Educ Stud Math
DOI 10.1007/s10649-012-9440-8
Teaching prospective teachers about fractions:
historical and pedagogical perspectives
Jungeun Park & Beste Güçler & Raven McCrory
# Springer Science+Business Media Dordrecht 2012
Abstract Research shows that students, and sometimes teachers, have trouble with fractions, especially conceiving of fractions as numbers that extend the whole number system.
This paper explores how fractions are addressed in undergraduate mathematics courses for
prospective elementary teachers (PSTs). In particular, we explore how, and whether, the
instructors of these courses address fractions as an extension of the whole number system
and fractions as numbers in their classrooms. Using a framework consisting of four
approaches to the development of fractions found in history, we analyze fraction lessons
videotaped 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 the
last two approaches—division and set theory—formalize fractions as numbers. Our results
show that the instructors only implicitly addressed fraction-as-number and the extension of
fractions from whole numbers, although most of them mentioned or emphasized these
aspects of fractions during interviews.
Keywords Whole numbers . Fractions . Rational numbers . Part–whole . Measurement and
ratio . Division . Pre-service teacher education . Elementary Mathematics . K–12
1 Introduction
The study of the teaching and learning of fractions has been an important area of mathematics education research for many years. Much of the research on fractions over the last
J. Park (*)
Department of Mathematical Sciences, University of Delaware, 402 Ewing Hall,
Newark, DE 19711, USA
e-mail: [email protected]
B. Güçler
Kaput Center for Research and Innovation in STEM Education, University of Massachusetts Dartmouth,
200 Mill Road, Suite 150B, Fairhaven, MA 02719, USA
e-mail: [email protected]
R. McCrory
Teacher Education, Michigan State University, 620 Farm Lane #114B, East Lansing, MI 48824, USA
e-mail: [email protected]
J. Park et al.
three decades has focused on student learning and misconceptions (e.g., Kieran, 1992;
Lamon, 2007; Post, Cramer, Lesh, Harel, & Behr, 1993), and on unpacking rational
numbers (represented as fractions) as mathematical and pedagogical objects (e.g., Ball,
1988; Ma, 1998; Mack, 1990; Tirosh, 2000). Recently, there also has been an increasing
interest in pre-service and in-service teachers’ knowledge about fractions. Various studies
have shown that many elementary in-service and pre-service teachers have difficulties
with 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 is
often weak. However, mathematics educators lack both a conceptualization of what preservice elementary teachers should learn as a requirement for certification and a clear
exposition 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 how
important mathematical aspects of fraction—fraction as a number and fraction as an
extension of whole numbers—are addressed in mathematics content courses for preservice 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 the
topic (Sowder, Philipp, Armstrong, & Schappelle, 1998; Lamon, 2007). They have reported the
dominance 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 extension
of 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 partitioning
to 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 and
fractions as an extension of whole numbers—are closely related; students are expected to use
their prior knowledge of whole numbers to make sense of fractions as well as the operations and
properties of fractions so that they conceive fractions as numbers that are connected to whole
numbers. Studies, however, have shown that some students, and teachers, conceptualize
fractions 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 early
mathematicians’ conceptualizations of fractions. Historical documents show that the idea of
fractions existed and was used for centuries with symbolic representations, without fractions
being considered as legitimate numbers on a par with whole numbers (Klein, 1968). This
similarity between students’ and early mathematicians’ conceptualizations of fractions
provided a motivation for us to use the mathematical development of fractions as numbers
to develop our framework to investigate how fractions are defined or developed from whole
numbers in mathematics classes for PSTs. In other words, we used a historical lens—a
distillation of the mathematical history of fractions—focusing particularly on the definition
of fractions and the extension of fractions from the whole number system as they play out in
a sample of required mathematics courses that include fractions. More specifically, we
address the following research questions:
1. Do instructors of undergraduate mathematics classes for PSTs address fractions as an
extension 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?
Teaching prospective teachers
In this paper, we first review the literature about students’ and teachers’ thinking about
fractions focusing on the two aspects—fraction as a number and as an extension of the whole
number system. Next, we provide our framework, an interpretation of the mathematical
history of fractions focusing on the development of fraction as a number. Finally, we present
data from interviews with instructors and classroom observations from six undergraduate
mathematics classrooms in which fractions are taught to PSTs, using the framework as a tool
for analysis.
