A NEW THEORETICAL MODEL FOR UNDERSTANDING
FRACTIONS AT THE ELEMENTARY SCHOOL
Aristoklis A. Nicolaou and Demetra Pitta-Pantazi
University of Cyprus
In the present study we propose a theoretical model for understanding fractions at
the elementary school. The point of departure for building this model was the theory
proposed by Sierpinska, Nnadozie and Oktaç (2002) for theoretical thinking; we then
integrated elements from other approaches. Our theoretical model consists of six
factors: inductive reasoning, explanations, justifications, conception for the
magnitude of fractions, representations and connections with other concepts. The fit
of the model was tested in a sample of 344 fifth and sixth grade elementary school
students using MPLUS software and it was found to have very good fit with the
empirical data. The results suggest that the proposed model is much comprehensive
of the factors suggested for understanding fractions at the elementary school.
Keywords: theoretical model, understanding fractions.
INTRODUCTION
Sierpisnka et al. (2002) offered a comprehensive model of what constitutes
theoretical thinking in the context of more advanced mathematics. What we attempt
in this article is to offer a suggestion of what constitutes mathematical understanding
at the elementary school. For this reason, we seek of crucial factors that constitute
understanding a mathematical concept at the elementary school level. Once these
factors are identified, then we may emphasize them during teaching. For the purpose
of this study, we consider the term “understanding” to include conceptual
understanding as well as procedural understanding.
The concept of fractions was selected for testing the theoretical model of
understanding that we tried to formulate. The concept of fractions was selected since
fractions represent an ideal venue for investigating mathematical understanding in
elementary school (Niemi, 1996). Fraction is an important concept of elementary
school mathematics and studies show students’ difficulties to understand fractions
(Niemi, 1996: Lamon, 1999, 2001).
Various theoretical models have been proposed for the learning and understanding of
fractions (Kieren, 1976: Behr, Lesh, Post, & Silver, 1983: Marshall, 1993: Lamon,
1999, 2001). For example, Kieren (1976) proposed a model with five interrelated
subconstructs of fractions: part-whole, ratio, operator, quotient and measure. Later on
Behr et al. (1983) further developed Kieren’s ideas, proposing a theoretical model
linking the different interpretations of fractions to operations on fractions, fraction
equivalence and problem solving. The abovementioned theoretical frameworks and
others treat fractions' understanding mainly from the perspective of fractions'
subconstructs. This study considers fractions' understanding from a different lens
compared to the previous research giving emphasis to the factors that constitute
understanding of fractions.
THEORETICAL BACKGROUND AND AIMS
The proposed theoretical model
The proposed theoretical model sets off from Sierpinska’s et al. (2002) theory and
borrows elements from other cognitive, epistemological, and semiotics theories. We
consider that the factors that constitute understanding of fractions are: inductive
reasoning, definitions and mathematical explanations, justification and
argumentation, students’ conception about the magnitude of fractions, representations
and connections of fractions with other concepts. In the space below, we provide
description of each factor with special reference to its importance for fractions’
understanding.
1. Inductive reasoning
Inductive reasoning is defined as the process that permits the extraction of general
conclusions or rules from specific cases (Demetriou, Doise, & van Lieshout, 1988).
Consequently, via inductive reasoning generalization can take place, the role of
which is essential for understanding mathematics and the world. Especially for
understanding fractions at the elementary school level, where students’ thought is at
the concrete level and they use a plethora of manipulatives and materials, inductive
reasoning is essential, since it permits the extraction of general rules and conclusions
from specific examples. For example, the equal partitioning of a piece of chocolate,
of a surface or of a group of objects can lead to the identification of the concept of
fraction (de Koning, Hamers, Sijtsma, & Vermeer, 2002).
2. Definitions and mathematical explanations
The ability of communication and use of language is very important according to
NCTM Standards (NCTM, 2000) for understanding fractions. Niemi (1996) argued
that students’ ability to explain issues regarding fractions is an indicator of “deep
understanding of fractions”. Literature review did not reveal any information as
regards definitions of fractions by elementary school students. We will consider for
the purpose of the present study that elementary school students can “define”, but by
defining we do not mean the formal definition that is required by elder students. We
will consider that students can define fractions if they can express in their own words
what is the meaning of the fraction, e.g. what a fraction is. Students can define either
verbally or with the use of drawings, symbols and diagrams.
Apart from those tasks that require definitions (e.g. what is a fraction), students might
use more than one ways to explain other issues regarding fractions, e.g. when two
fractions are equivalent. Moreover, students can explain using examples, e.g. when
they are asked how many fractions are situated between 0 and 1, they can answer by
using examples.
