Individual
differences in students’ knowing and learning about fractions:
Evidence from an indepth qualitative study
Maria Bempeni^{a,[*]}, Xenia Vamvakoussi^{b}
^{a} University
of Ioannina, Greece
^{b}University
of Ioannina, Greece
Article
received 22 November 2014 / revised 2 February 2015 /
accepted 11 February 2015 / available online 1 April 2015
Abstract
We present the results of an indepth qualitative
study that examined ninth graders’ conceptual and procedural
knowledge of fractions as well as their approach to
mathematics learning, in particular fraction learning. We
traced individual differences, even extreme, in the way that
students combine the two kinds of knowledge. We also provide
preliminary evidence indicating that students with strong
conceptual fraction knowledge adopt a deep approach to
mathematics learning (associated with the intention to
understand), whereas students with poor conceptual fraction
knowledge adopt a superficial approach (associated with the
intention to reproduce). These findings suggest that students
differ in the way they reason and learn about fractions in
systematic ways and could be used to inform future
quantitative studies.
Keywords: fractions; conceptual/procedural knowledge;
individual differences; learning approach
[1] Corresponding
author. Current address: Trempessinas 32, Athens, 12136,
Greece. Email address: mbempeni@gmail.com
1.
Theoretical background
The
distinction between procedural and conceptual knowledge has
elicited considerable research and discussion among
researchers in the fields of cognitivedevelopmental
psychology and mathematics education. Procedural knowledge is
defined as the ability to execute action sequences to solve
problems and is usually tied to specific problem types,
whereas conceptual knowledge is defined as knowledge of
concepts pertaining to a domain and related principles
(RittleJohnson and Schneider, in press).
The
relation between the two types of knowledge, particularly with
respect to their order of acquisition has elicited
considerable discussion, and there is evidence in favour of
contradictory views – in the words of RittleJohnson, Siegler,
and Alibali (2001), “conceptsfirst” and “proceduresfirst”
theories. According to conceptsfirst theories, children
develop (or are born with) conceptual knowledge in a domain
and then use this knowledge to select procedures for solving
problems. According to proceduresfirst theories, children
learn procedures for solving problems in a domain and later
extract domain concepts from repeated experience in solving
problems. In the area of mathematics education research, the
two types of knowledge (sometimes referred to by other terms)
are deemed practically inseparable (Gilmore &
PapadatouPastou, 2009; Hiebert & Wearne, 1996).
Nevertheless, it is assumed that procedural knowledge plays an
important role in the development of conceptual understanding
(Dubinsky, 1991; Gray & Tall, 1994; Sfard, 1991). More
specifically, it is suggested that mathematical concepts
develop out of related mathematical processes.
Such
accounts share two common background assumptions, namely that
there is a single developmental path and that this path is
independent of the particular domain considered.
RittleJohnson and Siegler (1998) challenged the latter
providing evidence that the order of acquisition many vary,
depending of the domain considered. In any case, the two types
of knowledge appear closely related. Thus, RittleJohnson et
al. (2001) argued for an iterative model, according to which
the two types of knowledge develop in a handoverhand process
and gains in one type of knowledge lead to improvements in the
other. This model is supported by empirical evidence and seems
to provide an adequate description of the relation between
conceptual and procedural knowledge (RittleJohnson &
Schneider, in press). Nevertheless, there is evidence that
sometimes the development of one type of knowledge does not
necessarily lead to the development of the other. Indeed, in
the area of fraction learning it has been shown that some
students have the ability to perform fraction procedures
without exhibiting comparable conceptual understanding or
without being able to explain why they are using these
procedures (Kerslake, 1986; Peck & Jencks, 1981). On the
other hand, Resnick (1982) presented evidence showing that
some children may exhibit conceptual understanding of
principles underlying subtraction without showing procedural
fluency.
Recently,
a different explanation for the contradictory findings has
been proposed, namely that not enough attention has been paid
to the individual differences in the way that students combine
the two types of knowledge (Gilmore & Bryant, 2008;
Gilmore & PapadatouPastou, 2009; Hallett, Bryant, &
Nunes, 2010; Hallett, Nunes, Bryant and Thrope, 2012). Hallett
and colleagues examined the procedural and conceptual fraction
knowledge of students at Grade 4 and 5 (2010) as well as at
Grade 6 and 8 (2012). They identified groups of students who
had strong (or weak) procedural as well as conceptual
knowledge. However, they also consistently traced two
substantial groups of students who demonstrated relative
strength with one form of knowledge and weakness with the
other, with differences between the two types of knowledge
becoming less salient with age. These findings challenge the
assumption that all children follow a uniform sequence in
gaining the two types of knowledge (see also Canobi, Reeve,
& Pattison, 2003).
In
their attempts to explain how such individual differences
arise, some researchers appealed to differences in students’
prior knowledge in the domain in question (Schneider, Rittle
Johnson, & Star, 2011); differences in students’ cognitive
profiles (Gilmore & Bryant, 2008; Hallett et al., 2012)
and differences in students’ educational experiences (Canobi,
2004; Gilmore & Bryant, 2008; Hallett et al., 2012).
However, empirical evidence in support of these assumptions is
so far lacking. For example, Schneider et al. (2011) found no
evidence supporting the hypothesis that the correlation
between the two kinds of knowledge might vary with different
levels of prior knowledge in the area of equation solving.
Hallett et al. (2012) investigated whether individual
differences in procedural and conceptual knowledge of
fractions can be explained by differences in students’ general
procedural and conceptual ability (measured by standardized
tests); they found no such evidence. In addition, Hallett et
al. (2012) examined the role of school experience, which they
measured as school attendance, that is, they investigated
whether attending different schools could explain the
individual differences in question; they found no such
relation.
Further
research, possibly with different measures, is necessary to
clarify the role of the above factors in individual
differences in procedural and conceptual knowledge, in
particular of fractions. We argue that a factor also worth
investigating is the individual student’s learning approach to
mathematics.
In
the literature there is an overarching distinction between the
deep approach to learning, associated with the individual’s
intention to understand; and the surface approach, associated
with the individual’s intention to reproduce. There are
several ways of characterizing each learning approach, mainly
adapted to tertiary education (Entwistle & McCune, 2004).
Stathopoulou and Vosniadou (2007) proposed a model, which was
tested with secondary students. They included three categories
for each learning approach, namely Goals, (study) Strategy
Use, and Awareness of Understanding. A deep approach to
learning involves goals of personal making of meaning, deep