2 Literature review
There has been a plethora of studies in teaching and learning about fractions, most of which
report K–8 students’ difficulties with fractions (e.g., Erlwanger, 1973; Mack, 1990; Post et
al., 1993). As mentioned earlier, studies have shown that the part–whole interpretation is
dominant in students’ understanding in K–8 classrooms, even though the part–whole
approach is not necessarily a good start for effective fraction teaching compared to other
approaches (Lamon, 2001, p. 163). This part–whole dominance is closely related to students’ failure to conceptualize fractions as numbers by introducing the denominator as a
whole 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 composition
of two numbers rather than a single entity. Similarly, Hart (1987) found that students may try
to find equivalent fractions by adding the same number to both numerator and denominator
(e.g., A=B ¼ ðA þ C Þ=ðB þ C Þ ); or multiplying only the numerator by a constant (e.g.,
A=B ¼ ðA % C Þ=B ) without considering equivalent fractions as numerals that represent the
same number. Post and colleagues (1993) also reported students’ difficulties locating a
fraction 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 different
from operations on whole numbers (Post et al., 1993). For example, when comparing two
fractions, students tend to compare two denominators instead of considering the sizes of the
two fractions (Post et al., 1993). In fraction addition, they often add numerators and
denominators 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 misunderstandings 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), even
though they bring considerable knowledge and experience with fractions to their
undergraduate mathematics classes (Mack, 1990; Tirosh, 2000). Studies have reported
PSTs’ difficulties understanding the relationship between whole numbers and fractions
and their incorrect applications of whole number properties in fraction operations.
Rizvi and Lawson (2007) reported that prospective teachers, who successfully represented whole number division word problems using various models, showed difficulties in developing representations of fraction division problems. As Rizvi and Lawson
(2007) pointed out, these difficulties might come from PSTs’ reliance on a repeated
subtraction understanding of division. The difficulties can also be seen as lack of
understanding of the relationship between whole numbers and fractions.
Newton (2008) found that one prevalent and persistent error among PSTs was adding
numerators 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
J. Park et al.
misconception that a smaller number (e.g., 1 in 1/3) should be subtracted from a larger
number (e.g., 2 in 2/7) in whole number subtraction (p. 1097). Post et al. (1993) also
mentioned that PSTs conceived of fractions as different from whole numbers, applying
concepts and properties of whole numbers incorrectly to understand fractions and compute
fraction operations.
There are similarities between K–8 students’ and preservice teachers’ difficulties conceiving of fractions as numbers and as an extension of whole numbers and the related
properties of fractions and early mathematicians’ conceptualizations of fractions. Fractions
were used in various computations without being considered as numbers by the Egyptians
by about 1600 BC and even earlier by the Babylonians (Cajori, 1928, p. 13; Klein, 1968;
Smith, 1923). Based on this similarity between current and historical conceptualizations of
fractions, we reviewed the historical development of fractions, identified four milestones
with respect to conceptualizing fractions as an extension of whole numbers, and used them
as our lens to explore whether and how fraction are addressed as numbers in mathematics
classes 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 instructors
of PSTs should teach the history of fractions. Instead, we use the history of the mathematics
as 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 mathematical similarities and differences across the instructors’ approaches in a sample of required
mathematics courses.
The history of mathematics has been used in learning and teaching of mathematical
concepts (Clark, 2011; Jankvist, 2009; Radford et al., 2002; Weil, 1978) based on similarities in the historical obstacles to development of a mathematical idea that seem to be
repeated 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 PSTs
improve 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 give
rise to “cognitive and socio-cognitive conflict” and “create favourable conditions for
students 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 studies
mentioned above that use history of the mathematics as a resource for teaching mathematical
concepts for students or teachers. In this study, we only use history of mathematics as a
source for developing an analytical lens to investigate and analyze the teaching of fraction
concepts. Whether such an analytical approach has implications for the teaching and
learning 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 in
analyzing 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 whole
numbers through four mathematical approaches: part–whole, measurement, division,
and set theory (Cajori, 1928; Heath, 1956; Klein, 1968). Our historical approach
focuses on how fractions were developed from the whole number system and ultimately, as part of the rational number system, accepted as numbers rather than
symbols with various interpretations and uses.
Teaching prospective teachers
One of the issues with investigating fraction-as-number is the elusiveness of the
idea and meaning of “number.” What does it actually mean to know that a fraction is
a 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 of
fractions as numbers was far from trivial, and entailed rethinking what it means for
something to be a number: “The shift from natural to rational numbers involved
changes in the status and meaning of the term ‘number’ that cannot be accounted
for 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-number
that fractions are part of a system that includes whole numbers, and that they inherit
the properties and definitions of the four basic operations from whole numbers. Our
definition of fraction-as-number prepares the way for what Wu (2010) calls the
Fundamental Assumption of School Mathematics, the coherent extension of whole
numbers 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-integer
fraction but is not necessarily considered as a number. In other words, a fractional
number can be represented as a fractional quantity in various contexts, but using
symbols to represent fractional quantities does not always imply that those symbols
are considered to be numbers. For purposes of this paper, we treat the terms “rational
number” 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 in
the mathematical history of fractions:
1. Part–whole approach: conceptualizing a part of a whole as a new unit. Historically, this
conceptualization of fraction grew from ancient times when “the one” was conceived as
“impartiable and indivisible” (Klein, 1968, p. 40). As early as 1650 BC, the Egyptians used
symbols to represent unit fractions as parts of the whole (Cajori, 1928; Smith, 1925;
Berlinghoff & Gouvea, 2004). A fractional quantity was not considered as a
number—on a par with whole or “natural” numbers—by the Egyptians or by
other ancient civilizations that had symbolic representations for parts of the
whole, for they had no common arithmetic for whole numbers and fractional
quantities. This historical development is similar to students’ difficulties in moving beyond the part–whole concept of fractions (e.g., Erlwanger, 1973; Mack,
1990). The part–whole conceptualization is an approach often used today. For
example, 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 quantity
can 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 area
models divided into equal parts, or mention that a part of a collection or object can be
expressed as a fraction with equal partitioning and counting.
