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Teaching prospective teachers about fractions:historical and pedagogical perspectives

Jungeun Park & Beste Güçler & Raven McCrory

Published online: 24 October 2012# Springer Science+Business Media Dordrecht 2012

Abstract Research shows that students, and sometimes teachers, have trouble with frac-tions, especially conceiving of fractions as numbers that extend the whole number system.This paper explores how fractions are addressed in undergraduate mathematics courses forprospective elementary teachers (PSTs). In particular, we explore how, and whether, theinstructors of these courses address fractions as an extension of the whole number systemand fractions as numbers in their classrooms. Using a framework consisting of fourapproaches to the development of fractions found in history, we analyze fraction lessonsvideotaped in six mathematics classes for PSTs. Historically, the first two approaches—part–whole and measurement—focus on fractions as parts of wholes rather than numbers, and thelast two approaches—division and set theory—formalize fractions as numbers. Our resultsshow that the instructors only implicitly addressed fraction-as-number and the extension offractions from whole numbers, although most of them mentioned or emphasized theseaspects of fractions during interviews.

Keywords Whole numbers . Fractions . Rational numbers . Part–whole . Measurement andratio . Division . Pre-service teacher education . ElementaryMathematics . K–12

1 Introduction

The study of the teaching and learning of fractions has been an important area of mathe-matics education research for many years. Much of the research on fractions over the last

Educ Stud Math (2013) 82:455–479DOI 10.1007/s10649-012-9440-8

J. Park (*)Department of Mathematical Sciences, University of Delaware, 402 Ewing Hall,Newark, DE 19711, USAe-mail: [email protected]

B. GüçlerKaput Center for Research and Innovation in STEM Education, University of Massachusetts Dartmouth,200 Mill Road, Suite 150B, Fairhaven, MA 02719, USAe-mail: [email protected]

R. McCroryTeacher Education, Michigan State University, 620 Farm Lane #114B, East Lansing, MI 48824, USAe-mail: [email protected]

three decades has focused on student learning and misconceptions (e.g., Kieran, 1992;Lamon, 2007; Post, Cramer, Lesh, Harel, & Behr, 1993), and on unpacking rationalnumbers (represented as fractions) as mathematical and pedagogical objects (e.g., Ball,1988; Ma, 1998; Mack, 1990; Tirosh, 2000). Recently, there also has been an increasinginterest in pre-service and in-service teachers’ knowledge about fractions. Various studieshave shown that many elementary in-service and pre-service teachers have difficultieswith fractions similar to the difficulties of K–8 students (Ball, 1988; Post et al., 1993;Post, Harel, Behr, & Lesh, 1988; Zhou, Peverly, & Xin, 2006; Sowder, Bedzuk, &Sowder, 1993). These results suggest that elementary teachers’ knowledge of fractions isoften weak. However, mathematics educators lack both a conceptualization of what pre-service elementary teachers should learn as a requirement for certification and a clearexposition of what we teach in their certification programs (Even, 2008; Wilson, Floden,& Ferrini-Mundy, 2001, pp. 6–11). Based on this observation, this study explores howimportant mathematical aspects of fraction—fraction as a number and fraction as anextension of whole numbers—are addressed in mathematics content courses for pre-service elementary teachers (PSTs).

Existing studies about teaching and learning fractions have identified five subconstructs—part–whole, measurement, ratio, operator, and quotient—that capture the complexity of thetopic (Sowder, Philipp, Armstrong, & Schappelle, 1998; Lamon, 2007). They have reported thedominance of part–whole interpretation in students’ and teachers’ thinking about fractions(Ball, 1988; Newton, 2008; Post et al., 1993, 1988; Sowder et al., 1993; Weller, Arnon, &Dubinsky, 2009; Zhou et al., 2006), and their failure to conceptualize fractions as an extensionof whole numbers (e.g., Post et al., 1993), and fractions as numbers (Kerslake, 1986; Pitkethly&Hunting, 1996). Researchers have pointed out that students need to move beyond partitioningto realize fractions as single entities, and thus as numbers (Behr, Harel, Post, & Lesh, 1993;Mack, 1993; Post et al., 1993). These two aspects of fractions—fractions as numbers andfractions as an extension of whole numbers—are closely related; students are expected to usetheir prior knowledge of whole numbers to make sense of fractions as well as the operations andproperties of fractions so that they conceive fractions as numbers that are connected to wholenumbers. Studies, however, have shown that some students, and teachers, conceptualizefractions as something separate from whole numbers, with a unique set of rules and procedures(Kerslake, 1986; Post et al., 1993, pp. 338–9).

Conceiving fractions as objects disconnected from whole numbers is also found in earlymathematicians’ conceptualizations of fractions. Historical documents show that the idea offractions existed and was used for centuries with symbolic representations, without fractionsbeing considered as legitimate numbers on a par with whole numbers (Klein, 1968). Thissimilarity between students’ and early mathematicians’ conceptualizations of fractionsprovided a motivation for us to use the mathematical development of fractions as numbersto develop our framework to investigate how fractions are defined or developed from wholenumbers in mathematics classes for PSTs. In other words, we used a historical lens—adistillation of the mathematical history of fractions—focusing particularly on the definitionof fractions and the extension of fractions from the whole number system as they play out ina sample of required mathematics courses that include fractions. More specifically, weaddress the following research questions:

1. Do instructors of undergraduate mathematics classes for PSTs address fractions as anextension of the whole number system and fractions as numbers, and if so, how?

2. In what ways is a historical lens helpful in analyzing and understanding the instructors’approaches to fractions as numbers and as an extension of whole numbers?

456 J. Park et al.

In this paper, we first review the literature about students’ and teachers’ thinking aboutfractions focusing on the two aspects—fraction as a number and as an extension of the wholenumber system. Next, we provide our framework, an interpretation of the mathematicalhistory of fractions focusing on the development of fraction as a number. Finally, we presentdata from interviews with instructors and classroom observations from six undergraduatemathematics classrooms in which fractions are taught to PSTs, using the framework as a toolfor analysis.

2 Literature review

There has been a plethora of studies in teaching and learning about fractions, most of whichreport K–8 students’ difficulties with fractions (e.g., Erlwanger, 1973; Mack, 1990; Post etal., 1993). As mentioned earlier, studies have shown that the part–whole interpretation isdominant in students’ understanding in K–8 classrooms, even though the part–wholeapproach is not necessarily a good start for effective fraction teaching compared to otherapproaches (Lamon, 2001, p. 163). This part–whole dominance is closely related to stu-dents’ failure to conceptualize fractions as numbers by introducing the denominator as awhole and the numerator as a part separately (Kerslake, 1986; Pitkethly & Hunting, 1996).Pitkethly and Hunting (1996, p. 10) stated that students consider a fraction as a compositionof two numbers rather than a single entity. Similarly, Hart (1987) found that students may tryto find equivalent fractions by adding the same number to both numerator and denominator(e.g., A=B ¼ Aþ Cð Þ= Bþ Cð Þ ); or multiplying only the numerator by a constant (e.g.,A=B ¼ A� Cð Þ=B ) without considering equivalent fractions as numerals that represent thesame number. Post and colleagues (1993) also reported students’ difficulties locating afraction on a number line, which suggests a lack of understanding fractions as numbers.Students also have trouble recognizing how operations on fractions are similar to or differentfrom operations on whole numbers (Post et al., 1993). For example, when comparing twofractions, students tend to compare two denominators instead of considering the sizes of thetwo fractions (Post et al., 1993). In fraction addition, they often add numerators anddenominators separately, ignoring the unit of addition (Erlwanger, 1973; Mack, 1990;Stafylidou & Vosniadou, 2004; Streefland, 1993; Tirosh, Fischbein, Graeber, & Wilson,1999).

