and it is important that we give them the time and

practice they need to move from decimals to percentages

with relative ease.

How to use referencing styles & TOC

Fractions: Some Common Misconceptions

Fractions: The difficulties children have with fractions

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**Assignment 2**– Developing your Subject and Pedagogical Knowledge

(Assignment undertaken in term 1 during School Experience)

Overview

In this assignment we want you to focus on a topic from the Mathematics National Curriculum that you wish to explore in some detail. You will be looking at children’s errors and misconceptions in that topic and you will need to consider how you can challenge these misconceptions and achieve progression.

First, we want you to explore what has been written about children’s learning of the topic, within mathematics educational research and professional literature.

Second, we want you to undertake some small-scale research with a small group of pupils at your School Experience school. You will need to assess the pupils’ prior knowledge of the topic, identifying misconceptions. You will then use this knowledge to plan and deliver a series of lessons that should challenge the pupils appropriately and achieve progression. Finally you will need to assess the pupils’ understanding after your teaching and, hence, review the effectiveness of the teaching strategies used.

To ensure that you develop an understanding of all the areas of the maths curriculum detailed below, you will get the opportunity to discuss your work with peers and present your findings to the maths group.

What you have to do

**1. Abstract**(~100 words)

The assignment should begin with an abstract summarising the content of your assignment.

2.

**Section 2 [ Number: Fractions ]**(1000 – 1200 words)

a. You should choose a topic from one of the Attainment Targets of the National Curriculum within the following five themes:

• Number: Fractions

*b(i) You should begin by identifying your chosen topic,*

(ii)then familiarise yourself with the way in which the topic is embedded in the National Curriculum;

(iii) what is included,

(iv) what precursors there are

(v) and how it is linked to other areas of the Mathematics National Curriculum.

(vi) Give a brief account of what you learn from this exercise.

(ii)then familiarise yourself with the way in which the topic is embedded in the National Curriculum;

(iii) what is included,

(iv) what precursors there are

(v) and how it is linked to other areas of the Mathematics National Curriculum.

(vi) Give a brief account of what you learn from this exercise.

c. You should then explore the academic and professional literature and give a detailed account of what is known – and not known – about children’s learning of your topic. In particular consider the common errors made by pupils and the logic behind these errors. Focus on what is known about how teaching approaches might reinforce these misconceptions and what might be done to challenge pupils’ understanding.

---------------- Literature Review----------------------------------------

Quantities represented by natural numbers are easily understood. We can count and say how many oranges are in a bag. But fractions cause difficulty to most people because they involve relations between quantities. What is 1/2? One half of what? If Ali and Jazmine both spent 1/2 of their pocket money on snacks, they may not have spent the same amount of money each (The teaching and learning research programme 2006).

The teaching and learning research programme ( http://www.tlrp.org) highlights in their reseach that Research on fractions has shown that many of the mistakes which pupils make when working with fractions can be seen as a consequence of their failure to understand that natural and rational numbers involve different ideas. One well-documented error that pupils make with fractions is to think that, for example, 1/3 of a cake is smaller than 1/5 because 3 is less than 5. Yet most children readily recognise that a cake shared among three children gives bigger portions than the same cake shared among five children. Because children do show good insight into some aspects of fractions when they are thinking about division, mathematics educators have begun to investigate whether these situations could be used as a starting point for teaching fractions.

Rational numbers take the form (a/b), where a and b are integers. It is well known that fractions also take the same form. Would it be in order to state that fractions and rational numbers are one and the same thing? Such statements need qualifications as they can easily be misunderstood and promote further misconceptions.

It is generally accepted that a lack of understanding of fractions in the earliest stages of children’s learning will result in difficulty that will follow them through to secondary years of schooling. Will children make sense of having twelve people sharing half a pizza or the idea of half a person sharing twelve pizzas? Such a luck of understanding can, for example, undermine their understanding of algebra since the rules that govern the manipulation of fractions are also important in solving many algebraic problems and equations (Nickson 2004, p.28-46).

