Learning and Instruction 14 (2004) 503–518
www.elsevier.com/locate/learninstruc
The development of students’ understanding
of the numerical value of fractions
Stamatia Stafylidou, Stella Vosniadou Cognitive Science Laboratory, Department of Philosophy and History of Science,
National and Kapodistrian University of Athens, Panepistimiopolis 15771, Ilissia, Athens, Greece
Abstract
An experiment is reported that investigated the development of students’ understanding of
the numerical value of fractions. A total of 200 students ranging in age from 10 to 16 years
were tested using a questionnaire that required them to decide on the smallest/biggest fraction, to order a set of given fractions and to justify their responses. Students’ responses were
grouped in categories that revealed three main explanatory frameworks within which fractions seem to be interpreted. The first explanatory framework, emerging directly from the
initial theory of natural numbers, is that fraction consists of two independent numbers. The
second considers fractions as parts of a whole. Only in the third explanatory framework,
students were able to understand the relation between numerator and denominator and to
consider that fractions can be smaller, equal or even bigger than the unit.
# 2004 Elsevier Ltd. All rights reserved.
1. Introduction
The purpose of the paper is to investigate the development of students’ understanding of the numerical value of fractions and to evaluate the usefulness of
approaching this question from a conceptual change point of view. The conceptual
change theoretical framework that will be adopted in the present study is the one
developed by Vosniadou (1994, 2001, 2002) in her attempts to explain how
students develop science concepts. The key definitional elements of this theoretical
framework to conceptual change are the following:
Corresponding author. Tel.: +30-210-7275506; fax: +30-210-7275504.
E-mail address: svosniad@compulink.gr (S. Vosniadou).
0959-4752/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.learninstruc.2004.06.015
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(a) The knowledge acquisition process is not always a process of enriching existing
conceptual structures. Sometimes the acquisition of new information requires
the radical reorganization of what is already known.
(b) Learning that requires the reorganization of existing knowledge structures is
more difficult and time consuming than learning that can be accomplished
through enrichment. Moreover, it is likely that in the process of reorganization
students will create misconceptions.
(c) Many misconceptions are synthetic models that reveal students’ attempts to
assimilate the new information to their existing knowledge base.
In the case of astronomy, Vosniadou and her colleagues (Vosniadou, 1994;
Vosniadou & Brewer, 1992, 1994) have argued that the information that the earth
is a sphere rotating around its axis and revolving around the sun in a heliocentric
solar system is very different from young children’s initial understanding of the
earth as a flat motionless, physical object supported by ground or water and located in the center of the universe. Cross-cultural developmental studies of children’s
representations of the earth reveal several synthetic models of the earth, such as the
model that the earth is round like disc, or that it is a hollow sphere with people living inside it on flat ground. These models reveal children’s gradual attempts to
reconcile the scientific information that the earth is a sphere with their initial model
of a flat, supported and motionless earth.
We believe that the conceptual change approach has implications for understanding the development of the concept of fraction. Many researchers coming
either from developmental psychology or mathematics education have noticed that
at least some of the difficulties children have in understanding fractions could be
explained to result from a conflict between the new information and their prior
knowledge (e.g. Carpenter, 1988; De Corte, Greer, & Verschaffel, 1996; De Corte,
Verschaffel, & Pauwels, 1990; Fischbein, Deri, Nello, & Marino, 1985; Greeno,
1991; Lehtinen, Merenluoto, & Kasanen, 1997; Resnick & Ford, 1981; Resnick
et al., 1989; Resnick and Omanson, 1987). In the specific area of fractions, Chi and
Slotta (1994) have argued that fractions represent a case of a concept the acquisition of which requires ontological change. This proposal has become more
explicit by Hartnett and Gelman (1998). The latter, following earlier research
(Behr, Harel, Post, & Lesh, 1992; Fischbein et al., 1985; Gallistel & Gelman, 1992;
Greer, 1992), show that one of the reasons why the mathematical notion of fraction is systematically misrepresented is because it is not consistent with the counting principles that apply to natural numbers. They conclude that early knowledge
about number may in fact serve as a barrier to learning about fractions, given children’s constructivist tendency to distort the new information (about fraction) to fit
their counting-based number theory (Hartnett and Gelman, 1998).
