1. Isaac’s comparison of a fourth and an eighth.

Isaac’s actions in response to the two tasks indicate that he could split, but that this way of operating was not yet wellestablished for him. For the first task, Isaac began with partitioning lines at the half and quarter marks, and seemed confusedas to how he should proceed. However, after adding a third partition, he knew he had produced a piece such that the givenwhole was four times as big, thus indicating that he understood partitioning and iterating as inverse operations. His responseto Kadyeisha’s task indicates that he was not simply acting procedurally, but could distinguish between producing a bar thatwas five times as big as the given bar, and producing a piece (through partitioning) such that the given bar was five times as big.

5.2. Session 2: April 26, 2010

Between the first two sessions, the teacher–researcher analyzed video from the first session to assess the students’ waysof operating. Having determined that Isaac was an emerging splitter, the teacher–researcher designed tasks to test hisconceptions of unit fractions. Did Isaac conceive of unit fractional parts as fractional sizes relative to a fixed whole? CouldIsaac iterate unit fractional parts as unit fractions, or would he treat them simply as units of 1? The teacher–researcher alsoplanned tasks for Kadyeisha, but she was absent for this session.

From the start of the session, Isaac consistently demonstrated his ability to relate unit fractional parts to the whole,through iteration of the part: He verified that a given fractional part was one-sixth of a given (unpartitioned) whole byiterating the part six times within the whole, and he produced a whole from a given one-fifth part by iterating that part fivetimes. Based on these actions, one might be tempted to say that Isaac could iterate unit fractions in a manner consistentwith PUFS and CCSSM 4.NF.4a. However, his actions in the following protocol (about 20 min into the session) indicate thatIsaac was not considering fractions like one-fifth and one-sixth as fractional sizes relative to a fixed whole.

Protocol 2:

T-R: What is bigger, a fourth or an eighth?”I: An eighth.T-R: Ok. Show me.I: [draws circle, partitions it evenly into eight parts and shades the four parts on the left-hand side, as shown in Fig. 1]T-R: If this was a pizza and you could have one fourth of the pizza, how many slices would you get?I: Four.T-R: Four looks like a half.I: Yeah. Half of eight.[a few minutes later in the session, the teacher–researcher returned to the topic]T-R: What does it mean to be a fourth? [after Isaac indicated some confusion about the meaning of the question] Doesone-fourth mean 1 out of 4?I: Sometimes.T-R: What else might one-fourth mean?I: Maybe a half?T-R: One fourth could be a half?I: [refers back to pizza drawing]



Novice learners of fractions commonly assume that a fraction such as one-eighth is bigger than a fraction such as one-fourth because eight is bigger than four. Such examples are used to justify the interference hypothesis (Behr et al., 1992).However, at least in Isaac’s case, this assumption seemed based on a particular way of operating with unit fractional parts.Consistent with his way of operating with unit fractions earlier in the session, Isaac established one-eighth as a part iteratedeight times within a whole. After producing this eight-part whole, he identified one-fourth as a part iterated four times withinthat whole so that his eighth was clearly bigger than his fourth. This was unproblematic for Isaac—who later went so far asto suggest that one-fourth could be a half (half of the whole, eight-part circle)—because one-fourth and one-eighth werenot fractional sizes relative to a fixed whole. They were based on iterations of a unit of 1, which he was not yet coordinatingwith a second unit—the fixed whole.

5.3. Sessions 3 and 4: April 28, 2010 and May 4, 2010

In Session 3, Isaac and Kadyeisha began solving tasks with fraction rods. Fraction rods are rectangular blocks ranging insize from a unit cube, to a rectangular rod that is ten times as long as the unit cube. We refer to these various rods based ontheir unit length (e.g., the unit cube is a 1-rod and the longest rod is a 10-rod); at times, we also mention their colors becausethe students often referenced the rods by color.

Isaac’s actions in Session 3, and at the beginning of Session 4, provide numerous affirmations that he conceptualizedunit fractions through iterations of a unit of 1. In the context of posing and solving tasks with Kadyeisha using fractionsrods, Isaac applied this way of operating to successfully reproduce the whole from given unit fractions (e.g., making thewhole from a one-fourth part) and to determine the fractional names of unit fractional parts when the whole was given(e.g., determining that a given red 2-rod was one-eighth of two brown 8-rods). His language provided further indicationof this way of operating, as he repeatedly referred to unit fractions in terms of how many times the “fit into” the whole.Protocol 3, in which Isaac poses a task to the teacher–researcher, occurs about 6 min into Session 4 and demonstrates thelimitations of Isaac’s associated conceptions for unit fractions.