This school of thought is further supported by Behr et al. (1992) who state that, ‘There is a great deal of agreement that learning rational number concepts remains a serious obstacle in the mathematical development of children’ (p.296). Some of this difficulty may be due to that fact that the idea of a fraction is one of the earliest abstract ideas with which children have to cope since there is no ‘natural’ context in which they automatically arise; for instance, a situation involving fractions that parallel the kind of everyday scenario in which children find themselves engaged in counting (Booker 1998).

Two tests on fractions set for 12 to 13 year olds and 14 to 15 year olds respectively by the Concepts in Secondary Mathematics and Science (CSMS) study reported in Hart (1981) revealed interesting observations; British primary schools emphasised the importance of notation of fraction and also expected the child to know the meaning of a half and a quarter at an early age even though addition and subtraction of fractions was not known and treated with less importance.

On the other hand, a lot of children in secondary schools appeared to have recognised the concept of ‘equivalent fractions’ in so far as ½ was concerned. According to the CSMS mathematics team, many children do not feel confident in the use of fractions and try whenever possible to apply the rules of whole numbers to operations on fractions. Children much prefer a remainder type answer than one which states a fraction and generally seem unaware that working within the set of fractions enables them to manipulate numbers in a far less restricted way than when they only had whole numbers within which to work. They still appear to be fixed within rules which apply to whole numbers e.g.

1. division of a small number by a lager one is impossible

2. multiplication makes bigger

Another interesting observation made by CSMS mathematics team is the way children find it easy to name a part (in fraction form) when the whole object is divided into exactly the same amount of parts as the denominator of a fraction being sought. It was discovered that children find difficulties in conceptualising problems which for instance, involve finding the number which is one third of twelve. For children, they expect to be asked about a third of three and not twelve. This brings cognitive conflict to what they think they have learnt. In the study, precisely 20% of younger 12-13 year olds could not answer problems of this nature; or when the problem involved finding two-thirds and the whole was divided into six parts, 36% of younger children failed.

It is quite evident that there is considerable complexity between naming fractions when no equivalence is needed and when it is.

It is quite evident that there is considerable complexity between naming fractions when no equivalence is needed and when it is.

Understanding of equivalent fractions is very important as it forms the basis on which operations of addition and subtraction of fractions hinges on. The CSMS mathematics team, cite more than 40% of the children tested had no understanding of equivalent fractions except when it involved less complex ones like doubling e.g. 1/2 = 4/8.

Many children seem to view the fraction as two unrelated whole numbers and deal with each separately as in the example below in figure 1;

Figure 1: treating numerator and denominator as unrelated numbers

It is apparent children concentrated on the two numbers in the ratio and judged the size of the entire fraction on the comparative size of the denominator or numerators; e.g. in the CSMS mathematics study, 20% of the 12-13 year olds stated that 4/14 is bigger than 2/7 clearly exhibiting the misconceptions.

Multiplication and division of fractions posed a lot of problems as well to the children as reported by the CSMA mathematics research. This problem arose due to the misconception which arises as the children try to deal with fractions in exactly the same way they are used to working with whole numbers. For instance; if a child sees 4 x 3 as four lots of three objects (which can be spread out and counted), the meaning he/she conjures from 1/3 x 6/7 is unclear and incoherent. The child finds it difficult to add meaning to; one-third part of six pieces out of seven of a pizza. Certainly, there is no easier way to teach this concept. If at all, the child tries to avoid the multiplication and use repeated addition, there is no way he/she can deal with with 1/3 x 6/7. Thus one is forced to use ‘rule learning’ methods rather than conceptualising what is happening and there is a danger of further misconceptions by using rules you do not fully understand. The learning process in such situations becomes difficult but does this mean we give up teaching the concept? Certainly not.

The use of area of a rectangle was found useful in the exploration, teaching and learning of multiplication and division of fractions. The CSMS team drew a rectangle with dimensions less than unit and asked the pupils to find the area and later on given the area to find the length. Figures 2 and 3 explain this task. The CSMS team concluded that fractions bring new rules and new possibilities and pose a lot of problems to children because they are rarely used in our everyday life.

Another difficulty children have in learning fractions and rational numbers is the fact that they take on numerous forms or constructs ( Kieren, 1988). Depending on the context one can be looking at measure or quotient or ratio number or possibly multiplicative operator. When do children know which construct is being referred to? This poses cognitive conflict in the early learning process of fractions in children.