Studies on PSTs’ thinking about fractions show that their errors and misunder-standings are similar to the problems children have with fractions (Ball, 1988; Osana& Royea, 2011; Post et al., 1993, 1988; Zhou et al., 2006; Sowder et al., 1993), eventhough they bring considerable knowledge and experience with fractions to theirundergraduate mathematics classes (Mack, 1990; Tirosh, 2000). Studies have reportedPSTs’ difficulties understanding the relationship between whole numbers and fractionsand their incorrect applications of whole number properties in fraction operations.Rizvi and Lawson (2007) reported that prospective teachers, who successfully repre-sented whole number division word problems using various models, showed difficul-ties in developing representations of fraction division problems. As Rizvi and Lawson(2007) pointed out, these difficulties might come from PSTs’ reliance on a repeatedsubtraction understanding of division. The difficulties can also be seen as lack ofunderstanding of the relationship between whole numbers and fractions.

Newton (2008) found that one prevalent and persistent error among PSTs was addingnumerators and denominators in fraction addition (p. 1096). In subtraction, A/B−C/D (e.g.,1/3–2/7), some PSTs attempted to subtract A/B from C/D (e.g., 1/3 from 2/7) suggesting the

Teaching prospective teachers 457

misconception that a smaller number (e.g., 1 in 1/3) should be subtracted from a largernumber (e.g., 2 in 2/7) in whole number subtraction (p. 1097). Post et al. (1993) alsomentioned that PSTs conceived of fractions as different from whole numbers, applyingconcepts and properties of whole numbers incorrectly to understand fractions and computefraction operations.

There are similarities between K–8 students’ and preservice teachers’ difficulties con-ceiving of fractions as numbers and as an extension of whole numbers and the relatedproperties of fractions and early mathematicians’ conceptualizations of fractions. Fractionswere used in various computations without being considered as numbers by the Egyptiansby about 1600BC and even earlier by the Babylonians (Cajori, 1928, p. 13; Klein, 1968;Smith, 1923). Based on this similarity between current and historical conceptualizations offractions, we reviewed the historical development of fractions, identified four milestoneswith respect to conceptualizing fractions as an extension of whole numbers, and used themas our lens to explore whether and how fraction are addressed as numbers in mathematicsclasses for PSTs. The details of each milestone are discussed in the following section.

It should be noted that our use of an historical approach does not suggest that instructorsof PSTs should teach the history of fractions. Instead, we use the history of the mathematicsas a lens for exploring whether and how the instructors addressed fractions as numbers and/or an extension of whole numbers while introducing fractions and to investigate mathemat-ical similarities and differences across the instructors’ approaches in a sample of requiredmathematics courses.

The history of mathematics has been used in learning and teaching of mathematicalconcepts (Clark, 2011; Jankvist, 2009; Radford et al., 2002; Weil, 1978) based on similar-ities in the historical obstacles to development of a mathematical idea that seem to berepeated by students learning the idea anew (Dorier, Robert, Robinet, & Rogalski, 2000;Sfard, 1995, p. 17). Some researchers have also used history to help students or PSTsimprove their understanding of concepts (e.g., Dorier, 1998; Radford, 1995; Clark, 2011)—e.g., by providing problem situations based on the history of mathematics, which could giverise to “cognitive and socio-cognitive conflict” and “create favourable conditions forstudents to reach a better understanding” (Fauvel & Maanen, 2000, p. 159).

Our use of the history of mathematics as an analytical tool is different from the studiesmentioned above that use history of the mathematics as a resource for teaching mathematicalconcepts for students or teachers. In this study, we only use history of mathematics as asource for developing an analytical lens to investigate and analyze the teaching of fractionconcepts. Whether such an analytical approach has implications for the teaching andlearning of mathematics is beyond the scope of this work and requires further investigation.We focus on whether such an analytical approach provides further insights for researchers inanalyzing teaching.

3 Theoretical background

As we elaborate below, historical documents and mathematics textbooks in algebra,real analysis, and set theory show that rational numbers were built up from wholenumbers through four mathematical approaches: part–whole, measurement, division,and set theory (Cajori, 1928; Heath, 1956; Klein, 1968). Our historical approachfocuses on how fractions were developed from the whole number system and ulti-mately, as part of the rational number system, accepted as numbers rather thansymbols with various interpretations and uses.

458 J. Park et al.

One of the issues with investigating fraction-as-number is the elusiveness of theidea and meaning of “number.” What does it actually mean to know that a fraction isa number? One can use a fraction symbol in operations—apparently as a number—without having a reasonable understanding of fraction-as-number (e.g., Erlwanger,1973). Researchers have noted that historically, the acceptance and understanding offractions as numbers was far from trivial, and entailed rethinking what it means forsomething to be a number: “The shift from natural to rational numbers involvedchanges in the status and meaning of the term ‘number’ that cannot be accountedfor in terms of the mere expansion of the natural number concept” (Vamvakoussi &Vosniadu, 2007, pp. 265–266). Here, we take as the definition of fraction-as-numberthat fractions are part of a system that includes whole numbers, and that they inheritthe properties and definitions of the four basic operations from whole numbers. Ourdefinition of fraction-as-number prepares the way for what Wu (2010) calls theFundamental Assumption of School Mathematics, the coherent extension of wholenumbers to rational numbers and finally to real numbers. In this paper, the term“fractional quantity” is used for a quantity that can be represented by a non-integerfraction but is not necessarily considered as a number. In other words, a fractionalnumber can be represented as a fractional quantity in various contexts, but usingsymbols to represent fractional quantities does not always imply that those symbolsare considered to be numbers. For purposes of this paper, we treat the terms “rationalnumber” and “fraction” interchangeably, always focusing on positive rational numbers.

3.1 Elements of the mathematical history of fractions

We identified four milestones with respect to the extension of whole numbers to fractions inthe mathematical history of fractions:

1. Part–whole approach: conceptualizing a part of a whole as a new unit. Historically, thisconceptualization of fraction grew from ancient times when “the one” was conceived as“impartiable and indivisible” (Klein, 1968, p. 40). As early as 1650BC, the Egyptians usedsymbols to represent unit fractions as parts of the whole (Cajori, 1928; Smith, 1925;Berlinghoff & Gouvea, 2004). A fractional quantity was not considered as anumber—on a par with whole or “natural” numbers—by the Egyptians or byother ancient civilizations that had symbolic representations for parts of thewhole, for they had no common arithmetic for whole numbers and fractionalquantities. This historical development is similar to students’ difficulties in mov-ing beyond the part–whole concept of fractions (e.g., Erlwanger, 1973; Mack,1990). The part–whole conceptualization is an approach often used today. Forexample, Beckmann (2008, p. 66) defines a fraction as follows:

If A and B are whole numbers and B is not zero, and if an object, collection, or quantitycan be divided into B equal parts, then the fraction

A=B

of an object, collection, or quantity is the amount formed by A parts (or copies of parts).

In today’s classrooms, the part–whole approach may play out when instructors use areamodels divided into equal parts, or mention that a part of a collection or object can beexpressed as a fraction with equal partitioning and counting.


2. Measurement approach: finding fractions from whole numbers through measurementand proportions, addressing the need for a common unit of measurement for twoquantities. Historically, the term encompassing measurement and proportion is “com-mensurability” which was defined by the Greek mathematician Euclid in 300BC asfollows: “Those magnitudes are said to be commensurable which are measured by thesame measure, and those incommensurable which cannot have any common measure”(Heath, 1956, p. 10). In modern sense, this statement can be rewritten as follows:

There is a real number C and integers n and m such that A0nC and B0mC. (In this case,Euclid said that A and B are commensurable.) (Austin, 2007).

The quantity C (when it is not a whole number) was not considered as a number byEuclid, but as “the part or parts of a number” (Klein, 1968, p. 43) since it can be seen as apart of A and of B. Even though measurement clearly evokes a need for numbers beyondwhole numbers, the ancients did not take the next step of conceiving of those units asnumbers on an equal footing with whole numbers. As in the part–whole approach, themeasurement approach did not define “the part or parts of a number” as a distinct number,but rather as a new unit.

