The development of the common fraction concept in Grade Three learners

This paper reports on a study undertaken on the development of the common fraction concept in Grade Three learners. According to the principles of a problem-centred approach to teaching and learning, learners were encouraged to discuss, reflect on and make sense of the mathematics they were doing while solving problems set in real-life contexts as vehicles for learning. The results indicate that learners are able to develop a stable common fraction concept and a deeper understanding of the concept.


Introduction
Recent research shows that some learners have great difficulty in solving problems involving common fractions (Pothier & Sawada, 1990;D'Ambrosio & Mewborn, 1994;Piel & Green, 1994).The common fraction concept appears to be one of the most difficult mathematical concepts to teach in the primary school, and instruction that emphasises applying rote procedures and following set rules has not been very effective.In such instructional programmes learners fail to develop the concept of a fraction and to gain a sense of fractional number (Kamii & Clark, 1995;Strang, 1990;Kieren, 1988, as cited in Steffe & Olive, 1991).
Many researchers (Murray, Olivier & Human, 1996a, 1998;D'Ambrosio & Mewborn, 1994;Empson, 1995;Kamii & Clark, 1995;Mack, 1995;Piel & Green, 1994) have been looked at alternative approaches to the teaching and learning of the common fraction concept.Understanding of common fractions is seen as an important foundation for building understanding of decimal fractions, percentages and our decimal measurement system, as well as (in later schooling) for algebra and algebraic manipulations, probability theory and statistics (Lebethe, Le Roux, Murray, Nkomo, Smith, Vincent & Williams, 1997).It is therefore of vital importance that this concept be soundly developed and that it become stable.

Theoretical Framework
The research intervention described here was based on a problem-centred approach to mathematics learning and teaching where learners were encouraged to construct their own knowledge by reflecting on, discussing and engaging in the mathematics they were doing (Murray et al. 1998).Learners were presented with problems that were meaningful and interesting to them, but that they could not solve with ease using routine procedures or drilled responses.The facilitator did not demonstrate a solution method, nor did she steer the activity in a direction that she had previously conceived as desirable.All the learners were expected to become involved in the problem and they were encouraged to attempt to solve the problem using their own invented methods.

Choice of Problems
The problems posed to learners covered the two subconstructs identified by Murray et al. (1996a) and D 'Ambrosio and Mewborn (1994), as well as the different ways in which fractions can be used.The two sub-constructs are the part-whole relationship between the fractional part and the unit, and the idea that the fractional part is that quantity which can be iterated a certain number of times to produce the unit.
Fractions are used in different ways and have different meanings, for example: • a part of a whole (half of an orange); • a part of a number of objects (half of the learners are boys); and • a relationship (a tenth of his income is given to the church).
Attempts were made to prevent limiting constructions from developing by posing problems that cover these different sub-constructs and meanings within a reasonable time.For the same reason, the fractions addressed also immediately included thirds, fifths etc., and not only halves and quarters (Murray, Olivier & De Beer, 1999;Empson, 1995).

Social Interaction
Social interaction created opportunities for learners to talk about their thinking and this encouraged reflection.As Von Glasersfeld (1991) points out, "[f]rom the constructivist point of view, there can be no doubt that the reflective ability is a major source of knowledge on all levels of mathematics … To verbalise what one is doing ensures that one is examining it.And it is precisely during such examination of mental operating that insufficiencies, contradictions, or irrelevancies are likely to be spotted."Also, "… leading students to discuss their view of a problem and their own tentative approaches, raises their self-confidence and provides opportunities for them to reflect and to devise new and perhaps more viable conceptual strategies " (p. xviii, xix).

Students' own constructions
Learners were expected to create their own representations of fractions.They were presented with sharing situations where a remainder also had to be shared out, for example sharing out four chocolate bars equally among three friends.In order to encourage learners to develop a need for and a concept of a fraction for themselves, prepartitioned materials and written fraction symbols for fractions were held back.By withholding prepartitioned materials and manipulatives learners were encouraged to allow their own thinking rather than the manipulatives to dictate the situation (Empson, 1995;Pothier & Sawada, 1990).