The purpose of the present experiment was to further investigate this proposal,
looking at students much older than the ones tested by Hartnett and Gelman
(1998), in an effort to understand how students gradually overcome the barriers
imposed by their knowledge of natural numbers and develop the concept of fraction under the influence of systematic instruction.
S. Stafylidou, S. Vosniadou / Learning and Instruction 14 (2004) 503–518
505
Table 1
Differences between natural numbers and fractions
Numerical value
Natural number
Fraction
Symbolic representation
One number (presupposition
of discreteness)
Supported by the natural numbers’
sequence (counting on)
Existence of a successor
or a preceding number
No number between two
different numbers
The unit is the smallest number
Two numbers and a line
(presupposition of density)
Not supported by the
natural numbers’ sequence
There is no unique successor
or a unique preceding number
Infinity
Ordering
Relationship to the unit
Operations
Addition–subtraction
Multiplication
Division
Supported by the natural
numbers’ sequence
Multiplication makes
the number bigger
Division makes the number smaller
No unique smallest number
Not supported by the natural
numbers’ sequence
Multiplication makes the
number either bigger or smaller
Division makes the number
either smaller or bigger
Building on the work of previous researchers (Behr et al., 1992; Fischbein et al.,
1985; Hartnett & Gelman, 1998; Ohlsson, 1988), we tried to specify exactly how
the explanatory framework for natural numbers, well developed by the age of 10
(which represented the younger students in our sample) differs from the explanatory framework for fraction. Table 1 shows some of these key differences.
First, fractions differ in their symbolic representation from natural numbers (one
number versus two cardinal numbers separated by a line). Second, with respect to
ordering, fractions differ from natural numbers in that one cannot use counting-based
algorithms for ordering them. Fractions do not have unique successors; there are infinitely many numbers between any two fractions. Third, with respect to the unit, while
the unit is the smallest natural number, there is no ‘‘smallest’’ rational number. Finally,
as Table 1 shows, operations (addition, subtraction, multiplication and division) on
natural numbers differ in important ways from operations performed on fractions. The
investigation of these differences is not among the purposes of the present study.
The hypothesis of the experiment was that the presuppositions of the explanatory
framework of natural numbers will inhibit the acquisition of fractions, causing systematic misconceptions. These misconceptions can be interpreted as attempts on the part of
the students to assimilate the new information into their existing conceptual structures.
2. Method
2.1. Participants
Two hundred students participated in this study: 40 fifth graders (mean age
10 years and 7 months); 40 sixth graders (mean age 11 years and 5 months); 40
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seventh graders (mean age 12 years and 4 months); 40 eighth graders (mean age 13
years and 8 months) and 40 first class of Lyceum students (mean age 15 years and
6 months). The students were approximately equally divided between boys and
girls and came from middle class public schools in the area of Thessaloniki.
2.2. Questionnaire
The questionnaire was designed to examine the development of children’s ideas
about the numerical value of a fraction. The questions were divided in two sets. In
the first set, students were asked to write the smallest and biggest fraction they
could think of and justify their answers (Table 2). In the second set, they were
asked to order and compare fractions of different kinds (Table 4).
2.3. Procedure
The questionnaire was administered to groups of 10 students by one of the
experimenters in their school. The students completed the questionnaire in the
presence of the experimenter and were free to discuss with her any clarification
problems they may have had in answering the questions.
3. Results
3.1. Set I: smallest/biggest fraction
Students’ responses in the first set of questions were grouped into six categories
(shown in Table 2) based on the examples they gave, the justifications of their
responses, and whether they made reference to infinity.
The students who were grouped in Category 1, usually focused on the size of the
numerator to compare the given fractions. They seemed to believe that the value of
a fraction increases when the numbers that comprise it (either the numerator or the
denominator) increase. They gave no indication that the fractions they gave as the
biggest (especially in the case where the numerator was equal to the denominator)
equalled the whole unit. Examples can be found in Table 2.