In practice, this measurement approach can play out when an instructor gives a measure-ment problem, measuring A units of a continuous quantity of B-unit size—in which the focusis the area, length, or volume of continuous quantity, and shows that the results of themeasurement are not always whole numbers.

3. Division approach: finding the algebraic solution for an equation Ax0B where Aand B are whole numbers and A is nonzero. This approach arises in the formaldefinition of a field, first conceived of by Galois in the early nineteenth centuryand formalized concretely by Dedekind in 1871 (Baumgart, 1969). We call this adivision approach since the need for the fraction B/A is a result of the need tohave a set of numbers that is closed under division. In order to talk about anumber system being closed under division, one has to think about the notion ofmultiplicative inverses, which are not contained in the whole number system. Inother words, one needs to extend the number system to include multiplicativeinverses of nonzero whole numbers. This set, with inclusion of the additiveinverses, forms a field with all the rational numbers. As a result, we can formgroups and rings in the rational number system, in which fractional quantitiesconstitute distinct elements of the number system. Therefore, fractions are con-sidered as numbers in this approach.1

An example of the division approach in practice could be performing divisionwith bare numbers or a partitive division, A÷B, in which A units of a continuousquantity are shared equally by B recipients and mentioning that the results ofdivision are not always whole numbers, and thus one needs fractions to expressthe results.

4. Set-theoretical approach: defining rational numbers as a set of ordered pairs consistingof whole numbers:

1 This development was preceded by many centuries of work by mathematicians who struggled with thenotation for fractions and with developing algorithms for operations with fractions. Attempts to integratefractions into the number system finally led to the formalization of decimal fractions by Stevin in theseventeenth century (Smith, 1925).

460 J. Park et al.

Take the set S of all ordered pairs (A, B) of integers, where B≠0. Partition the set S intosubsets by the rule: two pairs (A, B) and (C, D) are in the same subset if the ratio of Ato B is the same as the ratio of C to D, that is, if and only if AD0BC. (Childs, 1995, p. 3)

This approach can be found in the late nineteenth and early twentieth century efforts todevelop a rigorous foundation for mathematics based on arithmetic. In the late nineteenthcentury, Cantor developed set theory, which eventually led to formal, set theoretic defini-tions of rational numbers.

Fractions are considered as numbers in this set-theoretical approach, for numbers them-selves are defined by sets. By defining rational numbers as a set with elements that satisfycertain conditions, this approach provides a rigorous foundation for rational numbers in thenumber system. With added conditions, the set can be extended to the real numbers, forexample, including the limits of all sequences of rational numbers.

The set-theoretical approach was diluted over time to ignore the key idea of set of orderedpairs based on equivalence classes and reduce the definition to a set of symbols:

A=B : A; B 2 Z; B 6¼ 0f gwhere Z is a set of integers. This kind of definition is still seen in many textbooks, followedby explanations of what the symbols mean and how to manipulate them. In the analysis, weinitially labeled instances of this purely symbolic approach as “set-theoretical,” but changedit to “symbolic” because they were so far from the rigorous set-theoretical approach as not tobe recognizable as such. In fact, we saw no examples of a true set-theoretical approach tofractions, but many examples of a symbolic approach. Unlike the other three approaches, thesymbolic approach does not suggest a need or justification for fractions, but it doesincorporate whole numbers as fractions of the form N/1 where N is any whole number.

3.2 Framework summary

The push and pull of accepting new kinds of numbers into the realm of mathematics has occurredseveral times in the history of mathematics when other abstract ideas—zero, negative numbers,irrational numbers, and imaginary numbers—were investigated and debated (Dantzig, 1954;Fischbein, Jehiam, &Cohen, 1995; Pogliani, Randic, & Trinajstic, 1998; Seife, 2000). In the caseof fractions, accepting them as numbers increases the abstractness and generalizability of the ideaof fraction while supplanting or decreasing the intuitive meaning as part of a whole. Abstractionmakes it possible to include fractions in the number system and operate on them with all thearithmetic properties that whole numbers follow, letting them take their place in whatWu calls theFundamental Assumption of School Mathematics—that all information about operations onfractions can be extrapolated to real numbers (Wu, 2010).

4 Methods and data sources

Since the data of this study came from a larger project, the Mathematical Education ofElementary Teachers (ME.ET), we explain the process we went through to develop thiscurrent study in the context of the bigger project. The overall goal of the ME.ET project is toexplore PSTs’ learning in their undergraduate mathematics classes, with a particular focus onfractions. We collected data to analyze what is taught and learned in undergraduate


mathematics courses required for elementary certification at different institutions in twostates in the USA. Our data include pre- and posttests of over 1,000 PSTs in their mathe-matics classes, along with information about their instructors based on surveys and inter-views. Interviews were conducted individually in order to understand their general goals ofteaching this course. These data revealed various aspects of students’ knowledge aboutfractions and their relationships to characteristics of their instructors (see the project Web sitefor other articles and reports from the project, http://meet.educ.msu.edu/index.htm). We alsocollected videos of six mathematics instructors of these courses, focusing on fractionlessons, and it is these data that are used in the analysis for this paper.

In our initial analysis of the video data focused on teaching practices, we noticed thatinstructors rarely mentioned the idea of fraction-as-number or made explicit connections tothe ways that fractions fit into the number system. Even instructors who introduced a variety ofrepresentations and uses of fractions seemed to neglect making the strong connections thatwould tie fractions to whole numbers as an extension of a common idea. In seeking a way toexplain what we saw, we turned to the history of mathematics to see if the development offractions over timemight be a useful lens for understanding the disconnect wewere observing inthese classes. We also used the interview data to supplement what we observed during thelessons. Since the interviews were designed for other purposes, they do not provide completeinformation that would be relevant to this paper (e.g., the instructors’ assumptions about whattheir PSTs know about fractions as numbers). However, the instructors’ responses to some of theinterview questions (e.g., “what is the basic definition of fraction to you?” “do you have anykind of definition that you want your PSTs to have?” and “what aspects of fraction would bemost difficult for your students?”) provide indications of their foci and goals for fraction lessons.

Information about the instructors and their classes that we observed is given in Table 1. InTable 1, “Number of Fraction Lessons Observed” is the number of lessons in which fractionwas the focus, although fraction may have come up in other lessons not included in the table.

Researchers initially created rough transcripts for all videotaped lessons, noted parts whenthe definition of fractions or their extension from whole numbers were discussed, and mademore detailed transcripts of those parts. Two researchers analyzed these segments of video clips,transcripts, and field notes, and discussed the results until reaching agreement about theinterpretations of the instructors’ approaches to extend the number system. Specifically, welooked for mathematical elements of the four historical milestones in the instructors’ presenta-tion of fractions, both when (and if) they make explicit the extension of the whole numbersystem to include fractions and when they defined fractions. We used the four approaches todetermine how connections are made across conceptions of fractions with an eye toward aconceptualization of fraction-as-number and as an extension of the whole number system. Welooked for evidence of teaching fraction-as-number by identifying instances when instructorsexplicitly called attention to the fact that a fraction is a number, either by saying so or bydrawing analogies to whole number and/or real number properties that fractions possess. Table 2gives examples of the instructors’ explanations for each of the four approaches, with anexplanation of what we considered as an explicit extension of the whole numbers:

5 Results

This section reports how each instructor described their foci on fraction lessons during theinterview, and whether and how they extended the number system from whole numbers tofractions during the classes. For the interviews, we mainly analyzed their responses to thethree questions,

462 J. Park et al.

http://meet.educ.msu.edu/index.htm

1. What is the basic definition of fraction to you?2. What kind of definition of fractions do you want your PSTs to have?3. What aspects of fraction would be most difficult for your students?