3. Argumentation and justification
Reasoning and proof are considered important factors of what constitutes
understanding mathematical concepts (NCTM, 2000). According to NCTM (2000),
elementary school students should be able to develop and evaluate mathematical
arguments and proofs and select various types of reasoning and methods of proof.
Since formal proof cannot be the case for elementary school mathematics,
argumentation and justification could “substitute” what we call formal proof in the
upper level of education.
Argumentation can be defined as students’ ability to recognize the truth or the
falsehood of a mathematical statement (Duval, 1992/1993). At the same time,
students have to justify their answer. Argumentation and justification are very
important because they can reveal students’ conceptions about fractions, their
knowledge of fractions and their errors. For example, an argument referring to what
happens to the size of a fraction when increasing or decreasing the numerator and the
denominator could serve as an indicator of students’ understanding of fractions.
Students’ answers in this case will show if they understand that a fraction is a relation
and not see the nominator or the denominator as two different numbers having no
connection between them. We will consider that for the purpose of the present study
students can justify their answer by providing numerical examples or by means of a
“more general rule”. More sophisticated levels of justification would provide an
insight into students’ understanding of fractions.
4. Conception for the magnitude of fractions
Students’ sense for the magnitude of fractions is crucial for understanding, since in
1 smaller than , then
1 he/she probably
the case a student cannot perceive that is
4
3
does not understand the meaning of these fractional numbers and fractions in general.
It is very common for some students to consider the nominator and the denominator
of a fraction as two different numbers that do not constitute a unique entity, i.e. the
fraction. Conception about the magnitude of fractions is essential for comparing and
ordering. According to Clarke and Roche (2009), a number of researchers have
highlighted the importance of students being able to give meaning to the size of a
fraction and the many difficulties associated with doing so. An often-quoted result
from the National Assessment of Educational Progress (Carpenter, Kepner, Corbitt,
Lindquist, & Reys, 1980) supported the claim of student difficulty in understanding
the size of a fraction. Moreover, Post, Behr, and Lesh (1986) concluded that students’
sense of fraction magnitude is linked to better conceptual understanding of fractions.
5. Representations
Representations are very important for understanding the concept of fraction
(Newstead & Murray, 1998: Niemi, 1996: Gagatsis, Michaelidou, & Shiakalli, 2001).
In the context of teaching fractions, children come across a great variety of
representations. Lamon (2001) stresses the necessity of using different kinds of
representations in order to understand fractions.
Further to the recognitions and flexible use of various representational systems, a
basic goal of teaching and learning fractions should be the development of ability to
translate from one form of representation to another (Lesh, Post, & Behr, 1987).
Gagatsis et al. (2001) claimed that the ability to shift from one kind of representation
to another is especially important for fractions’ understanding.
For the purpose of developing the proposed model, we consider that a student
understands the concept of fraction if he/she is able to recognize fractions in iconic,
symbolic and verbal representations, if he/she is be able to translate to iconic,
symbolic and verbal representation and if he/she is able to construct drawings for
fractions.
6. Connections of fractions with other concepts
Students face serious difficulties in connecting the various forms of rational numbers
(Sweeny & Quinn, 2000). It is argued that students’ ability to convert from one kind
of rational number to the other is an indicator of understanding rational numbers
(Oppenheimer & Hunting, 1999). Moreover, it seems that students’ ability to see
fractions as division of the numerator by the denominator is an indicator of
understanding fractions (Newstead & Murray, 1998). Newstead and Murray (1998)
have reported students’ difficulties in doing so, thus their gap in understanding
fractions.
For the purpose of this study, we consider a student to be adequate in connections, if
he/she is able to link the concept of fraction with the concepts of decimal numbers,
percentage and division of integers.
Aim
The aim of this study was to develop and empirically test the theoretical model for
understanding fractions described in the previous section using a sample of students
at the upper level of elementary education (fifth and sixth grade students).
METHODOLOGY
Two tests were developed for measuring the six factors considered to constitute
understanding of fractions. The use of two tests was necessary, since students would
need a large amount of time to answer one test covering all the factors. The first test
comprised of 21 tasks and the second test of 44 tasks. The total of 65 tasks for
measuring all the factors were split to the two tests in such a way students needed
approximately the same time for answering the questions of each test. In Table 1
below, we give an example of an item for each of the six factors.
The first test included tasks for measuring inductive reasoning,
definitions/mathematical explanations and argumentation/justification. More
specifically tasks 1-7 (task 4 had two sub-tasks 4a, 4b) were used to measure
inductive reasoning and were similar to the tasks developed by Christou and
Papageorgiou (2007) for finding similarity, dissimilarity and integration attributes
and relations. Tasks 8-13 were used to measure definitions/mathematical
explanations and some of them were used by Niemi (1996) for measuring
explanations as regards the concept of fraction. In tasks 14-20 statements about
fractions were presented to students and they had to judge them as right or wrong and
explain their way of thinking. We considered that in this way students would provide
an argument about their choice and justify their choice to judge the statement as right
or wrong. Some of the tasks for argumentation/justification were similar to tasks
proposed by Lamon (1999) while discussing reasoning with fractions and some other
tasks were used by Niemi (1996).