The main idea behind students’ responses classified in Category 2, was that the
value of a fraction increases when the numbers that comprise it (either the numerator
or the denominator) decrease. For example, according to a fifth grader, ‘‘the smallest fraction is 90/950, because it has big numbers and the bigger the numbers it
has the smaller it is’’, while the biggest fraction is ‘‘1/2, because the smaller the
numbers a fraction has the bigger it is’’.
Category 3 was different from the Categories 1 and 2 in three basic respects.
First, the biggest fractions the children gave as examples did not consist of that
impressively big or small numbers. Usually the biggest fraction had just a bigger
numerator than the smallest fraction they gave as examples (i.e. 9/10 compared to
1/10). Second, students’ explanations made reference to part/whole relationships.
The most common interpretation was that the fraction is part of a unit (i.e. the
pieces of a pie or a cake). Third, the smallest fraction they chose always had a
The fraction 1/2
Because from the two parts we got only one
5. There is no biggest fraction (infinity only for big
fractions)
7. Mixed—could not be categorized
Fractions as 1/0, or 1 000 000 000/
+1 000 000 000, or ‘‘small as a lice’’
I don’t know
Fractions such as 1 000 000 000/+1 000 000 000,
or dinosaurs because they are old and big
I don’t know
6. There is no smallest/ biggest fraction (infinity for There is no unique smallest fraction or fractions There is no biggest fraction or fractions such as m/
all fractions)
as 1/m, or 1/1, or 1/10 000 000 000 000 000
1, or 1/1, or 10 000 000 000 000 000/1
Because the numbers are endless
Because its value becomes greater as the numerator
increases and the denominator decreases
Fractions such as 1000/100
I don’t believe that this is the biggest fraction
Fractions such as 1 000 000 000/1, or
999 999 999 999 999 999/1
Because the numerator is very big and the denominator is very small
1/1 000 000 000 000 000 000 or 1/
9 999 999 999 999 999
Because the numerator is very small and the
denominator is very big
The fraction 1/2, or 1/1
Because the smaller the numbers are the bigger the
fraction is
4. The value of the fraction increases as the size of
the numerator becomes bigger than the size of the
denominator
Fractions as 90/950, or 1 000 000/1 000 000
Because it has big numbers
2. The value of a fraction increases when the numbers that comprise it decrease
Fractions such as 95/100, or 1230/2000, or 100/
500, or 10 000/100 000 000, or 1000/1000
Because it is the biggest fraction I can think of
Fractions such as 1/2, or 99/100, or the fractions
1/1, or 8/8, or 100/100. Because the denominator
is divided in two big pieces
The fraction 1/1, or fractions like 1/2, 2/3, 1/6
Because it has small numbers
1. The value of a fraction increases when the numbers that comprise it increase
Question 2a: Write the biggest fraction that you
can think of
Question 2b: Why do you think this is the biggest
fraction?
3. The value of the fraction increases as the size of Fractions as 1/8 000 000, or 1/1000, or 1/100
the numerator approximates the size of the denomi- Because we divided it in and we only took one
nator
Question 1a: Write the smallest fraction that you
can think of
Question 1b: Why do you think this is the smallest fraction?
Categories of responses
Table 2
Numerical value of fraction—Set I: smallest/biggest fraction: categories of responses and examples
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numerator smaller than the denominator, while the biggest fraction always had a
numerator smaller or equal to the denominator (i.e. there was no use of improper
fractions). The main idea behind Category 3 is that a fraction is a part of a whole
and that the value of a fraction increases as the size of the numerator approximates
the size of the denominator.
Categories 4, 5 and 6 differed from the previous in two basic respects. First, the
biggest fraction was an improper fraction. Second, they either used the unit as a
point of reference, or interpreted the fraction as the quotient of the numerator divided by the denominator. More specifically, in Category 4, the students for the first
time conceptualized the biggest fraction to be the fraction whose numerator is a
bigger number than the denominator. In Categories 5 and 6, we had the appearance of the notion of infinity, first with respect to the biggest fraction (Category 5)
and then also with respect to the smallest fraction (Category 6). Category 7
included all the mixed responses that could not be categorized in an internally consistent category.