Derived from Kieren’s part-whole and quotient constructs, three ways have been identified in the learning of fractions and these being;

1. dividing a continuous quantity into equal quantities (e.g. capacity and length);

2. dividing a number of objects into equal sets (e.g. ten mangoes divided into two groups of five);

3. dividing single whole into a number of equal parts (e.g. a chocolate bar into three equal portions).

Whereas the part-whole approach looks at how many parts of a whole there are, the quotient looks at both kinds of division ( partitive and quotative) and takes into account the different meanings the numerator and the denominator can have (Nickson, 2004). There is a lot to comprehend about fractions and it is worth giving children some ample time to develop their learning process without prejudice.

Streefland (1991) identified five approaches to the teaching of the concept of fractions namely: structuralist, mechanistic, pragmatic structuralism, intermezzo and the realistic approach and the mechanist approach was classified as one exacerbating more problems in children learning fractions. On the whole, there are two sources of the fraction problem, identified as:

a) extreme underestimation of the complexity of this area of learning for children;

b) the mechanistic approach to fractions, detaching it from reality and focusing on rigid application of rules ( Streefland ,1991).

The difficulties that can arise in using illustrations with fraction work has been explored by Nickson (1998). In her study, she found out that most children were able to successfully identify one quarter of a pizza in a drawing and considerably fewer were able to identify one quarter when the illustration was a race track represented by a long thin rectangle. This was done at primary level and just showed how difficult it is at the initial stages of learning fractions.

The works of Alverez (1994), in Mexico primary schools which asked children to make sense out of 2/3 +2/4 and invent a problem that depicted the aforementioned sum often brought out strange results. They found it easy to solve the problem using the mechanistic approach as classified by Kieren (1988). In the same year, the research by Arnon et al. (1994) which was based heavily on the Theory of Learning Systems of Nesher (1989) which also is based on the Piagetian notion that children learn new concepts by reflecting on the physical handling of objects. This study used two classes of children as a sample; the group that was allowed to explore fractions on their own without algorithmic help exhibited more success rate than the set that was taught the algorithmic way. This was in support of Dubinsky (1991) theory that the learning of new concepts begins in the internalisation of an action which if left to develop on its own, enhances higher learning rates. Allow the children to make sense of their own learning process rather than showing pictures that might enhance further misconceptions in future.

This idea was also supported by Maher et al. (1994) in the study of 9 to 10 year olds as they worked on placing fractions ½, 1/3, ¼, ……1/10 in the interval 0 to 1 on the number line. The learning process was highly successful as the children argued on their own and in the end reached a consensus that allowed them to reconstruct their thinking so that their learning is meaningful ( Streefland 1991a).

Newstead and Murray (1998) in their research exposed a misconception in which when chidren were asked to solve the sun; 2 divide by ½ , they could not read it as ‘How many halves are there in 2?’. Had they known this fact, it would have been easy for them; but they looked at the sum as fractions and not normal division.

Some researchers are of the strong opinion that the introduction of written symbols and algorithms with fractions should not take place until children have experienced fractions as single quantities and not separate numbers split in numerator and denominator ( Barody and Hume 1991, Empson 1995). Also important is the fact that children should appreciate that fractions take up different forms depending on context.

This notion is also supported by Pirie et al. (1994) where 15 year olds was encouraged to have their own images from a questionnaire with six questions. No symbols or algorithms were used. Out of this exercise, children came up with four major images:

1. division e.g. a quantity divided by another quantity

2. part of a whole e.g. a fraction is a section of a whole

3. number – it is a number that has not been put into decimal

4. way of writing e.g. a number over another number

the general conclusion from this study was that the pupils who had a multiplicity of images and could move flexibly back and fourth amongst the images had least difficulties in understanding fractions and algebra later.

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3. Section 3 (300 – 500 words)

You should look at the textbook resource used in your SE school [or one of the text books used by the school] and critique its presentation of your chosen topic;

Consider all aspects of the textbook approach: style, content, explanations, coverage of concepts, style of questions, etc. Focus on how the textbook addresses, or potentially reinforces, the misconceptions known to be prevalent amongst pupils.

