MIND, BRAIN, AND EDUCATION
Developing Children’s
Understanding of Fractions:
An Intervention Study
Florence Gabriel1,2 , Frédéric Coché3 , Dénes Szucs2 , Vincent Carette3 , Bernard Rey3 , and Alain Content1
ABSTRACT—Fractions constitute a stumbling block in
mathematics education. To improve children’s understanding
of fractions, we designed an intervention based on learningby-doing activities, which focused on the representation of
the magnitude of fractions. Participants were 292 Grade 4
and 5 children. Half of the classes received experimental
instruction, while the other half pursued their usual lessons.
For 10 weeks, they played five different games using cards
representing fractions (e.g., Memory and Blackjack). Wooden
disks helped them represent and manipulate fractions while
playing games. Our results showed an improvement in the
conceptual understanding of fractions. The findings confirmed
that the usual practice in teaching fractions is largely
based on procedural knowledge and provides only minimal
opportunities for children to conceptualize the meaning and
magnitude of fractional notations. Furthermore, our results
demonstrate that a short intervention inducing children
to manipulate, compare, and evaluate fractions improves
their ability to associate fractional notations with numerical
magnitude.
1
Laboratoire Cognition, Langage et Développement, Faculté des Sciences
Psychologiques et de l’Education, Université Libre de Bruxelles
2
Department of Experimental Psychology, Centre for Neuroscience in
Education, University of Cambridge
3 Service des Sciences de l’Education, Faculté des Sciences Psychologiques
et de l’Education, Université Libre de Bruxelles
Address correspondence to Florence Gabriel, Laboratoire Cognition,
Langage et Développement, Faculté des Sciences Psychologiques
et de l’Education, Université Libre de Bruxelles, Ixelles, Belgium;
e-mail: [email protected], Dénes Szucs, Department of Experimental
Psychology, Centre for Neuroscience in Education, University of
Cambridge, Cambridge, UK; e-mail: [email protected], or Alain Content,
Laboratoire Cognition, Langage et Développement, Faculté des Sciences
Psychologiques et de l’Education, Université Libre de Bruxelles, Ixelles,
Belgium; e-mail: [email protected]
Volume 6—Number 3
Common fractions are quantities that can be represented by
a ratio of integers (e.g., 3/4) and are one of the most difficult
mathematical concepts to learn for primary school pupils.
Indeed, most Grade 5 children cannot place a fraction on a
graduated number line (Bright, Behr, Post, & Wachsmuth,
1988) and struggle when asked to order fractions (Mamede,
Nunes, & Bryant, 2005). Here, we report an intervention
programme designed with the objective to train the conceptual
understanding of fractions in a playful setting at the beginning
of fraction acquisition (Grades 4 and 5 in the French
Community of Belgium).
One crucial feature of numerical conceptual knowledge
is the ability to represent number magnitudes (Siegler,
Thompson, & Schneider, 2011). One of the major current
questions concerning fraction processing is whether the global
numerical magnitude of fractions (i.e., a fraction’s real value) is
directly available. That is, are educated adults able to process
a fraction as a whole? To date, behavioral and imaging studies
offer contradictory views. Some studies suggest that only the
individual integer components of a fraction are processed
(Bonato, Fabbri, Umiltà & Zorzi, 2007; Kallai & Tzelgov,
2009), whereas others suggest that the global magnitude of
fractions is also represented (Jacob & Nieder, 2009; Schneider
& Siegler, 2010). A conciliatory view is that adults and children
use a combination of both componential and global fraction
processing strategies, depending on task demands and the
type of fractions used (Ischebeck, Schocke, & Delazer, 2009;
Meert, Grégoire, & Noël, 2009, 2010a, 2010b; Schneider &
Siegler, 2010).
Another important issue to consider about children’s
difficulties in learning fractions is the distinction between
conceptual and procedural knowledge. Conceptual knowledge
can be defined as the explicit or implicit understanding of the
principles ruling a domain and the interrelations between the
different parts of knowledge in a domain (Rittle-Johnson &
Alibali, 1999). It can also be considered as the knowledge of
central concepts and principles and their interrelations in a
particular domain (Schneider & Stern, 2010). In the domain of
© 2012 the Authors
Journal Compilation © 2012 International Mind, Brain, and Education Society and Blackwell Publishing, Inc.
137
Developing Children’s Understanding of Fractions
fractions, conceptual knowledge refers to a combination of the
general properties of rational numbers (such as the principle
of equivalent fractions), the understanding of the roles of the
numerator and the denominator, and the understanding of
global fraction magnitudes.
Procedural knowledge can be defined as sequences
of actions that can be put to play to solve specific
problems (Rittle-Johnson & Alibali, 1999). Some authors
consider procedural knowledge as the knowledge of symbolic
representations, algorithms, and rules (Byrnes & Wasik, 1991).
In the domain of fractions, knowing how to calculate the
lowest common denominator to add or subtract fractions
with different denominators is a prime example of procedural
knowledge. However, in many instances, pupils seem to
apply procedures like these without fully understanding the
underlying concepts (Byrnes & Wasik, 1991; De Corte, 2004).
Kerslake (1986) observed that children would often perform
calculations without knowing why.
Some evidence from mathematics learning studies supports
the idea that conceptual knowledge plays a major role in the
generation and adoption of procedures. First, children who
have better conceptual understanding tend to have better procedural abilities (Byrnes & Wasik, 1991; Cauley, 1988). Second,
conceptual knowledge would precede procedural knowledge
in many domains. For example, Byrnes (1992) has shown that
prior conceptual knowledge of integer computations is a good
predictor of procedural knowledge. Third, instruction about
concepts and procedures could lead to improve procedural
abilities. Byrnes and Wasik (Experiment 2) assessed conceptual and procedural knowledge of Grade 5 children before
and after an instructional intervention on fraction addition.