Table 3 presents the frequency/percentage of students distributed in the various
categories described above. As expected, responses that belonged to Category 1
decreased with grade while responses that mentioned infinity (Categories 5 and 6)
increased with age. The Kruskal–Wallis test showed grade to be a statistically significant factor (v2 ð4Þ ¼ 20:21, p < 0:001). Comparing the means between the five
age groups of the participants, it was obvious that the older students were grouped
in categories that were closest to the scientific definition of fraction. Finally the
results of a Mann–Whitney test showed no gender difference (z ¼ 2:493,
p < 0:05).
3.2. Set II: comparison of fractions
Using similar criteria as those used in Set I students’, responses in Set II were
grouped into five categories, shown in Table 4. Category 1 was divided in three
subcategories. What was common in these three subcategories was that the students ordered the fractions either on the basis of the size of the numerator only or
of the denominator only.
Subcategory 1a included responses in which the ordering of fractions was made
usually on the basis of the size of the numerator. The underlying common belief
was that when the numerator of a fraction increases, the fraction itself increases
(ignoring whether the fractions in comparison had different denominators). These
students always considered the unit as the smallest of all fractions, in the third
question.
The students who ordered the fractions also on the basis of the size of the
numerator but considered the unit as the biggest of all fractions, in the third question, were grouped in Subcategory 1b. In this category, the fraction was interpreted
as two individual natural numbers whose value increases as they increase but never
exceeds the unit.
Category 2 included the students who, contrary to Category 1, believed that as
long as the numbers forming a fraction decrease, the value of the fraction increases
13 (32.5%)
4 (10%)
13 (32.5%)
6 (15%)
1 (2.5%)
2 (5%)
1 (2.5%)
40 (100%)
13 (32.5%)
3 (7.5%)
7 (17.5%)
9 (22.5%)
1 (2.5%)
3 (7.5%)
4 (10%)
40 (100%)
1. The value of a fraction increases when the
numbers that comprise it increase
2. The value of a fraction increases when
the numbers that comprise it decrease
3. The value of the fraction increases as
the size of the numerator approximates
the size of the denominator
4. The value of the fraction increases as
the size of the numerator becomes bigger
than the size of the denominator
5. There is no biggest fraction (infinity only
for big fractions)
6. There is no smallest/biggest fraction
(infinity for all fractions)
7. Mixed—could not be categorized
Total
6th Grade
5th Grade
Categories of responses
1 (2.5%)
40 (100%)
7 (17.5%)
2 (5%)
6 (15%)
13 (32.5%)
2 (5%)
9 (22.5%)
7th Grade
1 (2.5%)
40 (100%)
12 (30%)
3 (7.5%)
7 (17.5%)
10 (25%)
2 (5%)
5 (12.5%)
8th Grade
Table 3
Numerical value of fraction—Set I: smallest/biggest fractions frequency and percentage of responses per grade
4 (10%)
40 (100%)
17 (42.5%)
1 (2.5%)
6 (15%)
7 (17.5%)
1 (2.5%)
4 (10%)
1st Lyceum
11 (5.5%)
200 (100%)
41 (20.5%)
8 (4%)
34 (17%)
50 (25%)
12 (6%)
44 (22%)
Total
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Question 3a: order the numbers 5/6, 1, 1/7, 4/3
from the smallest to the biggest one
Question 3b: Why did you use this order?