study strategy use (e.g., integration of ideas), and high
awareness of understanding. A superficial approach involves
performance goals, superficial strategy use (e.g., rote
learning), and low awareness of understanding. Using these
categories, Stathopoulou and Vosniadou showed that students
with strong conceptual understanding of science concepts
adopted a deep approach to science learning, whereas students
with poor conceptual understanding adopted a superficial
approach. A similar association might be present in the case
of mathematics as well. Indeed, a student that follows a deep
learning approach to mathematics is more likely to pay
attention to the concepts and principles in the domain in
question, to be aware of conceptual difficulties, and to
invest the effort necessary to overcome them. On the contrary,
a student with a superficial approach is more likely to focus
on memorizing procedures, especially if procedures are
emphasized in instruction, as is often the case (Moss, 2005).
Before
we formulate our hypotheses, we turn to a methodological
issue, namely the difficulty to measure the two types of
knowledge validly and independently of each other (e.g.,
Gilmore & Bryant, 2006; Hiebert & Wearne, 1996;
RittleJohnson & Schneider, in press; Schneider &
Stern, 2010; Silver, 1986). The development of a procedural
test that would be conceptual free (and vice versa) is a
challenging task, since this type of tests may be person,
content and context sensitive (Haapasalo & Kadijevich,
2000; Schneider et al., 2011). Moreover, for tasks
administered in paperandpencil tests, it is often impossible
to decide how the student actually solved the task. For such
reasons, Hiebert and Wearne (1996) suggested that attention
should be also paid to students’ solution strategies (see also
Faulkenberry,
2013). A distinction between procedural
and conceptual strategies (Alsawaie, 2011; Clarke & Roche,
2009; Yang, Reys, & Reys, 2007) is relevant at this point:
Procedural strategies are related to rules and exact
computation algorithms learnt from instruction. Conceptual
strategies, on the other hand, are diverse, and tailored to
the specific problem at hand; they are mostly invented by
(some) students themselves that use them flexibly in order to
avoid lengthy computations as well as to deal with unfamiliar
problems (see also Smith, 1995).
In
this study, we examined ninth graders’ conceptual and
procedural fraction knowledge. Taking into account the
methodological issue mentioned above, we designed a
qualitative study in order to also monitor students’
strategies. Similarly to Hallett et al. (2010, 2012), we
hypothesized that there are individual differences in the way
students combine the two kinds of knowledge. We were
particularly interested in extreme cases, namely students with
strong conceptual knowledge and weak procedural knowledge, and
vice versa. Such cases are theoretically interesting, since
they are not compatible with the iterative model
(RittleJohnson et al., 2001). Moreover, tracing extreme cases
at grade 9 would indicate that individual differences may
persist, although the general tendency is for them to become
less salient with age (Hallett et al., 2012).
In
addition, we examined students’ learning approach to
mathematics learning, particularly fraction learning.
Following Stathopoulou and Vosniadou (2007), we explored
whether students with strong conceptual knowledge adopt a deep
learning approach to mathematics, whereas students with weak
conceptual knowledge adopt a superficial approach.
2. Methodology
2.1
Participants
The
participants were seven Greek students at grade nine (three
girls), from seven different schools in the area of Athens.
The selection of the participants was not random. First, based
on their school grades, all participants could be
characterized as medium level students in mathematics. Second,
they all had the same mathematics tutor, starting from the
last grades of the elementary school. Their tutor provided
information about their mathematical behaviour. Based on this
information, we had reasons to expect some variation in their
conceptual and procedural knowledge of fractions.
We
note that by grade seven Greek students are taught all the
material related to fractions as well as decimals, and are
introduced to the term “rational numbers”. We stress that at
the moment this study took place the mathematics curriculum as
well as the mathematics textbooks, were “traditional’, in the
sense that they emphasized general, computationintensive
procedures for dealing with fraction tasks (Smith, 1995).
Consider, for example, that mental calculations and estimation
strategies were not among the curricular goals. Based on
information provided by our participants’ tutor, who had
extensive knowledge about their homework assignments as well
as their assessment tests on a longterm basis, we had good
reasons to believe that instruction relied heavily on the
textbooks, at least with respect to what students were
expected to do.
2.2
Materials
We
used thirty fraction tasks grouped in four categories (see
Appendix A). Category A included five procedural tasks, that
is, tasks that for which a standard procedure was taught at
school: four tasks that examined operations with fractions
(Q.1.1Q.1.4); and one task that required conversion to an
equivalent fraction (Q.1.5).
Category
B, consisting of eight tasks, targeted on conceptual
knowledge. Four tasks involved fraction representations
(Q.1.6Q.1.9); one task required recognizing fraction as a
ratio (Q.1.10); one item focused in the role of the unit of
reference (Q.1.11); and two tasks targeted on the
understanding of the effect of multiplication and division
with fractions (Q.1.12, Q.1.13). There were no tasks similar
to Q.1.10Q.1.13 in the textbooks, either at the elementary,
or at the secondary level. On the other hand, the area model
for the representation of fractions was salient in the
elementary school textbooks, but unlike Q.1.8., the shape was
typically given, already equally partitioned; examples of
improper fraction representations were scarce (Q.1.9), and
there was no task similar to Q.1.7.
Category
C consisted of seven comparison (Q.1.14Q.1.17, Q.1.20Q.1.22)
and two ordering tasks (Q.1.18, Q1.19). Although these tasks
could be solved by standard methods taught at school, they
could also be solved by a variety of conceptual strategies.
Finally,
the tasks of the Category D required deep conceptual
understanding or the combination of conceptual understanding
and procedural fluency. More specifically, there were two
tasks regarding locating fractions on the numberline (Q.1.23,
Q.1.24); one problem that involved an intensive quantity and
required the comparison of ratios (Q.1.25); one task regarding
estimation of a fraction sum (Q.1.26); one task that required
substituting variables with nonnatural numbers (Q.1.27); one
task that tested the
use of the inverse relationship between addition and
subtraction, as well as between multiplication and division
with fractions (Q.1.28); and two tasks targeting the dense
ordering of rational numbers (Q.1.29, Q.1.30). There were no
tasks similar to Q.1.27Q.1.30 in the mathematics textbooks.
Locating fractions on the number line was presented at the
secondary level (Grade 7), albeit not particularly emphasized.
The
selection and categorization of the tasks was based on
relevant literature (e.g., Clarke & Roche, 2009; Hallett
et al., 2010, 2012; McIntosh, Reys, & Reys, 1992; Moss
& Case, 1999; Smith, 1995). We note that we included items
targeting students’ awareness of the differences between
natural and rational numbers (e.g., Q.1.12, Q.1.13, Q.1.29,
Q.1.30) which is considered an important aspect of conceptual