The lesson focused on the distinction between natural number addition and fraction addition (conceptual knowledge).
Pupils were also taught the lowest common denominator procedure (procedural knowledge). Grade 5 children were more
accurate in the execution of the fraction addition procedure
after the instruction but did not gain any conceptual understanding. Similarly, Perry (1991) studied the mathematical
equivalence concept with Grade 5 and 6 children. Pupils were
more likely to succeed on mathematical equivalence problems when taught principles than when taught algorithms.
Thus, conceptual knowledge can facilitate the acquisition of
procedural knowledge. Another kind of relations between
conceptual and procedural knowledge can be considered.
Procedural and conceptual knowledge could develop in an
interactive and iterative way; a small improvement of the first
kind of knowledge would lead to a small improvement of the
other (Rittle-Johnson & Alibali, 1999).
In our intervention programme, we decided to focus on
the conceptual understanding of fractions, an area which
is poorly grasped by primary school children. Here, we
decided to train the numerical concept of fractions, that
is, the representation of the global magnitude of fractions.
138
Several studies on natural numbers suggest that the basic
ability to process and compare numerical magnitudes is
linked to general achievement in mathematics (De Smedt,
Verschaffel, & Ghesquière, 2009; Holloway & Ansari, 2009;
Lonneman, Linkersdorfer, Hasselhorn, & Lindberg, 2011).
The precision of these core numerical representations could
provide the necessary foundations to build more complex
mathematical abilities, such as arithmetic skills. Moreover,
Siegler et al. (2011) showed that accurate representations of
fraction magnitude play a key role in both fraction arithmetic
and general mathematical abilities. Hence, we investigated
whether a training of the better understanding of the
magnitude of fractions is beneficial for more complex fraction
skills. Similar training programmes have been implemented
for integers but never for fractions (Kucian et al., 2011;
Siegler & Ramani, 2009; Wilson, Revkin, Cohen, Cohen,
& Dehaene, 2006).
We designed an intervention aiming at developing and
consolidating the representation of fraction magnitude. The
intervention was based on several educational principles:
learning-by-doing, starting with a concrete support to
progressively build more abstract representations, playfulness,
and collaboration. First, it was based on a learning-bydoing programme and followed a constructivist approach,
focusing on the meaning of the magnitude of a fraction and
how it is related to fractional symbols, challenging pupils’
preexisting knowledge about natural numbers (Ni & Zhou,
2005; Stafylidou & Vosniadou, 2004). That is, children were
confronted with obstacles that could only be overcome by
conceptual change. Many authors agree that difficulties in
learning fractions are linked to the whole number knowledge.
This has been referred to as the whole-number bias (Ni
& Zhou, 2005). Some conceptual change theories insist
that the whole number bias will interfere with learning
about fractions. But, as Siegler et al. (2011) noted, one of
the difficulties in learning fractions originates more from
drawing inaccurate analogies between fractions and whole
numbers than from drawing parallels between these two
types of numbers. Therefore, this intervention fostered correct
analogies between fractions and whole numbers. Indeed, we
focused on the fact that fractions have magnitudes that can be
ordered and compared, just like whole numbers.
Second, our intervention was called ‘‘from pies to numbers’’
as children were invited to use concrete support at first and
then build more abstract representations. A common concrete
reference support (wooden pieces), based on what pupils
know best (disk representation and their parts) was provided.
This reference support allowed them to manipulate concrete
materials and helped them find answers to problems included
in the different games and solve conflicts occurring during
the games. The wooden pieces could be superposed so that
children could also get a concrete representation of fractions
larger than 1. Children usually feel more comfortable when
Volume 6—Number 3
Florence Gabriel et al.
using concrete aids when dealing with fractions (Arnon,
Nesher, & Nirenburg, 2001), and concrete aids, such as
the wooden reference support, should help them develop
the corresponding abstract concepts (Arnon et al., 2001). In
addition, constructivist theories suggest that development
goes from the concrete operational stage to a more abstract
stage, and therefore, learning should follow this trajectory
(Santrock, 2009). Moreover, concrete material may be more
engaging and appealing to pupils and thus engage them more
in the learning process (Ball, 1992).
A circular support can be seen as a controversial choice as
children already have stereotyped representations of fractions
as parts of a pie (Nunes & Bryant, 1996). On the contrary,
we considered that it would be a good starting point as
it corresponds to something that pupils know very well and
thus provides a very natural reference. Other representations of
fractions such as various symbolic representations, collections
of items as well as other continuous shapes were progressively
introduced along with the game progression. Children should
be using the reference less and less as far as the games
progress as more and more diverse representations were
proposed throughout the intervention. They should become
less dependent of that circular representation and develop a
more abstract representation of the magnitude of fractions.
Third, we developed a playful intervention, associating
fraction learning with fun. Playfulness is known to have
positive effects on the learning process (Sawyer, 2006)
and contributes to advances in verbalization, concentration,
curiosity, problem-solving strategies, cooperation, and group
participation (Kangas, 2010). It can increase children’s
motivation and their involvement in the learning process.
Finally, pupils were placed in groups of 3–5 and each group
was given a game to play collectively. Several authors conclude
that collaborative work provides a deeper understanding of
taught concepts as pupils exchange views and work toward
identifying the right solution (Cobb, 1995; Forman & Cazden,
1985; Palincsar, 1998).
METHOD
Participants
We chose to work with Grades 4 and 5 as this corresponds
to the first fraction lessons in the French Community of
Belgium. In Grade 4, pupils learn how to read and represent
the value of a fraction. They start placing fractions on a
graduated segment; they learn about equivalent fractions and
are trained in the addition and subtraction of fractions with
small common denominators. In Grade 5, children learn that
fraction represents quantity, are trained to convert fractions
into decimal numbers and vice versa, continue to use addition
and subtraction of fractions with different denominators, and
Volume 6—Number 3
learn how to simplify fractions. Improper fractions are also
introduced in Grade 5.