Protocol 3:

I: [holds a white 1-rod] This is your candy bar. What’s your whole candy bar? . . . Wait. Does that make sense?I: [after K helps him clarify that it is “a piece of your candy bar”] Yeah. A piece!T-R: Now, this could be a really small piece of my candy bar. . .I: It’s a really small piece. . . a fifth.T-R: [places five white 1-rods together]I: Darn it! [apparently upset that the teacher–researcher solved the task so readily]

In posing his task for the teacher–researcher, Isaac did not seem to recognize the importance of designating the fractionalsize of the part relative to the whole. He began by simply showing the teacher–researcher a piece that represented “yourcandy bar” or “a piece”, as if the teacher–researcher could determine the whole from that information. Only after a fewprompts from Kadyeisha and the teacher–researcher did he define the fractional relationship between the given piece andthe whole.

On the other hand, immediately after posing this task, Isaac solved a task involving the reproduction of the whole from agiven unit fraction—one-sixth. Once again, he justified this production by referencing the number of times the given part fitinto the produced whole: “Well, it fits in there six times.” So it seems that Isaac interprets a unit fraction, 1/n, as a prompt foriterating the given part n times, but he does not conceive of it as an a priori size relation relative to the whole. As we see inProtocol 4, this way of reasoning becomes especially problematic when Isaac attempts to generalize it to non-unit fractions.

Protocol 4:

T-R: This is one-seventh of your candy bar [holding a single white 1-rod]. If this is one-seventh of your candy bar, whatwould two-sevenths of your candy bar look like?I: [immediately pulls out seven red 2-rods and lines them up]T-R: What does two-sevenths of the candy bar look like?I: [points to left end of the line of 2-rods] Right there.T-R: This whole thing is two-sevenths?I: Yeah.T-R: So if that’s one-seventh of your candy bar [picking up the 1-rod], that’s two-sevenths of your candy bar?I: Yeah, because. . .T-R: Can you explain, Kadyeisha? Do you agree, first of all?K: [indicates that she agrees] because these are two, and you have seven of them.I: That’s exactly what I was thinking because it fits in there twice [holding 1-rod next to 2-rod].

Recall that Isaac conceived of unit fractions, such as one-seventh, as one of seven iterations of the given unit. In Protocol4, Isaac seemed to have generalized this way of operating to non-unit fractions. Knowing that the red rod represented twoone-rods, Isaac iterated this 2-rod seven times and identified the line of 2-rods as two-sevenths. The teacher–researcher



did not immediately recognize Isaac’s production as a manifestation of his way of operating with unit fractions. Still, theteacher–researcher had a goal of engendering cognitive conflict by asking the students to compare this production to one thatwould result from their use of their part-whole schemes. The part-whole scheme relies upon a visible whole, and becausethe teacher–researcher knew that Isaac could produce the whole from a given unit fractional part, he picked up there.

Protocol 4 continued:

T-R: If this is one-seventh of the candy bar, show me what the whole candy bar looks like.I: So, it’s kind of like what we did earlier.T-R: [repeats task] What does your candy bar look like if this is a one-seventh piece of it.I: Oh. Okay. I’ve got it.K: [began working on the problem as Isaac watched her, but then indicated that she wanted help]I: [laid down a handful of 1-rods] He said this is one-seventh [holding a 1-rod, lines up and counts six 1-rods].K: [lays another 1-rod at the end of the other six, just as Isaac reached for another one]I: [stops and looks at the teacher–researcher]T-R: Kadyeisha, can you explain?K: You told him that this was one-seventh, so he added six more and that was the whole.T-R: [to Isaac] Is that what you would have said?I: I would have said the same thing.T-R: So this was one-seventh and this is the whole. [Pulls out a black 7-rod and lines it up behind the seven 1-rods] Can Isay that this is the whole?KI: [nod]T-R: So this [black rod] is the whole. Can you show me what 3/7 looks like?K: [pulls out 3 of the 7 and lines them up on the left side of black rod]T-R: [to Isaac] Can you explain what she did and whether you think she’s right?I: I think she’s right because you said three-sevenths. So three-sevenths of this [pointing to black rod]. So there were sevenall together, so you just. . . [unintelligible] put three.[The students go on to similarly produce five-sevenths, seven-sevenths, and two-sevenths.]T-R: A few minutes ago, I said “this [holding 1-rod and looking at KI] is one-seventh of the whole and I said show metwo-sevenths. And when I first asked that question, you made this [pointing at seven 2-rods, lined up] was two-sevenths.But just now, you both agreed that this [pointing at two 1-rods lined up with a black 7-rod] was two-sevenths.I: [smiles] Ahhh!T-R: So, I need y’all to think about it and figure out. This one you said was two-sevenths [pointing to the seven red rods], andthis one you said was two-sevenths [pointing to the two 1-rods]. I want y’all to try to work that out and explain whateveryou think.I: You said this was seven-sevenths [pointing to black rod] and make two-sevenths of this.T-R: I never said that. All I ever said [picking up a spare 1-rod] was this is one-seventh and then I asked you to make. . . Iasked you to make the whole. You’re right. I asked you to make seven-sevenths and all these others, but you did it all.KI: [both students think silently for about 15 s until there is an interruption from another student passing by the room]T-R: So, why don’t y’all tell me, is this one right? Is this one right? Are they both right? And I’d like to get reasons. . .K: I think they’re both right.T-R: Ok. What do you think, Isaac?I: I think this one’s right [pointing to seven 2-rods].T-R: Ok. Tell me why you think that one’s right, Isaac?I: Well, because you said this [holding a single] is one-seventh. If this is one-seventh, you have to, like, go higher to makeit to two sevenths. Ok. Like, um. It’s half [holding a white rod next to a red rod]. This is half [puts another white rod next tothe red rod]. . . It’s hard to explain.T-R: Ok, Kadyeisha, now, you said they’re both right, so you explain. . .K: I think you can get two ways, because. . . You take two out of seven [pointing to two 1-rods], and like. . .I: Wait. I kinda think that they’re both right because of what she said. . .. [adding another 1-rod to the end of the two 1-rods,lined up with the black rod] Well, you said this was three-sevenths.T-R: I never said that.I: Yeah. We said that.T-R: It made sense. . .I: Yeah. It made sense. And so you just take one away [taking the one away] and it will make two sevenths.T-R: If you could only pick one that was right, would you think this one [pointing to the seven red rods] or this one [pointingto the two white rods], for two-sevenths?K: [after about 10 s, points to two white rods]I: [after another 10 s, also points to two white rods]T-R: You both picked that one. Why do you believe that 1 more than that?T-R: [after about 15 s] Ok. Well, I just have one more question. If this was one-seventh of a candy bar [holding up a whiterod], and I said, “hey, here’s one seventh of the candy bar. One-seventh of the candy bar is this big.” And I said, “I want two