1. The value of the fraction increases as either the
a. 1, 1/7, 4/3, 5/6 because the smallest goes first and
numerator or the denominator increase. The unit is the other numbers follow
the smallest of all fractions
b. 1/7, 4/3, 5/6, 1: 4=3 < 5=6 < 1=7 < 1, because 1
is a whole unit and it is the biggest. The numerator
or the denominator orders the other fractions
c. 1/7, 4/3, 5/6: 4=3 < 5=6 < 1=7, because the first
fraction has the smallest denominator while the others have bigger
2. The value of the fraction increases as either the
5=4 < 4=3 < 1=7 < 1; 5/6,4/3,1/7,1; 1/7,5/6,4/3,1
numerator, or the denominator decrease. The unit is because the big numbers are for the small fractions,
bigger than all fractions
or for as fewer the parts we divide a unit are, the
biggest the parts are
3. Correct ordering only for the fractions, not for
a. Not right ordering of the unit, or ordering of the
the unit
fractions without the unit. 1, 1/7, 5/4, 4/3; 1/7, 5/4,
4/3
b. The unit is the biggest number 1/7, 5/4, 4/3, 1
because number 7 is divided into more pieces than
the other numbers
4. Correct ordering for the fractions and the unit
1=7 < 5=6 < 1 < 4=3 because
6=42 < 35=42 < 42=42 < 56=42, or we do the divisions to find out which is the smallest
50 6
10 7
4 3
20 0:83 30 0:142 10 1:3 so the fractions from
2
2
1
the smallest to the biggest are: 1/7, 5/6, 1,4/3
5. Mixed—could not be categorized
1, 1/7, 4/3, 5/6, or I don’t know it is so, or I don’t
know
Categories of responses
Table 4
Numerical value of fraction—Set II: comparison of fractions categories of responses and examples
I don’t know
The fraction 4/5 because out of the five we get
four, while in the fraction out of five we get
two, or because these fractions have common
denominator and 4/5 has the biggest numerator
The fraction with the smallest numerator, or
sometimes (as in Question 5) the smallest
denominator. Because it has the smallest numbers
The fraction 4/5 because we get four out of the
five parts of a circle, or because it has more pieces, or because it has the biggest numerator
when we convert them into fractions with common denominator
Usually the fraction with the biggest numerator, or sometimes (as in Question 5) the biggest denominator, because it has the biggest
numbers
Questions 4a, 5a, 6a, 7a and 8a had the form:
Which of these fractions is bigger? The fractions used were respectively the following: 4/5–
2/5; 4/15–4/7; 5/8–4/3; 2/7–5/6; and 2/3–4/9
Questions 4b, 5b, 6b, 7b and 8b asked students
to justify their respective orderings
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and that the unit is bigger than all fractions. In the same category, we also grouped
the responses of children who ordered the numbers as above, although they interpreted the fraction as part of a unit. These students compared the parts a fraction
represents—the pieces—and mentioned that the smaller the parts in which a unit is
divided, the bigger the fraction is. This interpretation of the fraction is compatible
with their belief that a fraction is a part of a unit; therefore, it is always smaller
than the unit.
Category 3 included responses that gave the correct ordering of the fractions
with the exception of the unit. Some of our participants ordered the fractions correctly, but put the unit in the beginning believing that it is smaller than any fraction. These students were grouped in Subcategory 3a. Others put the unit at the
end, saying that it is bigger than all fractions. They were grouped in Subcategory
3b.
Students who gave the correct ordering of all the fractions, as well as the unit,
were grouped in Category 4. The processes they used to do so included rules of
comparison of fractions, interpretation of the fractions as parts of a whole unit,
and conversion of the fractions and the unit to fractions with a common denominator. They also used the unit as a point of reference for comparing fractions, and
interpreted of the fraction as the quotient of two numbers.
Table 5 shows the frequency/percentage of students’ responses in the various
categories. As can be seen, 22% of our participants expressed the idea that the
value of a fraction increases as the size of the numbers of the fraction increase
(Category 1). This response was mostly present in the Primary school children and
declined in the High school and Lyceum students. The number of students that
ordered the fractions and the unit correctly in all the questions increased in the
High school and the Lyceum, as expected.
The results of the Kruskal–Wallis test showed that there is a statistically significant effect of grade (v2 ð4Þ ¼ 13:62, p < 0:01), while the Mann–Whitney test
showed that gender does not differentiate the answers of students (z ¼ 2:2,
p < 0:05).