Eight Grade 4 and 5 classes took part in the intervention
programme. These classes came from four schools of the
French Community of Belgium. Every school included two
classes of Grade 4 and 5. For each grade, one of the classes
was the intervention group and the other the control group.
Control groups received their usual lessons about fractions.
The teachers of control classes were asked to give a 1-hr
lesson on fractions per week to match with the time
spent by the intervention group on learning fractions. The
intervention group did not receive their regular lessons during
the intervention and only played the games we designed. We
allocated classes to intervention conditions at random.
A total of 292 children (133 Grade 4 and 136 Grade 5
children) took part in the activities. Data for 23 pupils had to
be discarded because they did not take part in either the preor the posttest.
Intervention
We created and adapted five card games involving fractions:
Memory, War, Old Maid, Treasure Hunt, and Blackjack. The
activities were organized twice a week for 30 min during the
usual classes for a period of 10 weeks. Pupils played games in
small groups of 3 up to 5 children. Every week a new game was
introduced. The second session of the week implied playing
the same game at the next level of difficulty (Table 1).
Wooden disks cut into pieces from halves to twelfths
were manufactured to help children manipulate fractions and
represent fractional magnitudes. That teaching aid allowed
children to manipulate concrete equipment and progressively
construct a representation of the magnitude of fractions. It also
allowed them to compare different fractions during the games.
One set of wooden disks (two for the larger class) was available
in each classroom and served as reference during the games. At
the beginning of the intervention, children were encouraged to
use the teaching aid by the experimenter or the teacher every
time they were not sure about the outcome of a game. Even
Table 1
Organization of the Intervention
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week 10
1st session (30 min)
2nd session (30 min)
Presentation of wooden disks
Memory—Level 1
War—Level 1
Old Maid—Level 1
Treasure Hunting—Level 1
Blackjack—Level 1
Memory—Level 3
War—Level 3
Old Maid—Level 3
Treasure Hunting—Level 3
Memory—Level 2
War—Level 2
Old Maid—Level 2
Treasure Hunting—Level 2
Blackjack—Level 2
Memory—Level 4
War—Level 4
Old Maid—Level 4
Blackjack—Level 3
139
Developing Children’s Understanding of Fractions
if there was one set per classroom, it never was a problem as
groups of pupils progressed at different rhythms. Moreover,
they did not have to compare the same sets of fractions as
children from another group, so the reference support could
be used by different small groups at the same time. The teacher
and the experimenter were always supervising the games to
make sure that pupils got to the correct outcome at the end of
each game.
Games
Memory
The goal of the Memory game was to associate pairs of fractions
that represented the same quantity. This activity implied to
match two fractions, transcode symbols and numbers, and
memorize the representations of fractions. Cards were laid
down in a grid face down. Players took turns flipping pairs
of cards over. If the cards matched, the player said ‘‘bingo’’
and showed the pair to the other pupils to make sure they
agreed. In case of disagreement between players, they could
use the wooden disks to compare both fractions. If the pair
was correctly matched, the player kept both cards. The aim of
the game was to match more pairs of fractions than the other
players.
any pairs they had faced up. Each player took turns offering his
hand face down to the pupil next to him. That pupil selected
a card and added it to his hand. This player then saw whether
the selected card made a pair with their original cards. If so, the
pair was discarded face up as well. The player who just took
a card then offered his or her hand to the person sitting next
to him and so on. In case of disagreement between players,
they were encouraged to use the wooden disks to compare
fractions. The game went on until all players except one had no
cards left. The player who was left with the one unmatchable
card lost the game.
Treasure Hunt
This activity involved comparison between fractions, discovery of improper fractions, and addition of fractions by
estimation or using the wooden disks. This way of adding
fractions focused on the representation of the magnitude of
fractions and did not require the understanding of the lowest
common denominators procedure. For this game, the deck of
cards was laid out face down on the table. Two cards were
turned over. The first player took the card that he thought represented the larger magnitude. Another card was then flipped
over and the next player chose the card representing the larger
magnitude. When every player had two cards (or three for
more experienced players), they were asked to count which
War
player had the highest magnitude in total. The player with the
This activity involved comparison of fractions, which led highest total magnitude won a bonus card. The game started
pupils to get a representation of the magnitude of fractions. over again until every player had two (or three) cards and
A total of 40 cards were equally distributed between players. so on.
Cards were placed faced down on the table. In unison, players
turned over their top card and played it face up. The player
who turned the card representing the largest fraction could Blackjack
collect all the cards in the middle of the table and move them to This game involved the ability of estimating the result of an
the bottom of his stack. When cards were of equal value, each addition of fractions and the relation between fractions and
player laid down another card and the higher valued fraction units. The rule of Blackjack was to add the value of cards to
won. In case of disagreement between players, they were approach two (or three for more experienced players) without
allowed to use the wooden disks to compare fractions. When exceeding it. A referee was chosen for each team. He first
a player did not have any cards left, he was eliminated. When dealt two cards by player. Afterward, each player could decide
there were only two players left, they were both declared whether he wanted another card or would rather stop. When
winners and a new game could start again.
every player had stopped, they all showed their card and
checked who was closer to two. The winner was the player
with a total of fractions closer to two, without exceeding that
Old Maid
quantity. In case of disagreement between players, they could
The goal of this game was to form pairs of fractions representing use the wooden disks to compare fractions. When the first
the same magnitude. This activity involved matching two game was over, another referee was chosen and they could
fractions. Children had to compare symbolic as well as figural start over again.
representations of fractions and associate them with their
corresponding magnitudes. Unlike Memory, Old Maid did not
involve memorization. In this game, children compared more Difficulty Levels
than just two cards to match fractions. Indeed, they had to Four difficulty levels were defined. Each level was composed
compare all the cards they had in their hand. The first step of of 40 cards of different colors. The first level (yellow cards)
the game was to remove one card, resulting in one unmatchable was only composed of familiar fractions with denominators up
card. This card became the ‘‘Old Maid.’’ Players had to discard to 5. Some fractions were represented by numbers (e.g., 1/5),
140
Volume 6—Number 3
Florence Gabriel et al.
words (e.g., one-fifth), or drawings (e.g., circles or squares).