J. Park et al.
2. Measurement approach: finding fractions from whole numbers through measurement
and proportions, addressing the need for a common unit of measurement for two
quantities. Historically, the term encompassing measurement and proportion is “commensurability” which was defined by the Greek mathematician Euclid in 300 BC as
follows: “Those magnitudes are said to be commensurable which are measured by the
same 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 by
Euclid, but as “the part or parts of a number” (Klein, 1968, p. 43) since it can be seen as a
part of A and of B. Even though measurement clearly evokes a need for numbers beyond
whole numbers, the ancients did not take the next step of conceiving of those units as
numbers on an equal footing with whole numbers. As in the part–whole approach, the
measurement 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 measurement problem, measuring A units of a continuous quantity of B-unit size—in which the focus
is the area, length, or volume of continuous quantity, and shows that the results of the
measurement are not always whole numbers.
3. Division approach: finding the algebraic solution for an equation Ax0B where A
and B are whole numbers and A is nonzero. This approach arises in the formal
definition of a field, first conceived of by Galois in the early nineteenth century
and formalized concretely by Dedekind in 1871 (Baumgart, 1969). We call this a
division approach since the need for the fraction B/A is a result of the need to
have a set of numbers that is closed under division. In order to talk about a
number system being closed under division, one has to think about the notion of
multiplicative inverses, which are not contained in the whole number system. In
other words, one needs to extend the number system to include multiplicative
inverses of nonzero whole numbers. This set, with inclusion of the additive
inverses, forms a field with all the rational numbers. As a result, we can form
groups and rings in the rational number system, in which fractional quantities
constitute distinct elements of the number system. Therefore, fractions are considered as numbers in this approach.1
An example of the division approach in practice could be performing division
with bare numbers or a partitive division, A÷B, in which A units of a continuous
quantity are shared equally by B recipients and mentioning that the results of
division are not always whole numbers, and thus one needs fractions to express
the results.
4. Set-theoretical approach: defining rational numbers as a set of ordered pairs consisting
of whole numbers:
1
This development was preceded by many centuries of work by mathematicians who struggled with the
notation for fractions and with developing algorithms for operations with fractions. Attempts to integrate
fractions into the number system finally led to the formalization of decimal fractions by Stevin in the
seventeenth century (Smith, 1925).
Teaching prospective teachers
Take the set S of all ordered pairs (A, B) of integers, where B≠0. Partition the set S into
subsets by the rule: two pairs (A, B) and (C, D) are in the same subset if the ratio of A
to 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 to
develop a rigorous foundation for mathematics based on arithmetic. In the late nineteenth
century, Cantor developed set theory, which eventually led to formal, set theoretic definitions of rational numbers.
Fractions are considered as numbers in this set-theoretical approach, for numbers themselves are defined by sets. By defining rational numbers as a set with elements that satisfy
certain conditions, this approach provides a rigorous foundation for rational numbers in the
number system. With added conditions, the set can be extended to the real numbers, for
example, 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 ordered
pairs based on equivalence classes and reduce the definition to a set of symbols:
fA=B : A; B 2 Z; B 6¼ 0g
where Z is a set of integers. This kind of definition is still seen in many textbooks, followed
by explanations of what the symbols mean and how to manipulate them. In the analysis, we
initially labeled instances of this purely symbolic approach as “set-theoretical,” but changed
it to “symbolic” because they were so far from the rigorous set-theoretical approach as not to
be recognizable as such. In fact, we saw no examples of a true set-theoretical approach to
fractions, but many examples of a symbolic approach. Unlike the other three approaches, the
symbolic approach does not suggest a need or justification for fractions, but it does
incorporate 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 occurred
several 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 case
of fractions, accepting them as numbers increases the abstractness and generalizability of the idea
of fraction while supplanting or decreasing the intuitive meaning as part of a whole. Abstraction
makes it possible to include fractions in the number system and operate on them with all the
arithmetic properties that whole numbers follow, letting them take their place in what Wu calls the
Fundamental Assumption of School Mathematics—that all information about operations on
fractions 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 of
Elementary Teachers (ME.ET), we explain the process we went through to develop this
current study in the context of the bigger project. The overall goal of the ME.ET project is to
explore PSTs’ learning in their undergraduate mathematics classes, with a particular focus on
fractions. We collected data to analyze what is taught and learned in undergraduate