knowledge (Vamvakoussi & Vosniadou, 2010; McMullen,
Laakkonen, HannulaSormumen, & Lehtinen, 2014). We also
used a considerable number of tasks related to fraction
magnitude (e.g., Category D tasks, Q.1.23, Q.1.24) (for the
importance of accessing fraction magnitude in students’
developing knowledge see Siegler & Pyke, 2013). We stress,
however, that this categorization was tentative, since we also
looked into students’ strategies. This consideration is
particularly important for Category C tasks, but relevant for
all tasks.
In
addition, we developed twelve items so as to explore students’
learning approach (deep/superficial) to fraction and, more
generally, to mathematics learning (see Appendix B). The items
were presented as scenarios describing a situation that the
student had to react to.
2.3
Procedure
In
the first phase of the study each student was asked to solve
the fraction tasks, thinking aloud and explaining their
answers. No time limit was imposed. In the second phase three
participants were selected to participate in an indepth,
semistructured individual interview about their learning
approach to mathematics. Because this was a first attempt to
explore a potential relation between individual differences in
conceptual and procedural fraction knowledge and the
individual’s learning approach, we selected one student with
strong procedural, but weak conceptual knowledge; one student
with strong conceptual but weak procedural knowledge; and one
student who combined both procedural and conceptual knowledge.
These students were additionally asked to comment on the
responses of the first questionnaire (certainty about the
solution, awareness of their performance in the tasks). The
second interview took place about one week later. Each
interview lasted about one hour. All interviews were recorded
and transcribed.
2.4
Data Analysis
First,
we assessed the accuracy of students’ responses in all tasks.
Second, we examined the strategies used. We categorized a
strategy as procedural, if it was based on instructed rules
and procedures related to our research tasks. Based on
mathematics textbooks, as well as information by our
participants’ mathematics tutor, we categorized as procedural
strategies the standard algorithms for fraction operations;
and transformation strategies (Smith, 1995), namely converting
to equivalent fractions, similar fractions, decimals, or mixed
numbers. Transformation strategies are relevant to operations
as well as comparison, and they were overemphasized in the
textbooks. We also categorized as procedural the instructed
method for Q.1.25, namely the construction of a 2x2 table
placing the like quantities one below the other, and forming
and comparing the ratios. Regarding the placement of fractions
on the number line, the instructed method involved segmenting
the unit in the appropriate number of parts. Finally, given
the salience of the area model for the representation of
fractions, particularly in the elementary grades, we reasoned
that it had the status of definition for fractions. We thus
did not consider that students used a strategy, either
conceptual, or procedural in the related tasks (Q.1.6–Q.1.9).
We
categorized as conceptual the strategies that were not based
on instructed procedures. For comparison tasks, such
strategies involved, for example, the use or reference
numbers, such as the unit and one half; and also residual
thinking, that is, comparing the complementary fractions
(Alsawaie, 2011; Clarke & Roche, 2009; Smith, 2005; Yang
et al., 2007). In a more general fashion, we categorized as
conceptual strategies the ones that relied on estimation of
fraction magnitudes, on spontaneous use of representations,
and on spotting and employing the multiplicative relations
present in the task at hand (e.g., in Q.1.25).
We
categorized a strategy as conceptual/procedural if it involved
conceptual and procedural features, such as adjusting a
procedural strategy to deal with a novel task. A prominent
example was the use of a transformation strategy, namely
converting to equivalent fractions, as a first step to deal
with Q.1.30, combined with the idea that this process can be
repeated infinitely many times.
We
also note that in certain cases students provided immediate
responses that were not based on a specific strategy; rather,
they relied on a holistic understanding of the situation at
hand. This was the case mainly for tasks targeting the
differences between natural numbers and fractions (Q.1.12,
Q.1.13, Q.1.29, Q.1.30). For example, some students answered
immediately that there is no other number between 2/5 and 3/5,
directly drawing on their natural number knowledge. We
categorized the strategy of relying on natural number
knowledge as conceptual.
For
the second phase of the study, the categories (i.e., Goals,
Strategy Use, and Awareness of Understanding) and the related
indicators used by Stathopoulou and Vosniadou (2007) were our
starting point for the analysis. We reviewed all transcripts
and coded them when possible. We selected sentences as unit of
analysis, but in some cases we used paragraphs so as to obtain
a sense of the whole. We looked for utterances that included
keywords pertaining to the indicators of each category (e.g.,
remember, memorize, memory and similar expressions for the
indicator ‘‘rotelearning’’ as a superficial Strategy Use). We
placed the sentences in the coding categories according to the
initial indicators and developed new indicators when needed.
After coding, data that could not be coded were identified and
analyzed to determine if they represented a new category. One
new category emerged, namely Engagement Factors, consisting of
two subcategories: Preferred Tasks/Strategies
(conceptual/procedural), and also Motivation (intellectual
challenge/coping). In addition, we replaced the category
Awareness of Understanding with the more general category
Awareness with indicators pertaining to awareness of
understanding (high/low) as well as to awareness of the
effectiveness of one’s personal study strategies (high/low).
The categories are presented in Table 5.
3.
Results of the 1^{st} phase of the study
Tables
14 present how students performed in the tasks of Categories
AD, respectively; and the type of strategy (conceptual,
procedural, or a combination of both) they used in each task.
As
shown in Tables 14, students 1, 2, and 3 were rather
successful across all task categories. Students 4, 5, and 6
were successful in Categories A and C, but not in Categories B
and D. Student 7 failed in Category A, but was rather
successful in Categories B, C, D. We placed the students in
three profiles: a) ConceptualProcedural (Students 1, 2, and
3); b) Procedural (Students 4, 5, and 6); and c) Conceptual
(Student 7). In the following we present these profiles in
more detail.
3.1
Conceptual  Procedural Profile
The
conceptualprocedural students succeeded in all tasks of
Category A using procedural strategies, that is, standard
algorithms (Table 1). Student 1 and Student 3 (hereafter,
Kosmas) also succeeded in all tasks of Category B (Table 2).
All three students relied heavily on conceptual strategies
(reference numbers, residual thinking) to deal with the tasks
of Category C (Table 3). All three performed well in the tasks
of this category, with Kosmas responding correctly to all
tasks.
Table 1
Students’
Performance (Success, Failure) and Type of Strategy Used
(Conceptual, Procedural, Or ConceptualProcedural) in the
Tasks of Category A
Student 
Q.1.1 
Q.1.2 
Q.1.3 
Q.1.4 
Q.1.5 
Profile 
1 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
Conceptual/Procedural 
2 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 

