Journal of Mathematical Behavior 28 (2009) 150–161
Contents lists available at ScienceDirect
The Journal of Mathematical Behavior
journal homepage: www.elsevier.com/locate/jmathb
A quantitative analysis of children’s splitting operations
and fraction schemes
Anderson Norton a,∗ , Jesse L.M. Wilkins b
a
b
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, United States
School of Education, Virginia Tech, Blacksburg, VA 24061-0313, United States
a r t i c l e
i n f o
Article history:
Available online 3 August 2009
Keywords:
Fractions
Mathematical development
Rational number
Scheme theory
Splitting
Teaching experiment
a b s t r a c t
Teaching experiments with pairs of children have generated several hypotheses about
students’ construction of fractions. For example, Steffe (2004) hypothesized that robust
conceptions of improper fractions depends on the development of a splitting operation.
Results from teaching experiments that rely on scheme theory and Steffe’s hierarchy of fraction schemes imply additional hypotheses, such as the idea that the schemes do indeed form
a hierarchy. Our study constitutes the first attempt to test these hypotheses and substantiate
Steffe’s claims using quantitative methods. We analyze data from 84 students’ performances
on written tests, in order to measure students’ development of the splitting operation and
construction of fraction schemes. Our findings align with many of the hypotheses implied
by teaching experiments and, additionally, suggest that students’ construction of a partitive
fraction scheme facilitates the development of splitting.
Published by Elsevier Inc.
1. Introduction
Teaching experiments with pairs of students provide opportunities for teachers to closely analyze students’ problem
solving activities while taking into account student–student interaction (Steffe & Thompson, 2000). Based on students’
actions (including verbalizations) teachers can build models of students’ reasoning using schemes: hypothetical ways of
operating that explain the students’ actions. Such models have produced numerous hypotheses with regard to students’
learning of fractions (e.g., Hackenberg, 2007; Olive, 1999; Steffe, 2002; Tzur, 1999). Our study aims to test some of these
hypotheses in a quantitative study that affords a much larger sample. We recognize that assessing student knowledge with
written tests always comes with limitations, but Kilpatrick (2001) reminded us that, as mathematics education researchers,
we are obliged to quantitatively test our hypotheses whenever possible.
We examine the fraction schemes available to grade 5 and grade 6 students, as indicated by their performance on a written
test. We designed the test to measure whether students had constructed splitting operations and the following schemes:
part–whole fraction scheme, partitive unit fraction scheme, partitive fraction scheme, reversible partitive fraction scheme,
and iterative fraction scheme. We describe the schemes in more detail in a subsequent section. For now, we note that Steffe
(2002) and Olive (1999) learned about these ways of operating from their work with children in the Fractions Project. They
theorized that the schemes form a hierarchy (respective to the order in which we list them) in which each preceding scheme
is reorganized to form the succeeding scheme. Based on this idea and our own work with children, we posit the research
hypotheses listed in Table 1.
Steffe’s research program calls on teachers and researchers to use the mathematics of children as “the basis on which
to teach” (1991, p. 181); this mathematics can be understood as the operation and coordination of schemes. So research
∗ Corresponding author. Tel.: +1 540 231 6942; fax: +1 540 231 5960.
E-mail addresses: norton3@vt.edu (A. Norton), wilkins@vt.edu (J.L.M. Wilkins).
0732-3123/$ – see front matter. Published by Elsevier Inc.
doi:10.1016/j.jmathb.2009.06.002
A. Norton, J.L.M. Wilkins / Journal of Mathematical Behavior 28 (2009) 150–161
151
Table 1
Hypotheses tested.
Number
Hypothesis
1a
1b
In part–whole reasoning there are no operational differences between situations involving unit and proper fractions.
In partitive reasoning there are operational differences between situations involving unit and proper fractions; success with situations
involving unit fractions precedes success in situations involving proper fractions.
In order for children to operate successfully in situations involving a reversible partitive fraction scheme (from a researcher’s perspective), they
must have a splitting operation available.
In order for children to operate successfully in situations involving an iterative fraction scheme (from a researcher’s perspective), they must
have a splitting operation available (Steffe, 2004).
Children develop partitive (unit) fraction schemes and splitting independently.
2a
2b
3
on children’s fraction schemes provides a foundation for educational decision-making. Such research generally occurs in
teaching experiments with one or two children, in order to build models of individual cognition (Steffe & Thompson, 2000).
For example, the CoSTAR project combined results from single-student teaching experiments with results from whole-class
teaching experiments, in order to closely examine effects of classroom instruction on individual student learning. As part of
that project, in working with a sixth-grade student named Olive and Vomvoridi (2006) found that modeling the student’s
fraction schemes was crucial to understanding how he and other students were conceptualizing unit fractions. “The study
illustrated the importance of coming to understand a student’s mathematical activity in terms of possible conceptual schemes
and modifying instructional strategies to build on those schemes” (p. 18). Such studies often yield general hypotheses about
children’s mathematical development, such as those addressed here. Previous research using teaching experiments motivates
our study and informs its results, but our approach differs considerably. Teaching experiments involve personal interactions
between a teacher and a small number of students, whereas our study involves assessments from a one-time written test
taken by many students.