This first level was mainly aimed at introducing simple and
familiar fractions. The second level (red cards) was composed
of fractions with denominators up to 12. Moreover, equivalent
fractions were introduced, with the result that a given
magnitude could be represented by different fractions. Thus,
there were different ways to match pairs of fractions. Improper
fractions (i.e., numerator > denominator) were introduced in
the third level (blue cards). Pictorial representations were
multiplied (e.g., circles, squares, octagons, and collections of
objects). As the main focus in level three was put on improper
fractions, equivalent fractions were set aside, leaving them for
the last level. The fourth and last level (green cards) involved
fractions with denominators up to 12, improper fractions,
various pictorial representations, equivalent fractions, and
less familiar fractions.
Treasure Hunt and Blackjack were not used with Level 4
cards because the difference with Level 3 consists of equivalent
fractions. This parameter was not relevant in both these games.
The use of equivalent fractions was not needed to perform
better in these games. Nonetheless, two more sessions could
have been added with Level 4 cards. But they would have been
very similar to Level 3 in terms of learning processes.
of reference was marked underneath one of the other 12 tick
marks. The fraction of reference and its position changed from
trial to trial. One point was given if they placed the fraction
at the right graduation. The use of graduated number lines
was particularly interesting to analyze different types of errors
that pupils would make. Cronbach’s α indicated a high level
of internal consistency for the conceptual category (α = .84).
Procedural skills were assessed by items that could be
easily solved by applying a procedure or an algorithm,
such as arithmetic operations or simplification. Arithmetic
operations involved additions and subtractions of symbolic
fractions with the same denominator (n = 4), additions and
subtractions of symbolic fractions with different denominators
(n = 6), multiplication of a symbolic fraction by a natural
number (n = 4), and multiplication of a symbolic fraction by
another symbolic fraction (n = 4). We were also particularly
interested in analyzing the proportion of errors linked to the
whole number bias, such as 1/2 + 1/4 = 2/6 (Ni & Zhou, 2005).
The simplification category (n = 4) involved the denominator
to be either divided by 2 or by 3. Cronbach’s α was .79 for the
procedural category.
The maximum length of the test was 50 min.
Tests
Pre- and posttests were identical. They were divided into
five different categories of questions assessing on one hand
conceptual aspects (three categories: estimation, comparison,
and number lines) and on the other hand procedural aspects
(two categories: arithmetic operations and simplification).
Diverse concepts relative to fractions were included in the
tests, such as improper fractions, equivalent fractions, and
unfamiliar fractions.
Conceptual items assessed a comparison between quantities or ordering. Other studies have used those criteria to
assess conceptual understanding (Byrnes & Wasik, 1991; Hallett, Nunes, & Bryant, 2010). For the estimation category
(n = 4), pupils had to place a symbolic fraction on a nongraduated number line going from 0 to 1. One point was given
if they placed the mark at a point situated within 1 cm of
the right location, and no point was attributed if the mark
was further away. For the comparison category, children had
to choose the larger symbolic fraction out of two (n = 16).
Fractions used in the comparison task were fractions with
the same numerator (n = 4), the same denominator (n = 4),
different numerators and denominators (n = 4), or a fraction
and one (n = 4). One point was attributed for each correct
answer. For the graduated number line category (n = 9), children were asked to place a symbolic fraction on a graduated
number line. Each line was 13 cm long with one tick mark
every 1 cm. The digit 0 was always indicated below the first
tick mark at the left end of the line, and one other fraction
EXPERIMENTAL DESIGN
Volume 6—Number 3
We ran Grade (two levels: Grade 4 and Grade 5) × Condition
(two levels: experimental group vs. control group) analyses
of covariance (ANCOVAs) using the posttest scores as
dependent variable, with the corresponding pretest score
as a covariate. This was necessary to interpret our findings,
given the differences between groups at pretest. By doing
this, we could test that the experimental group improved
more than the control group. We also introduced grade as a
between-subject factor to test whether the intervention was
more efficient in Grade 4 or 5. The ANCOVAs were run for
both conceptual and procedural factors and also for each task
separately. Similar analyses were also run on comparisons of
fractions with the same denominators and same numerators,
as well as comparison of a fraction to the unit and placing the
unit on a graduated number line. Common patterns of errors
are also reported for graduated number lines and arithmetic
operations.
RESULTS
To account for varying pretest scores, the posttest scores were
compared by running an ANCOVA using the corresponding
pretest score as the covariate. The Grade × Condition
ANCOVA with pretest score as the covariate was run for
each measure.