J. Park et al.
mathematics courses required for elementary certification at different institutions in two
states in the USA. Our data include pre- and posttests of over 1,000 PSTs in their mathematics classes, along with information about their instructors based on surveys and interviews. Interviews were conducted individually in order to understand their general goals of
teaching this course. These data revealed various aspects of students’ knowledge about
fractions and their relationships to characteristics of their instructors (see the project Web site
for other articles and reports from the project, http://meet.educ.msu.edu/index.htm). We also
collected videos of six mathematics instructors of these courses, focusing on fraction
lessons, 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 that
instructors rarely mentioned the idea of fraction-as-number or made explicit connections to
the ways that fractions fit into the number system. Even instructors who introduced a variety of
representations and uses of fractions seemed to neglect making the strong connections that
would tie fractions to whole numbers as an extension of a common idea. In seeking a way to
explain what we saw, we turned to the history of mathematics to see if the development of
fractions over time might be a useful lens for understanding the disconnect we were observing in
these classes. We also used the interview data to supplement what we observed during the
lessons. Since the interviews were designed for other purposes, they do not provide complete
information that would be relevant to this paper (e.g., the instructors’ assumptions about what
their PSTs know about fractions as numbers). However, the instructors’ responses to some of the
interview questions (e.g., “what is the basic definition of fraction to you?” “do you have any
kind of definition that you want your PSTs to have?” and “what aspects of fraction would be
most 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. In
Table 1, “Number of Fraction Lessons Observed” is the number of lessons in which fraction
was 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 when
the definition of fractions or their extension from whole numbers were discussed, and made
more 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 the
interpretations of the instructors’ approaches to extend the number system. Specifically, we
looked for mathematical elements of the four historical milestones in the instructors’ presentation of fractions, both when (and if) they make explicit the extension of the whole number
system to include fractions and when they defined fractions. We used the four approaches to
determine how connections are made across conceptions of fractions with an eye toward a
conceptualization of fraction-as-number and as an extension of the whole number system. We
looked for evidence of teaching fraction-as-number by identifying instances when instructors
explicitly called attention to the fact that a fraction is a number, either by saying so or by
drawing analogies to whole number and/or real number properties that fractions possess. Table 2
gives examples of the instructors’ explanations for each of the four approaches, with an
explanation 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 the
interview, and whether and how they extended the number system from whole numbers to
fractions during the classes. For the interviews, we mainly analyzed their responses to the
three questions,
Teaching prospective teachers
Table 1 Information about the instructors and their courses
Instructora Position
Times
previously
taught this
course
Number Type of
school
of
fraction
lessons
observed
Type of
course
Number Textbook used
of
students
in
section
Edie
20
Associate
Professor,
Mathematics
9
Mathematics 12
Large
& methods
public,
Masters for
elementary
PSTs
N/A
Eliot
0
Assistant
Professor,
Mathematics
7
Medium Mathematics 35
for
public,
Masters elementary
PSTs
Department
generated
materials
Jamie
1
Graduate
Student
Instructor,
Mathematics
Education
4
Large
public,
PhD
Pat
15–20
Assistant
Professor,
Mathematics
10
Mathematics 23
Large
& methods
public
Masters for
elementary
PSTs
Sam
0
Graduate
Student
Instructor,
Mathematics
Education
4
Large
public,
PhD
Mathematics 34
for
elementary
PSTs
Elementary
Mathematics for
Teachers (Parker
& Baldridge,
2003)
Terry
2
Assistant
Professor,
Mathematics
19
Large
public,
PhD
Mathematics 23
for
elementary
PSTs
Mathematics for
Elementary
Teachers
(Beckmann, 2005)
a
Mathematics 29
for
elementary
PSTs
Elementary
Mathematics for
Teachers (Parker
& Baldridge,
2003)
Children’s
Mathematics:
Cognitively
Guided
Instruction
(Carpenter,
Fennema, Franke,
Levi, & Empson,
1999)
Names used in this paper are pseudonyms
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 were
introduced. We examined the approaches instructors used, whether fractions were
named explicitly as numbers, and whether properties of whole numbers were explicitly
extended 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
J. Park et al.
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: A whole number cannot express any part of a whole which is smaller than
the whole
Measurement Performing measurement division, A÷B—measuring A unit of a continuous quantity with a
quantity of B-unit—in which the focus can be the area, length, or volume of continuous
quantity
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 a
continuous quantity is shared equally by B recipients
Explicit extension: The results of division are not always whole numbers; we need different
numbers (fractions) to represent the result
Symbolic
Providing the definition of fraction to a set of symbols: fA=B : A; B 2 Z; B 6¼ 0g
Explicit extension: The symbolic definition implies that any whole number N can be expressed
as N/1
Note that this approach does not necessarily suggest a need for fractions, but rather includes
whole numbers in the definition of fractions
quantity and be able to visualize it, to represent it…to develop a number sense and
comfortableness with fractions as quantities.” She did not explicitly mention fraction-asnumber during the interview. Later in the interview, she mainly talked about the part–whole
interpretation 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; she
connected composing and decomposing fractional parts to composing and decomposing
whole numbers. Specifically, she said, “I…try to connect with whole numbers…in terms of
discrete quantities in the sense of like a fourth of 36 is 9 so going back to whole numbers I
can imagine the quantity 36 and I can imagine 9 so I can see some sense of proportion of 9
and 36,” and “everything I do about fractions currently connects explicitly back to our work
with whole numbers.”