3.3. Explanatory frameworks for the numerical value of a fraction
As is apparent from the categories of responses, a large number of students,
particularly in the lower grades, were guided by the belief that the numerical value
of a fraction was represented by two independent natural numbers. We called this the
Explanatory Framework of Fraction as Two Independent Natural Numbers and we
assigned in this explanatory framework all students whose responses were consistently categorized in Response Categories 1 and 2 in Set I, and in Response Categories 1, 2 and 3 in Set II. Students’ responses in this explanatory framework
formed two subcategories: Subcategory A1 was guided by the beliefs that (a) The
numerical value of a fraction increases when either the numerator or the denominator
increase and (b) The unit is the smallest natural number, therefore, it is smaller than
any fraction. Subcategory A2 was guided by the belief that ‘‘in fractions the smaller
is the bigger’’. We interpreted this belief as a transitory phase in the process of
8 (20%)
4 (10%)
3 (7.5%)
4 (10%)
7 (17.5%)
13 (32.5%)
1 (2.5%)
40 (100%)
7 (17.5%)
2 (5%)
2 (5%)
6 (15%)
5 (12.5%)
13 (32.5%)
5 (12.5%)
40 (100%)
1. The value of the fraction increases as either
the numerator, or the denominator increase—the
unit is the smallest of all fractions
2. The value of the fraction increases as either the
numerator, or the denominator increase—the unit
is bigger than all fractions
3. The value of the fraction increases as either the
numerator, or the denominator increase—ordering
without the unit
4. The value of the fraction increases as either the
numerator, or the denominator decrease—the unit
is bigger than all fractions
5. Correct ordering only for the fractions
6. Correct ordering for the fractions and the unit
7. Mixed—could not be categorized
Total
6th Grade
5th Grade
Categories of responses
9 (22.5%)
18 (45%)
0 (0%)
40 (100%)
4 (10%)
0 (0%)
2 (5%)
7 (17.5%)
7th Grade
Table 5
Numerical value of fraction—Set II: comparison of fractions frequency and percent of responses per grade
9 (22.5%)
21 (52.5%)
0 (0%)
40 (100%)
5 (12.5%)
0 (0%)
2 (5%)
3 (7.5%)
8th Grade
7 (17.5%)
23 (57.5%)
3 (7.5%)
40 (100%)
3 (7.5%)
0 (0%)
0 (0%)
4 (10%)
1st Lyceum
37 (92.5%)
88 (44 %)
9 (4.5%)
200 (100%)
22 (11%)
5 (2.5%)
10 (5%)
29 (14.5%)
Total
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513
fraction learning and as a result of students’ effort to take on the new information
regarding the ordering of fractions.
The second Explanatory Framework we identified seemed to be guided by the
belief that A Fraction is a Part of a Whole. In this explanatory framework, the children face the numerator and the denominator of a fraction as two numbers connected with a relationship of part/whole of a natural object (which they consider
to be the unit of reference). In this explanatory framework, we assigned all students who belonged to Response Categories 3 and 4 in Set I, and Response Categories 4 and 5 in Set II. This explanatory framework was again divided in three
subcategories: In Subcategory B1: Naı̈ve Part of a Unit, we observed that the students, while they seemed to have conquered some ideas of the concept of a fraction
as part of a unit, still maintained some beliefs from the base-theory which reflected
upon their ordering of fractions. In Subcategory B2: Advanced Part of a Unit, the
students interpreted the fraction as the part (piece) of a quantity (unit) where the
numerator indicates the parts ‘‘that we take’’ and the denominator indicates the
parts in which the unit is divided. Therefore, these students believed that the fraction represents always a quantity smaller than a unit. In Subcategory B3: Sophisticated Part of a Unit, we included the students who had developed more
sophisticated processes of handling fractions. While these students considered the
fractions as representing a value smaller than the whole unit, they had developed
processes which helped them correctly order the fractions and the unit. For
example, use of the algorithm of converting dissimilar to similar fractions in order
to compare them, does not necessarily imply that they know what equivalent fractions are, as shown in their justifications.