3
(Kosmas) 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 

4 
S,
P 
S,
P 
S,
P 
S,
P 
F,
P 
Procedural 
5 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 

6
(Stella) 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 

7
(Filio) 
F,
P 
F,
P 
F,
P 
F,
P 
S,
P 
Conceptual 
Kosmas
was the only student who responded correctly to all tasks of
Category 4. In general, however, all three students performed
well in Category D tasks, showing a rather strong conceptual
understanding, combined with procedural fluency. A good
indicator of their conceptual understanding is their responses
to the density tasks (Q.1.29, Q.1.30), in particular to the
first that is the most challenging. Student 2 and Kosmas
provided an impressively sophisticated answer, stating
explicitly that there is no such number and explained that,
given any number, no matter how small, one can always find a
smaller one. Student 1, on the other hand, assumed that such a
number exists, thus typically his answer is incorrect;
however, he stated that this number cannot be found, not even
by a computer; and described it as “zero point zero, followed
by infinitely many zeroes, and one unit in the end”.
These
students’ tendency to prefer conceptual over procedural
strategies manifested itself in the tasks of Category D as
well. None of them applied the instructed method to solve
Q.1.25; instead, they focused on the relations between the
quantities involved. In the words of Student 2: “Stella’s milk
tastes sweeter, because George dissolved the double quantity
of chocolate in the triple quantity of milk”.
The
data presented in Tables 14 show that Kosmas was the only one
who succeeded in all tasks. Moreover, Kosmas’s responses were
more elaborated than his peers’ in terms of completeness as
well as of the explanations he provided. Consider, for
example, Q.1.26 that asked for the estimation of 7/15 and
5/12. All three students noticed that each addend was smaller
than 1/2 and concluded that the sum was smaller than the unit.
Kosmas, however, went farther to notice that “This sum equals
the unit minus 0.5/15+1/12. The missing part is close to 0.1;
more precisely, a bit bigger than 0.1”. He reached this close
estimate of the missing part mainly via mental calculations,
writing down some of the intermediate results.
Table 2
Students’
Performance (Success, Failure) and Type of Strategy Used
(Conceptual, Procedural, Or ConceptualProcedural) in the
Tasks of Category B
Student 
Q.1.6 
Q.1.7 
Q.1.8 
Q.1.9 
Q.1.10 
Q.1.11 
Q.1.12 
Q.1.13 
Profile 
1 
S 
S 
S 
S 
S 
S 
S,
C 
S,
C 
Conceptual/Procedural 
2 
S 
S 
S 
S 
S 
F 
S,
C 
S,
C 

3
(Kosmas) 
S 
S 
S 
S 
S 
S 
S,
C 
S,
C 

4 
S 
F 
F 
F 
F 
F 
F,
C 
F,
C 
Procedural 
5 
S 
F 
F 
F 
F 
F 
F,
C 
F,
C 

6
(Stella) 
F 
F 
F 
F 
F 
F 
F,
C 
F,
C 

7
(Filio) 
S 
S 
S 
S 
S 
S 
S,
C 
S,
C 
Conceptual 
Table 3
Students’
Performance (Success, Failure) and Type of Strategy Used
(Conceptual, Procedural, Or ConceptualProcedural) in the
Tasks of Category C
Student 
Q.1.14 
Q.1.15 
Q.1.16 
Q.1.17 
Q.1.18 
Q.1.19 
Q.1.20 
Q.1.21 
Q.1.22 
Profile 
1 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
F,
C 
S,
C 
S,
C 
Conceptual/Procedural 
2 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
F,
C/P 
S,
C 
S,
C 
S,
C 

3
(Kosmas) ssss) 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 

4 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
Procedural 
5 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 

6
(Stella) 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 
S,
P 

7
(Filio) 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
S,
C 
Conceptual 
Table 4
Students’
Performance (Success, Failure) and Type of Strategy Used
(Conceptual, Procedural, Or ConceptualProcedural) in the
Tasks of Category D
Student 
Q.1.23 
Q.1.24 
Q.1.25 
Q.1.26 
Q.1.27 
Q.1.28 
Q.1.29 
Q.1.30 
Profile 
1 
F,
C 
F,
C 
S,
C/P 
F,
C 
S,
C 
S,
C/P 
F,
C 
S,
C/P 
Conceptual/Procedural 
2 
S,
C/P 
S,
C/P 
S,
C 
S,
C 
S,
C 
F,
C 
S,
C 
S,
C/P 