For the classroom data, we examined in detail the lessons when fractions wereintroduced. We examined the approaches instructors used, whether fractions werenamed explicitly as numbers, and whether properties of whole numbers were explicitlyextended to fractions.

5.1 Edie

During the interview, Edie defined fraction as “a/b, where b is not equal to zero.” Instead of“emphasiz[ing] this definition,” she wanted her PSTs “to conceptualize fractions as a

Table 1 Information about the instructors and their courses

Instructora Position Timespreviouslytaught thiscourse

Numberoffractionlessonsobserved

Type ofschool

Type ofcourse

Numberofstudentsinsection

Textbook used

Edie AssociateProfessor,Mathematics

20 9 Largepublic,Masters

Mathematics& methodsforelementaryPSTs

12 N/A

Eliot AssistantProfessor,Mathematics

0 7 Mediumpublic,Masters

MathematicsforelementaryPSTs

35 Departmentgeneratedmaterials

Jamie GraduateStudentInstructor,MathematicsEducation

1 4 Largepublic,PhD


29 ElementaryMathematics forTeachers (Parker& Baldridge,2003)

Pat AssistantProfessor,Mathematics

15–20 10 LargepublicMasters

Mathematics& methodsforelementaryPSTs

23 Children’sMathematics:CognitivelyGuidedInstruction(Carpenter,Fennema, Franke,Levi, & Empson,1999)

Sam GraduateStudentInstructor,MathematicsEducation

0 4 Largepublic,PhD


34 ElementaryMathematics forTeachers (Parker& Baldridge,2003)

Terry AssistantProfessor,Mathematics

2 19 Largepublic,PhD


23 Mathematics forElementaryTeachers(Beckmann, 2005)

a Names used in this paper are pseudonyms


quantity and be able to visualize it, to represent it…to develop a number sense andcomfortableness with fractions as quantities.” She did not explicitly mention fraction-as-number during the interview. Later in the interview, she mainly talked about the part–wholeinterpretation of fractions, and mentioned that parts of the whole and units for fractions (e.g.,composing and decomposing fractional parts) would be difficult for her PSTs to understand.In this context, she emphasized a connection between whole numbers and fractions; sheconnected composing and decomposing fractional parts to composing and decomposingwhole numbers. Specifically, she said, “I…try to connect with whole numbers…in terms ofdiscrete quantities in the sense of like a fourth of 36 is 9 so going back to whole numbers Ican imagine the quantity 36 and I can imagine 9 so I can see some sense of proportion of 9and 36,” and “everything I do about fractions currently connects explicitly back to our workwith whole numbers.”

Her emphasis on the connection between whole numbers and fractions was also identifiedin her introductory fraction lessons. Edie extended the number system from whole numbersto fractions through a division approach using word problems involving equal sharing andmulti-digit division. The word problem Edie introduced to the class was: “If there are 3 subsandwiches and 4 kids, how much did each child get?” (Edie, June 10, 2008). When workingon this problem, Edie first asked what kind of operation this problem represented, and thencontinued to discuss a need for fractions in the context of the problem (Fig. 1).

Edie connected this new type of number to a whole number division problem, which theyhad discussed before in their classes. Moreover, Edie explicitly talked about a need forfractions as “a new kind of number” resulting from whole number division.

Edie followed up on this idea in the following lesson where they talked about multi-digitdivision of whole numbers. While representing 151 divided by 7 with a rectangular areamodel, Edie explained the representations on the board, ending with the string of equationsshown in Fig. 2. Pointing to the last equation of the solution process, she explicitly

Table 2 Descriptions of each approach in instructors’ explanations

Approaches Descriptions

Part–whole Partitioning of a continuous object or a set of discrete objects and iterating them, for example,by shading

Explicit extension: Awhole number cannot express any part of a whole which is smaller thanthe whole

Measurement Performing measurement division, A÷B—measuring A unit of a continuous quantity with aquantity of B-unit—in which the focus can be the area, length, or volume of continuousquantity

Explicit extension: Not all results of measuring are whole numbers; we need different numbers(fractions) to represent/conceptualize the result

Division Performing division with bare numbers or a partitive division, A÷B, in which A units of acontinuous quantity is shared equally by B recipients

Explicit extension: The results of division are not always whole numbers; we need differentnumbers (fractions) to represent the result

Symbolic Providing the definition of fraction to a set of symbols: A=B : A; B 2 Z; B 6¼ 0f gExplicit extension: The symbolic definition implies that any whole number N can be expressedas N/1

Note that this approach does not necessarily suggest a need for fractions, but rather includeswhole numbers in the definition of fractions

464 J. Park et al.

mentioned the need for a fraction, saying “this gets me into thinking about parts of numbersand fractions,” and noting that the “box” would be four sevenths.

During the interview, she mainly interpreted fraction as part of a whole, and emphasizedthe connection between whole numbers and fractions. Edie extended whole numbers tofractions through division and mentioned explicitly that what is obtained from wholenumber division, in this case a fraction, is a number. In this approach, she emphasized partsand the whole in a whole number division, and connected it to a need for fraction. Thisdivision approach was the only extension Edie used.

5.2 Eliot

Eliot’s response to the interview question about the basic definition of fractions she wouldlike her students to learn, was as follows:

I defined rational number as something that can be written in the form of a fraction…talked about how that included whole numbers and integers and what they tradition-ally considered to be a fraction like three fourths, seven eighths, and… I wouldn’t callit a really a formal definition…though I do think that it’s important that they have thatconcept [of] the rational number system and how it includes the whole number system.

Her definition of a rational number as a number that has a fraction form and her emphasison its connection to whole number systemwas consistent with her fraction lessons. On the firstday of the fraction unit (February 13, 2008), Eliot extended the number system from wholenumbers to fractions using two approaches: symbolic and division. Unlike the other instruc-

Edie: Now using your number sense, what is going to happen if you have 4 children and 3

subs? Do we have enough for each one to get one?

Students: No.

Edie: So now we suddenly introduce a new kind of number. We have got parts of subs

going on here and that leads us to fractions.

Fig. 1 Edie’s explanation about a need for fractions, June 10, 2008

Fig. 2 Reproduction of Edie’swriting about “151 divided by7,” June 12, 2008


tors, Eliot avoided general use of the term “fraction,” instead defining rational numbers andusing the term “rational number” throughout her teaching. In this way, she continuallyemphasized that these are numbers. She started with a symbolic definition of rational numbers,A/B where A and B are integers and B is not zero (Fig. 3). Then, she used this definition toexplain any whole number, n, is a rational number because it can be written as n/1 (Fig. 4).

Eliot explicitly mentioned that a rational number is “a number” in the definition, and thenpointed out that a whole number, n, is a rational number because it satisfies the definition; inother words, it can be expressed as n/1. On the same day, Eliot also used the divisionapproach to explain the extension from whole numbers to rational numbers (Fig. 5):

In this excerpt, Eliot explained a need for fractions to include the result of whole numberdivision in the number system. Specifically, she illustrated the need to solve a divisionproblem with bare numbers without a context, 5÷4, making it similar to solving an algebraicequation using the definition of division, 504x.

In her symbolic approach, Eliot explicitly stated that the rational numbers are numbers,and explained the relationship between whole numbers and rational numbers—any wholenumber is a rational number—based on the definition she gave. Her interview responsesignified fractions as examples of rational numbers, supporting Eliot’s view of fractions asnumbers including whole numbers. In the division approach, she explicitly extended thenumber system from whole numbers to rational numbers by explaining why rationalnumbers are necessary for whole number division.

5.3 Jamie

During the interview, Jamie defined fraction as part of a whole. She emphasized the concept ofthe whole in the definition because she said that her PSTs would have difficulty with finding thewhole of a fraction in the problem context. Mentioning her textbook use, she said, “the textbookdid not have much explanation about the concept of the whole…[so] I have to find and usemany examples from Singapore and Connected Mathematics Project [student] textbook.” Herpart–whole definition and emphasis on whole were also found in her fraction lessons.