The students placed in the third explanatory framework Relation between
Numerator/Denominator radically changed their beliefs about the concept of fraction, considering it is a relation between the numerator and the denominator. They
believed that the order-relation of a fraction and the unit is connected immediately
with the order-relation between numerator and denominator: they considered a
fraction smaller than the unit when its numerator was smaller than its denominator, and bigger than the unit when the numerator was bigger than the denominator. The third explanatory framework was divided into Subcategory C1: Relation
of Two Numbers Without Infinity, in which we grouped the students who did not
refer to the unbounded infinity of the fractions at all (they believed that there is a
unique smallest and biggest fraction, respectively) and Subcategory C2: Relation of
Two Numbers With Infinity, in which we grouped the students who had developed
a more complete knowledge of the expressions of the fraction. These students
believed that the fraction is an unbounded infinite number, resulting from the quotient of the numerator divided by the denominator.
We were able to assign 89% of our participants in one of the above Explanatory
Frameworks shown in Table 6. In order to check the reliability of our classification, another judge checked the answers of 10 participants from each age group
both at the first level of scoring and their assignment to explanatory framework.
Agreement between two independent judges making use of the same scoring key to
5 (12.5%)
3 (7.5%)
9 (22.5%)
40 (100%)
C. Relation between numerator/denominator
(1) Relation of two numbers without Infinity
(2) Relation of two numbers with Infinity
D. Mixed
Mixed—could not be categorized
Total
2 (5%)
40 (100%)
5 (12.5%)
3 (7.5%)
7 (17.5%)
5 (12.5%)
5 (12.5%)
2 (5%)
3 (7.5%)
4 (10%)
3 (7.5%)
1 (2.5%)
8 (27.5%)
6th Grade
12 (30%)
5th Grade
B. Part of a whole
(1) Naı̈ve part of a unit
(2) Advanced part of a unit
(3) Sophisticated part of a unit
A. Two independent numbers
(1) Two independent numbers whose value increases
as the numerator (or the denominator) increase
(2) Two independent numbers whose value increases
as the numerator (or the denominator) decrease
Explanatory frameworks
Table 6
Explanatory frameworks for the numerical value of a fraction
2 (5%)
40 (100%)
6 (15%)
9 (22.5%)
4 (10%)
5 (12.5%)
5 (12.5%)
2 (5%)
7 (17.5%)
7th Grade
4 (10%)
40 (100%)
6 (15%)
13 (32.5%)
4 (10%)
5 (12.5%)
3 (7.5%)
2 (5%)
3 (7.5%)
8th Grade
5 (12.5%)
40 (100%)
5 (12.5%)
17 (42.5%)
2 (5%)
5 (12.5%)
1 (2.5%)
1 (2.5%)
4 (10%)
1st Lyceum
22 (11%)
200 (100%)
27 (13.5%)
45 (22.5%)
21 (10.5%)
23 (11.5%)
15 (7.5%)
10 (5%)
37 (18.5%)
Total
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515
score all the responses was high (about 96%). All disagreements were resolved after
discussion.
From the distribution of children in the various explanatory frameworks, we can
conclude that the children progressively reorganize their initial interpretations of
fraction as they approach the scientific explanatory framework. We checked our
results with the Kruskal–Wallis non-parametric test. Control v2 of independence
showed that the effects of grade are statistically significant (v2 ð4Þ ¼ 21:13,
p < 0:001).
4. Discussion
The results were consistent with our hypothesis that the children, in their effort
to approach the concept of fraction, will not adopt the scientific concept of the
rational numbers immediately. As expected, they interpreted fractions in ways that
revealed their efforts to reconcile their initial ideas about number with the new
information coming from instruction. In this process, the students formed synthetic
models revealing their misconceptions.
In the introduction, we argued that the development of the concept of fraction is
different from the concept of natural number in its symbolic representation. As
expected, the younger students in our sample had difficulty understanding the
relationship between the numerator and the denominator and regarded the fraction
as consisting of two independent numbers. The initial explanatory framework
Fraction as Two Independent Natural Numbers is consistent with the presupposition
of the base-theory of number according to which ‘‘each number corresponds to a
symbol’’ (see Table 1). This presupposition appears to constrain students’ representation of fractions, as it is revealed from their answers concerning the smallest/
biggest fraction and the ordering of fractions in which the numerators and the
denominators were treated as if they were natural numbers. The students that
interpret fractions in this way can use improper fractions only intuitively.
