INTRODUCTION:

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any

number of equal parts. When spoken in everyday English, a fraction describes how many

parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common,

vulgar, or simple fraction (examples: and 17/3) consists of an integer numerator, displayed

above a line (or before a slash), and a non-zero integer denominator, displayed below (or

after) that line. Numerators and denominators are also used in fractions that are not common,

including compound fractions, complex fractions, and mixed numerals.

The numerator represents a number of equal parts, and the denominator, which cannot be

zero, indicates how many of those parts make up a unit or a whole. For example, in the

fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the

denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates or

3/4 of a cake. Fractional numbers can also be written without using explicit numerators or

denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and

10−2 respectively, all of which are equivalent to 1/100). An integer such as the number 7 can

be thought of as having an implied denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and to represent division.[1] Thus the fraction

3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3

÷ 4 (three divided by four).

In mathematics the set of all numbers which can be expressed in the form a/b, where a and b

are integers and b is not zero, is called the set of rational numbers and is represented by the

symbol Q, which stands for quotient. The test for a number being a rational number is that it

can be written in that form (i.e., as a common fraction). However, the word fraction is also

used to describe mathematical expressions that are not rational numbers, for example

algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational

numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational)

CONCEPTS OF FRACTIONS AND DECIMALS AND LEARNING DIFFICULTIES

The decimal numeral system (also called base ten or occasionally denary) has ten as its base.

It is the numerical base most widely used by modern civilizations. Decimal notation often

refers to a base-10 positional notation such as the Hindu-Arabic numeral system; however, it

can also be used more generally to refer to non-positional systems such as Roman or Chinese

numerals which are also based on powers of ten.

Decimals also refer to decimal fractions, either separately or in contrast to vulgar fractions. In

this context, a decimal is a tenth part, and decimals become a series of nested tenths. There

was a notation in use like 'tenth-metre', meaning the tenth decimal of the metre, currently an

Angstrom. The contrast here is between decimals and vulgar fractions, and decimal divisions

and other divisions of measures, like the inch. It is possible to follow a decimal expansion

with a vulgar fraction; this is done with the recent divisions of the troy ounce, which has three

places of decimals, followed by a trinary place.

We have found it useful to use the following model from Derek Haylock and Anne Cockburn

(Haylock and Cockburn; 1989) to consider the different mathematical elements that need to

be experienced and connected in order to create full understanding of concepts.

Haylock and Cockburn suggest that effective learning takes place when the learner makes

cognitive connections.

Let us consider a particular example in early fractions. Two children are cooking, filling a tray

of 12 cake cases. They are told they can fill half each. One child looks at the tray and says,

“We can do two lines each”. The other child looks at the lines and says “That’s six because

three and three is six, like on a dice”. The children fill six cake cases each. The cooking is the

context, the tray and dice the images, the language of fractions, division and multiplication is

used and there is the opportunity to model both 12 x 1/2 = 6, 12 ÷2 = 6 and 6x 2 = 12

Problems can arise when not all the four elements are experienced or, if they are all

experienced, but they are not connected in a meaningful way. The role of classroom

talk/dialogue is to help the children make the connections themselves. This talk/ dialogue can

take the form of teacher questioning, children questioning, talk between children, explanation

of points of view…etc. The verbal accompaniment to the children’s experiences is what

allows them to frame their understanding. You can imagine that classroom talk/dialogue is the

arrows on the model that connect the four fields of experience.

Questions on proportion, in all its forms, are often answered poorly in national tests. There are

some common misconceptions that seem to be partly responsible. These include the belief

that: fractions are always parts of 1, never bigger than 1; fractions are parts of shapes and not

numbers in their own right;

a fraction such as ¾ is only ‘three lots of a quarter’, never ‘a quarter of three’; decimals with

more digits are bigger; and percentages can never be bigger than 100%.

Fractions objectives

List of images for teaching fractions

Fractions

One of the things we need to ensure, as teachers, is that children are given a variety of

experiences that allow them to engage with fractions as both the names of numbers and also

as operators. They can then consider where different fractions fit into our number system as

well as how to find a fraction ‘of’ something.

As there is such a gap between how confident children are with whole numbers as compared

with fractions, it is useful to look at how children first learn about whole numbers. Children

are exposed to whole numbers, very early on, through the counting ‘rhyme’. This exposure is

something that can happen on an almost daily basis, other people around counting out loud in

different situations and the children are immersed in it. It can be quite sometime before they

join in with parts of the ‘rhyme’ or learn the ‘rhyme’ for themselves. So at the start it is about

learning a rhyme, a list of words in a particular sequence that seem to have a certain rhythm to

them.

Through learning this rhyme children become familiar with the number names, (the language

of numbers) and the order of numbers. When the rhyme is used in contexts and children count

objects they learn about how the change of number name indicates an increase of one. Later

they learn to match these to the symbols (link language to symbols. Without a background of

being immersed in counting, can you imagine how hard it would be to calculate? Children

who arrive in reception classrooms without four to five years of counting immersion are at a

serious disadvantage and struggle as a consequence.