141
Developing Children’s Understanding of Fractions
Table 2
Mean Percentage of Correct Responses and Standard Errors for the Intervention and Control Groups in Grades 4 and 5 for the Conceptual
Factor and Each Conceptual Task
Control G4
Experimental G4
Control G5
Experimental G5
Conceptual
pretest
Conceptual
posttest
Estimation
pretest
Estimation
posttest
Comparison
pretest
Comparison
posttest
Number lines
pretest
Number lines
posttest
31.6 ± 1.6
39.3 ± 2.3
44.9 ± 2.2
44.3 ± 2.6
36.7 ± 2.2
53.2 ± 3.2
49.5 ± 2.2
60.1 ± 2.9
20.2 ± 6.5
31 ± 8.2
35 ± 8.1
33.1 ± 8.5
27.1 ± 7.3
44 ± 9.3
38.1 ± 9.7
52.1 ± 8.5
53.4 ± 3.5
58.9 ± 4.1
70.7 ± 4.2
65.2 ± 5.2
56.7 ± 4.3
74.9 ± 4.1
75.1 ± 3.1
81 ± 3.1
21.3 ± 2.2
28.9 ± 3.1
28.8 ± 3.1
34.7 ± 3.4
26.2 ± 2.6
41 ± 3.9
35.2 ± 3.5
47.2 ± 3.9
Table 3
Mean Percentage of Correct Responses and Standard Errors for the Intervention and Control Groups in Grades 4 and 5 for the Procedural
Factor and Each Procedural Task
Control G4
Experimental G4
Control G5
Experimental G5
Procedural pretest
Procedural posttest
Operations pretest
Operations posttest
Simplification
pretest
Simplification
posttest
15 ± 1.8
17 ± 1.6
31.9 ± 2.7
36.7 ± 2.7
35 ± 2.9
27.8 ± 2.5
46.8 ± 3.4
36.5 ± 2.9
15 ± 1.2
22.7 ± 1.7
35.7 ± 2.2
32.6 ± 2.2
23.6 ± 2.2
27.1 ± 2
38.6 ± 3.1
30.5 ± 2.2
14.8 ± 6.8
11 ± 6.3
28.1 ± 9.1
40.8 ± 9.9
46.3 ± 9.4
28.5 ± 8.2
55 ± 9.9
42.6 ± 9.6
Comparison
In the comparison task, pupils were asked to decide
which of two fractions was the largest. The Grade ×
Condition ANCOVA showed a significant effect of pretest,
F(1, 264) = 35.6, p < .001, η2p = 0.12, a significant effect of
condition, F(1, 264) = 29.9, p < .001, η2p = 0.10, indicating
a significant difference between experimental classes and
control groups in the comparison task and a significant effect
of grade, F(1, 264) = 14.1, p < .001, η2p = 0.05, indicating that
Grade 5 performed better than Grade 4. Pretest results in
Grade 4 were similar in experimental and control groups.
In Grade 5, results for experimental classes were lower
than that for control classes (−5%). Experimental groups
improved more at posttest (+16% in Grade 4, +15% in
Grade 5) than control groups (+3% in Grade 4, +5% in
Grade 5).
Estimation
The same pattern of results was obtained for the
In the estimation task, children were asked to place a fraction
comparison of fractions with the same denominators, the
on a nongraduated number line going from 0 to 1. The mean
same numerators, and the comparison of a fraction to the unit
percentage of correct responses was different for pretest in
(Table 4).
control and experimental groups in Grade 4. Nevertheless,
experimental classes (+13%) improved more than control
classes (+6.9%). In Grade 5, results for pretest were similar in Number Lines
both groups. Experimental classes (+19%) improve more than Children had to place a fraction on a graduated number
control classes (+3.1%). The Grade × Condition ANCOVA line. The Grade × Condition ANCOVA showed a significant
showed a significant effect of pretest, F(1, 264) = 75.4, effect of pretest, F(1, 264) = 31.8, p < .001, η2p = 0.11, and
p < .001, η2p = 0.22, and a significant effect of condition, a significant effect of condition, F(1, 264) = 10, p = .002,
F(1, 264) = 13.1, p < .001, η2p = 0.05, indicating a significant η2p = 0.04. There was no significant effect of grade (p = .13).
difference between experimental classes and the control group Experimental groups (+12% in Grade 4, +13% in Grade 5)
for the estimation task. There was no significant effect of grade progressed more than the control groups (+5% in Grade 4,
+6% in Grade 5).
(p = .14).
Conceptual and Procedural Factors
The ANCOVA run on the conceptual factor showed a
significant effect of pretest, F(1, 264) = 134.7, p < .001, η2p =
0.34, and a significant effect of condition, F(1, 264) = 21.7,
p < .001, η2p = 0.076, indicating that the experimental group
scored higher on the conceptual factor (Table 2). We ran the
same analysis for the procedural factor. Results showed a
significant effect of pretest, F(1, 264) = 60.2, p < .001, η2p =
0.19, and a significant effect of condition, F(1, 264) = 15.6,
p < .001, η2p = 0.056, indicating that the control group scored
higher on the procedural factor (Table 3).
We also analyzed results for each category, namely
estimation, comparison, number lines, arithmetic operations,
and simplification.
142
Volume 6—Number 3
Florence Gabriel et al.
Table 4
Mean Percentage of Correct Responses and Standard Errors for the Intervention and Control Groups in Grades 4 and 5 for the Comparison
of Fractions With Same Denominators, Fractions With Same Numerators, and Fractions Referring to 1
Control G4
Experimental G4
Control G5
Experimental G5
Comparison same
denominator pretest
Comparison same
denominator posttest
Comparison same
numerator pretest
Comparison same
numerator posttest
Comparison unit
pretest
Comparison unit
posttest
69.3 ± 8.8
76.6 ± 8.9
85.8 ± 6.1
76.5 ± 9.6
71.3 ± 9.1
87.2 ± 7.1
89.3 ± 5.5
91.6 ± 6.1
34.4 ± 9.6
41.7 ± 11.2
67.9 ± 9.1
59.1 ± 10.2
34.3 ± 7.9
66.1 ± 8.9
71.5 ± 7.4
78.3 ± 8.2
53.2 ± 5.3
63.7 ± 6.4
73.7 ± 7.9
68 ± 6.4
54.1 ± 6.3
81.7 ± 5.9
84.4 ± 5.3
90.9 ± 4.3
Placing the Unit on the Number Line
Some items required placing 1 on the number line. The
Grade × Condition ANCOVA run on posttest scores revealed
a significant effect of pretest, F(1, 264) = 27.7, p < .001, η2p =
0.095, and a significant effect of condition, F(1, 264) = 11.6,
p < .001, η2p = 0.042, indicating a significant difference
between experimental classes and control classes. There
were large differences in Grade 4 between control (15.7%)
and experimental classes (31.5%) for the pretest. However,
experimental classes improved more than control classes
(+16.9% and +7.8%, respectively). In Grade 5, there were
also differences between control (30.9%) and experimental
classes (36.8%) for the pretest. Experimental classes improved
more than control classes (+19.6% and +8.5%, respectively).