We report our results with one important caveat: Although we designed written tasks to elicit student responses that
would provide indication or counter-indication for particular schemes and operations; and whereas written responses enable
us to assess larger numbers of students in a relatively objective manner; such tests are inferior to teaching experiments in
building models of students’ ways of operating. In other words, the extensive interactions teaching experiments entail provide
more opportunity to generate and test hypotheses about students’ ways of operating, but they also severely limit the number
of students for which those hypotheses can be tested.
Our study establishes a means, as well as some preliminary results, to quantitatively test hypotheses generated by teaching
experiments. We pose tasks similar to those employed in teaching experiments and use students’ written responses as
indication of their ways of operating. In the next two sections we describe some of these theorized ways of operating. We
can then revisit our hypotheses in more depth.
2. Schemes and operations
Following Glasersfeld (1995), we define schemes as three-part structures, which include a set of perceived situations that
activate the scheme (a trigger, or recognition template), a set of operations to act on the situation, and an expected result
from operating. Glasersfeld defined operations—the active and therefore most important component of schemes—as mental
actions abstracted through reflection on previous experience. For example, students might abstract partitioning operations
from experiences in forming equal shares, folding paper, or otherwise creating equal parts from an existing whole. The
partitioning operation results from reflection on the cognitive processes involved completing such physical activities. Once a
student has abstracted the operation, she can apply it to new situations, such as determining how to break a whole candy bar
into five equal parts. Iteration—another operation important to working with fractions—involves creating copies of a unit.
For example, a student might create three-fifths from one-fifth by iterating a one-fifth piece three times.
According to Steffe (2003, p. 240), the splitting operation is the simultaneous composition of partitioning and iterating,
in which partitioning and iterating are understood as inverse operations.1 For example, a student with a splitting operation
can solve tasks like the one illustrated in Fig. 1. Finding an appropriate solution requires more than partitioning; the student
must anticipate that she can use partitioning to resolve a situation that is iterative in nature. Namely, by partitioning “my”
bar into three parts, the student obtained a part that could be iterated three times to reproduce the whole bar.
3. Fraction schemes
3.1. The part–whole fraction scheme
Students who have constructed a part–whole fraction scheme conceive of fractions as so many pieces in the partitioned
fraction out of so many pieces in the partitioned whole. This scheme relies upon operations of identifying (unitizing) a whole,
1
We acknowledge that other researchers (e.g., Confrey, 1988) have developed equally useful definitions of “splitting,” based on their own interactions
with children.
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Fig. 1. Task response providing indication of a splitting operation (alternative number: “five”).
Fig. 2. (a and b) Task responses providing indication/counter-indication of part–whole fraction scheme (alternate fractions: “1/5” and “4/6”).
partitioning the whole into equal pieces, and disembedding some number of pieces from the partitioned whole. Although
the pieces in the fraction and the whole are all the same size, they may not be identical, in the sense that students may
not be able to substitute one piece for another. “A child who has constructed [only] a part–whole fraction[al] scheme is yet
to construct unit fractions as iterable fractional units” (Steffe, 2003, p. 242). In other words, a fraction such as three-fifths
means three pieces out of five equal pieces in the whole, but it is not yet understood as three iterations of one of the fifths.
Consider the tasks illustrated in Fig. 2a and b.
Note that one of the tasks involves a unit fraction, whereas the other involves a non-unit proper fraction. The part–whole
fraction scheme, as abstracted from researcher interactions with children, theoretically operates the same way in either
situation. The response to the first task correctly indicates one out of six equal parts. On the other hand, the response to
the second task (by a different student) serves as counter-indication of a part–whole fraction scheme because the parts in
the fraction are not equal in size to the unshaded parts of the bar. Although the part–whole fraction scheme provides a way
of producing and conceptualizing any proper fraction, it provides no meaning for improper fractions because part–whole
fractions are taken out of the whole (taking nine parts out of seven makes no sense).
3.2. The partitive unit fraction scheme
Both a part–whole fraction scheme and a partitive unit fraction scheme generate fraction language, but the difference
between the powers of the schemes is evident in resolving the task illustrated in Fig. 3. Students with only a part–whole
fraction scheme cannot determine the fraction because the whole is unpartitioned.
The response in Fig. 3 indicates a partitive unit fraction scheme because the student seems to understand (note the mark
on the left side of the large bar) that the unit fractional part (small bar) could be iterated four times to re-produce the whole
(large bar) and that this number of iterations (four) determines the size of the unit fraction relative to the whole (one-fourth).
The partitive unit fraction scheme, “establishes a one-to-many relation between the part and the partitioned whole” and
involves “explicit use of fractional language to refer to that relation” (Steffe, 2002, p. 292). However, a partitive unit fraction
scheme cannot be used to determine the fractional size of a non-unit fraction, because the iterations will not reproduce the
whole (unless, of course, the fraction in question simplifies to a unit fraction; e.g., two-sixths).
Fig. 3. Task response providing indication of a partitive unit fraction scheme (alternative unit stick length: 1/5).
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Fig. 4. Task response providing indication of a partitive fraction scheme (alternative stick length: 4/5).