The students in this explanatory framework develop two almost contradictory
beliefs. One is the belief that ‘‘the value of the fraction increases as the value of the
numerator (or denominator) increases’’ and the other is the belief that ‘‘the value
of the fraction increases as the value of the numerator (or the denominator)
decreases’’. The first is totally consistent with the base-theory of natural numbers.
The second appears to be a synthetic model, a misconception, created as a transitional phase in the process of understanding fractions. It appears that children
start by understanding fraction in terms of a unit divided into parts. When the
number of the parts a unit is divided into increases, each part becomes progressively smaller in relation to the whole. In this sense, it could be argued that ‘‘the
more parts the less value’’ a fraction has. This belief is shared by students who
belong to the first and the second explanatory framework. The above analysis constitutes an example of how a synthetic model can develop in the case of fractions.
Children’s beliefs about the relationship between the numerator and denominator of a fraction also provide important information about the development of
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the scientific concept of fraction. As we have shown, only the children grouped in
the third explanatory framework Fraction as a Relation Between Two Numbers
have understood that there can be fractions with the numerator equal or even bigger than the denominator, as their initial beliefs are that the fraction 1/1 is smaller
than the fractions 100/100.
The development of children’s understanding of the relationship between the
numerator and the denominator is directly related to the third presupposition of
the initial base-theory of natural numbers, according to which ‘‘the unit is the smaller number’’ (Table 1). The children adopting this presupposition believe that a
fraction is always bigger than the unit. Moreover, by generalizing this belief, they
come to the conclusion that the fraction 1/1 is the smallest fraction. This belief
defines the initial explanatory framework we identified.
In the second explanatory framework Fraction as Part of a Unit, children’s
beliefs regarding the relationship between a fraction and a unit change radically.
The students who adopt the second explanatory framework believe that a fraction
always represents a quantity smaller than the unit; in many cases, they use examples of natural objects in order to express the unit. This idea is strengthened by the
way fractions are usually taught initially. It is also consistent with the original
ontology of the concept of fraction—as a part of something, the quantieme of the
Ancient Egyptians (Caveing, 1992). Our results indicate that a radical change is
required in students’ beliefs to form a concept of fraction as a number that can be
smaller or even bigger than the unit. Only the children who adopt the explanatory
frameworks B3, C1 and C2, have understood that a fraction can be smaller, equal,
or even bigger than a unit, that is, they recognize the improper fractions.
Many researchers (e.g. Hart, 1987; Hiebert & Wearne, 1988; Nesher & Peled,
1986; Resnick and Ford, 1981; Resnick et al.,1989; Resnick and Omanson, 1987)
have examined the systematic errors and children’s misconceptions, in the case of
fractions. Our analysis adds on these efforts and has the additional advantage that
it explains how such misconceptions can be generated from children’s attempts to
relate the information they receive about fractions with their prior knowledge. It
also reveals the dynamic aspect of such synthetic models, or misconceptions, as
they can change in the process of learning.
The present results agree with the results of previous researches that a large
number of students adopt the second explanatory framework ‘‘Fraction as Part of
a Unit’’. The students in this explanatory framework believe that a fraction with a
numerator equal to the denominator is equivalent to the unit. This belief is
revealed by their answers in the first set of questions as well as in the fraction
ordering questions. Some researchers (Behr et al., 1992; Lamon, 1999) have
focused their attention on the concept of the unit believing that it connects integers
with rational numbers. They stress the need for more practice on partitioning and
measurement activities, which they believe will help children develop the concept of
the unit. Our results also agree with the idea that the understanding of the concept
of the unit is important and suggest that possibly the concept of the unit best
develops through activities that relate the unit to fractions.
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517
Finally, our results are consistent with those by Hartnett and Gelman (1998),
and Carey and Spelke (1994) in that children’s acceptance of the idea that decimals
and fractions as numbers demands a reconsideration of the identification of the
number with counting and with the successor principal. Also required is a new
understanding of division as an operation different from continuous subtraction.
The presence of different explanatory frameworks within which the concept of the
fraction can be interpreted, shows that a radical reorganization is occurring in
children’s conceptual structure enabling them to adopt the scientific model.
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