Her emphasis on the connection between whole numbers and fractions was also identified
in her introductory fraction lessons. Edie extended the number system from whole numbers
to fractions through a division approach using word problems involving equal sharing and
multi-digit division. The word problem Edie introduced to the class was: “If there are 3 sub
sandwiches and 4 kids, how much did each child get?” (Edie, June 10, 2008). When working
on this problem, Edie first asked what kind of operation this problem represented, and then
continued 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 they
had discussed before in their classes. Moreover, Edie explicitly talked about a need for
fractions 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-digit
division of whole numbers. While representing 151 divided by 7 with a rectangular area
model, Edie explained the representations on the board, ending with the string of equations
shown in Fig. 2. Pointing to the last equation of the solution process, she explicitly
Teaching prospective teachers
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
mentioned the need for a fraction, saying “this gets me into thinking about parts of numbers
and fractions,” and noting that the “box” would be four sevenths.
During the interview, she mainly interpreted fraction as part of a whole, and emphasized
the connection between whole numbers and fractions. Edie extended whole numbers to
fractions through division and mentioned explicitly that what is obtained from whole
number division, in this case a fraction, is a number. In this approach, she emphasized parts
and the whole in a whole number division, and connected it to a need for fraction. This
division approach was the only extension Edie used.
5.2 Eliot
Eliot’s response to the interview question about the basic definition of fractions she would
like 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 traditionally considered to be a fraction like three fourths, seven eighths, and … I wouldn’t call
it a really a formal definition…though I do think that it’s important that they have that
concept [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 emphasis
on its connection to whole number system was consistent with her fraction lessons. On the first
day of the fraction unit (February 13, 2008), Eliot extended the number system from whole
numbers to fractions using two approaches: symbolic and division. Unlike the other instrucFig. 2 Reproduction of Edie’s
writing about “151 divided by
7,” June 12, 2008
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Fig. 3 Eliot’s definition of a rational number, February 13, 2008
tors, Eliot avoided general use of the term “fraction,” instead defining rational numbers and
using the term “rational number” throughout her teaching. In this way, she continually
emphasized 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 to
explain 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 then
pointed out that a whole number, n, is a rational number because it satisfies the definition; in
other words, it can be expressed as n/1. On the same day, Eliot also used the division
approach 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 number
division in the number system. Specifically, she illustrated the need to solve a division
problem with bare numbers without a context, 5÷4, making it similar to solving an algebraic
equation 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 whole
number is a rational number—based on the definition she gave. Her interview response
signified fractions as examples of rational numbers, supporting Eliot’s view of fractions as
numbers including whole numbers. In the division approach, she explicitly extended the
number system from whole numbers to rational numbers by explaining why rational
numbers are necessary for whole number division.
5.3 Jamie
During the interview, Jamie defined fraction as part of a whole. She emphasized the concept of
the whole in the definition because she said that her PSTs would have difficulty with finding the
whole of a fraction in the problem context. Mentioning her textbook use, she said, “the textbook
did not have much explanation about the concept of the whole…[so] I have to find and use
many examples from Singapore and Connected Mathematics Project [student] textbook.” Her
part–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 to
fractions using the division and part–whole approaches. Jamie started the fraction unit with
Fig. 4 Eliot’s connection to whole numbers, February 13, 2008
Teaching prospective teachers
Fig. 5 Eliot’s extension from whole numbers to fractions using division, February 13, 2008
two whole number division word problems involving equal sharing: one had a whole
number, 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, and
one 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). After
drawing the diagrams on the board, Jamie explained the meaning of the denominator and
numerator 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 the
denominator 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 as
part of a whole and emphasized the concept of the whole. She, however, did not mention
fraction as an extension of whole numbers or as a number itself. This was consistent with her
introductory lessons. Jamie extended fractions from whole numbers using the division and
part–whole approaches. In the division approach, although Jamie provided a situation where it
was possible to talk about the need for fraction in the number system, she did not explicitly
explain 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 that
this resulted in the extension of the whole number system. She neither mentioned fractions as
numbers nor how fractions are extended from whole numbers.
5.4 Pat
During the interview, Pat defined fraction as “a particular representation of a rational
number, a rational number being any number that can be expressed as a quotient of two
integers a/b.” He, however, said that this was not the way he “want[s] them [PSTs] to
define,” but rather he was “more interested in that at least they recognize that a fraction is a
number within itself, a rational number, and then can be operated on it across multiple
representations.” 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
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Fig. 7 Jamie’s discussion about the result of division, October 30, 2007
During class, Pat extended the number system from whole numbers to fractions based on the
division and measurement approaches using word problems involving equal sharing or measurement; some have divisible remainders, others indivisible remainders.2 The class had
worked on whole number division in earlier lessons, including the exploration of the difference
between 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 a
number; 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 a
real life problem is not a whole number, something else is needed. His arguments were
similar to the historical development based on measurement and commensurable quantities,
using whole number division (equal sharing) and measurement problems that do not yield
whole number solutions.