This Study
The primary aim of the research was to investigate how learners develop their own conceptual and procedural knowledge of common fractions, using real-life situations.It took the form of a longitudinal teaching intervention conducted with Grade 3 learners in two schools in Fort Beaufort, Eastern Cape.
Over a period of nine months in 1999, 41 lessons were presented to the Grade 3 learners.The researcher took the class once or twice a week, while the teacher continued teaching the class during the other mathematics periods.The aspects covered included the development of the common fraction concept through equal-sharing situations, introducing realistic problem situations for operations involving fractions, comparison of fractions, equivalence, and introducing the fraction symbol.The problems covering the abovementioned aspects were used as vehicles for learning and were based on real-life situations so as to encourage learners to use and build on their informal and previous knowledge.
The learners worked in approximately equal ability groups and they were encouraged to solve the problems at whatever level they felt comfortable with.The main reason for using equal ability grouping was that those learners who were still on Van Hiele's Level 1 of understanding needed a longer time to solve the problems, whereas the learners who were working on a more abstract level could proceed at their own pace.Learners were also encouraged to share their ideas with all members of their group and class, so that the groups were not isolated.In order for the learners to develop an understanding of a particular structure, the materials repeated certain problem structures at regular intervals.This provided repeated opportunities for learners to make sense of a particular structure.
It must be noted that learners at both schools had been exposed to a minimum amount of prior learning of the common fraction concept in Grade 2.

Results
The data was collected and analysed according to the five aspects covered during the research.

Development of the common fraction concept
Four solution categories emerged from the data collected: a.The learners used economic sharing.They shared out the maximum number of whole units, divided the remaining unit into an appropriate number of fractional parts and then shared them out.They recorded their answers in a variety of ways, namely drawings, numbers and drawings, numbers and no drawings or drawings and words.
For 7 ÷ 2, Caroline shared out the units and the remaining unit was divided into halves.She gives each friend a half.Caroline recorded her answer using the correct fraction notation.b.The second category that emerged was partitioning all the units into an appropriate number of fractional pieces and then sharing out the parts from each in turn.Learners only used drawings to record their solutions.
For 7 ÷ 3, Kate divided each unit into thirds and said that each friend would receive seven pieces.

Kate explained as follows:
Teacher: Kate, can you explain what you did here?Kate: I took each chocolate and divided it into three pieces.Then I gave each child a piece.I did that with each chocolate.Teacher: So Kate, how much did each child get?Kate: Seven Teacher: Seven whole chocolates?Kate: No, seven pieces c.In the third category, the learners shared out the maximum number of whole units, but divided the remaining unit into an inappropriate number of fractional parts.Some learners chose to ignore the remaining part, while others chose to share the remaining part.

Whe
or the ame problem, these answers had to be compared and inv where the learner had shared out the incorrect number of units or where the researcher could not identify the mistake that the learners had made.tially, some learners had difficulty in 'cutting up' th example, if learners had to cut a bar into fifths, they needed to make four 'cuts'.The learners would sometimes make five 'cuts' which would result in six pieces and not five.One possible reason for this could be the fact that in everyday life, when slicing bread, a person who wants four slices will cut four times.Some learners took quite a few attempts to master this, which concurs with the findings of Ball (1993), as cited in Pitkethly and Hunting (1996).
By the end of the research learners had developed the concept of a common fraction.Th ntify the part-whole relationship between the fractional part and the unit.All the learners were able to solve the equal-sharing problems correctly.