3
(Kosmas) 
S,
C/P 
S,
C/P 
S,
C/P 
S,
C 
S,
C/P 
S,
C/P 
S,
C 
S,
C/P 

4 
F,
P 
F,
P 
F,
P 
F,
C 
F,
P 
F,
P 
F,
C 
F, C 
Procedural 
5 
F,
P 
F,
P 
F,
P 
F,
C 
F,
P 
F,
P 
F,
C 
F,
C 

6
(Stella) 
F,
P 
F,
P 
F,
C 
F,
C 
F,
P 
F,
P 
F,
C 
F,
C 

7
(Filio) 
S,
C 
S,
C 
S,
C/P 
S,
C 
S,
C 
F,
C/P 
F,
C 
S,
C 
Conceptual 
3.2
Procedural Profile
As
shown in Table 1, the students of this profile performed very
well in the tasks of Category A (Table 1). On the contrary,
their performance was very law in the tasks of Category B
(Table 2). In particular, Student 3 (hereafter, Stella) failed
in all the tasks of this category. She stated that “the
nominator shows how many pieces to take” to justify her answer
in Q.1.6, and she drew a circle and partitioned it in three
unequal parts in Q.1.8 (Figure 1). None of these students
exhibited any understanding of the fundamental principle that
the fractional parts of the unit should be equal, as also
evidenced by their performance in Q.1.7 (Table 2).
Figure 1. Stella’s response to Q.1.6, Q.1.8:
Representations for the Fractions 1/4 and 2/3, respectively.
(see pdf)
All
three students failed to represent the improper fraction 5/3
(Q.1.9). Figure 2 presents S5 and Stella’s attempts to deal
with this task. S4 gave no answer to the problem.
Figure
2. Procedural Profile:
Student 5 and Stella’s’ Attempt to Represent the Fraction 5/3.
(see pdf)
In
addition, all three students failed in Q.1.10, explaining that
the denominator shows how many pieces the pizza had, and the
nominator how many pieces were eaten. They also failed in
Q.1.11, since they did not consider that the units of
reference might be different. Moreover, they all insisted on
executing the calculations in Q.1.12 and Q.1.13. When they
were explicitly instructed not to do it, they came up with the
rule “multiplication makes bigger, whereas division makes
smaller”.
All
students of this profile were flawless in the tasks of
Category C, using only procedural strategies. They were,
however, very reluctant to try without using paper and pencil,
when they were asked to. In case they tried, their responses
reflected severe lack of understanding. For example, Stella
claimed that 123/220 is greater than 6/5 because the numbers
123 and 220 are greater than 6 and 5, respectively.
The
students of this profile failed in all tasks of Category D
(Table 4). Again, they relied heavily on procedural
strategies, in particular transformation strategies. For
example, they all converted fractions into decimals in Q.1.23
and Q.1.24. They also attempted to use this strategy or to
perform the calculation in the estimation task Q.1.26,
although they were specifically asked not to. Stella, in
particular, explicitly stated that it is impossible to solve
the task without converting to similar fractions or to
decimals first.
Students
4 and 5 applied the instructed method Q.1.25. However, they
were not able to interpret the result. Consider, for example,
the answer and the explanation provided by Student 5: “George’
s milk tastes sweeter, because his proportion 600/100=6 is
better than Stella’s 200/50=4”. On the other hand, Stella’s
answer indicated that she neglected the multiplicative
relations defining the relative quantities that are involved
in the situation: “The girl’s quantities are rather small
compared to the boy’s. So I believe that her milk tasted
sweeter”.
These
students’ responses to the tasks on dense ordering (Q.1.29,
Q.1.30) were immediate and reflected the idea that fractions
(or decimals, in case they had converted them) are discrete,
like the natural numbers. Stella stated that “there are no
other numbers between 2/5 and 3/5, because 3 comes right after
2”. According to
Stella, one was the smallest positive number, while Students 4
and 5 proposed 0.1.
3.3
Conceptual Profile
As
mentioned above, there was only one student placed in this
profile, namely Filio. As shown in Table 1, Filio failed in
all tasks of Category A, except for Q.1.5, since she was quite
competent with equivalent fractions (see also her solution in
Q1.25 below). On the contrary, she succeeded in all tasks of
Category B (Table 2). She was able to explain adequately her
responses. For example, to explain her disagreement with Maria
in Q.1.10, Filio said that “I don’t know how many pieces this
pizza had. Kostas could have eaten 3 pieces, only if the pizza
was cut in four”. Similarly, in Q.1.11, she exclaimed: “Where
are the pizzas? I need to see the pizzas. Are they the same or
not?” While dealing with Q.1.12 and Q.1.13, she explicitly
stated that the outcome is not necessarily bigger, just
because there is multiplication involved. She tried with
several numbers, and eventually came up with a generalization:
“when we multiply a number a by a fraction smaller than the
unit, the product is smaller than the number a”.
Filio
succeeded in all tasks of Category C (Table 3) using
consistently only conceptual strategies. Interestingly, she
also succeeded in most of the tasks of Category D (Table 4).
Her responses in Q.1.23, Q.1.24, were based on estimation of
the fraction magnitudes and a rough approximation of their
location on the numbers line. Unlike the students of the
ConceptualProcedural Profile, she didn’t attempt to find the
exact locations by partitioning the line segments. Quite
similar to these students, however, she focused on the
relations between the quantities in Q.1.25, employed a
transformation strategy, and came up with a solution that is
not taught at school: “The 50gr of chocolate powder that
Stella put in 200gr milk is half the quantity that George put