Figure 9 shows two of the problems the class worked on. Here, he explained the result of a
measurement division problem, emphasizing the need for fractional quantities and “fractionbased 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 indivisible
remainder refers to an object that cannot be divided like a school bus or a person. That is, a result can include
half a candy bar, but not half a person.
Teaching prospective teachers
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
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assumed that a quantity implies a number. Except for the single mention of mixed number
quoted above, he did not make explicit that the results—fractional quantities—are numbers.
In summary, during the interview, Pat defined fractions as representations of rational
numbers and emphasized fraction as numbers. During class, Pat used division and measurement to address the extension of the number system. In both approaches, he justified
explicitly the need for fractions. Although he stated that he wanted his PSTs to understand
fractions as numbers during the interview, he did not make explicit that fractions are
numbers 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 and
fractions 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] two
things or objects, or…a ratio.” She explained the goal of this course by saying, “I think this
course focused more on [fractions as] numbers, but I like to…see what they [PST]s know
about 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 it
means to add two numbers, the fractions with different denominators.”
During class, Sam used two approaches to extending the whole numbers to fractions. On the
first day of the fraction unit, she used a part–whole approach. On the second day, Sam used the
division 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 smaller
parts. Students named the parts with fractions. After the activity, she defined a fraction as
part 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 with
respect to the whole thing. (Sam, November 5, 2007)
Then, Sam stated that a fraction is a new type of number different from whole numbers
and showed this by placing 3/16 on the number line (Fig. 10). As shown in the excerpt, Sam
explained how to place a fraction (3/16) on the number line by establishing 1/16 as a unit
fraction and counting three 1/16-ths. Although she used the number line to show 3/16, she
put more emphasis on interpreting a fraction as part of a whole than as a number as implied
from the following excerpt shown in Fig. 11.
In the second day of the fraction unit, Sam explained the extension from whole numbers
to 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 to
an 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] we
had a perfect whole number. …[A] fraction, 19 over 7…is division, but not necessar[ily], we get a nice looking whole number. How about 3 over 2? We have three
triangles 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 that
the 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, “Think
about 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
Teaching prospective teachers
Fig. 10 Sam’s explanation about fraction as a number, November 5, 2007
19/1. However, she did not connect this explanation to the extension of the number system
which could be addressed by explaining the set of fractions is a bigger system which
includes whole numbers.
In summary, during the interview, Sam interpreted fraction as a number, proportion, and
ratio between two quantities. She addressed each of these three aspects of fractions during
the class. Sam expanded the number system from whole numbers to fractions using two
approaches: part–whole and division. Sam did not explain that the part–whole definition of
fraction 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 also
emphasized the concept of the whole with additional examples involving proportions and
ratio between two quantities (see Fig. 11). In the division approach, although she mentioned
that whole number division does not always result in “a nice looking whole number,” she did
not make explicit why this division approach leads to a fraction and that the result of whole
number division, which can be a fraction, is a number. She also did not explicitly state that a
whole 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 real
number system, “we started this course with the real number system, various subsets, various
main subsets of numbers that we use, and so we followed the book definition of a fraction as
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Fig. 11 Sam’s explanation about importance of the Whole, November 5, 2007
anything of the form A over B where B is non-zero.” Then she said, “We’re going to focus on
the parts of the whole. Numerators, bottoms, how many parts the whole is, how many equal
parts it is going to be subdivided into.” She emphasized the concept of the whole in the
definition, “the definition in Beckman’s [book]…really emphasizes that it is a fraction of
something and there is a whole involved there. So I really like that aspect and then one thing I
will really emphasize later is word problems. I really emphasize them being able to write a
valid word problem representing A divided by B or when we work on the standard one.” Her
introductory fraction lessons were consistent with her responses during the interview.
Terry taught from a book focusing on four basic operations rather than the extension of
number system (Beckmann, 2005). Fractions were included in most lessons after their introduction in the third week of class. For this reason, we videotaped her class for every lesson
beginning with the introduction of fractions, a total of 19 videotaped 50-min lessons, 8 of which
were mostly devoted to the discussion of fractions. Terry extended the number system from
whole numbers to fractions using both symbolic and part–whole approaches. She first defined a
fraction 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 whole
numbers, she did not explain whether and how the definition of fraction includes whole
numbers. In other words, the connection between whole numbers and fractions was not
explicitly 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 stands
for 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
Teaching prospective teachers
Fig. 12 Terry’s definition of a fraction, February 13, 2008
emphasized the importance of the word “of” saying that fractions “are of something. So that
is really a key idea and it can be of one.” Then, she explicitly mentioned that a fraction is a
number while explaining 2/3 on the number line as an example: “When you think of
fractions as just a number on the number line…two thirds would be two thirds of the
number one…so that of is absolutely essential there.” She, however, did not draw the
number line to place 2/3 on it or explain further about fraction-as-number through gestures
or other visual means. In other words, although her references to fraction-as-number were
based on the number line, they did not include explanation of unit fractions as parts of the
whole or as means of measuring non-integral parts. In the remaining lessons during which
Terry 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 an
extension 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 of
a whole during the interview. Although she emphasized the meaning of the parts and whole
during the interview, it was not explicit in her introductory fraction lessons. She did not
explain why the word “of is essential.” She mentioned that fraction is a number using the
number 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 predominant
extensions were division and part–whole, employed by five and three instructors, respectively.