Fig. 2. Isaac’s comparison of one-sixth and one-seventh.

sevenths of the candy bar,” do you think I’d get this piece [bracketing seven red rods with both hands] or do you think I’dget this piece [bracketing two white rods with both hands]?I: [immediately points to the two white rods]K: [immediately follows with same gesture]I: Yeah. Because this [pointing to the seven red rods] is huge!

By asking the students to produce two-sevenths in a second way—based on first producing the seven-part whole—theteacher–researcher engendered a conflict between two ways of operating: a generalization of Isaac’s previous way of oper-ating with unit fractions (through iteration), and his part-whole scheme. The teacher–researcher hoped to leverage Isaac’spart-whole scheme so that Isaac would recognize the need for further modification of his previous way of operating withunit fractions. At first, it seemed that Isaac’s previous way of operating was dominant. The turning point occurred whenKadyeisha explained her own part-whole reasoning: “You take two out of seven.” This explanation seemed to resonate withIsaac’s own part-whole scheme so that he began to question his initial production of two-sevenths. He even began to reflecton the size of that production: “this is huge!”

5.4. Session 6: May 12, 2010

Isaac was absent for the Session 5 (May 10, 2010); Kadyeisha was absent for the Session 6. The teacher–researcher tookKadyeisha’s absence as an opportunity for a closer investigation of Isaac’s understanding of unit fractions. Isaac’s responsesto the tasks posed in Protocol 5 affirm that Isaac had not been considering unit fractions as size relations relative to a fixedwhole.

Protocol 5:

T-R: Just think about this in your head, and tell me which one you think is bigger, one-sixth or one-seventh?I: One-sixth or one-seventh? One. . . seventh.T-R: One-seventh is bigger than one sixth?I: Yeah.T-R: Okay. Why do you think that?I: Well, because, if I took one seventh and put it in a row [spreading his hands apart to trace a long thin rectangle] and putone sixth [same motion but shorter], one seventh would be bigger.T-R: Okay. Can you show me what you mean? [sliding two index cards to Isaac] Would these help?I: [writes “1/7” and “1/6” on the right side of one of the index cards and then draws the two figures shown in the top left ofFig. 2]T-R: [just as Isaac finishes his drawings] Okay. Let me ask you a different question. Which one is bigger, one-seventh of acandy bar or one sixth of a candy bar?I: [after about 8 s of silence] Can I use this? [pointing to same index card]T-R: Yeah, but what’s your initial guess?I: One seventh.T-R: You think one-seventh of a candy bar is bigger than one sixth?I: [nods]T-R: Okay.I: Wait. One and one-seventh or just one-seventh?