Social knowledge
communicating abo fractions and writing of fractions using fraction notation (Murray et al. 1996a).
Although some fraction notation had been taught to learners in both schools, there w orrect naming of the fractional parts.It was evident from the way the learners named their fractional parts that this social knowledge had not yet become stable.Some learners referred to fractional parts as 'pieces,' while others called all fractional parts 'halves'.These limiting constructions could be due to the teaching and teaching materials that had been used in Grade 2 at School B. It cannot be determined to what extent these limiting constructions could have originated from the learners' pre-school or outside school experiences.
In this study, the learners were first exposed to the fraction names in words.The symbols were introduc er.
The learners did not have any difficulty in applyi posed to.Some learners did, however, switch between using words and fraction symbols.Early on in the research learners were able to name certain fraction parts on a fraction wall when asked to do so, but when solving equal-sharing problems where the solution included a fraction, a few of the learners still referred to the fractional part as a 'piece'.This could have been because the social knowledge had not yet become stable or the terminology of fraction names had not yet become part of their vocabulary.It could have been that the learners did not see the necessity of naming the fractional part correctly as they simply regarded it as 'a piece'.By the end of the research, the majority of learners were able to use the social knowledge correctly.

Equiv
ractions are as the four arithmetic operations are taught (Kamii & Clark, 1995).By using problems as the vehicles for learning, discussions around the different solutions produced by learners resulted in equivalence and comparison of fractions occurring early on in the study without learners being given a 'rule' to be applied by rote with little understanding on their part.
The following types of problems were presented to the learners: What would n learners produced different answers f s estigated for possible equivalence.For example, when four sausages have to be shared equally among six children, the following answers might appear: ving common e 4: Elaine -equivalent fractio ners were, however, able to compare fr ed which would be the bigger piece of chocolate bar or, they explained that would be the bigger piece because quarters are bigger than fifths therefore if one takes two quarters it must be bigger than two fifths.
Towards the end of the research, when solving problems where the solution could be simplified, som rners were able to form equivalent fractions without the use of the fraction wall.One learner talked about "seeing it in her head".Murray et al., 1999).Learners are frequently presented with rules and rote procedures to follow without understanding.This, according to Mack (1990), causes misconceptions to occur.Many learners are simply taught to "invert the fraction and multiply".This rule might then be partially or incorrectly used, for example: 20 ÷ How many cars can they make from 20 metres of wire?

Division basic operations invol fractions (see
d to draw the solution, while thers were able to solve the problems using numbers

Addition and iteration of fract
As iteration is closely linked with the concept addition of fractions, these two sectio together.Some of the problems included in the worksheets provide exposure to two of the multiplicative structures, namely repeated addition and rate.
Problems such as the following were presented to the lea Two netb each child 2 1 of an orange.How many oranges does she need?difficult.The researcher repeatedly found that the learners' inadequate number concept development and lac study showed that learners were able to achieve a significant degree of success in le common fraction concept and that elopment of the fraction concept can be suc r.Some lea the operations example the ad Some learners who were unable to work on an abstract level had difficulty in completing these.T set the r o a c n arne chocolate that had been eaten.They were then able to work out the total amount of chocolate that had been eaten.By doing this, the learners were able to make sense of the situation.Mack (1990) and Piel and Green (1994) found that learners who could not solve problems abstractly were able to do so once the problems had been put into a real-life context.

Part of a number of objects
The second aspect of the first sub-construct, tween the fractional part objects, was introduced as early as the third lesson.Murray, Olivier, and Human (1996b) felt that if learners did not meet fractions used in this way, they might find this concept difficult to manage at a later stage.
Here is an example of a problem that was posed to the learners in the eighth month of the study (Septem Vusi's book has 88 pages.He says: "I have read more than ha Many of the learners found this section very k of experience with different problem types, especially division, hindered them when they had to deal with the problems in the later part of the fractions programme.For example, for some learners finding half or a third of 60 minutes was impossible unless they drew 60 objects, in spite of the fact that they understood the concept and could find half and a third of smaller numbers.