Other approaches used were symbolic (used by 2) and measurement (used by 1). When multiple
extensions 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 an
explicit 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 fractions
as 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 to
fractions varied considerably from class to class. However, the idea of extension in general
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Table 3 Summary of instructors’ extensions and definitions of fractions
Instructor Type of
extension
Definition statement
Type of
definition
Edie
Division
No explicit definition
N/A
Eliot
Division and
symbolic
“A rational number is a number that can be written as a traditional Symbolic
fraction…A over B … A and B are integers B … is not zero.”
Jamie
Division and
part–whole
“The denominator” as “total number of parts in a whole unit” and
“the numerator” as “number of parts shaded or we count.”
Pat
Division and
No explicit definition
measurement
Sam
Division and
part–whole
Part–whole
“The bottom number is how we cut this whole thing into pieces,
and the total pieces, and 3 [numerator] is gonna [sic] be the
number of pieces we are talking about with respect to the whole
thing.”
Terry
Symbolic and
part–whole
“A over B, where A and B are whole numbers, and B is non-zero.” Symbolic
Part–whole
N/A
was not fully elaborated in the classes that we observed. Even when the instructors provided
a means of deriving fractions from whole numbers (e.g., by division), not all of them
unpacked the ideas and integrated them into the discussion of fractions. Connections across
interpretations and ideas about fractions were similarly unelaborated, including connections
among any of the four approaches instructors used. For example, Sam used both the division
and part–whole approaches to introduce fractions and connect them to whole numbers, but
she did not make clear, or help her students understand, how these two approaches led to a
single, coherent system of numbers.
Fraction-as-number was also not emphasized or fully justified even when it was mentioned. In many cases, the mention of “fraction as number” was separated from the
discussion of the extension from whole numbers to fractions, and from discussion of the
need for new numbers other than whole numbers. For example, Jamie and Pat both used
division word problems to address the need for a fraction, or fractional quantity, beyond
whole numbers, but they did not address that the results of their division problems were also
numbers or have the same properties that the whole number have. We conclude that in our
sample of classes, instructors are missing many possible instances to intervene in some of
the 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 in
which fractions were introduced and defined. This lens enabled us to compare mathematical
aspects of fractions in relation to whole numbers across the classes. It also highlighted the
instances in which opportunities for the instructors to make explicit links between whole
numbers and fractions, and across different interpretations of fractions, were or were not
taken up by the instructors.
The mathematical approach based on history emphasizes the connectedness and coherence of fractions and whole numbers as numbers in a system. For example, the part–whole
approach in the framework emphasized the similarity in the property between whole
numbers and fractions (e.g., Sam’s explanations on how to place 3/16 on the number line
Teaching prospective teachers
by partitioning the segment between 0 and 1 into 16 equal segments and iterating the smaller
unit three times). The measurement approach also addresses not only how to interpret A/B as
measuring A units of a quantity with a B-unit quantity but also how to deal with a remainder
in terms of the divisor, which Pat explained with divisible remainders. Using the division
approach makes explicit that a fraction is a result of the whole number division such as
Eliot’s example of five subs shared by four recipients. The symbolic approach also makes
the connection between whole numbers and fractions clear by making observable the
instances in which instructors emphasized that a whole number, n, can be expressed by a
fraction form, n/1. In summary, the four approaches provided by the historical lens allowed
us to point out the instances where these aspects were addressed in the classrooms, and
analyze those cases focusing on similarities between whole numbers and fractions and their
mathematical relationships as two sets of numbers.