Table 5
Error Analysis for Items Involving Placing 1 on a Graduated Number
Line
Pretest
Beginning End
Control group
Grade 4
Experimental
group Grade 4
Control group
Grade 5
Experimental
group Grade 5
Posttest
10th
grading
Beginning End
10th
grading
11.7
13.7
15.3
5.1
12.5
17.6
13.1
34.7
10.3
11
26.9
19.4
9.9
13.4
14.1
8.4
8.5
12.7
10.2
12.3
17.3
9.6
6.2
21.9
Note. Errors are expressed in percentages.
Error Analysis
We were particularly interested to see the representation
that children have of the relationship between the unit and a
fraction. When pupils were asked to place 1 on the number
line, common errors could be pinpointed. When they made a
mistake, they often placed 1 at the very beginning of the line as
if 1 could only be representing the beginning of the counting
sequence. Many pupils also placed the unit at the end of the
number line, showing their poor understanding of improper
fractions. For those children, a fraction can only be smaller
than one. Another common error was to place the unit at the
10th grading, showing the influence of the decimal numeral
system (Table 5).
Table 6
Error Analysis in Arithmetic Operations: Proportion of Errors Linked
to the Whole Number Bias (Expressed in Percentage)
Control group Grade 4
Experimental group Grade 4
Control group Grade 5
Experimental group Grade 5
Whole number bias
errors pretest
Whole number bias
errors posttest
54.8 ± 4
63.8 ± 3.6
46.6 ± 3.3
60.3 ± 3.4
58.6 ± 3.6
59.8 ± 3.7
44.2 ± 3.7
48.1 ± 3.8
percentage of errors linked to the whole number bias for
each grade and condition. Interestingly, errors linked to the
whole number bias only dropped in the experimental group in
Arithmetic Operations
Arithmetic operations included additions, subtractions, and Grade 5 (−12%).
multiplications of fractions. The Grade × Condition ANCOVA
run on posttest scores revealed a significant effect of pretest,
Simplification
F(1, 264) = 10.4, p = .001, η2p = 0.038. In general, the control
The Grade × Condition ANCOVA revealed a significant
groups progressed more than experimental classes both in
effect of pretest, F(1, 264) = 38.1, p < .001, η2p = 0.13, and
Grades 4 and 5. However, there were no significant effect of
a significant effect of condition, F(1, 264) = 15.8, p < .001,
condition (p > .2), nor grade (p = .81).
η2p = 0.06, showing a significant difference between control
and experimental classes in the simplification task. There was
no significant effect of grade (p = .54). Grade 4 children had
Whole Number Bias Errors
Errors linked to the whole number bias were, for example, equivalent scores at pretest, that is, between 11% and 15%
1/3 + 1/4 = 2/7 (Ni & Zhou, 2005). Table 6 shows the of correct responses (Table 3). However, in Grade 5, control
Volume 6—Number 3
143
Developing Children’s Understanding of Fractions
groups had poorer performance than experimental classes at mistakes observed in the number line subtest consisted in
the pretest. Globally, control groups progressed more than placing one at the beginning or end of the line or placing it
experimental groups for the simplification task.
at the 10th grading. At posttest, Grade 5 children from the
experimental classes made fewer of these typical mistakes.
We also examined the potential effects of the intervention
DISCUSSION
on procedural skills. One current hypothesis is that
We designed an educational intervention to improve the improvements of conceptual knowledge could facilitate
learning of fractions in Grades 4 and 5. The development of the mastery of procedures (Byrnes & Wasik, 1991). Our
this intervention was based on the premise that most of the results do not support this hypothesis as experimental
difficulties in fraction learning stem from the fact that pupils classes showed less improvement than control classes for
have a poor representation of the global magnitude of fractions arithmetic operations and simplification tasks. Nevertheless,
and that they do not grasp the conceptual meaning of fractions. two interesting results were found. First, there was a drop
Our data showed that pupils who received the interven- in errors linked to the whole-number bias in arithmetic
tion improved more than children from the control groups operations in Grade 5 experimental classes. Second, children
for conceptual items of the test (estimation, comparison, and from the experimental group showed some improvement
number lines). Even if these results do not allow us to con- for additions and subtractions with the same denominator.
clude that pupils developed the numerical representation of It is possible that conceptual knowledge (in this case, the
fractions, we can still posit that it allows children to develop construction of the mental representation of the magnitude
conceptual knowledge about fractions and compensate for of fraction) allowed children to answer questions that were
considered as procedural items in the design of the tests.
major difficulties.
The intervention had a positive impact on pupils’ However, the use of conceptual knowledge as a substitute for
motivation. In line with research showing that motivation procedural knowledge seems to be limited. More complex
can be fostered through the introduction of playfulness into operations, such as additive operations with fractions of
teaching (Kangas, 2010; Smilansky & Shefatya, 1990), pupils different denominators, remain inaccessible for children
actively took part in the learning process and showed great who have not explicitly learned the procedure. Conceptual
enthusiasm when playing the fraction games. Even if we did knowledge thus appears insufficient to allow pupils to invent
not explicitly measure motivation levels in the classroom, appropriate procedures to solve such problems. However,
teachers were positively surprised to see their pupils enjoying longer and reinforced conceptual learning might perhaps lead
learning fractions. Motivation in the classroom is an important to improved procedural performance.