3.3. The partitive fraction scheme
The partitive fraction scheme is a generalization of the partitive unit fraction scheme. Students can use the more general
scheme to conceive of a proper fraction, such as three-fourths, as three of one-fourth of the whole. This involves producing
composite fractions from unit fractions through iteration, while maintaining the relation between the unit fraction and the
whole. It also involves units coordination at two levels (Hackenberg, 2007; Steffe, 2002), because the student must coordinate
three-fourths as three iterations of the fractional unit and the whole as four iterations of the fractional unit. In order words,
three-fourths is a unit of three fractional units, and the whole is a unit of four fractional units. Consider the coordinations
indicated by the student response in Fig. 4.
The response provides some indication of a partitive fraction scheme because the student identified the smaller bar as
three iterations of a part that, when iterated four times, would reproduce the larger bar (the whole). However, the response
also illustrates the limitations of assessing students’ schemes through written responses (as mentioned in the introduction
of this paper): We have witnessed students solving similar problems by guessing a number in which to partition the whole
and then seeing how many of those partitions are taken up by the fraction in question; this is essentially a part–whole way
of operating that does not rely on iterating at all. We cannot say with certainty that the student responding in Fig. 4 operated
one way or the other, although we took the response as indication that the student had constructed a partitive fraction
scheme for the reasons mentioned above.
3.4. The reversible partitive fraction scheme
Theoretically, the reversible partitive fraction scheme is the first scheme in the hierarchy to rely on the splitting operation
(Hackenberg, 2007). The scheme serves to produce an implicit whole from a proper fractional part of it. For example, consider
the task in Fig. 5 and the student response that provides counter-indication of the scheme.
The student seemed to interpret the task as a partitioning task, calling for the production of one-fifth of the given bar. We
recognize this interpretation and resulting action as a necessary error (Steffe, 2002) resulting from limitations in the student’s
available ways of operating. The student seemed constrained to operating within a given whole, so that the denominator of
a fraction always indicates the number of equal parts that a given bar needs to be partitioned into. This way of operating is
sufficient for resolving most partitive tasks, but in order to reverse that way of operating, the student would need to posit a
whole beyond the given bar. Theoretically, the student would need to understand four-fifths as four of one-fifth and partition
four-fifths into four parts, with the goal of iterating that part five times, in order to produce the whole. Partitioning with the
simultaneous goal of iterating implies the need for splitting.
3.5. The iterative fraction scheme
Students can use (reversible) partitive fraction schemes to estimate the size of any proper fraction or reproduce the whole
from it. However, students cannot use such schemes to produce improper fractions, because when the fraction exceeds the
Fig. 5. Task response providing counter-indication of a reversible partitive fraction scheme (alternative fraction: “3/4”).
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Fig. 6. Task response providing indication of an iterative fraction scheme (alternative fraction: “5/4”).
whole, the whole is lost. Students with a partitive fraction scheme will often refer to a fractions stick like four-thirds (from the
teacher’s point of view) as four-fourths or even three-fourths (Tzur, 1999), redefining the whole as the larger stick. Moving
beyond this limitation requires an iterative fraction scheme (Steffe, 2002, p. 299), as indicated by the student response in
Fig. 6.
Although there are no supporting marks on either bar, the student response in Fig. 6 is uncannily accurate, indicating
that the student purposefully produced the three-thirds bar. Presumably, this would involve partitioning the given bar (fourthirds) into four parts, with the understanding that one of those parts (one-third) could be iterated three times to produce
the whole (three-thirds). Note that such a way of operating would, again, involve splitting, but, this time, the student must
also understand that a unit fraction can be iterated beyond the whole to produce improper fractions, such as four-thirds.
Steffe (2002) has described this attribute as an iterable unit fraction, which is “freed from its containing whole” (p. 299).
Hackenberg (2007) argued that the iterative fraction scheme also relies on coordinating three levels of units prior to activity.
For example, four-thirds would be understood as a unit of four units, any of which can be iterated three times to produce the
whole, another unit.
4. Revisiting our hypotheses
Table 2 summarizes the theoretical hierarchy of schemes (Olive, 1999; Steffe, 2002) outlined in the previous section. The
schemes follow a general progression from part–whole reasoning, with which fractions are treated as a ratio between part
and whole; to partitive reasoning, with which fractions become comparisons of relative size within the whole; and finally to
iterative reasoning, with which fractions are freed from the whole (though still understood in relation to it). We refer to this
progression and the schemes outlined in Table 2 in order to elaborate on our hypotheses and the importance of testing them.
4.1. Research on the progression of fraction schemes
We focus on eight research papers that relied on teaching experiments to report on the progression of children’s construction of fraction schemes. Three of these papers resulted from Steffe and Olive’s three-year teaching experiment, known
as the Fractions Project. In a pair of papers, Steffe (2002, 2004) reported on two students’ constructions of fraction schemes.
He found that one of the students, Jason, constructed a partitive unit fraction scheme during third grade; he constructed a
partitive fraction scheme during fourth grade; and he constructed splitting and a reversible partitive fraction scheme during
fifth grade. An iterative fraction scheme remained beyond his ways of operating. The other student, Laura, constructed a
part–whole fraction scheme during fourth grade, but did not progress to the other ways of operating. Along with Olive,
Steffe also reported on Joe, who constructed a partitive fraction scheme during third grade, and constructed splitting and
an iterative fraction scheme during fourth grade (Olive & Steffe, 2002). These studies are consistent with the hierarchy of
schemes presented here.
In separate projects, several other studies provide further affirmation of the hierarchy. For example, Olive and Vomvoridi
(2006) reported that Tim came into sixth grade with a part–whole fraction scheme and subsequently constructed a partitive
Table 2
Fraction schemes.