7 Discussion
The results of this study show that key mathematical aspects of fraction, including fractionas-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 PSTs
in 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 in
their K-12 education, one of the challenges faced by their instructors is finding out how
much, and what, to take for granted in these students’ prior knowledge. Do they already
know what a fraction is, and in particular, that fractions are part of a coherent number
system? One might assume that they know these things, although pushing on what it means
to “know that fractions are numbers” could reveal some serious problems with making this
assumption. 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 number
line. Instructors may make different assumptions about their students’ prior knowledge,
which could affect their approaches to fractions in these particular classes. Although
investigating such assumptions was beyond the scope of our study, we have evidence that
some of the instructors in our study assume that their PSTs have knowledge about fractions
from the interviews and classroom observation. For example, Terry said, “theoretically, they
already know the concept or they are exposed to fractions” during the interview, and Jamie
asked her PSTs “You already know the concept of fractions, right?” in her first fraction
lesson. During the interview, Eliot elaborated further on her assumption about her PSTs
knowing “how” but not “why” and her goal in her fraction lessons:
They [PSTs] have seen so much addition and multiplication and stuff in high school
but I’m not sure they truly understand it but they can definitely do it…In terms of me
figuring out what level of understanding they really have, I haven’t been able to [find
out]…And we talk about that a lot in class, and they know how to do just about
everything…I think that’s true they rarely know why. So, I do try to point that out
whenever possible and sometimes they are reluctant because they already know how
to 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 grades
that they wanted…So now I’m trying to undo some of that and not show them tricks
and shortcuts and try to really teach them the real reason why things work like they do.
(Eliot, February 14, 2008)
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Exploring how instructors’ teaching goals are based on their assumptions about what
their PSTs know in fraction lessons would be an interesting topic for a future study that
would provide useful information about instructors’ decisions in practice. A follow-up study
can be designed to explore the possible explanations for explicitness/implicitness of classroom discussion based on the instructors’ beliefs and assumptions about PST’s knowledge
on these aspects.
What we observed in our sample classes shows the instructors’ implicit approaches to key
aspects of fractions: the conceptual links and mathematical developments that underlie the
definition and use of fractions. As mentioned earlier, studies have reported that although
PSTs bring considerable knowledge about fractions to the mathematics courses for PSTs
(Ball, 1990), their knowledge includes various incorrect notions about fractions, most of
which 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 results
from existing studies, we argue that addressing fractions as numbers through at least one
approach, which extends whole numbers to factions, may be a route toward improving PSTs’
mathematical knowledge for teaching fractions. The mathematics behind the historical
development of fractions shows that neither the part–whole approach—the most dominant 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 of
fraction as a number. Although fractional quantities had been common in different
cultures over hundreds of years, it took mathematicians centuries to accept fractions as
numbers, until Stevin (1548–1620) defined a fractional number as “a part of the parts
of a whole number” (Cajori, 1928; Klein, 1968, p. 290). It is logical that some PSTs
may have similar difficulty incorporating the idea that fractions are numbers into their
dominant notion of fractions as parts of a whole. If PSTs did not come to one of these
courses with sound knowledge about fraction as number, their instructors’ implicit and
limited discussion may leave the PSTs with incomplete understanding of fractions even
after completing the course.
Addressing how different interpretations of fractions—whether part–whole, division, or
measurement which were the primary interpretations we saw—are related to each other is
also important because these relations help PSTs see that fractions derived from the different
approaches can be the same fractional number. The connections across these four ways of
defining and conceptualizing fractions are not mathematically trivial in the historical development of fractions. There exist considerable temporal and conceptual gaps between the first
two approaches (the part–whole and measurement approaches) and the last two approaches
(the division and symbolic approaches). In the first two, fractions were conceived intuitively
as a part of some entity or as the result of measurement, whereas in the last two, they were
defined as abstract mathematical objects based on operations and arithmetic properties.
Thus, even in the history of mathematics, there is a disjunction between the intuitive ideas
of fractions and the formalization of fraction as a number. This implies that the connections
among 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 for
extending from whole numbers to fractions, they did not connect these paths to emphasize
that they lead to the same outcome: fractions as numbers. A lack of discussion about such
connections may lead PSTs to choose and use one dominant interpretation of fraction, and
possibly teach students fraction in a limited context in the future.
Addressing the mathematics behind the different approaches to extend whole numbers to
fractions implicitly in the mathematics courses for PSTs may lead them to reproduce this
implicitness in their K–8 classrooms. PSTs, when they become K–8 teachers, may also teach
Teaching prospective teachers
fractions without addressing important aspects of fractions such as the need for a new kind of
number, whole numbers as fractions, and fractions as numbers. Many K–8 students conceive
of fraction as part of a whole without appreciating it as a number (e.g., Lamon, 2007; Post et
al., 1993). Teaching fractions without appreciation of and attention to the justification for
fractions as numbers and the conceptual difficulty of seeing fractions as numbers may
contribute to K–8 students’ inaccurate realizations of fractions as objects disconnected from
whole numbers rather than as part of a coherent number system.
In this study, we observed that fraction-as-number went unproblematized; it was taken as
given in mathematics courses for PSTs. In conclusion, we argue that addressing the
mathematical ideas behind the extension of fractions from whole numbers explicitly is
important 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 of
lessons of mathematics courses for PSTs. We do suggest, however, that it is important
for the instructors of these courses, and PSTs in their classes in the long run, to be
aware that understanding fractions as numbers is not trivial either to mathematicians
in the past or to today’s K–8 students. As part of their teacher education, mathematics
content course for PSTs need to address these issues, and thus they could provide an
adequate opportunity to develop their sound content knowledge, and knowledge for
their future K–8 students’ thinking about fractions.
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