in 600gr. So I double the quantities 50/200 and I get 100/400.
Then, 100 in 400 means more chocolate powder in the milk than
100 in 600! So, Stella’s milk tastes sweeter.”
Similarly
to Kosmas, Filio explicitly stated that there are infinitely
many pairs whose product is 3 (Q.1.27). Moreover, she also
stated that there are infinitely many numbers between 2/5 and
3/5 (Q.1.30). Unlike all other participants, she justified her
answer using spontaneously a rather sophisticated
representation: “If we locate them on the numberline, there
is definitely a gap in between. In this gap, there are
infinitely many numbers”.
We
note that Filio explicitly expressed her discomfort with tasks
in which she could not avoid using procedures (e.g., Category
A tasks, Q.1.28). We also note that Filio was monitoring her
performance during the solution process. She explicitly
expressed doubt about responses that were actually incorrect;
she also revised certain answers herself. For example, when
solving Q.1.18, she initially answered that the fractions 3/4
and 6/7 are equal, because for both one fractional unit is
needed to complete the unit. She revised this answer after
locating the two fractions on the number line.
3.4
Conclusions
The
first phase of the study revealed three different student
profiles: The ConceptualProcedural Profile consisted of three
students with quite strong conceptual knowledge of fractions,
combined with procedural fluency. These students appeared to
prefer conceptual strategies over procedural strategies, when
this was possible. One of these students, namely Kosmas
(Student 3), was exceptionally strong: not only did he succeed
in all tasks, but he also gave the most complete and
elaborated answers.
The
Procedural Profile consisted of three students who were
capable of applying instructed procedures. This capability
allowed them to deal very successfully with the tasks that
could actually be solved by an instructed procedure. However,
these students failed in most tasks that required conceptual
knowledge, exhibiting lack of understanding for even the most
fundamental fraction ideas. Stella, in particular, failed in
the simplest conceptual tasks. These students relied heavily
on procedural strategies and avoided consistently to try
otherwise. When they did try, they typically failed.
Finally,
the Conceptual Profile consisted of one student, namely Filio
(Student 7). Filio consistently avoided applying procedures
throughout the interview, and she failed when she had to do
it. She nevertheless exhibited a firm understanding of
fundamental fraction ideas; and thus she managed to deal quite
successfully with many tasks by applying consistently
conceptual strategies.
Thus,
in line with recent discussions regarding the relation between
conceptual and procedural knowledge of fractions (e.g.,
Hallett et al., 2010, 2012), we found individual differences
in the way that students combine the two kinds of knowledge.
Moreover, we showed that these differences can be extreme –
consider, for example, Stella and Filio.
4.
Results of the 2nd phase of the study
Table
5 presents the categories that describe the Deep Learning
Approach and the Superficial Learning Approach to mathematics,
and their indicators. In the following we present excerpts
from transcribed interviews of Kosmas, Filio, and Stella, in
order to highlight the similarities and the differences in
their learning approaches to mathematics, along these
categories.
4.1
Goals
Kosmas
and Filio repeatedly referred to the importance of learning
with understanding in mathematics, which they both juxtaposed
with rote learning. For them, learning with understanding
meant personal making of meaning. This point is illustrated in
the following excerpts, in which Kosmas and Filio explain how
they would help a hypothetical younger student that is
challenged by the comparison of fractions:
“Perhaps
I
could try to explain fractions as I understand them. He has
to find a personal way of thinking though. He could study
the rules. In fact, there are two ways: In the case of
fractions, the first one is to memorize the rules and apply
them. For example, between two fractions with the same
numerator 3/5 and 3/7 the bigger is this one with the
smaller denominator. Alternatively, he would compare the two
fractions to the unit, that is, notice that 3/7 is closer to
1 than 3/5. There is a difference: In the second case you
have understood exactly what happens with fractionsthe
first is rote learning. You can reach a conclusion regarding
which of two fractions is bigger but you don’t understand
why. Personally, if I saw these two fractions, I would
compare the fractions to the unit so as to check the
validity of the rule.” (Kosmas, Q.2.11)
“I
would
help him understand the concept of fraction. But, you know,
everyone has their own way of thinking. Mathematics is not
rote learning, you have to put your mind to the work. […] I
could explain to him how to compare fractions based on the
rules, but if he wants be really able to compare fractions,
I think that he should understand the concept of fraction.
He must understand what fractions are and then he will do
well in fractions.” (Filio, Q.2.11)
Consider
also the following excerpts:
“The
most
important thing is to understand. Knowing the rules will
also help you, there is no doubt about it. But understanding
is the most important thing.” (Kosmas, Q.2.11)
“If
I
understand the meaning of what I do, then I can solve the
exercises.” (Filio, Q.2.2)
Table 5
Deep
vs. Superficial Learning Approach to Mathematics: Categories
and Indicators
Categories 
SubCategories 
Indicators 