Scheme
Associated actions
Part–whole fraction scheme
Partitive unit fraction scheme
Producing m/n by partitioning a whole into n parts and disembedding m of those parts.
Determining the size of a unit fraction relative to a given, unpartitioned whole, by iterating the unit
fraction to produce a continuous, partitioned whole.
Determining the size of a proper fraction relative to a given, unpartitioned whole, by partitioning the
proper fraction to produce a unit fraction and iterating the unit fraction to reproduce the proper fraction
and the whole.
Producing an implicit whole from a proper fraction of the whole (no referent whole given), by partitioning
the fraction to produce a unit fraction, and iterating that fraction the appropriate number of times.
Producing an implicit whole from any fraction (including improper fractions) in the manner described
above.
Partitive fraction scheme
Reversible partitive fraction scheme
Iterative fraction scheme
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unit fraction scheme. Saenz-Ludlow (1994) conducted a teaching experiment with a student named Michael who began third
grade with a part–whole fraction scheme and constructed a partitive unit fraction scheme during the school year. Hunting
(1983) witnessed the same development in a fourth-grade student named Alan. In results from yet another study, Nabors
(2003) described four students who entered seventh grade with part–whole fraction schemes and partitive fraction schemes;
they constructed reversible partitive fraction schemes during seventh grade, but never constructed iterative fraction schemes.
Finally, Hackenberg (2007) found that the four students in her teaching experiment entered sixth grade with reversible
partitive fraction schemes (and splitting operations), and two of them subsequently constructed iterative fraction schemes.
Working with three sixth-grade students in a study on students’ conjecturing, Norton and D’Ambrosio (2008) found
that one of them, Will, began the school year with a part–whole fraction scheme and a partitive unit fraction scheme, and
subsequently constructed a partitive fraction scheme. Will’s partner in the teaching experiment, Hillary, began the school
year with a splitting operation and a partitive fraction scheme; she subsequently constructed a reversible partitive fraction
scheme. A third student, named Josh, began the year with splitting and a part–whole fraction scheme, and subsequently
constructed a partitive unit fraction scheme and a partitive fraction scheme (Norton, 2008).
Collectively, student progress reported in these studies affirms the theoretical hierarchy of schemes and provides some
indication of development by grade level. First of all, students seem to develop part–whole fraction schemes early, but other
schemes often lag far behind or do not develop at all. Although part–whole conceptions are fundamental to understanding
fractions (Pitkethly & Hunting, 1996), “teaching efforts have focused almost exclusively on the part–whole construct of a
fraction” (Streefland, 1991, p. 191) and this can lead to misconceptions. For example, until he constructed a partitive unit
fraction scheme, Tim conceived of 1/n and n/n as the same thing because he could not consider 1/n apart from the whole
(Olive & Vomvoridi, 2006). “Sparse conceptual structures limit students’ understanding; once these conceptual structures
had been modified and enriched, Tim was able to function within the context of classroom instruction” (p. 44).
Saenz-Ludlow (1994) emphasized the importance of developing a partitive conception of fractions when she referred to
the need for students to understand fractions as quantities. This conception also aligns with Mack’s description of fractions
as “multiplicative size transformations” (2001, p. 269). Still, partitive conceptions have their own limitations, particularly
in working with improper fractions. For example, after iterating a unit fraction beyond the whole, students relying on a
partitive fraction scheme often interpret the result as its reciprocal (e.g., nine iterations of 1/8 becomes 8/9). Tzur (1999)
concluded that, “conceptualizing improper fractions is not a simple extension of iterating a unit fraction within the whole
(e.g., by iterating 1/8 six times)” (p. 409). So previous research indicates at least two developmental hurdles with regards to
fractions: moving from part–whole to partitive conceptions, and moving from multiplicative conceptions of proper fractions
to multiplicative conceptions of improper fractions.
4.2. Hypotheses concerning fraction schemes and operations
In addition to outlining the progression of fractions development, teaching experiments have also resulted in hypotheses
about the operational relationships between particular schemes and operations. Concerning the development and role of
splitting, there is a history of conflicting hypotheses to resolve. Steffe previously hypothesized that the splitting operation
results from a reorganization of the reversible partitive fraction scheme (2004, p. 161). He also hypothesized that “upon the
emergence of the splitting operation” the partitive fraction scheme is reorganized as an iterative fraction scheme (2002,
p. 299), but he revised these hypotheses in light of Hackenberg’s (2007) teaching experiment. In the case of the second
hypothesis, Hackenberg countered (and Steffe agreed, personal communication, October 2006) that “although the splitting
operation still seems to be instrumental in the construction of an iterative fraction scheme, it does not appear to be sufficient
for it” (p. 46). Steffe’s first hypothesis was replaced with the following: “constructing a splitting operation is what allows
students with partitive fraction schemes to reverse the operations of their partitive fractional schemes” (Hackenberg, 2007,
p. 46).