T-R: One-seventh of a candy bar or one-sixth of a candy bar, which is bigger?I: It’s the same as this [pointing to his previously drawn figures]T-R: Okay. Let me ask you a different question. [holding second index card] This is a candy bar. If it were a real candy bar,which would you rather have, one-seventh of it or one-sixth of it?I: [after about 5 s of silence] One-seventh.T-R: Okay. Why?I: Because it’s bigger.

Despite the varying contexts in which the question was posed, Isaac insisted that one-seventh was bigger than one-sixth.Isaac’s drawing justified his reasoning by illustrating that one-seventh was seven iterations of a unit, and one-sixth was onlysix iterations of that unit. We could take this as evidence for the interference hypothesis, but it can also be explained as anecessary error based on Isaac’s ways of operating. Isaac’s responses were not simple mistakes, but rather were correct (fromhis perspective) based on the functioning of his current schemes (cf., Steffe, 2001). Isaac’s actions are consistent with hisway of operating with unit fractions across the four previous sessions. The teacher–researcher’s role is to challenge theseways of operating with appropriate tasks so that, through the activity of resolving the tasks, the student might reorganizehis ways of operating. In what follows, the teacher–researcher again attempted to engender conflict by appealing to Isaac’spart-whole scheme, beginning with a given whole and asking Isaac to produce each unit fraction from that whole.


T-R: All right. I’m going to get two copies of this candy bar [grabbing two blank 5 × 8 index cards]. I’d like you to makeone-seventh of the candy bar using this one, and one-sixth of the candy bar using this [other] one.I: [draws seven equally-spaced vertical lines down the length of the index card, then counts to find that he produced eightparts] Oh. That’s eight. [flips the card over and draws five lines, producing six equal parts] Okay. That’s one-sixth.T-R: Which one is one-sixth? If I said I was going to give you one-sixth of this candy bar, what would I give to you?I: Right there [pointing to left-most part].T-R: Now, show me what I would give you if I was going to give you one seventh of the candy bar.I: [draws six vertical lines across the length of a new 5 × 8 index card] Right there [pointing to rightmost part].T-R: Okay. Which one is bigger, the one-seventh piece or the one-sixth piece?I: Wait, like, which one has more pieces or. . .T-R: I said I will give you either one-sixth of the candy bar, which you said was this [pointing to leftmost most of index cardwith six parts] or I will give you one-seventh of the candy bar [laying hand on index card with seven parts].I: Oh! It’s going to be one-sixth is bigger [smiling].T-R: Explain.I: Because you’re going to have to make this smaller [tracing lines across the index card with seven parts]T-R: Oh! So, what’s wrong with this [pointing to the two drawings in the top left of Fig. 1].I: [very quietly] Well, that’s right, but this is right too.T-R: What does the whole candy bar look like in this picture?I: That one [points to top drawing in Fig. 1, labeled “1/7”]T-R: What about that one [pointing to bar labeled “1/6”]? Is that not the whole candy bar?I: I don’t know.T-R: So, this one might be the whole candy bar, but what is this then [pointing to “1/6” bar again]?I: Six?T-R: But you’re pretty sure that one-sixth is bigger than one-seventh. . .I: Yeah.T-R: And how do you know that?I: Well, because, wait for this one [pointing to one of the partitioned index cards] or this one [pointing to the card shownin Fig. 1]?T-R: Uh, for both.I: For this one [pointing to index card with six parts], this is bigger than this [pointing to card with seven parts] because ittakes up less space.T-R: Ok. And what about for this [pointing to card with drawings]?I: I don’t know.

In the absence of a whole on which to operate, Isaac had produced one-sixth and one-seventh by iterating a unit withfixed size. When given a whole on which to operate, Isaac produced one-sixth and one-seventh by partitioning that fixedwhole. Reflection on this latter activity seemed to help him conceptualize the inverse relationship between the number ofparts in the whole and the sizes of those parts, because when you use more parts, “you’re going to have to make [the parts]smaller.” Yet, Isaac remained uncertain about the validity of his initial productions [the ones illustrated in Fig. 2].

As the session continued, Isaac’s actions indicate that he had begun modifying his ways of operating to resolve theconflicts he had experienced in Protocols 4 and 5. In particular, when given a 1-rod that represented one-eighth of a candybar and asked to produce three-eighths, Isaac readily lined up three 1-rods (in contrast to his previous response, in Protocol



4, when asked to make two-sevenths given one-seventh). He was even able to operate in reverse, producing one-fifth giventhree-fifths, as indicated by his actions in Protocol 6.

Protocol 6:

T-R: This [placing a red 2-rod in front of Isaac] is a fifth of a candy bar. If that’s a fifth of a candy bar, what does the wholelook like?I: [pulls out four more red rods and lines them up]T-R: If that [placing a green 3-rod in front of Isaac] is three-fifths of a candy bar, can you show me what one fifth of thecandy bar looks like?I: [lines up three 1-rods and then removes two of them, leaving one 1-rod next to the green rod] Right there.T-R: How’d you do that? That was a tough one, I thought.I: Well, you said this was three [picking up the green rod], so you just put. . . [pulling out another 1-rod] These are thesmallest pieces, so you just put three in there, and then take away two.