Deep Approach 
Superficial
Approach 
Goals 

Understanding / Personal
making of meaning 
Focus on what is required /assessed at
school 
Study Strategies 

Combining theory and practice Systematic, longterm time investment 
Memorizing and Rehearsing More rehearsing 
Awareness
of 
Understanding 
High 
Low 
Effectiveness
of Own Study Strategies 
High 
Low 

Engagement
factors 
Task/Strategy
Preferences 
Conceptual 
Procedural 
Motivation

Intellectual challenge 
Coping 
On
the contrary, Stella repeatedly referred, explicitly or
implicitly, to the importance of complying with what is
assessed at school and appeared to focus exclusively on the
material taught at school. This is summarized nicely in the
following excerpt:
“What
I
would advise a younger student is to look at the exercises
solved at school, to focus on what is likely to be asked in
the exams, and to pay attention to what the teacher has
emphasized on.” (Stella, Q.2.1)
4.2
Study Strategies
Kosmas
and Filio both stressed that in order to study efficiently in
mathematics one needs to combine studying theory in depth and
extensive practice with exercises. They also expressed their
conviction that solving unfamiliar problems is important as a
study strategy as well as an indicator of understanding.
“You
have
to know the theory very well so as to understand
mathematics. If you only solve exercises, your competence is
very limited. One has to understand the theory in depth
before trying to solve exercises.” (Kosmas, Q.2.2)
“If
you
give me any problem and I can solve it, then it means I have
understood well.” (Kosmas, Q.2.9)
“One
should
understand the theory very well and practice a lot as well;
and solve exercises beyond the ones in the textbook.”
(Filio, Q.2.2)
In
contrast, Stella’s study strategies were limited to memorizing
and rehearsing:
“Studying
what
is needed for solving the exercises is pretty much
sufficient.” (Stella, Q.2.2)
“Studying
the
theory is good, because you have to know some theory to be
able to solve the exercises. But I think that it is better
to focus on exercises. Personally, I look at what we have
done at school, so as to remember how the exercises are
solved. I solve them again and again, and then I check if
they are correct.” (Stella, Q.2.3)
In
addition, unlike Stella, Kosmas and Filio appeared to value
the hypothetical students’ study strategies in Q.2.3, although
they both admitted that they don’t study like this.
“There
is
no doubt that this is the appropriate way of studying the
theory. […] This is how I should study but, unfortunately, I
don’t. That’s why I am not strong in mathematics.” (Kosmas,
Q.2.3)
“What
she
does is just fine. I don’t study like this, but I wish I
did.” (Filio, Q.2.3)
Moreover,
Kosmas and Filio referred to the importance of investing time
on mathematics studying. They distinguished between merely
spending time on studying, and studying systematically and in
depth.
“Mathematics
is
a course that has to do a lot with understanding, so you
have to study a lot. You have to start systematically in
mathematics from the beginning. Gaps are difficult to cover,
one needs to dedicate lots of time for both theory and
exercises.” (Kosmas, Q.2.2)
“I
was
preparing for a mathematics test and I spent lots of time,
but only during the last two days before the exam. I believe
that studying in depth results to success. If you study
superficially, you are not prepared appropriately. When we
talk about mathematics, you can’t prepare at the last
minute. If you do it, you will fail. It is impossible to
learn mathematics two days before the exams.” (Kosmas,
Q.2.4)
“It’s
not
only the time spent on studying, it’s also the way you
study. […] You may feel wellprepared for a test because you
have spent lots of time on solving exercises and fail in the
end. For example, what has happened to me is to face
unfamiliar problems in a test and fail. In that test, our
teacher tested whether we can think for ourselves, so he
examined us in different tasks than the ones we had solved
in the classroom. […] In order to succeed, you must have
understood the concepts and have practiced a lot.” (Filio,
Q.2.4)
Stella
also mentioned time as an important factor of success in
mathematics. For Stella, however, spending more time on
studying meant more rehearsing:
“[One
of
my classmates] is a very good at math. I believe that I am
good too, but not exactly at the same level. […] I think he
spends more hours studying than I do. […] Perhaps he solves
the exercises more times than I do.” (Stella, Q.2.5)
4.3
Awareness
4.3.1
Awareness of understanding
Kosmas
felt confident that he was able to assess his performance in
mathematical tasks in general. In fact, he was very accurate
in assessing his performance regarding the fraction tasks.
As
already mentioned, Filio was monitoring her performance in the
fraction tasks and corrected several mistakes herself in the
process. She also detected practically all the tasks that she
had answered incorrectly. In addition, she was aware that she
lacked procedural fluency:
“I
don’t
remember rules and procedures regarding fractions. However,
if someone reminded me of them, I could apply them.” (Filio,
Q.2.9)
Filio
acknowledged that fractions require “a lot of thinking” and
recalled that she was challenged by fractions at the
elementary school. Interestingly, she mentioned that she
managed to grasp the meaning of fractions, by connecting the
“school fractions” with the fractions she met at her music
courses. (Filio, Q.2.6)
Stella,
on the other hand, was confident that she had answered pretty
much all fraction tasks correctly. She appeared to detect her
mistake in Q.1.9, and she revised her answer. However, her
second attempt was again incorrect, since it was based on the
assumption that 5/3 is “a bit bigger than 0.5”. Nevertheless,
Stella believed that she had a firm understanding of fractions
in general:
“I
believe
that I understand everything about fractions. I never had
any difficulty with fractions. I found them very easy at the
elementary school, too. In general, I have never had any
problems with mathematics, as far as I can remember.”
(Stella, Q.2.8)
4.3.2
Awareness of the effectiveness of own study strategies
As
mentioned before, Kosmas and Filio both admitted that they did
not follow effective study strategies in mathematics, although
they recognized and appreciated them. In addition, they both
attributed the fact that they didn’t excel in mathematics to
their own way of studying.
“[One
my
classmates] is really strong in mathematics. I am at a
considerably lower level. This is because I don’t invest
enough time to study seriously in mathematics. [...] Often I
only solve the exercises that I have as homework and stop
there. [...] I could be as strong as my fellow student,
provided that I would be determined to study seriously
(Kosmas, Q.2.5)
“I
could
be as good as him [my fellow student]. How? The
oldfashioned way: Putting time and effort in studying as I
should.” (Filio, Q.2.5)
On
the
contrary, not once did Stella question her study strategies:
“Every
time
something went wrong, this happened because I was not so
careful. […] Or I thought I knew the material and that there
was no need to look at it again, but in fact I did not
remember it well. But in cases that I had studied as I
should, I believe that stress was responsible for my
failure.” (Stella, Q.2.4)
4.4
Engagement Factors
4.4.1
Task/Strategy Preferences
As
already mentioned, during the first phase of the study it was
more than obvious that Filio resented the tasks that she
perceived as procedural. For instance, she grew impatient with
Q.1.28 and quitted trying, exclaiming “I’ve had enough! I
spent too much time on this already. I can’t do it, I won’t do
it!”.
Kosmas,