On the first day of the fraction lessons, Jamie explained extension from whole numbers tofractions using the division and part–whole approaches. Jamie started the fraction unit with

Fig. 3 Eliot’s definition of a rational number, February 13, 2008

Fig. 4 Eliot’s connection to whole numbers, February 13, 2008

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two whole number division word problems involving equal sharing: one had a wholenumber, and the other a fraction as the result of division (Fig. 6).

Then, Jamie asked students to think about the difference between the two problems, andone student pointed out that the result of division in the first example is a whole number.Jamie represented the result of the second example, 2/3, using a pie diagram (Fig. 7). Afterdrawing the diagrams on the board, Jamie explained the meaning of the denominator andnumerator in 2/3 using the part–whole approach as shown in Fig. 8.

As seen from the transcript, Jamie defined a fraction as a part of a whole mentioning that thedenominator represents the “total number of parts in a whole unit,” and the numerator represents“the number of parts shaded or [that] we count”. During the interview, Jamie defined fraction aspart of a whole and emphasized the concept of the whole. She, however, did not mentionfraction as an extension of whole numbers or as a number itself. This was consistent with herintroductory lessons. Jamie extended fractions from whole numbers using the division andpart–whole approaches. In the division approach, although Jamie provided a situation where itwas possible to talk about the need for fraction in the number system, she did not explicitlyexplain that the result of the whole number division leads to a new type of number. Similarly,when defining a fraction using the part–whole based approach, she did not state explicitly thatthis resulted in the extension of the whole number system. She neither mentioned fractions asnumbers nor how fractions are extended from whole numbers.

5.4 Pat

During the interview, Pat defined fraction as “a particular representation of a rationalnumber, a rational number being any number that can be expressed as a quotient of twointegers a/b.” He, however, said that this was not the way he “want[s] them [PSTs] todefine,” but rather he was “more interested in that at least they recognize that a fraction is anumber within itself, a rational number, and then can be operated on it across multiplerepresentations.” His definition and emphasis on fraction as a number in his fraction lessons,however, were not as explicit as in this interview.

Fig. 6 Jamie’s examples for extension, October 30, 2007

Fig. 5 Eliot’s extension from whole numbers to fractions using division, February 13, 2008


During class, Pat extended the number system fromwhole numbers to fractions based on thedivision and measurement approaches using word problems involving equal sharing or mea-surement; some have divisible remainders, others indivisible remainders.2 The class hadworked on whole number division in earlier lessons, including the exploration of the differencebetween measurement and equal sharing problems. Explaining one of the problems, Pat said,“When we use the term, divisible remainder, it’s implied [that the result is] a mixed number.”Other than this reference to number, Pat left implicit that the solution to each word problem is anumber; instead, he used the term “fractional quantity” to describe the results.

His approach gave explicit justification for the need for fractions: when the solution to areal life problem is not a whole number, something else is needed. His arguments weresimilar to the historical development based on measurement and commensurable quantities,using whole number division (equal sharing) and measurement problems that do not yieldwhole number solutions.

Figure 9 shows two of the problems the class worked on. Here, he explained the result of ameasurement division problem, emphasizing the need for fractional quantities and “fraction-based thinking.” Pat did not explicitly equate quantity and number, although he may have

2 A divisible remainder occurs when an object can be divided, for example, a candy bar. An indivisibleremainder refers to an object that cannot be divided like a school bus or a person. That is, a result can includehalf a candy bar, but not half a person.

Fig. 7 Jamie’s discussion about the result of division, October 30, 2007

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Fig. 8 Jamie’s discussion about the numerator and denominator, October 30, 2007

Fig. 9 Pat’s description of a need for fractions, April 3, 2008


assumed that a quantity implies a number. Except for the single mention of mixed numberquoted above, he did not make explicit that the results—fractional quantities—are numbers.

In summary, during the interview, Pat defined fractions as representations of rationalnumbers and emphasized fraction as numbers. During class, Pat used division and measure-ment to address the extension of the number system. In both approaches, he justifiedexplicitly the need for fractions. Although he stated that he wanted his PSTs to understandfractions as numbers during the interview, he did not make explicit that fractions arenumbers in either approach; instead, he used “fractional thinking” or “fractional quantity”to refer to fractions. He also did not explain the connections between whole numbers andfractions or that every whole number is a fraction in either approach.

5.5 Sam

During the interview, Sam described a fraction as “a number or…a proportion [between] twothings or objects, or…a ratio.” She explained the goal of this course by saying, “I think thiscourse focused more on [fractions as] numbers, but I like to…see what they [PST]s knowabout other things about fractions.” She also mentioned that her focus is for PSTs to know“what fractions are from the textbook,” and PSTs “might have trouble understanding what itmeans to add two numbers, the fractions with different denominators.”

During class, Sam used two approaches to extending the whole numbers to fractions. On thefirst day of the fraction unit, she used a part–whole approach. On the second day, Sam used thedivision approach to explain the extension while explaining the division-fraction equivalence.

Sam started the fraction unit with an activity that used a square subdivided into smallerparts. Students named the parts with fractions. After the activity, she defined a fraction aspart of a whole using 3/16:

The bottom number is how we cut this whole thing into pieces, and the total pieces,and 3 [the] numerator is gonna [sic] be the number of pieces we are talking about withrespect to the whole thing. (Sam, November 5, 2007)

Then, Sam stated that a fraction is a new type of number different from whole numbersand showed this by placing 3/16 on the number line (Fig. 10). As shown in the excerpt, Samexplained how to place a fraction (3/16) on the number line by establishing 1/16 as a unitfraction and counting three 1/16-ths. Although she used the number line to show 3/16, sheput more emphasis on interpreting a fraction as part of a whole than as a number as impliedfrom the following excerpt shown in Fig. 11.

In the second day of the fraction unit, Sam explained the extension from whole numbersto fractions using the division approach when discussing the fraction-division equivalence(A÷B0A/B). She started this discussion without a specific context (19/7) and then moved toan equal sharing situation of 3 divided by 2. She said:

A fraction is a special kind of division. Up to now, when we do division, most[ly] wehad a perfect whole number. …[A] fraction, 19 over 7…is division, but not neces-sar[ily], we get a nice looking whole number. How about 3 over 2? We have threetriangles and I want to divide into two persons equally. (Sam, November 7, 2007)

Here, Sam justified the need for a fraction in whole number division but left implicit thatthe result of the division is always a number, and thus a fraction is a number. Additionally,Sam mentioned that a whole number, n, can be expressed as a A/B form by saying, “Thinkabout any whole number like 7. I can always [express that] 7 is equal to 7 divided by 1(Writing “7/1”). In this case, 7 is A and this (pointing out 1) is B.” She also expressed 19 as

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19/1. However, she did not connect this explanation to the extension of the number systemwhich could be addressed by explaining the set of fractions is a bigger system whichincludes whole numbers.

In summary, during the interview, Sam interpreted fraction as a number, proportion, andratio between two quantities. She addressed each of these three aspects of fractions duringthe class. Sam expanded the number system from whole numbers to fractions using twoapproaches: part–whole and division. Sam did not explain that the part–whole definition offraction implies an expanded set of numbers that includes whole numbers. She did, however,show explicitly that the part of a whole is a number by placing it on the number line. She alsoemphasized the concept of the whole with additional examples involving proportions andratio between two quantities (see Fig. 11). In the division approach, although she mentionedthat whole number division does not always result in “a nice looking whole number,” she didnot make explicit why this division approach leads to a fraction and that the result of wholenumber division, which can be a fraction, is a number. She also did not explicitly state that awhole number is a fraction in either of the two approaches.