Table 1: Examples of items for each of the six factors
Inductive reasoning
One of the following fractions differs from the
others. Find that fraction and circle it.
2
7
3
2
14
49
10
35
4
14
Definitions/mathematical explanations Imagine that your teacher asked you to explain
to one of your classmates what a fraction is. Use
as many different ways you can.
If I double both the numerator and the
Argumentation/Justification
denominator of a fraction, then the formed
fraction has twice value compared to the initial
one.
T
F
Justify your answer:
…………………………………………………...
………………………………………………….
Conception about the magnitude of Put the fractions 1 , 4 , 2 , 1 in order starting
2 3 3
4
fractions
from the smallest one.
Representations
Write a problem that could be solved by the
equation
1
1
+ =n
2
4
Connections of fractions with other Convert the following fractions to decimals.
concepts
1
2
3
1
a) =
b) =
c)
=
d)
=
4
5
10
20
In the second test tasks 1a-1f and task 2 were used to measure conception about the
magnitude of fractions. Tasks 1a-1f referred to fraction comparison, whereas in task 2
students had to put four fractions in the right order starting from the smallest one.
Similar tasks were used by Clarke and Roche (2009) for comparing and ordering
fractions. Tasks 3, 5 and 6 were previously used by Niemi (1996) and involved
recognizing fractions in iconic form. Tasks 4, 10, 12 and 13 referred to writing
problems that have a fraction as an answer, from an equation or on the basis of a
drawing (translating to verbal representation). Tasks 7, 11, 15 and 17 asked students
to construct their own drawings to show a fraction, for an equation of adding
fractions and for two problems involving fractions. Tasks 8a-8f asked students to
select the right fraction that could be represented by pictures (translation to symbolic
form, similar tasks were used by Niemi, 1996). In tasks 9a-9c number lines were
presented to students and they had to select the right fraction for each. In tasks 14 and
16 students had to solve two problems of addition and multiplication of fractions and
they had to write the equation for each. Finally, in tasks 18a-18f students were asked
to convert fractions to decimals, in 19a-19f they had to convert fractions to
percentages, while tasks 20a-20c and 21 were about the relation of fractions with the
division of integers.
The two tests were administered to 344 fifth and sixth grade students (119 fifth grade
and 225 sixth grade) from 11 different schools in Cyprus (both urban and rural areas
were represented). The time period for administration was the end of the school year,
so that both fifth and sixth grade students had covered the notions included in the
model. Both the tests were administered by the classroom teachers who were
requested to provide no further clarification and ask students to work on their own;
students had sixty minutes to work on each test.
The answers were coded on the basis of a coding scheme (due to lack of space we
cannot provide more details). Some tasks were marked either with 0 or 1 (categorical
variables), whereas in other tasks the score could vary from 0 to 1.
For testing the fit of the proposed model MPLUS software was used with WLSMV
estimator, since this kind of estimator is the most appropriate for categorical variables
(Muthén & Muthén, 2004). More than one fit indices were used to evaluate the extent
to which the data fit the theoretical model under investigation. More specifically, the
fit indices and their optimal values were: (a) the ratio of chi-square to its degrees of
freedom, which should be less than 1.96, since a significant chi-square indicates lack
of satisfactory model fit, (b) the Comparative Fit Index (CFI), the values of which
should be equal to or larger than 0.90, and (c) the Root Mean Square Error of
Approximation (RMSEA), with acceptable values less than or equal to 0.06 (Muthén
& Muthén, 2004).
RESULTS
After subsequent model tests and the withdrawal of eight tasks (the three of the tasks
referred to number line), the model shown in Figure 1 proved to have very good fit to
the data (x2=338.478, df =198, x2/df =1.71, CFI = 0.971, and RMSEA = 0.045)
From Figure 1, we verify that the factors we consider to constitute understanding of
fractions do so very well (the fit indices are very good).Three of the factors and more
specifically inductive reasoning, definitions/mathematical explanations and
argumentation/justification seem to constitute a second order factor which contributes
to understanding fractions. We will call this second order factor as “reasoning and
informal proof” at the elementary level, since all three factors are related to
“reasoning and informal proof” at the elementary school. More specifically,
argumentation and justification refer to reasoning with regard to mathematical
statements, definitions and mathematical explanations are required for reasoning and
informal proof, and inductive reasoning is necessary for reasoning and informal proof
at the elementary school since students start from specific examples to arrive at a
general rule, thus they can provide reasoning about how they were lead to the
generalization and a kind of informal proof. Figure 1 also confirms that verbal,
symbolic, iconic representations and students’ ability to construct their own drawings
for fractions constitute the factor “representations” which in turn constitutes
understanding fractions. In the same manner connections of fractions with decimals,
percentages and division constitute the factor “connections of fractions with other
concepts” which also constitutes understanding of fractions. Figure 1 shows that
representations have the highest contribution towards understanding fractions,
followed by the conception about the magnitude of fractions, connections of fractions
with other concepts and “reasoning and informal proof”. The coefficients could serve
as an indicator of the importance of each factor for understanding fractions.