These ways of operating—forward and reverse—indicate that Isaac had constructed a splitting operation and that he hadbegun to reconcile the results of iterating and partitioning with his part-whole scheme. As we saw in Protocol 5, he alsobegan to consider unit fractions as relative sizes, indicating a PUFS. However, he did not conceive of non-unit fractions asiterations, or multiples, of unit fractions, as required by CCSSM 4.NF.4a. This is indicated in Protocol 6 when Isaac referred tothe green rod as three, rather than three fifths. He still seems to be treating unit fractions as units of one without consideringtheir relation to the whole. The limitation of this way of operating becomes apparent in Session 7.

5.5. Session 7: May 17, 2010

The teacher–researcher met with Isaac and Kadyeisha once more during their sixth-grade year. This final meeting beganwith several unit fraction tasks. Although the teacher–researcher intended these tasks as opportunities to assess and supportKadyeisha’s reasoning with unit fractions, Isaac’s responses provide opportunity to assess what he learned from Session 6. Inparticular, the teacher–researcher asked the students which was bigger, one-eighth or one-ninth, to which Isaac respondedone-eighth, because “one-ninth has littler squares if you divide it up.” This response indicates that Isaac had made a relativelypermanent modification to his conception of unit fractions, on the basis of his experiences in the previous session. He clearlyunderstood the inverse relationship between the number of parts indicated by a unit fractional name and the size of eachpiece relative to the given whole, which provides affirmation that Isaac had constructed a PUFS.

About 17 min into the session, the teacher–researcher began investigating the students’ conceptions of non-unit properfractions. Isaac’s responses in Protocol 7 illustrate that, despite significant progress with unit fractions, his conceptions forproper fractions had not progressed to size relations with the whole.

Protocol 7:

T-R: [places another longer rectangular index card in front of the students] Suppose this is a candy bar, and I said. . . Again,I want you to do this in your heads. This is a candy bar, and I said you can have three-fifths of this candy bar. Imagine inyour heads how much you would get if I said you could have three-fifths of that candy bar.I: [places hand on index card] Three-fifths of this?T-R: Yeah. Just imagine in your heads first.I: [about 2 s later] I got it.K: [thinks for another 7 s and then indicates she has it too]I: Can I cut it out?T-R: I’ll let you later, but make sure you can imagine it in your head. What’s it look like?I: Huh?T-R: How big of a piece?I: [uses thumb and forefinger to span leftmost third of the index card] Probably about that big.T-R: Kadyeisha, do you think the piece is going to be bigger or smaller than that, or the same size?K: [spans the same width with her fingers] same size.T-R: So you agree? He got it exactly right?K: [nods head]I: Right there [spans the same region, and then moves his fingers to the next third]. Right there.T-R: How did you come up with that, Isaac? How did you do that in your head?I: I don’t know. I just figured this candy bar [placing hand across whole index card] was pretty big.T-R: Ok. And when you get three-fifths of it, what does that mean?I: Oh! Wait! I thought you meant like. . . I thought you said one fifth.T-R: No. No. No. Three fifths. [to Kadyeisha] Did you think one-fifth too, or did you think three-fifths?K: [points to left side of card] Three. . .I: I mean, I thought you said one third.



T-R: Ok. So you thought I said either one-third or one-fifth. What were you thinking of, Kadyeisha? . . .three-fifths . . .orwere you thinking one-fifth or one-third?I: Oh, it’s going to be way smaller than that, but I think I got it.T-R: [to Isaac] Hold on a second. I need to see what she was thinking. [to Kadyeisha] Do you still think what he had wasthree-fifths, because he now says he was thinking of one-fifth or one-third?I: One-third.

It’s important to note that Isaac interpreted my request for three-fifths as a request for one-third, because this assimilationindicates that Isaac could not operate with non-unit fractions in the way he could operate with unit fractions. Furtherindication arises as the protocol continues.


K: [produces an accurate estimate by taking three of five equal parts]T-R: Isaac, what do you think? How big do you think three-fifths is?I: [spans fingers across about one-fifth of the index card] Right there.T-R: Now, you think this [spanning one-fifth] is going to be three-fifths of the candy bar, and Kadyeisha thinks this [spanningthe section Kadyeisha did, which included Isaac’s part but went about three-fifths across] is going to be three-fifths of thecandy bar.I: Oh! Oh! Oh! Oh! Oh! I thought we were doing like. . . I’m conf. . . I thought we were doing like five [using fingers toindicate five equal sections that span the length of the index card].T-R: But what does three-fifths mean?K: Three out of five.I: [using thumb and forefinger to span approximately the same region Kadyeisha had] Yeah. That’s probably about righthere.T-R: Ah! That’s pretty close to what Kadyeisha said.