5.6 Terry

During the interview, Terry mentioned a symbolic definition of fraction as a subset of the realnumber system, “we started this course with the real number system, various subsets, variousmain subsets of numbers that we use, and so we followed the book definition of a fraction as

Fig. 10 Sam’s explanation about fraction as a number, November 5, 2007


anything of the form A over B where B is non-zero.” Then she said, “We’re going to focus onthe parts of the whole. Numerators, bottoms, how many parts the whole is, how many equalparts it is going to be subdivided into.” She emphasized the concept of the whole in thedefinition, “the definition in Beckman’s [book]…really emphasizes that it is a fraction ofsomething and there is a whole involved there. So I really like that aspect and then one thing Iwill really emphasize later is word problems. I really emphasize them being able to write avalid word problem representing A divided by B or when we work on the standard one.” Herintroductory fraction lessons were consistent with her responses during the interview.

Terry taught from a book focusing on four basic operations rather than the extension ofnumber system (Beckmann, 2005). Fractions were included in most lessons after their intro-duction in the third week of class. For this reason, we videotaped her class for every lessonbeginning with the introduction of fractions, a total of 19 videotaped 50-min lessons, 8 of whichwere mostly devoted to the discussion of fractions. Terry extended the number system fromwhole numbers to fractions using both symbolic and part–whole approaches. She first defined afraction as A over B, where B is nonzero, and A and B are whole numbers (Fig. 12).

As shown in the excerpt, aside from mentioning that A and B in the definition are wholenumbers, she did not explain whether and how the definition of fraction includes wholenumbers. In other words, the connection between whole numbers and fractions was notexplicitly addressed while she explained this symbolic definition of fraction.

She next explained the meaning of A and B using the part–whole approach: “So A standsfor the number of parts; and what does B stand for? This is directly in your book essentially,the type or name of the parts.” Using the definition from the Beckmann book, she

Fig. 11 Sam’s explanation about importance of the Whole, November 5, 2007

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emphasized the importance of the word “of” saying that fractions “are of something. So thatis really a key idea and it can be of one.” Then, she explicitly mentioned that a fraction is anumber while explaining 2/3 on the number line as an example: “When you think offractions as just a number on the number line…two thirds would be two thirds of thenumber one…so that of is absolutely essential there.” She, however, did not draw thenumber line to place 2/3 on it or explain further about fraction-as-number through gesturesor other visual means. In other words, although her references to fraction-as-number werebased on the number line, they did not include explanation of unit fractions as parts of thewhole or as means of measuring non-integral parts. In the remaining lessons during whichTerry discussed fractions, she continued to use the symbolic and part/whole interpretations,without emphasizing fraction-as-number or how fractions can and should be seen as anextension of the whole numbers that eventually lead to the real number system.

In summary, Terry defined fractions as a subset of the real number system and as parts ofa whole during the interview. Although she emphasized the meaning of the parts and wholeduring the interview, it was not explicit in her introductory fraction lessons. She did notexplain why the word “of is essential.” She mentioned that fraction is a number using thenumber line, but did not visualize it or make a connection to whole numbers.

5.7 Summary of types of extensions and definitions presented by the instructors

Table 3 summarizes results by each instructor. As shown in the table, the predominantextensions were division and part–whole, employed by five and three instructors, respectively.Other approaches used were symbolic (used by 2) andmeasurement (used by 1).Whenmultipleextensions were used, the connections between or among extensions were not made explicit.Regarding the definition of fraction (or rational number), two of the instructors did not use anexplicit definition; two used a part–whole definition; and two a symbolic definition.

6 Conclusions

Based on the analysis above, this section addresses the two research questions.

6.1 Do instructors of undergraduate mathematics classes for PSTs address fractionsas an extension of the whole number system and fractions as numbers, and if so, how?

The instructors’ approaches to the extension of the number system from whole numbers tofractions varied considerably from class to class. However, the idea of extension in general

Fig. 12 Terry’s definition of a fraction, February 13, 2008


was not fully elaborated in the classes that we observed. Even when the instructors provideda means of deriving fractions from whole numbers (e.g., by division), not all of themunpacked the ideas and integrated them into the discussion of fractions. Connections acrossinterpretations and ideas about fractions were similarly unelaborated, including connectionsamong any of the four approaches instructors used. For example, Sam used both the divisionand part–whole approaches to introduce fractions and connect them to whole numbers, butshe did not make clear, or help her students understand, how these two approaches led to asingle, coherent system of numbers.

Fraction-as-number was also not emphasized or fully justified even when it was men-tioned. In many cases, the mention of “fraction as number” was separated from thediscussion of the extension from whole numbers to fractions, and from discussion of theneed for new numbers other than whole numbers. For example, Jamie and Pat both useddivision word problems to address the need for a fraction, or fractional quantity, beyondwhole numbers, but they did not address that the results of their division problems were alsonumbers or have the same properties that the whole number have. We conclude that in oursample of classes, instructors are missing many possible instances to intervene in some ofthe problematic ways that PSTs understand these mathematical aspects of fractions.

6.2 In what ways is a historical lens helpful in analyzing and understanding the instructors’approaches to fractions as numbers and as an extension of whole numbers?

The mathematical history of fractions provided a useful lens for analyzing the episodes inwhich fractions were introduced and defined. This lens enabled us to compare mathematicalaspects of fractions in relation to whole numbers across the classes. It also highlighted theinstances in which opportunities for the instructors to make explicit links between wholenumbers and fractions, and across different interpretations of fractions, were or were nottaken up by the instructors.

The mathematical approach based on history emphasizes the connectedness and coher-ence of fractions and whole numbers as numbers in a system. For example, the part–wholeapproach in the framework emphasized the similarity in the property between wholenumbers and fractions (e.g., Sam’s explanations on how to place 3/16 on the number line

Table 3 Summary of instructors’ extensions and definitions of fractions

Instructor Type ofextension

Definition statement Type ofdefinition

Edie Division No explicit definition N/A

Eliot Division andsymbolic

“A rational number is a number that can be written as a traditionalfraction…A over B … A and B are integers B … is not zero.”

Symbolic

Jamie Division andpart–whole

“The denominator” as “total number of parts in a whole unit” and“the numerator” as “number of parts shaded or we count.”

Part–whole

Pat Division andmeasurement

No explicit definition N/A

Sam Division andpart–whole

“The bottom number is how we cut this whole thing into pieces,and the total pieces, and 3 [numerator] is gonna [sic] be thenumber of pieces we are talking about with respect to the wholething.”

Part–whole

Terry Symbolic andpart–whole

“A over B, where A and B are whole numbers, and B is non-zero.” Symbolic

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by partitioning the segment between 0 and 1 into 16 equal segments and iterating the smallerunit three times). The measurement approach also addresses not only how to interpret A/B asmeasuring A units of a quantity with a B-unit quantity but also how to deal with a remainderin terms of the divisor, which Pat explained with divisible remainders. Using the divisionapproach makes explicit that a fraction is a result of the whole number division such asEliot’s example of five subs shared by four recipients. The symbolic approach also makesthe connection between whole numbers and fractions clear by making observable theinstances in which instructors emphasized that a whole number, n, can be expressed by afraction form, n/1. In summary, the four approaches provided by the historical lens allowedus to point out the instances where these aspects were addressed in the classrooms, andanalyze those cases focusing on similarities between whole numbers and fractions and theirmathematical relationships as two sets of numbers.