on the other hand, never expressed any discomfort when he had
or chose to apply procedural strategies. In spite of this
important difference, these students both expressed their
preference for conceptual over procedural tasks, when they
were explicitly asked to chose:
“This
is
an easy choice! I would choose the second one, because I do
not like using methods. I do know, however, that the first
one is easier. At any moment you can open your book and
remember how it is solved.” (Kosmas, Q.2.10)
“Not
the
first one, for sure. It’s better to think something new,
instead of constantly doing the same. I find no meaning in
the application or rules and procedures. It is not
interesting. It is like rote learning, you know, a method to
solve exercises.” (Filio, Q.2.10)
Unlike
Kosmas and Filio, Stella showed a clear preference for
procedural strategies during the first phase of the studies;
and she explicitly stated that she would prefer the standard,
procedural task in Q.2.10.
4.4.2
Motivation
As
it may be evident by their responses to Q.2.10, Kosmas and
Filio were motivated by novel and challenging tasks. There
were clear such indications about Kosmas already in the first
phase of the study. For instance, when he first saw Q.1.29,
his immediate reaction was the following: “The smallest
positive number! This is a nice question, isn’t it?” In fact,
Kosmas was the only participant who chose to deal with the
most demanding and unfamiliar tasks first. When asked why, he
replied:
“I
like
challenging tasks much more. I find no interest in solving
exercises similar to the ones I have met before. The point
is to think of something new.”
Similarly,
Filio explained her choice of the unfamiliar task in Q.2.10 as
follows:
“When
you
try to solve an exercise and you finally discover that
something that you thought for yourself is correct, you get
a very nice feeling.” (Filio, Q.2.10).
On
the contrary, Stella’s main concern was to stay on the safe
side. As may be evident by her responses presented above, she
was mainly interested in good school performance. When she
explained why she would prefer the “standard”, procedural,
task in Q.2.10, she indicated that she was minding the
possible failure that guided her choice:
“I
would
choose the first one because it involves operations, which I
already know. So I would be sure that I can respond
correctly. The second one may involve something I don’t know
or never met before.”
Finally,
we note that for Kosmas and Filio learning with understanding,
besides being an important goal in mathematics learning, also
had a motivational aspect. Consider, for example, the
following excerpts:
“If
you
are to study mathematics, you should understand what you’re
doing. You should find meaning in what you do.” (Filio,
Q.2.2)
“[My
classmate
who excels in mathematics] has a special interest in math,
he loves it. He finds meaning in what he does. That’s why he
dedicates so many hours to studying.” (Filio, Q.2.5)
Both
students mentioned that they felt they understood mathematics
at the elementary level, but not so at the secondary level.
This was due to the fact that procedures are overemphasized
at the secondary level and this appeared to be demotivating
for them.
“Instruction
on
fractions is based on algorithms and students do not
understand the concept of fraction. For example, in the
addition of fractions we learn a priori that fractions must
have the same denominator without understanding why.
Something similar happens to mathematics teaching in
general. We should understand mathematics deeper and I think
that teachers must help us. How? I don’t know.” (Kosmas,
Q.2.7).
4.5
Conclusions
As
evidenced by their interview excerpts, Kosmas and Filio
exhibited similar features along the categories Goals, (Study)
Strategy use, Awareness, and Engagement Factors. Specifically,
they both appeared to value understanding and personal making
of meaning in mathematics learning; they were convinced that
the study of mathematics requires combining deep understanding
of theory as well as extensive practice; systematic and
longterm time investment was a key issue for them, as they
appeared aware that merely spending time on mathematics
studying is not enough to succeed in mathematics. Kosmas and
Filio showed high awareness of understanding in the domain of
fractions; they were also highly aware of their limitations as
students in mathematics. Finally, they showed a clear
preference for tasks that require conceptual understanding and
present an intellectual challenge, which appeared to be
motivating for them.
On
the contrary, Filio differed across all categories.
Specifically, Filio’s goal was to cope successfully with what
was required at school; her study strategies were limited to
memorizing rules and procedures as well as solving similar or
even the same exercises repeatedly; she preferred procedural
tasks because she was confident that she would succeed.
Finally, she showed practically no awareness of her (extremely
limited) conceptual understanding of fractions, and no
awareness of the limitations of her study strategies.
5.
Discussion
Our
results support the hypothesis that there are individual
differences in the way that students develop conceptual and
procedural knowledge of fractions. Similarly to Hallett et al.
(2010, 2012), we identified students who were strong with
respect to one type of knowledge, but weak with respect to the
other. Although the findings of Hallet et al. (2012) indicate
that such individual differences become less salient with age,
we showed that for some students they remain extreme, even at
Grade 9. Consider, for example, Stella and Filio: It appears
that for these students conceptual and procedural knowledge of
fractions have not developed in a handoverhand process, as
predicted by the iterative model (RittleJohnson et al.,
2001).
In
addition, our study provides preliminary evidence indicating
that the individual student’s learning approach to mathematics
is worth investigating in relation to individual differences
in conceptual and procedural knowledge. Similarly to
Stathopoulou and Vosniadou (2007), we found that Kosmas and
Filio, who exhibited strong conceptual knowledge of fractions,
both valued a deep approach to mathematics learning; whereas
Stella, who exhibited poor conceptual knowledge of fractions,
appeared to follow a superficial approach. This finding
cannot, of course, be generalized, given that it comes from a
qualitative study, with small sample. Moreover, it is based on
“extreme” cases of individuals. Nevertheless, this qualitative
evidence can inform the hypotheses and the design of future
quantitative studies.
Investigating
individual differences in conceptual and procedural knowledge
is important for understanding mathematical development
(Canobi, 2004; Hallett et al., 2010, 2012). From an
educational perspective, however, encouraging the symmetrical
development of the two kinds of knowledge is an important
goal, since they are both considered essential for students’
mathematical competence (RittleJohnson & Schneider, in
press). To this end, probably the first step would be to
foster learning environments in which both conceptual and
procedural knowledge are valued – and also assessed.
Keypoints
There are individual differences, even extreme, in
the way students combine conceptual and procedural knowledge of
fractions.
The individual student’s learning approach to
mathematics is a factor worth investigating with respect to
individual differences in conceptual and procedural fraction
knowledge.
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[*] Corresponding
author. Current address: Trempessinas 32, Athens, 12136,
Greece. Email address: mbempeni@gmail.com
Doi http://dx.doi.org/10.14786/flr.v3i1.132