Because we did not include items to separately assess students’ units coordination, we could not test Hackenberg’s
hypothesis that, along with splitting, three levels of units coordination is necessary for the construction of the iterative
fraction scheme. However, we can test her hypothesis that splitting precedes a reversible partitive fraction scheme. The latter
hypothesis also aligns with Norton’s teaching experiment with Josh—the student who seemed to have constructed splitting
even before he had constructed a partitive fraction scheme (Norton, 2008). So, we hypothesize that the partitive fraction
scheme and splitting can be constructed independently (see Hypothesis 3 in Table 1), and that splitting is a prerequisite for
the reversible partitive fraction scheme and the iterative fraction scheme (Hypothesis 2). Our first hypothesis (Hypothesis
1) relates to the previous discussion about part–whole and partitive conceptions of fractions. The quantitative methods
described in the next section address all three hypotheses, thus contributing to the resolution of the theoretical issues raised
here, while informing the general progression of students’ construction of fractions.
5. Methods
5.1. Data and assessment
We administered two versions of the test during each fall as part of a two-year professional development study, which
took place from 2005 to 2007 in a low- to middle-income small-town school in the mid-western United States (Norton &
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McCloskey, 2008). The study involved one fifth-grade classroom and one sixth-grade classroom for each of the two years,
with no students involved in both years; a total of 84 students completed tests.
Each year, we administered the tests to one fifth-grade class and one sixth-grade class, simultaneously, at the beginning
of the school year. The classes were chosen in such a way that no student took the test both years. The classes had adopted
the Everyday Mathematics curriculum, which uses a spiral approach so that students study fractions (along with other topics,
such as probability and geometry) each year, beginning in third grade. Prior to taking the tests, none of the students had been
involved in (nor had been taught by teachers involved in) the professional development study. The students were provided
50 min to complete the tests, but all students completed their tests within 35 min.
Each test contained nine randomly-ordered items: including two like the splitting item in Fig. 1, two like the part–whole
items in Fig. 2, and the items illustrated in Figs. 3–6. A ninth item tested for a simple partitioning operation not relevant to
this report. The two versions of the test differed only by the numbers used in them; numbers from one version of the test are
illustrated in the figures, with the alternate numbers shown in parentheses. We randomized item ordering for each student
so that no two students received identical tests, even if they took the same version. The randomization procedure eliminates
the possibility of ordering effects.
We designed the items as indicators of particular schemes. That is, responses to each item provided opportunities for
students to enact particular ways of operating. When assessing student responses, two raters looked at the students’ work
on each single item, and inferred from all markings (e.g., written answers, drawn partitions, shading, calculations) whether
there was indication that the student had operated in a way that is compatible with the particular scheme. The raters scored
responses to each item in the following way:
0: There was counter-indication that the student could operate in a manner compatible with the theorized scheme or
operation. Counter-indication might include incorrect responses and markings that are incompatible with actions that
would fit the scheme. For example, the response illustrated in Fig. 2b indicated that the student did not understand the
importance of creating equal parts in the fraction and the whole.
1: There was strong indication that the student operated in a manner compatible with the theorized scheme or operation.
Indications might include correct responses and partitions. For example, the partitions drawn in Fig. 1 indicate that the
student has used partitioning to resolve a task that was iterative in nature—a strong indication of splitting.
Some items were initially scored as .5 and required further inference on the part of the scorers to select a final score of
0 or 1. For example, in response to the task illustrated in Fig. 6, a student might have drawn a bar that was slightly shorter
or longer than the correct size and with no supporting marks. The two raters reexamined such responses on a case-by-case
basis to come to a consensus, and then used that consensus to inform decisions on similar subsequent cases.
5.2. Measures
5.2.1. Splitting
The two items involving splitting operations were combined to create an indicator of splitting (e.g., see Fig. 1). This resulted
in a score for this measure ranging from 0 to 2, with students getting only one item correct getting a score of 1. This measure
was used to test the relationships between the existence of a splitting operation and situations involving partitive, reversible
partitive, and iterative fraction schemes (see Hypotheses 2 and 3 in Table 1).
5.2.2. Part–whole fraction scheme (PWFS)
Two items were used as indicators of students’ part–whole fraction schemes. One item involved unit fractions (e.g., see
Fig. 2a) and the other item involved non-unit fractions (e.g., see Fig. 2b). These items were used to test for students’ operational
differences between situations involving unit and proper fractions (see Hypothesis 1a).
5.2.3. Partitive fraction scheme
Two items were used as indicators of students’ partitive fraction schemes. One item involved unit fractions (e.g., see
Fig. 3) and was used as an indicator of a partitive unit fraction scheme (PUFS) and the other involved non-unit fractions (e.g.,
see Fig. 4) and was used as an indicator of a partitive fraction scheme (PFS). The two items were used separately to test for
students’ operational differences between situations involving unit and proper fractions (Hypothesis 1b). These two items
were combined to create a single indicator of partitive reasoning, resulting in a score ranging from 0 to 2. This measure was
used to test for a relationship between splitting and partitive (unit) fraction schemes (Hypothesis 3).
5.2.4. Reversible partitive fraction scheme (RPFS)
One item was used as an indicator of the existence of a reversible partitive fraction scheme (e.g., see Fig. 5). This measure
was used to test whether the existence of a splitting operation is a developmental prerequisite for a reversible partitive
fraction scheme (Hypothesis 2a).
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5.2.5. Iterative fraction scheme (IFS)
One item was used as an indicator of the existence of an iterative fraction scheme (e.g., see Fig. 6). This measure was
used to test whether the existence of a splitting operation is a developmental prerequisite for an iterative fraction scheme
(Hypothesis 2b).