This time, Isaac seemed to interpret the request for three-fifths as a request for one-fifth. This was the second time that heattempted to assimilate the task into his PUFS, but three-fifths was not a relative size for Isaac, in the way that one-third andone-fifth were. Neither was three-fifths three iterations of a unit that was one-fifth of the whole. Thus, in his assimilationof the task, he focused on either the numerator or denominator and used that number to establish a unit fraction. When hedid finally construct three-fifths, he seemed to rely on his PWS, as Kadyeisha had (“three out of five”), by first identifyingthe five equal parts in the whole, which is something he did not need to do in order to identify one-fifth or one-third.

6. Conclusions

The purpose of this paper was to illustrate cognitive challenges introduced by the Common Core State Standards forMathematics (2010) with regard to conceptualizing fractions. Isaac’s case illustrates these challenges and their relation tohis available ways of operating with fractions (i.e., his fraction schemes). In this section, we summarize our findings, whichconnect fractions standards, fractions sub-constructs, fractions schemes, and the ability to coordinate various levels of units.

The three fractions standards discussed here (see Table 1) roughly address three of Kieren’s (1979) subconstructs: part-whole (3.NF.1), measure (4.NF.4a), and operator (5.NF.5b).1 These three standards also represent an intended learningprogression, from part-whole conceptions of fractions to more powerful conceptions. The intent of 4.NF.4a—transcendingpart-whole conceptions by treating non-unit fractions as iterations of unit fractions (i.e., measures)—is especially laudablebecause research has demonstrated that this progression is critical to understanding improper fractions (Tzur, 1999), equiva-lent fractions (Steffe, 2003), fractional sizes (Hackenberg, 2007), as well as fractions multiplication and division (Hackenberg& Tillema, 2009). International comparisons indicate that, relative to many developed countries, the United States currentlylacks such a focus, and that this disparity might partially explain the poor performance of US students, relative to peers inthose countries (Charalambous et al., 2010; Li et al., 2009; Moseley, Yukri, & Ishida, 2006; Watanabe, 2007). Moreover, whenwe consider students’ underlying ways of operating with fractions and how these ways of operating can be reorganized fromone domain to another, we can understand how iterating unit fractions supports a more general progression in students’multiplicative structures that spans whole number, rational number, and even algebraic reasoning (Hackenberg, 2010).

From a scheme theoretic perspective, in order to meet standards specified by CCSSM, students need to construct waysof operating (schemes) in which to meaningfully assimilate fractional situations (Steffe & Olive, 2010). With his part-wholescheme, Isaac was able to assimilate part-whole tasks, in line with 3.NF.1, but this scheme proved inadequate for solvingtasks aligned with 4.NF.4a. Over the course of the teaching experiment (and similar to findings by Olive & Vomvoridi,2006), tasks and activities involving the iteration of unit fractional pieces did seem to support Isaac’s construction of a PUFS,with which to conceive of unit fractions as measures. However, Isaac continued to struggle with measure tasks involving

1 A fourth standard, 5.NF.3, addresses the quotient sub-construct.



non-unit fractions. The need for a more powerful scheme is best demonstrated in Protocol 7, in which he repeatedlyassimilated three-fifths as a unit fraction within his PUFS, with unsuccessful results.

Limitations among Isaac’s fractions schemes seemed to emanate from limitations in his units coordinating activity. Forexample, when iterating unit fractional parts, it became clear that Isaac was iterating them as units of 1. At first, he did notseem to coordinate these iterations with a fixed whole, unless the whole was physically present, where he could assimilatethe situation within his part-whole scheme. This limitation explains sessions in Protocols 2 and 6, in which Isaac argued thata fourth was bigger than an eighth, and that a seventh was bigger than a sixth. Although Isaac eventually established theinverse relation between number of parts in the whole and the size of each part, along with the construction of a PUFS, heseemed to have no structure available for coordinating non-unit fractions, a/b, as a iterations of 1/b, as explicitly called forby standard 4.NF.4a. In fact, in attempting to produce two-sevenths from one-seventh, he iterated a two-stick seven times!

The reorganization hypothesis posits that students can reorganize their ways of operating with whole numbers toconstruct fractions schemes (Olive & Steffe, 2001), and units coordinating plays a particularly important role in this reor-ganization (Hackenberg, 2007; Steffe & Olive, 2010). Isaac’s case supports the reorganization hypothesis by demonstratinghow his units coordinating activity—particularly his iterations of unit fractional parts as units of 1—both afforded and con-strained his assimilations of fractional situations. At the same time, Isaac’s case supports Lamon’s (2007) contention that,beyond subconstructs, it is multiplicative structures—as determined by units coordinating activity (Hackenberg, 2007)—thatmediate students’ development of rational number concepts.