So when we expose children to new parts of the number system we need to think about how

we can immerse them in the numbers so that the language becomes familiar and can be

connected to the symbols, contexts and images. One way to do this is through counting. With

fractions this often doesn’t happen, children are introduced to the idea of fractions and are

expected to make sense of language and symbols without any immersion. There is a page in

the ‘images’ section on counting in fractions.

Counting can help to address the gap in understanding fractions as numbers in their own right.

What children tend to experience early on is a fraction as an operator – cut it in half, get half

of those. Understanding fractions as operators is not something particular to fractional

numbers. Children do also experience this with whole numbers – give me two of those – and

this is then linked very clearly to multiplication and division. Fractions are another piece to

this puzzle, another way of talking about a situation involving multiplication and division

PROPOSED STRATEGIES FOR LEARNING THE CONCEPTS OF FRACTIONS AND

DECIMALS

In this study, we investigated the difficulties encountered by primary school children when

learning fractions. One of the main goals of this study was to clarify the relationships between

conceptual and procedural understanding of fractions. In order to do so, a test was

administered in Grade 4–6 in classes of the French Community of Belgium. The test was

based on the different conceptual meanings of fractions, namely part-whole/partition, number,

proportion, as well as on procedural questions involving arithmetical operations and

simplification of fractions.

Globally, the results showed large differences between categories. Pupils seemed to master

the part-whole concept, whereas numbers and operations posed tremendous problems. Some

conceptual meanings, such as numbers, were less used in primary school classes. Part-whole

seems to be a concept that is widely used in the classrooms. Indeed, children performed well

in the part-whole/partition category. However, they seem to have a stereotypic representation

of fractions. Indeed, when they were asked to represent a given fraction, they mostly used a

circle or a square, even when drawing collections could have been easier (e.g., 1/7).

Moreover, when asked to select a figure representing a certain fraction, they performed better

for continuous than discrete quantities. Pupils performed well with proportion items. These

results contrast with textbooks and lessons given by teachers. In fact, the connection between

proportions and fractions is rarely made in textbooks and formal lessons, even if some aspects

of fractions are based upon proportional reasoning (e.g., the rule of three).

In the proportion category, most errors were linked to additive reasoning. For example, when

pupils are asked questions such as “3 cakes cost €12, 6 cakes cost €24, 8 cakes cost €?” the

most common error would be the answer €36. In this case, children built their answer on only

a subset of the given information and they applied additive strategies where multiplicative

strategies should be used. Mistakes linked to additive reasoning are commonly reported

during early stages of children's understanding of proportional reasoning (Lesh et al., 1988).

This kind of mistakes was common in Grade 4, but could still be observed in Grade 6.

Pupils performed poorly in the numerical category. Even if children are trained to deal with

number lines from grade 4, results showed major difficulties when they were asked to place a

fraction on a graduated number line. They do not seem to have an appropriate representation

of the quantities of fractions. Other studies have reported that many pupils experience

difficulties when asked to locate a fraction on a number line. Pupils often view the whole

number line, irrespective of its magnitude as a single unit instead of a scale (Ni, 2001). When

they are asked to place a fraction between 0 and 1, pupils often place fractions disregarding

any other reference point or known fractions. Pearn and Stephens (2004) pointed out that the

incorrect location of fractions could also be the consequence of a lack of accuracy when

dividing segments.

The lack of accuracy in children's mental representations of the magnitude of fractions seems

to be confirmed by the weak percentage of correct response for questions involving sorting

out a range of fractions in ascending order. Furthermore, mean percentage of correct

responses for comparison of fractions were very low for fractions with common numerators

and fractions no common components. When fractions share the same denominator (e.g.,

2/5_4/5), the global magnitude of fractions is congruent with the magnitude of the numerators

(e.g., 4 is larger than 2). In this case, pupils could only compare the numerators in order to

choose the larger fraction. When fractions share the same numerator, the global magnitude of

fractions is incongruent with the magnitude of denominators. Thus, pupils might not take the

incongruity into account and their judgment might have been influenced by the whole number

bias (Ni and Zhou, 2005). For fractions with no common components, pupils probably only

compared numerators and denominators separately. This strategy led to larger error rates.

Focusing now on operations, children performed well in addition and subtraction of fractions

with the same denominator, while performance dropped dramatically in addition and

subtraction of fractions with different denominators. The most common errors were dictated

by the whole number bias (Ni and Zhou, 2005). For example, when asked 3/4 + 2/5 = ?, the

majority of pupils answers 5/9. Surprisingly, results were poorer for items involving the

multiplication of an integer by a fraction, than for multiplication of two fractions. In the last

case, pupils could successfully apply procedures based on natural numbers knowledge, which

would explain higher percentage of correct response. Another surprising result was the better

performance in Grade 4 than Grade 5 when children were asked to multiply an integer by a

fraction. There might be a contamination of procedures applied to addition and subtraction

with different denominators learnt in Grade 5.