7 Discussion

The results of this study show that key mathematical aspects of fraction, including fraction-as-number, were not explicitly addressed in introductory lessons of the classes we observed.These implicit discussions may come from the instructors’ assumptions about what the PSTsin their classroom already know. Because the students in these classes are young adults,usually in their second or third year of college, who have had much exposure to fractions intheir K-12 education, one of the challenges faced by their instructors is finding out howmuch, and what, to take for granted in these students’ prior knowledge. Do they alreadyknow what a fraction is, and in particular, that fractions are part of a coherent numbersystem? One might assume that they know these things, although pushing on what it meansto “know that fractions are numbers” could reveal some serious problems with making thisassumption. Similarly, one could ask if they already know how to operate with fractions,how to determine if fractions are equivalent, or how to represent fractions using a numberline. Instructors may make different assumptions about their students’ prior knowledge,which could affect their approaches to fractions in these particular classes. Althoughinvestigating such assumptions was beyond the scope of our study, we have evidence thatsome of the instructors in our study assume that their PSTs have knowledge about fractionsfrom the interviews and classroom observation. For example, Terry said, “theoretically, theyalready know the concept or they are exposed to fractions” during the interview, and Jamieasked her PSTs “You already know the concept of fractions, right?” in her first fractionlesson. During the interview, Eliot elaborated further on her assumption about her PSTsknowing “how” but not “why” and her goal in her fraction lessons:

They [PSTs] have seen so much addition and multiplication and stuff in high schoolbut I’m not sure they truly understand it but they can definitely do it…In terms of mefiguring out what level of understanding they really have, I haven’t been able to [findout]…And we talk about that a lot in class, and they know how to do just abouteverything…I think that’s true they rarely know why. So, I do try to point that outwhenever possible and sometimes they are reluctant because they already know howto do it…A lot of these students were either told or decided they weren’t good at math,and they were taught tricks to get by. And they liked it because it got them the gradesthat they wanted…So now I’m trying to undo some of that and not show them tricksand shortcuts and try to really teach them the real reason why things work like they do.(Eliot, February 14, 2008)


Exploring how instructors’ teaching goals are based on their assumptions about whattheir PSTs know in fraction lessons would be an interesting topic for a future study thatwould provide useful information about instructors’ decisions in practice. A follow-up studycan be designed to explore the possible explanations for explicitness/implicitness of class-room discussion based on the instructors’ beliefs and assumptions about PST’s knowledgeon these aspects.

What we observed in our sample classes shows the instructors’ implicit approaches to keyaspects of fractions: the conceptual links and mathematical developments that underlie thedefinition and use of fractions. As mentioned earlier, studies have reported that althoughPSTs bring considerable knowledge about fractions to the mathematics courses for PSTs(Ball, 1990), their knowledge includes various incorrect notions about fractions, most ofwhich are related to lack of conceiving of fraction as numbers (Ball, 1988; Ma, 1998;Stafylidou & Vosniadou, 2004). Based on our analysis with the historical lens and the resultsfrom existing studies, we argue that addressing fractions as numbers through at least oneapproach, which extends whole numbers to factions, may be a route toward improving PSTs’mathematical knowledge for teaching fractions. The mathematics behind the historicaldevelopment of fractions shows that neither the part–whole approach—the most dom-inant idea in today’s K–8 classrooms in the USA (e.g., Lamon, 2007; Post et al.,1993)—nor the measurement approach automatically supports the consideration offraction as a number. Although fractional quantities had been common in differentcultures over hundreds of years, it took mathematicians centuries to accept fractions asnumbers, until Stevin (1548–1620) defined a fractional number as “a part of the partsof a whole number” (Cajori, 1928; Klein, 1968, p. 290). It is logical that some PSTsmay have similar difficulty incorporating the idea that fractions are numbers into theirdominant notion of fractions as parts of a whole. If PSTs did not come to one of thesecourses with sound knowledge about fraction as number, their instructors’ implicit andlimited discussion may leave the PSTs with incomplete understanding of fractions evenafter completing the course.

Addressing how different interpretations of fractions—whether part–whole, division, ormeasurement which were the primary interpretations we saw—are related to each other isalso important because these relations help PSTs see that fractions derived from the differentapproaches can be the same fractional number. The connections across these four ways ofdefining and conceptualizing fractions are not mathematically trivial in the historical devel-opment of fractions. There exist considerable temporal and conceptual gaps between the firsttwo approaches (the part–whole and measurement approaches) and the last two approaches(the division and symbolic approaches). In the first two, fractions were conceived intuitivelyas a part of some entity or as the result of measurement, whereas in the last two, they weredefined as abstract mathematical objects based on operations and arithmetic properties.Thus, even in the history of mathematics, there is a disjunction between the intuitive ideasof fractions and the formalization of fraction as a number. This implies that the connectionsamong different approaches that lead to a fraction would not be obvious for PSTs to see.However, in the classes we observed, when instructors made explicit one or more routes forextending from whole numbers to fractions, they did not connect these paths to emphasizethat they lead to the same outcome: fractions as numbers. A lack of discussion about suchconnections may lead PSTs to choose and use one dominant interpretation of fraction, andpossibly teach students fraction in a limited context in the future.

Addressing the mathematics behind the different approaches to extend whole numbers tofractions implicitly in the mathematics courses for PSTs may lead them to reproduce thisimplicitness in their K–8 classrooms. PSTs, when they become K–8 teachers, may also teach

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fractions without addressing important aspects of fractions such as the need for a new kind ofnumber, whole numbers as fractions, and fractions as numbers. Many K–8 students conceiveof fraction as part of a whole without appreciating it as a number (e.g., Lamon, 2007; Post etal., 1993). Teaching fractions without appreciation of and attention to the justification forfractions as numbers and the conceptual difficulty of seeing fractions as numbers maycontribute to K–8 students’ inaccurate realizations of fractions as objects disconnected fromwhole numbers rather than as part of a coherent number system.

In this study, we observed that fraction-as-number went unproblematized; it was taken asgiven in mathematics courses for PSTs. In conclusion, we argue that addressing themathematical ideas behind the extension of fractions from whole numbers explicitly isimportant in these classes because it is related to various students’, and sometimes teachers’,incorrect notions of fractions (Kerslake, 1986; Pitkethly & Hunting, 1996; Post et al., 1993).We are not suggesting that historical development of fractions should be part oflessons of mathematics courses for PSTs. We do suggest, however, that it is importantfor the instructors of these courses, and PSTs in their classes in the long run, to beaware that understanding fractions as numbers is not trivial either to mathematiciansin the past or to today’s K–8 students. As part of their teacher education, mathematicscontent course for PSTs need to address these issues, and thus they could provide anadequate opportunity to develop their sound content knowledge, and knowledge fortheir future K–8 students’ thinking about fractions.

References

Austin, D. (2007). Pulling digits out of pi. Feature column: Montly Essays on Mathematical Topics. http://www.ams.org/samplings/feature-column/fcarc-pi. Accessed 18 Sept 2010

Ball, D. (1988). Knowledge and reasoning in mathematical pedagogy: examining what prospective teachersbring to teacher education. Unpublished dissertation, Michigan State University, East Lansing.

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. TheElementary School Journal, 90(4), 449–466.

Baumgart, J. K. (1969). Historical topics for the mathematics classroom. Reston, VA: National Council ofTeachers of Mathematics.

Beckmann, S. (2005). Mathematics for elementary teachers. Boston: Addison Wesley.Beckmann, S. (2008). Mathematics for elementary teachers (2nd ed.). Boston, MA: Addison Wesley.Behr, M., Harel, G., Post, T., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis—emphasis on

the operator construct. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: Anintegration of research (pp. 13–47). Hillsdale: Lawrence Erlbaum.

Berlinghoff, W. P., & Gouvea, F. Q. (2004). Math through the ages, Expanded edition. Washington, DC:Mathematical Association of America, Farmington, ME: Oxton House Publishers.

Cajori, F. (1928). A history of mathematical notation—Notations in elementary mathematics. Illinois: TheOpen Court Publishing Company.

Carpenter, T., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics:Cognitively guided instruction. Portsmouth: Heinemann.