One of the surprising results of this study was that pupils
factor as it has been shown to develop feelings of mastery and
from
the control group improved more than pupils from the
competence and is related to school achievement (Gottfried,
experimental
groups on most of the procedural tasks. This
1982). Future studies examining the role of learning-by-doing
and playful interventions should assess more objectively the seems related to the lessons that teachers gave children of
impact of motivation in learning fractions. We would have the control group. Teachers from the control classes put
liked to also control for the possibility that fraction skills more emphasis on procedural knowledge than on conceptual
improved due to an increase in motivation through the knowledge. Children of the control group did not show better
playing of games rather than any fraction-specific training conceptual understanding after 10 weeks. It seems that they
but did not have the resources to do this. However, children have been learning procedures in a mechanical way and still
from our control group selectively improved in procedural have poor representations of the magnitude of fractions.
As mentioned earlier, another characterization of the
skills, while the experimental group selectively improved in
relationship
between conceptual and procedural knowledge
conceptual understanding, thus suggesting that the effect of
(Rittle-Johnson & Alibali, 1999) is the suggestion of iterative
the intervention was not solely a global motivational effect.
Understanding the relationship between a fraction and the development with interactive and reciprocal influences.
unit is well known to be a major difficulty when learning Children would gradually improve both their conceptual
fractions (Case, 1985; Smith, Solomon, & Carey, 2005). and procedural knowledge, and each type of improvement
However, the performance of the experimental classes suggests would permit progress on the other in turn. If this view is
that class activities focusing on comparing and manipulating correct, better conceptual understanding could be fruitful for
fraction magnitudes allow children to grasp the link between the rest of the curriculum when pupils learn procedures
fractions and the concept of unity. Indeed, experimental classes with their teacher. Their increased understanding of the
had better performance in understanding the link between meaning of fractions would help them make sense of the
fractions and one as a ‘‘unit.’’ They performed better than procedures and hence gain faster mastery and better control
children of the control group when comparing fractions to of their application. Longitudinal studies would be necessary
the unit and when placing the unit on a number line. Typical to further investigate this question.
144
Volume 6—Number 3
Florence Gabriel et al.
Further studies should investigate long-term advantages
of these types of teaching methods. As conceptual learning
is considered to lead to longer and deeper understanding
(Sawyer, 2006), children may benefit from learning-by-doing
style interventions in the long term. Pupils could use their
conceptual understanding to develop a more appropriate
representation of fractions, which might then help them
create further procedural knowledge. As shown in this
study, experimental classes showed improvement in certain
procedures, such as addition and subtraction of fractions with
the same denominator. Their conceptual understanding of
numerical fraction helped them deal with procedural skills.
Long-term advantages should be examined in future research.
Most theories about conceptual change underline how
knowledge about whole numbers interferes with learning
fractions (e.g., Ni & Zhou, 2005). However, Siegler et al. (2011)
offered another viewpoint on the whole number bias. They
argue that difficulties in learning fractions do not come from
comparisons between fractions and whole numbers as such
but rather in inaccurate comparisons between the two types of
numbers. Teaching methods should focus more on the common
characteristics between fractions and whole numbers, such as
both fractions and whole numbers have magnitudes that can
be ordered, compared, and represented on number lines.
In conclusion, this study showed clear improvements
in conceptual understanding of fractions in Grades 4
and 5 after an educational intervention aimed at helping
children associate fractional notations with magnitudes. The
conceptual improvements allowed children to develop some
elementary arithmetic procedures with fractions, but further
research is needed to assess the influence of conceptual
understanding on procedural learning. Further research
should investigate through longitudinal studies whether such
interventions have long-lasting positive effects and whether,
as we would hypothesize, such conceptual improvements
facilitate children’s acquisition and use of procedures and
algorithms and help avoid mathematical blockage at later
stages of the curriculum.
Acknowledgments—This research was supported by a research
grant from the Service général du Pilotage du système éducatif
du Ministère de la Communauté Française de Belgique to A.C.,
V.C., and B.R. and a travel grant from the Wiener-Anspach
Fund to F.G. Professor Vincent Carette, who helped initiate
this research project, died suddenly in January 2011. We
would like to dedicate the present publication to his memory.
We thank the reviewers for their helpful and constructive
comments.
Volume 6—Number 3
REFERENCES
Arnon, I., Nesher, P., & Nirenburg, R. (2001). Where do fractions
encounter their equivalents? Can this encounter take place
in elementary-school? International Journal of Computers for
Mathematical Learning, 6, 167–214.
Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of
math education. American Educator, 16, 14–18.
Bonato M., Fabbri S., Umiltà, C., & Zorzi M. (2007). The mental
representation of numerical fractions: Real or integer? Journal
of Experimental Psychology: Human Perception and Performance, 33,
1410–1419.
Bright, G., Behr, M., Post, T., & Wachsmuth, I. (1988). Identifying
fractions on number lines. Journal for Research in Mathematics
Education, 19, 215–232.
Byrnes, J. P. (1992). The conceptual basis of procedural learning.
Cognitive Development, 7, 235–257.
Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in
mathematical procedural learning. Developmental Psychology, 27,
777–786.
Case, R. (1985). Intellectual development: Birth to adulthood. New York,
NY: Academic Press.
Cauley, K. M. (1988). Construction of logical knowledge: Study of
borrowing in subtraction. Journal of Educational Psychology, 80,
202–205.
Cobb, P. (1995). Continuing the conversation: A response to Smith.