5.3. Analysis
We conducted a chi-square test of association for each pair of items across the two versions of the test, in order to
examine possible performance bias associated with differences in the numbers and respective sizes of the fractions used
for corresponding fraction pairs. We then entered frequency of student scores into contingency tables (see Tables 4–8) and
analyzed them using appropriate measures of association for ordinally scaled variables.
Our hypotheses test both symmetrical and asymmetrical associations. For example, Hypothesis 3 (Table 1) reflects a
symmetrical association in that neither variable is hypothesized to precede the other or be a developmental prerequisite.
In this case, we are just interested in testing whether there is an association between splitting and the existence of a PFS.
The gamma statistic, G, is an appropriate index for testing symmetrical associations between two ordered variables (Siegel &
Castellan, 1988) and was used for Hypotheses 3. Gamma compares the number of order agreements to the number of order
disagreements, and has values from −1 to 1 with values closer to −1 and 1 indicating a stronger relationship. If the variables
are independent then gamma will be 0. However, the converse is not true: If gamma is 0, this does not imply that the variables
are independent. Visually, a direct relationship will manifest itself in the contingency table as frequencies concentrated along
the diagonal or concentrated in cells indicating order agreements.
Hypothesis 2b describes an asymmetrical situation in which splitting is hypothesized to precede or be a developmental
prerequisite to an IFS. In other words, one variable is an independent variable (e.g., iterative fraction scheme) and the other is
a dependent variable (e.g., splitting). Although splitting is hypothesized as a developmental prerequisite, its existence does
not necessarily lead to the existence of an IFS. Instead, given our hypothesis, the existence of an IFS suggests that a student
must first have had a splitting operation. Thus the variable associated with splitting is considered the dependent variable
and the variable associated with the IFS is considered the independent variable. Somer’s D is an appropriate statistic for
testing asymmetrical associations between two ordered variables (Siegel & Castellan, 1988) and was used for Hypotheses 2a,
2b, and 1b. Somer’s D is a measure comparing the number of agreements in order between two variables with the number
of disagreements in order. Its value ranges from −1 to 1 with values closer to −1 and 1 indicating a stronger asymmetrical
relationship. In the perfect relationship there would be no disagreements. Similar to gamma, visually this would be seen in
a contingency table as a staircase.
Finally, Hypothesis 1a will be analyzed using a Binomial Test. In this case, a test for no operational differences is concerned
with the distribution of students who did not get both items correct. In other words, if there are no operational differences
then students should get both items correct or both items incorrect, and the distribution of those students missing one item
should not reflect a bias toward either the unit or proper fraction items.
6. Results
6.1. Scheme hierarchy
The chi-square test of association for each item pair found no statistical difference in success across the nine item pairs,
which suggests that success on the items was not influenced by the context of number and respective size of the fractions
used for corresponding fraction pairs. Further, this finding provides evidence of the reliability of the specified item types in
measuring the underlying schemes.
In Table 3, we present descriptive statistics for each variable representing the different schemes by grade. We investigated
the scheme hierarchy by examining the overall percent correct for each variable representing the different schemes. In this
case, higher percentages were considered to be indicative of those schemes that are, in general, developmentally more stable
for the particular grade level students considered; that is, the scheme develops earlier in the hierarchy. Vice versa, schemes
with lower percent correct would represent those schemes that develop later. Based on the percentages, the schemes tend
Table 3
Percent correct for each of the items associated with the schemes by grade.
Scheme
Part–whole (2)
Partitive unit fraction (1)
Partitive fraction (1)
Reversible partitive fraction (1)
Iterative fraction (1)
Splitting (2)
Grade 5 (N = 44)
Grade 6 (N = 40)
M
SD
M
SD
.58
.52
.27
.14
.14
.34
.39
.51
.45
.35
.35
.40
.78
.70
.48
.18
.20
.36
.22
.46
.51
.38
.41
.45
Note: Number in parentheses represents the number of items used to calculate mean scores for scheme.
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A. Norton, J.L.M. Wilkins / Journal of Mathematical Behavior 28 (2009) 150–161
Table 4
Success in non-unit part–whole situations by success in unit part–whole situations.
Unit
Non-unit
0
1
Total
0
1
11
16
13
44
24
60
Total
27
57
84
Note: Binomial Test, p = .71.
to align as hypothesized, with the part–whole fraction scheme developing earlier, followed by partitive fraction schemes,
and the iterative and reversible partitive schemes developing later. Although the RPFS is hypothesized to develop prior to
the IFS, the data do not clearly differentiate those schemes.
6.2. Part–whole reasoning
We hypothesized that when working in part–whole situations no operational differences would exist between unit and
proper fractions (see Hypothesis 1a in Table 1). In other words, the PWFS develops for both situations at the same time. In
order to investigate the relationship between unit and non-unit fractions as part–whole fractions, we used a Binomial Test
to determine whether there was a bias toward one item type for those students who were incorrect for one of the items.
In essence, we tested whether the cell frequencies for the lower left and upper right cells of the contingency table were
statistically the same (see Table 4). The number of students missing the unit fraction item compared to the non-unit item
was statistically found to be the same (Binomial Test, p = .71) indicating that there is not sufficient evidence to reject the null
hypothesis that there are no operational differences between situations involving unit and proper fractions. Because most
students were successful with the part–whole items it would be good to replicate this investigation with a younger sample
of students in which an increased number of students would not have developed part–whole reasoning.