Childs, L. (1995). A concrete introduction to higher algebra. New York: Springer.Clark, K. M. (2011). History of mathematics: illuminating understanding of school mathematics concepts for

prospective mathematics teachers. Educational Studies in Mathematics. http://www.springerlink.com/content/k5mv6814246755l5/. Accessed 26 Nov 2011

Dantzig, T. (1954). Number: The language of science. New York: Macmillan.Dorier, J. (1998). The role of formalism in the teaching of the theory of vector spaces. Linear Algebra and Its

Application, 275–276, 114–160.Dorier, J., Robert, A., Robinet, J., & Rogalski, M. (2000). On a research programme concerning the teaching

and learning of linear algebra in the first-year of a French science university. International Journal ofMathematical Education in Science and Technology, 31(1), 27–35.


http://www.ams.org/samplings/feature-column/fcarc-pi

http://www.ams.org/samplings/feature-column/fcarc-pi

http://www.springerlink.com/content/k5mv6814246755l5/

http://www.springerlink.com/content/k5mv6814246755l5/

Erlwanger, S. H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’sMathematical Behavior, 1(2), 7–26. Reprinted in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.).(2004). Classics in mathematics education research. Reston, VA: National Council of Teachers ofMathematics.

Even, R. (2008). Facing the challenge of educating educators to work with practicing mathematics teachers. InB. Jaworski & T. Wood (Eds.), The international handbook of mathematics teacher education, vol 4. Themathematics teacher educator as a developing professional (Vol. 4, pp. 57–74). Rotterdam: Sense.

Fauvel, J., & van Maanen, J. (Eds.). (2000). History in mathematics education: The ICMI study. Dordrecht:Kluwer.

Fischbein, E., Jehiam, R., & Cohen, C. (1995). The concept of irrational number in high-school students andprospective teachers. Educational Studies in Mathematics, 29, 29–44.

Hart, K. (1987). Children’s mathematics frameworks: part 2. What are equivalent fractions? Mathematics inSchool, 16(4), 5–7.

Heath, T. L. (1956). The thirteen books of Euclid’s elements, vols. 1–3 (2nd ed.). New York: Dover.Jankvist, J. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education.

Educational Studies in Mathematics, 71, 235–261.Kerslake, D. (1986). Fractions: Children’s strategies and errors. Windsor: NFER-Nelson.Kieren, T. E. (1992). Rational and fractional numbers as mathematical and personal knowledge: Implications

for curriculum and instruction. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmeticfor mathematics teaching (pp. 323–372). Hillsdale: Erlbaum.

Klein, J. (1968). Greek mathematical thought and the origin of algebra. Cambridge: M.I.T. Press.Lamon, S. L. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco & F.

Curcio (Eds.), The roles of representations in school mathematics (2001), Yearbook, Reston: NCTM, (pp.146–165).

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework forresearch. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning(2007), Information Age Publishing, Charlotte NC, (pp. 629–667).

Ma, L. (1998). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamentalmathematics in China and the United States. Mahwah: Lawrence Erlbaum.

Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal forResearch in Mathematics Education, 21(1), 16–32.

Mack, N. K. (1993). Learning rational numbers with understanding. The case of informal knowledge. In T. P.Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 85–105). Hillsdale: Lawrence Erlbaum.

Newton, K. J. (2008). An extensive analysis of elementary preservice teachers’ knowledge of fractions.American Educational Research Journal, 45(4), 1080–1110.

Osana, H. P., & Royea, D. A. (2011). Obstacles and challenges in preservice teachers’ explorationswith fractions: A view from a small-scale intervention study. The Journal of MathematicalBehavior, 30, 333–352.

Parker, T. H., & Baldridge, S. J. (2003). Elementary mathematics for teachers. Okemos: Sefton-AshPublishing.

Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts.Educational Studies in Mathematics, 30, 5–38.

Pogliani, L., Randic, M., & Trinajstic, N. (1998). Much ado about nothing—an introductory inquiry aboutzero. International Journal of Mathematical Education in Science and Technology, 29, 729–744.

Post, T. R., Harel, G., Behr, M. J., & Lesh, R. (1988). Intermediate teachers knowledge of rational numberconcepts. In E. Fennema, T. P. Carpenter, & S. J. Lamon (Eds.), Integrating research on teaching andlearning mathematics (pp. 177–198). New York: State University of NY Press.

Post, T. R., Cramer, K. A., Lesh, R., Harel, G., & Behr, M. (1993). Curriculum implications of research on thelearning, teaching and assessing of rational number concepts. In T. P. Carpenter, E. Fennema, & T. A.Romberg (Eds.), Rational numbers: An integration of research (pp. 327–362). Hillsdale: LawrenceErlbaum Associates Publishers.

Radford, M. (1995). Before the other unknowns were invented: Didactic inquires on the method and programsof medieval Italian algebra. For the Learning of Mathematics, 15(3), 28–38.

Radford, M., Barolini-Bussi, M. G., Bekken, O., Boero, P., Dorier, J., Katz, V., Rogers, L., Sierpinska, A., &Vasco, C. (2002). Historical formation and student understanding of mathematics. New ICMI StudySeries, 6, 143–170.

Rizvi, N. F., & Lawson, M. J. (2007). Prospective teachers’ knowledge: Concept of division. InternationalEducation Journal, 8(2), 377–392.

Seife, C. (2000). Zero, the biography of a dangerous idea. New York: Penguin.

478 J. Park et al.

Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. TheJournal of Mathematical Behavior, 14, 15–39.

Smith, D. E. (1923). History of mathematics, volume I. New York: Dover.Smith, D. E. (1925). History of mathematics, volume II. New York: Ginn & Company.Sowder, J. T., Bedzuk, N., & Sowder, L. K. (1993). Using principles from cognitive psychology to guide

rational number instruction for prospective teachers. In T. P. Carpenter, E. Fennema, & T. A. Romberg(Eds.), Rational numbers: An integration of research (pp. 239–259). Hillsdale: Lawrence Erlbaum.

Sowder, J., Philipp, R., Armstrong, B., & Schappelle, B. (1998). Middle-grade teachers’ mathematicalknowledge and its relationship to instruction: A research monograph. Albany: State University of NewYork Press.

Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value offractions. Learning and Instruction, 14, 503–518.

Streefland, L. (1993). Fractions: A realistic approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.),Rational numbers: An integration of research (pp. 289–325). Hillsdale: Lawrence Erlbaum AssociatesPublishers.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of divisionof fractions. Journal for Research in Mathematics Education, 31(1), 5–25.

Tirosh, D., Fischbein, E., Graeber, A. O., & Wilson, J. W. (1999). Prospective elementary teachers’conceptions of rational numbers. http://jwilson.coe.uga.edu/texts.folder/tirosh/pros.el.tchrs.html.Accessed 11 Jun 2004

Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational number interval?Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X.Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 265–282). Amsterdam: Elsevier.

Weil, A. (1978). History of mathematics: Why and how. In Proceedings of the International Congress ofMathematicians, Helsinki (pp. 227–236) 1978. Helsinki: Academia Scientiarum Fennica.

Weller, K., Arnon, I., & Dubinsky, E. (2009). Preservice teachers’ understanding of the relation between afraction or integer and its decimal expansion. Canadian Journal of Science, 9(1), 5–28.

Wilson, S., Floden, R. E., & Ferrini-Mundy, J. (2001). Teacher preparation research: Current knowledge,gaps, and recommendations. Seattle: Center for the Study of Teaching and Policy.

Wu, H. (2010). Learning school algebra in the U.S. In Y. Li & Z. Huan (Eds.), Mathematics education:Perspectives and practices in the East and West. Special Issue of Mathematics Bulletin (pp. 101–114).Beijing, China: Chinese Mathematical Society, Beijing Normal University.

Zhou, Z., Peverly, S. T., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study ofAmerican and Chinese mathematics teachers. Comtemporary Educational Psychology, 31, 438–457.


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