Figure 1: The proposed model for understanding fractions
Definitions/
Mathematical
explanations
Inductive
Reasoning
0.856
Argumentation
/Justification
0.875
0.960
Reasoning and
informal proof
0.729
0.843
UNDERSTANDING
FRACTIONS
Connections of
fractions with other
concepts
0.934
0.938
Representations
0.701
0.959
0.978
0.965
Decimals
0.885
Conception about
the magnitude of
fractions
Division
Construct
a drawing
0.858
Verbal
0.873
Percentages
Symbolic
Iconic
DISCUSSION
The results of the statistical analysis confirmed our theoretical model and suggested
that the six factors constitute understanding of fractions. Moreover, three of these
factors constitute a second order factor which we call “reasoning and informal proof”
at the elementary school. Therefore, we claim that for a student to understand
fractions he/she should be able to engage to “reasoning and informal proof” as
regards fractions, he/she should possess a conception about the magnitude of
fractions, he/she must be fluent in representations and the translation from one kind
of representation to the other and he/she must be able to connect fractions with other
concepts. From a teaching perspective, it is suggested that with appropriate teaching
of the factors, students’ understanding of fractions can be improved. Therefore, a
teaching intervention can be carried out aiming at improving students’ ability in the
factors mentioned above.
The contribution of the factors was high (all loadings were greater than 0.7) showing
that all factors had a considerable contribution towards understanding fractions.
However, the contribution of representations was the highest (0.938), showing the
great importance of this factor for understanding fractions, as other researchers have
also claimed (Lamon, 2001; Lesh et al., 1987). The second factor in importance was
conception about the magnitude of fractions with also very high loading, followed by
connections of fractions with other concepts with a little bit less contribution.
Therefore, both factors are considered much important for understanding fractions.
“Reasoning and informal proof” had the lowest contribution among the factors but it
was high enough to stress the importance of this factor as well.
The findings of the present study are also important for assessment purposes as we
have developed validated tests for assessing fraction understanding. Niemi (1996) in
a similar way developed instruments for assessing conceptual understanding of
fractions. Nevertheless, Niemi (1996) did not confirm the fit of a model with factors
constituting understanding; he hypothesized that representations, mathematical
explanations, problem solving and justification of solutions are indicators of
conceptual understanding of fractions and he then assessed conceptual understanding
by measuring the aforementioned dimensions.
In a similar way, the present study provides a way for assessing representations and
connections of fractions with other concepts. As shown in Fig. 1, translation to
verbal, iconic, and symbolic representations and the ability to construct drawings to
show fractions comprise representations. In a similar manner, Panaoura, Gagatsis,
Deliyianni, and Elia (2009) developed a model for understanding fraction addition.
As regards connections of fractions with other concepts, it can be claimed that for the
elementary level, this can be assessed by asking students to convert fractions to
decimals and percentages and by the relation of fractions with the division of
integers.
The results indicated that inductive reasoning, definitions/mathematical explanations
and argumentation/justification are highly correlated forming a second order factor.
The relation of definitions/mathematical explanations and argumentation/justification
was expected since terms such as explanations and justification are used
interchangeably for reasoning at the elementary school and a relation of the two has
been reported from other studies (Niemi, 1996). Explanations and justification can
also provide a kind of “informal proof” at the elementary school, but by this we do
not refer to formal proof required at higher levels of education. The relation of
inductive reasoning to the other two factors can be explained by the fact that
inductive reasoning is also necessary for reasoning at the elementary school.
The importance of the present study is situated in that it proposes a new theoretical
framework with factors that constitute understanding of fractions. We believe that we
have “decomposed” understanding of fractions in factors that can be directly
measured. However, it must be stressed that the proposed theoretical model lays in
the cognitive, the epistemological and the semiotics domains, and in this manner
social or affective factors that may influence understanding of fractions are neglected.
Although the sample is high enough to reach solid conclusions, in future the proposed
theoretical model could be tested again, enhancing the validity and the reliability of
the results. Moreover, from a teaching perspective, an intervention could take place
for improving students’ ability in the factors found to constitute understanding of
fractions. Finally, the proposed theoretical model can be tested for other concepts
beyond fractions.
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