Results showed massive familiarity effects in every category. Children performed

significantly better on questions including familiar fractions, such as 1/2, 1/4, or 3/4 than on

items with less familiar fractions. This could be due to the fact that the magnitude of 1/2 is

known better than other fractional magnitudes. We do not know precisely when children start

to quantify continuous quantities in informal contexts. Bryant (1974) suggests that children

are able to understand part/part relations before part/whole relations. Relations such as “larger

than/smaller than” and “equals to” could be the first logical relationships used at the

beginning of fraction learning. Spinillo and Bryant (1991) designed experiments to analyse

how 4- to 7-year-olds use the concept of “half” in equivalence judgment tasks. Their results

suggest that using the concept of half would be the first step in relationships used by children

to quantify fractions.

Desli (1999) also investigated the role of half by examining part/whole relationships. 6- to 8-

year-olds were told that two parties had been organized and that chocolate bars would be

equally distributed among children. They had to judge if they would receive the same amount

of chocolate bars in both parties, and if not, in which party they would get more chocolate

bars. Children had ceiling performance when they could use half as a reference. In the

condition where they could not use half as a reference, only 8-year-olds had performance

above chance. Desli (1999) also showed the importance of the concept of half in the

construction of fractions quantifications. In a recent study using a fraction-based judgment

task, Mazzocco et al. (2013) showed that fractions equivalent to 1/2 were easier to

conceptualize. Moreover, children as young as 3 and 4 years old already have a good

representation of the half boundary (Singer-Freeman and Goswami, 2001). As children are

frequently exposed to 1/2 quite early in life, the familiarity of that quantity might induce a

different type of mental representations compared to other less familiar fractions. Pupils might

benefit from lessons including a larger pool of fractions. Teaching programs mostly insist on

quantities that can be divided by 2. This limited vision of fractions seems to generate

difficulties when it comes to generalization. Teachers could diversify the number of fractions

used during lessons.

Improper fractions represented another major difficulty for primary school children (Bright et

al., 1988; Tzur, 1999). The main difficulty appeared in the test when pupils were asked to

graphically represent an improper fraction or when an improper fraction was presented in an

ordering task. When pupils were asked to order 1 in a sequence involving fractions, the most

common error was to put it at the end of the sequence, even if there was an improper fraction.

This could mean that some children cannot imagine fractions can be larger than 1. This is

consistent with the results found by Kallai and Tzelgov (2009) who showed that adults have a

mental representation of what they called a “generalized fraction.” A “generalized fraction

corresponds to an “entity smaller than one” emerging from the common notation of fraction

(Kallai and Tzelgov, 2009).

References

1. Arnon I., Nesher P., Nirenburg R. (2001). Where do fractions encounter their equivalents? Can this encounter take place in elementary-school? Int. J. Comput. Math. Learn. 6, 167–214 10.1023/A:1017998922475 [Cross Ref]

2. Baroody A. J. (2003). “The development of adaptive expertise and flexibility: the integration of conceptual and procedural knowledge,” in The Development of Arithmetic Conceptsand Skills: Constructing Adaptive ExpertiseI, eds Baroody A. J., Dowker A., editors. (Mahwah, NJ: Erlbaum; ), 1–33

3. Behr M. J., Harel G., Post T., Lesh R. (1992). “Rational number, ratio, and proportion,” in Handbook of Research on Mathematics Teaching and Learning, ed Grouws D. A., editor. (New York, NY: Macmillan; ), 296–333

4. Behr M. J., Lesh R., Post T. R., Silver E. A. (1983). “Rational numbers concepts,” inAcquisition of Mathematics Concepts and Processes, eds Lesh R., Landau M., editors. (New York, NY: Academic Press; ), 91–125

5. Bonato M., Fabbri S., Umiltà C., Zorzi M. (2007). The mental representation of numerical fractions: real or integer? J. Exp. Psychol. Hum. Percept. Perform. 33, 1410–1419 10.1037/0096-1523.33.6.1410 [PubMed] [Cross Ref]

6. Bright G., Behr M., Post T., Wachsmuth I. (1988). Identifying fractions on number lines, J. Res. Math. Educ. 19, 215–232 10.2307/749066 [Cross Ref]

7. Brissiaud R. (1998). “Les fractions et les décimaux au CM1. Une nouvelle approche,” in Actes du XXVème Colloque des Formateurs et Professeurs de Mathématiques chargés de la Formation des Maîtres, (IREM de Brest), 147–171

8. Brousseau G., Brousseau N., Warfield V. (2004). Rationals and decimals as required in the school curriculum. Part 1: rationals as measurements. J. Math. Behav. 23, 1–20 10.1016/j.jmathb.2003.12.001 [Cross Ref]

9. Bryant P. (1974). Perception and Understanding in Young Children: An Experimental Approach, Vol. 588 London: Methuen