Conclusions
The outcomes of the developing a stab they had the ability to solve problems involving common fractions with confidence.By stable, we mean that a learner can be depended on to produce the same correct response in different situations.He/she is able to explain his/her thinking and is not led astray by conflicting suggestions from other learners.The problem-centred approach afforded learners the opportunity to construct their own ideas and to develop a deeper understanding of the concepts involved.The approach encouraged learners to make sense of realistic problems and to invent their own procedures in an atmosphere of discussion and argument.
Initially, learners based their methods on their informal knowledge of sharing.We are of the opinion that the dev cessfully based on this informal knowledge and recommend that this be considered as a starting point for the teaching of common fractions.Learners fully understood equal-sharing and had the ability to deal with equal-sharing situations, although at the outset a few learners had to be encouraged to share out the remaining unit.The development of the fraction concept was successfully initiated using equal-sharing situations, as learners were able to construct their own idea of fractions through their own thinking and actions and in collaboration with their peers.
To begin with, knowledge of the fraction names and symbols (with the exception of halves and sometimes quarters) was generally poo rners called all fractional parts 'pieces' while others referred to any fractional part as 'a half' or 'quarter'.This limiting construction could have been the result of previous teaching.It became evident that social knowledge relating to the common fraction concept took a long time to become stable, and for the terminology to become part of the learners' vocabulary it needed to be repeated often.It also needed to be transmitted to the learners in a variety of ways.

EM ildren Learn Fractions" in Teaching
A tional LE has been the traditional practice, but can be successfully introduced in the lower grades in the way it was done in this study.The concept was developed gradually by posing many real-life practical experiences of comparing fractions and putting together fractional parts.Learners, to a lesser or greater extent, were able to grasp the idea of equivalence, albeit gradually.Most learners were able to identify equivalent fractions by the end of the intervention.They usually solved the problems either by consulting the fraction wall or by drawing the solution, and not by memorisation of an equivalence 'rule' ("what you do to the top, you do to the bottom").
Learners were able to compare fractions successfull ction wall to do so.At times, though, learners' drawings were not accurate enough causing solutions to be incorrect.Later on in the intervention, some of the learners were able to compare unit fractions by reasoning.
Traditional methods of teaching operations involving fr rote procedur s and r les with lit e or no understanding.These operations were usually presented in a particular sequence.However, during the intervention, learners were challenged with a variety of problem situations.The problems covered the different meanings of fractions as well as the different ways in which fractions can be used.The problems were set in real-life contexts to facilitate understanding, to provide a sound basis for building fraction knowledge and to enable learners to build connections to existing knowledge.The researcher found the problems to be powerful vehicles for preparing the way for operations involving fractions.Learners used their own diagrammatic representation of the physical situations to solve the problems.They worked co-operatively, spending much time discussing, arguing and reflecting on the methods of others.
By the end of the intervention learners were able to alternate K confidence.The learners, working informally, were generally able to add fractions, both with like and unlike denominators, and to subtract fractions with like denominators.They successfully solved repeated addition and rate problems, two of the multiplication types.Division problems that are traditionally considered the most difficult basic operation were solved successfully.By making use of diagrams, the learners made sense of the situation and invented their own procedures to work out the problems accordingly.Learners were also able to construct solutions for problems where the fractions were part of a collection of objects.The concept of iteration assisted learners to convert improper fractions to proper fractions and vice versa.
The Grade 3 learners made great strides with respect without being given rote procedures to follow and rules to learn and apply.They coped relatively well with all aspects of common fractions dealt with during the research, and for this reason the researcher strongly suggests that this method of teaching fractions be tried more readily at all levels of primary school.This may, however, require a major reorientation in the thinking of many teachers.
From the research it is evident that learners have the ability to think and construct knowledge fractions, both individually and through interacting with others.Solving real-life problems set in a context that makes sense to them afforded learners the opportunity of making sense of and understanding the mathematics that they were doing.

Figure 3 :
Figure 3: Martin (7 ÷ 2) For 7 ÷ 3, Martin shared out the maximum number of units and then divided the remaining unit into the inappropriate number of fractional parts (quarters).He gave one quarter to each of the friends, and chose to ignore the remaining part.

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Figur ns study, learners we e bl t solve division by a fraction using real-li .r a e o fe situations This cor ren are making different animals and cars responds with the findings of Mack (1990) and Murray et al. (1999).Problems such as the following were presented to the learners: wire.The children have 20 metres of wire.

Figure
Figure 6: Gail Later on, activities involving the addition of fractions without embedding in a word problem were introduced, for dition chains: that the teaching of equivalent fractions need not be delayed until the higher grades, as y.Initially, they used drawings or the fra actions usually result in learners applying e