Meaningful attainment of standard 4.NF.4a calls for an iterative fraction scheme, with which to iterate unit frac-tions as fractional units, rather than as units of 1. Although Isaac had constructed the requisite splitting operation, theteacher–researcher worked in vain to support his construction of the iterative fraction scheme because Isaac lacked a struc-ture for coordinating the three levels of units involved; e.g., 3/5 as a unit of three units of 1/5, which has an invariant1-to-5 size relation with the whole (Hackenberg, 2007; Tzur, 1999). Standard 5.NF.5b calls for yet more powerful schemes,even further beyond Isaac’s existing ways of operating (see Hackenberg, 2010 for a description of schemes requisite forunderstanding fractions as operators).

7. Implications

Isaac’s case highlights the need for curricular standards to account for more comprehensive learning progressions, espe-cially ones that attend to students’ ways of operating and that cross content domains, such as whole number and fractions.To justify their decision to focus on learning progressions as the basis for sequencing mathematics standards, the authors ofthe CCSSM cited Confrey (2007), who cautioned against standards that are logically correct, but exclude “insights of meaningthat result from a careful study of learning” (p. 39). Particularly germane to Isaac’s difficulties, careful studies of learningindicate that coordinating three levels of units plays a critical role in students’ abilities to meet standard 4.NF.4a and, likemany other state standards, CCSSM does not specify such operational connections or prerequisites.

Over the next few years, as the new standards are implemented and tested across 46 states and three US territories,teachers and researchers will be evaluating their effect. Early evidence suggests what many mathematics educators havesuspected—that CCSSM will present new challenges for students and teachers. For example, when Kentucky modifiedits test to align with CCSSM, the state experienced a steep decline in student performance (http://www.edweek.org/ew/articles/2012/11/02/11standards.h32.html?tkn=LTUFfOpEpCAWut48lfLCsU4FHbuNRdCD%2F0qa&cmp=clp-edweek). Onepossible response is to reorder standards across grade levels. The present study suggests that students’ available levels ofunits coordination should be a contributing factor in determining the appropriate placement of fractions standards.

With regard to the three fractions standards considered here (and outlined in Table 1), the relative order need not change,but educators and policy makers should recognize that the latter two standards (4.NF.4a and 5.NF.5b) might require moretime to satisfy because of the underlying cognitive demands. Rather than simply moving the standards back another year,CCSSM might be framed as developmental standards that educators explicitly promote beginning in fourth and fifth grades(respectively) but that might take more than one year to achieve. Furthermore, in recognizing that underlying cognitivedemands require students to coordinate three levels of units, curriculum and instruction should support such ways of oper-ating in order to enable students to meaningfully achieve those standards. And recent research indicates similar challengesfor professional development, as many middle school teachers do not reason with three levels of units—a distinction thatdefines an important subgroup in developing mathematical knowledge for teaching (Izsák, Jacobson, de Araujo, & Orrill,2012).

The role of units coordination in mathematical development is not limited to standards 4.NF.4a and 5.NF.5b, nor is itlimited to fractions. Recent research, from disparate theoretical perspectives, implicates units coordination as a way ofoperating that undergirds meaningful attainment of mathematical knowledge across content domains, including wholenumber (e.g., Steffe, 1992), integers (e.g., Ulrich, 2012), fractions (e.g., Hackenberg, 2010; Izsák et al., 2012), and algebra (e.g.,Ellis, 2007; Olive & C aglayan, 2008). Thus, existing research suggests that an instructional and curricular focus on the levelsof units that students coordinate could prove a wise investment in achieving CCSSM. The present study contributes to suchwork by demonstrating particular challenges associated with students’ available ways of operating and Common Core StateStandards for fractions. Similar studies are needed within other content domains (especially integers and algebra) in orderto better align associated standards with their particular cognitive challenges.



References

Behr, M., Harel, G., Post, T. A., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook for research on mathematics teachingand learning (pp. 296–333). New York: Macmillan Publishing Company.

Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts andprocesses (pp. 91–126). New York: Academic Press.

Brenner, M. E., Herman, S., Ho, H., & Zimmer, J. M. (1999). Cross-national comparison of representational competence. Journal for Research in MathematicsEducation, 30(5), 541–557.

Charalambous, C. Y., Delaney, S., Hsu, H.-Y., & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from threecountries. Mathematical Thinking and Learning, 12(2), 117–151.

Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies inMathematics, 64(3), 293–316.