6.3. Partitive reasoning
We present frequencies for student success on the PUFS and PFS items in Table 5. We hypothesized that, for partitive
reasoning, there would be operational differences between situations involving unit and proper fractions, and further, we
would expect successful operations on unit fractions to precede operations on proper fractions (Hypothesis 1b). Consistent
with this hypothesis, we were interested in testing the asymmetrical association between PUFS and PFS. In this case, PFS was
considered the independent variable and PUFS was considered the dependent variable, and a direct relationship between the
variables was hypothesized. We found a statistically significant direct relationship between PUFS and PFS (Somer’s D = .37,
p < .001, one-tailed) indicating that, in general, for partitive reasoning there are operational differences between situations
involving unit and proper fractions; and moreover, in general a PUFS tends to develop prior to a general PFS.
6.4. Splitting and the reversible partitive fraction scheme
We present student splitting scores and success on the RPFS item in Table 6. Consistent with the hypothesis, we were
interested in testing the asymmetrical association between splitting and students success with items involving an RPFS. The
RPFS was considered the independent variable and splitting was considered the dependent variable, and we hypothesized
a direct relationship between the variables. Somer’s D was equal to .33 (p = .027, one-tailed) indicating a consistent direct
association between the existence of a RPFS and the pre-existence of a splitting operation. The magnitude and direction of
the index represents a trend that is consistent with the hypothesized relationship. It is evident from the percent correct that
developmentally this item is a difficult item for the particular sample of students in this study, suggesting that perhaps an
older sample of students might provide a more accurate picture of the developmental relationship.
Table 5
Success in non-unit partitive situations by success in unit partitive situations.
Unit
Non-unit
0
1
Total
0
1
28
25
5
26
33
51
Total
53
31
84
Note: D = .37 (p < .001, one-tailed).
A. Norton, J.L.M. Wilkins / Journal of Mathematical Behavior 28 (2009) 150–161
159
Table 6
Student success in situations involving a reversible partitive fraction scheme by existence of a
splitting operation.
Splitting
Reversible partitive
0
1
Total
0
1
2
42
14
15
4
3
6
46
17
21
Total
71
13
84
Note: D = .33 (p = .027, one-tailed).
Table 7
Student success in situations involving an iterative fraction scheme by existence of a splitting
operation.
Splitting
Iterative
0
1
Total
0
1
2
41
13
16
5
4
5
63
17
21
Total
70
14
84
Note: D = .23 (p = .071, one-tailed).
Table 8
Student success in situations involving a partitive fraction scheme by existence of a splitting operation.
Splitting
Partitive
0
1
2
Total
0
1
2
21
4
3
14
11
5
11
2
13
46
17
21
Total
28
30
26
84
Note: Partitive is combined unit and proper; G = .46 (p = .002, two-tailed).
6.5. Splitting and the iterative fraction scheme
We present frequencies for student splitting scores and success on the IFS item in Table 7. Given our hypothesis, we were
interested in testing the asymmetrical association between splitting and student success with items involving an IFS. In
this case, an IFS was considered the independent variable and splitting was considered the dependent variable and a direct
relationship between the variables was hypothesized. Somer’s D was equal to .23 (p = .071, one-tailed) indicating a small
direct association between the existence of an IFS and the pre-existence of a splitting operation. Although the test did not
reach statistical significance based on the strict criterion (i.e., p < .05), the magnitude and direction of the index is consistent
with the hypothesis and suggests that further investigation of the relationship could be fruitful. Again, similar to the RPFS,
an older sample of students might provide a better representation of the relationship.
6.6. Splitting and the partitive fraction scheme
We present student splitting scores and success on the combined partitive fraction scheme indicator in Table 8. We were
interested in investigating the relationship between splitting and PFS but had no a priori hypothesis about the relationship.
Thus, the gamma statistic, G, was appropriate to test this symmetrical relationship. We found a statistically significant direct
relationship between splitting and PFS (G = .46, p = .002, two-tailed). Further examination of the contingency table suggests
that, in general, a PFS develops prior to the development of a splitting operation. This finding is evidenced by the concentration
of occurrences in the cells in the upper right of the table as well as along the diagonal.
7. Conclusions
We consider the study preliminary—a necessary prelude to continued research—for the following reasons: First, no previous studies have quantitatively tested hypotheses resulting from teaching experiments; second, the limited power of the tests
render our conclusions tentative; finally, the results of this study inform the direction for more targeted quantitative studies
of students’ fraction schemes. The results from our study, however tentative, provide affirmation for several of the hypotheses
listed in Table 1 and inform the direction for more targeted teaching experiments. Before sharing conclusions—in terms of
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implications for research and teaching—we briefly discuss the nature of our approach to testing hypotheses, as compared to
that of teaching experiments.
Teaching experiments already contain a form of hypothesis testing through the use of superceding models (Steffe &
Thompson, 2000). These models develop in the same way as scientific theories in general. Existing models of students’ ways
of operating inform task design and predictions of student responses. To the degree that student responses fit predictions, the
models are confirmed. Otherwise, models are revised in order to explain newly observed activities, as well as the activities the
old models explained. Likewise, quantitative tests of models formed during teaching experiments contribute to hypothesis
building in the same way they do for scientific theories in general.