Educational Researcher, 24, 25–27.
De Corte, E. (2004). Mainstreams and perspectives in research on
learning (mathematics) from instruction. Applied Psychology, 53,
279–310.
De Smedt, B., Verschaffel, L., & Ghesquière, P. (2009). The predictive
value of numerical magnitude comparison for individual
differences in mathematics achievement. Journal of Experimental
Child Psychology, 103, 469–479.
Forman, E. A., & Cazden, C. B. (1985). Exploring Vygotskian
perspectives in education: The cognitive value of peer
interaction. In J. V. Wertsch (Ed.), Culture, communication and
cognition: Vygotskian perspectives (pp. 323–347). Cambridge, UK:
Cambridge University Press.
Gottfried, A. (1982). Relationships between academic intrinsic
motivation and anxiety in children and young adolescents.
Journal of School Psychology, 20, 201–215.
Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in
conceptual and procedural knowledge when learning fractions.
Journal of Educational Psychology, 102, 395–406.
Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes
onto symbols: The numerical distance effect and individual
differences in children’s math achievement. Journal of Experimental
Child Psychology, 103, 17–29.
Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing
and representation of fractions within the brain. An fMRI
investigation. NeuroImage, 47, 403–413.
Jacob, S., & Nieder, A. (2009). Notation-independent representation
of fractions in the human parietal cortex. The Journal of
Neuroscience, 29, 4652–4657.
Kallai, A., & Tzelgov, J. (2009). A generalized fraction: An entity
smaller than one on the mental number line. Journal of
Experimental Psychology: Human Perception and Performance, 35,
1845–1864.
145
Developing Children’s Understanding of Fractions
Kangas, M. (2010). Creative and playful learning: Learning through
game co-creation and game play in a playful learning
environment. Thinking Skills and Creativity, 5, 1–15.
Kerslake, D. (1986). Fractions: Children’s strategies and errors. London,
UK: NFER-Nelson.
Kucian, K., Grand, U., Rotzer, S., Henzi, B., Schönemann, C.,
Plangger, F., et al. (2011). Mental number line training in children with developmental dyscalculia. NeuroImage. doi:10.1016/
j.neuroimage.2011.01.070
Lonneman, J., Linkersdorfer, J., Hasselhorn, M., & Lindberg, S. (2011).
Symbolic and non-symbolic distance effects in children and their
connection with arithmetic skills. Journal of Neurolinguistics. doi:
10.1016/j.jneuroling.2011.02.004
Mamede, E., Nunes, T., & Bryant, P. (2005). The equivalence and ordering
of fractions in part-whole and quotient situations. Proceedings of the
29th Conference of the International Group for the Psychology of
Mathematics Education. Melbourne, Australia: DSME—University
of Melbourne.
Meert, G., Grégoire, J., & Noël, M.-P. (2009). Rational numbers:
Componential versus holistic representation of fractions in a
magnitude comparison task. The Quarterly Journal of Experimental
Psychology, 62, 1598–1616.
Meert, G., Grégoire, J., & Noël, M.-P. (2010a). Comparing 5/7 and
2/9: Adults can do it by accessing the magnitude of the whole
fractions. Acta Psychologica. doi: 10.1016/j.actpsy.2010.07.014
Meert, G., Grégoire, J., & Noël, M.-P. (2010b). Comparing the
magnitude of two fractions with common components: Which
representations are used by 10- and 12-year-olds? Journal of
Experimental Child Psychology, 107, 244–259.
Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and
rational numbers: The origins and implications of whole number
bias. Educational Psychologist, 40, 27–52.
Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford, UK:
Blackwell.
Palincsar, A. S. (1998). Keeping the metaphor of scaffolding fresh—A
response to C. Addison Stone’s ‘‘The metaphor of scaffolding:
Its utility for the field of learning disabilities.’’ Journal of Learning
Disabilities, 31, 370–373.
146
Perry, M. (1991). Learning and transfer: Instructional conditions and
conceptual change. Cognitive Development, 6, 449–468.
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other?
Journal of Educational Psychology, 91, 175–189.
Santrock, J. W. (2009). Life-span development (12th ed.). New York, NY:
McGraw-Hill.
Sawyer, R. K. (2006). Educating for innovation. Thinking Skills and
Creativity, 1, 41–48.
Schneider, M., & Siegler, R. S. (2010). Representations of the
magnitudes of fractions. Journal of Experimental Psychology: Human
Perception and Performance, 36, 1227–1238.
Schneider, M., & Stern, E. (2010). The developmental relations
between conceptual and procedural knowledge: A multimethod
approach. Developmental Psychology, 46, 178–192.
Siegler, R. S., & Ramani, G. B. (2009). Playing linear number
board games—but not circular ones—improves low-income
preschoolers’ numerical understanding. Journal of Educational
Psychology, 101, 545–560.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated
theory of whole number and fractions development. Cognitive
Psychology, 62, 273–296.
Smilansky, S., & Shefatya, L. (1990). Facilitating play: A medium for
promoting cognitive, socio- emotional, and academic development in
young children. Gaithersburg, MD: Psychological and Educational
Publications.
Smith, C. L., Solomon, G. E. A., & Carey, S. (2005). Never getting to
zero: Elementary school students’ understanding of the infinite
divisibility of number and matter. CognitivePsychology, 51, 101–140.
Stafylidou, S., & Vosniadou, S. (2004). The development of student’s
understanding of the numerical value of fractions. Learning and
Instruction, 14, 508–518.
Wilson, A. J., Revkin, S. K., Cohen, D., Cohen, L., & Dehaene, S.
(2006). An open trial assessment of ‘‘The Number Race’’,
an adaptive computer game for remediation of dyscalculia.
Behavioral and Brain Functions, 2, 1–19.
Volume 6—Number 3