Cobb, P., & Jackson, K. (2011). Assessing the quality of the Common Core State Standards for Mathematics. Educational Researcher, 40(4), 183–186.Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved from:. http://www.corestandards.org/assets/

CCSSI Math%20Standards.pdfConfrey, J. (2007, February). Tracing the evolution of mathematics content standards in the United States: Looking back and projecting forward. Paper presented

at the K12 mathematics curriculum standards conference: What should students learn and when should they learn it. Arlington, VA. Retrieved from:http://ced.ncsu.edu/sites/default/files/Confrey.pdf

Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth-and fifth-grade students: A comparison of the effects of using commercialcurricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144.

Cramer, K. A., Wyberg, T., & Leavitt, S. (2009). Fractions operations and initial decimal ideas. Retrieved from:. http://www.cehd.umn.edu/rationalnumberproject/rnp2.html

Ellis, A. B. (2007). The influence of reasoning with emergent quantities on students’ generalizations. Cognition and Instruction, 25, 439–478.Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. The Journal of Mathematical

Behavior, 26(1), 27–47.Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432.Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes.

The Journal of Mathematical Behavior, 28(1), 1–18.Hartnett, P., & Gelman, R. (1998). Early understandings of numbers: Paths or barriers to the construction of new understandings? Learning and Instruction,

8(4), 341–374.Izsák, A., Jacobsen, E., de Araujo, Z., & Orrill, C. H. (2012). Measuring mathematical knowledge for teaching fractions with drawn quantities. Journal for

Research in Mathematics Education, 43(4), 391–427.Kieren, T. E. (1979). The development in children and adolescents of the construct of rational numbers as operators. Alberta Journal of Educational Research,

25(4), 234–247.Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of

research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age Publishing.Li, Y., Chen, X., & An, S. (2009). Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese, and U.S. mathematics

textbooks: The case of fraction division. ZDM – The International Journal on Mathematics Education, 41(6), 809–826.Moseley, B. J., Yukari, O., & Ishida, J. (2006). Comparing US and Japanese elementary school teachers’ facility for linking rational number representations.

International Journal of Science and Mathematics Education, 5, 165–185.Mullis, I. V. S., Martin, M. O., & Foy, P. (2008). TIMMS 2007 international mathematics report: Findings from IEA’s trend in international math and

science study at the fourth and eighth grades. Chesnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College. Retrieved from:.http://timss.bc.edu/TIMSS2007/mathreport.html

Norton, A., & Wilkins, J. L. M. (2010). Students’ partitive reasoning. The Journal of Mathematical Behavior, 29(4), 181–194.Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis. Mathematical Thinking and Learning, 1(4), 279–314.Olive, J., & C aglayan, G. (2008). Learner’s difficulties with quantitative units in algebraic word problems and the teacher’s interpretations of those difficulties.

International Journal of Science and Mathematics Education, 6, 269–292. http://dx.doi.org/10.1007/s10763-007-9107-6Olive, J., & Steffe, L. P. (2001). The construction of an iterative fractional scheme: The case of Joe. The Journal of Mathematical Behavior, 20(4), 413–437.Olive, J., & Vomvoridi, E. (2006). Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim. The Journal

of Mathematical Behavior, 25(1), 18–45.Piaget, J. (1970). Genetic epistemology. New York: Columbia University Press.Porter, A., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core Standards: The new U.S. intended curriculum. Educational Researcher, 40(3), 103–116.Saenz-Ludlow, A. (1994). Michael’s fraction schemes. Journal for Research in Mathematics Education, 25(1), 50–85.Spillane, J. P. (2000). Cognition and policy implementation: District policymakers and the reform of mathematics education. Cognition and Instruction, 18(2),

141–179.Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5),

503–518.Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309.Steffe, L. P. (2001). A new hypothesis concerning children’s fractional knowledge. The Journal of Mathematical Behavior, 20(3), 267–307.Steffe, L. P. (2003). Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5. The Journal of Mathematical

Behavior, 22(3), 237–295.Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer: New York.Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly, & R. A. Lesh (Eds.),

Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Lawrence Erlbaum Associates.Streefland, L. (1993). Fractions: A realistic approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp.

289–326). Hillsdale, NJ: Erlbaum.Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research

in Mathematics Education, 30(4), 390–416.Ulrich, C. L. (2012). Additive relationships and signed quantities (Doctoral dissertation). Retrieved from: http://www.libs.uga.edu/etd/von Glasersfeld, E. (1995). A constructivist approach to teaching. In L. P. Steffe, & J. Gale (Eds.), Constructivism in education (pp. 3–15). Hillsdale, NJ: Erlbaum.Watanabe, T. (2007). Initial treatment of fractions in Japanese textbooks. Focus on Learning Problems in Mathematics, 29(2), 41–60.Wilkins, J., & Norton, A. (2011). The splitting loope. Journal for Research in Mathematics Education, 42(4), 386–416.

author's personal copy - virginia tech · a major change in the intended mathematics...

Documents