Quantitative tests, such as ours, provide at least three important contributions. First, they allow for assessments of much
larger samples of students. Second, they provide further confirmation of hypotheses developed in teaching experiments.
Finally, they provoke new studies. In our case, the larger sample provided indication of grade-level development; the confirmation of hypotheses also provided validation of the teaching experiment itself—a kind of triangulation in assessing students’
ways of operating; and our findings inform the need for further teaching experiments, especially regarding the relationship
between PFS and splitting.
7.1. Implications for research
We based our first hypothesis (Table 1) on theoretical distinctions between part–whole and partitive reasoning with
fractions. These distinctions arose from and are supported by several teaching experiments with children (Hunting, 1983;
Norton, 2008; Saenz-Ludlow, 1994; Steffe, 2004; Olive & Vomvoridi, 2006), and they are indicated by the two names given to
the partitive schemes. The present study affirms those distinctions by measuring significant differences between students’
performance on unit and non-unit partitive items, and by indicating no significant difference between students’ performance
on unit and non-unit part–whole items.
Our second hypothesis addressed a less-established theory concerning the development of splitting and higher-level
fraction schemes. Hackenberg (2007) revised Steffe’s initial hypothesis about splitting to claim that splitting is necessary,
but not sufficient, for the construction of an iterative fraction scheme, and even a reversible partitive fraction scheme. Tests
of this hypothesis yielded indices consistent with this hypothesis, and reached statistical significance for the RPFS. Overall,
our results support Hackenberg’s hypothesis but suggest the need for targeted testing among older students.
Based on the case of Josh—the student who had constructed splitting without even a partitive unit fraction scheme—we
further hypothesized (Hypothesis 3) that the partitive fraction scheme and the splitting operation could be constructed
independently. Tests of this hypothesis yielded our most surprising result: We found an unexpected, statistically significant,
relationship between splitting and the PFS, suggesting that the PFS generally develops before the splitting operation. So,
Josh’s case (Norton, 2008) may be more the exception than the rule. We interpret the result as an indication that students’
experiences in working with partitive fractions supports their operational development of splitting.
Our findings suggest further studies of students’ development, such as the need to investigate a possible causal relationship between children’s PFS and splitting. Thus, we enter a feedback loop between teaching experiments that generate
and internally test hypotheses, and quantitative studies that provide external validation of those hypotheses while also suggesting directions for future teaching experiments. Such feedback exemplifies the kind of productive relationship between
methodologies that might fulfill the promise of mathematics education research, to “generate hypotheses that could be
tested” (Kilpatrick, 2001, p. 424).
7.2. Implications for teaching and curricular design
Our research findings reinforce those from other studies (e.g., Mack, 2001) that suggest teachers should distinguish
between part–whole reasoning and partitive reasoning among their students. Furthermore, teachers should recognize that
progressing from partitive reasoning with unit fractions to partitive reasoning with non-unit fractions may pose a developmental hurdle on par with the more obvious one of developing improper fractions. Our findings also emphasize the
importance of the splitting operation in the latter development, while suggesting the value of utilizing students’ partitive
reasoning in supporting the development of splitting. Finally, our results provide some indication of developmental patterns
for fraction schemes, which might inform curricular design.
Percentages of successful performance by students on the items (Table 3) support the theoretical hierarchy of schemes
proposed by Steffe (2002) and Olive (1999) and affirmed by numerous teaching experiments, except we found no clear
distinction between student performance on the RPFS item and the IFS item. These percentages indicate that most students
have developed part–whole reasoning before entering fifth grade and that most have not constructed the reversible partitive
or iterative fraction schemes by sixth grade. Coordinating this finding with those already mentioned suggests that fifth grade
provides fertile ground for fostering the development of partitive reasoning and splitting, which would subsequently support
students’ development of the more advanced fraction schemes.
In general, our quantitative assessments of fraction schemes indicate both the operational norms and the conceptual
diversity of students by grade level (see Table 3). These assessments should inform curricular design and instructional practice. Curricula should be developed based on students’ existing ways of operating, rather than benchmarks set irrespective
of students’ mathematics. The developmental norms indicated by our assessments should inform curricular design by sug-
A. Norton, J.L.M. Wilkins / Journal of Mathematical Behavior 28 (2009) 150–161
161
gesting appropriate tasks in which students can engage, while provoking development toward the next level in the fraction
scheme hierarchy. In sixth grade, for example, our assessments imply that curricular materials should engage students in
iterating unit fractions to produce non-unit fractions: 70% of the sixths graders seemed to have developed PUFS, but only 48%
seemed to have developed PFS. At the same time, teachers should recognize that students in their classrooms will operate
at different levels. This is most apparent in our assessments of the fifth-grade students: it seems that many of them had yet
to develop PWFS, and about half had already developed PUFS. Such conceptual diversity could be challenging for teachers,
especially when procedural responses might mask operational differences between students. The tasks we used in our tests
might provide teachers with some guidance in assessing operational differences and setting educational goals for supporting
students’ operational development, but certainly we have more work to do in that area too.
Acknowledgments
We thank Angie Conn for her help in preparing this paper, Andrea McCloskey for her work in the underlying study, and
Amy Hackenberg and Erik Tillema for their thoughtful responses to earlier drafts.
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