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4. Section 4 (1000 - 1200 words)

This section is about working with a small group (no more than 4 - 6) of pupils focusing on your chosen topic.

a. You should give an account of how you established the state of the children’s understanding of the topic, which will involve you undertaking some form of assessment and evaluation. It should involve the use of good diagnostic assessment, which should be supported by discussions with the pupils. Do your findings support what you discovered in your research? What are the implications for your teaching?

b. As a result of the assessment you will need to plan a series of activities, tasks and further assessments for your group. You will need to provide a commentary explaining what you were trying to do at each stage and how these built upon your findings in Section 2. This may be in the form of individual plans for each pupil based on your assessment of their needs.**[http://www.partnership.mmu.ac.uk/cme/Student_Writings/CDAE/SarahMayson/SarahM.html]- Typical series of lessons **

c. You will need to include all plans and provide copies of all the activities, tasks and assessment materials you used in appendices. The plans need to be clearly annotated to describe your rationale for the activities that you decided upon. If available, include any written feedback from your mentor or class teacher.

d. You should include some examples of pupils’ work that illustrates their thinking, misconceptions and learning.

e. You should include an account of how you assessed the pupils’ understanding after the various activities, this might be similar to the initial diagnostic assessment. One way of recording the progress of the pupils might be to keep mini reports of each pupil and your conversations with them and how their understanding does (or does not) develop over the time you spend with them.

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5. Section 5 (~300 words)

You should end the assignment with a conclusion/summary section.

You should collate the material for this assignment in an appropriately structured portfolio.

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Submission

You will need to complete this assignment so that it can be handed in to Reception by Friday 14th December 2006.

--------------------------------------------------------------**My references**

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Books (Author, year, title with edition: subtitle .Place, publishers)

Alverez, M. E. (1994) Various representations of the fraction through a case study. Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education.2, 16-23

Arnon, I., Dubinsky, E. & Nesher, P. (1994). Actions which can be performed in the learner's imagination: The case of multiplication of a fraction by an integer. In J. P. da Ponte & J. F. Matos (Eds.), *Proceedings of the Eighteenth International Conference for the Psychology of Mathematics Education II, (pp. 32-39). Lisbon, Portugal.*

Behr, M. J., Harel, G., Post, T. and Lesh, R. (1992) Rational number, ratio and proportion.

Behr, M. J., Harel, G., Post, T. and Lesh, R. (1992) Rational number, ratio and proportion in Nickson, M. (2004) Teaching and Learning Mathematics 2nd Edition: a guide to recent research and its applications. London, Continuum. 28-46.

Booker, G. (1998) Children’s construction of initial fraction concepts. Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education. 2,128-35.

CSMS mathematics team, In PGCE mathematics 2005-6;Assignment 2-Reader; Nottingham

CSMS mathematics team, In Hart, K. (1981) Children's Understanding of Mathematics: 11-16, London: John Murray

Dickson L, Brown M and Gibson O (1984) Children Learning Mathematics: a teacher’s guide to recent research, The Alden Press, Oxford pp. 274-320

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall, (Ed.) Advanced Mathematical Thinking, Dordrecht: Kluwer, 95-126.

Grouws, D. (ed) Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan Publishing Company. 296-333.

Hart, K. (1981) Children's Understanding of Mathematics: 11-16, London: John Murray

Keiren, T. (1988) Personal knowledge of rational numbers: its intuitive and formal development.

Nesher, P. (1989). Microworlds in Mathematical Education: A Pedagogical Realism. Knowing Learning and Instruction. L. B. Resnick. Hillsdale, NJ, Lawrence Erlbaum Associates: 187-215

Nickson, M. (1988) What is multicultural mathematics? In Enerst, P. (ed) Mathematics Teaching: The state of the art. Lewes, Sussex; Falmer Press.

Nickson, M. (2004) Teaching and Learning Mathematics 2nd Edition: a guide to recent research and its applications. London, Continuum

Streefland, L. (1991) Fractions in Realistic Mathematics Education, Kluwer Academic Publishers, London**Websites**

http://www.tlrp.org/pub/documents/no13_nunes.pdf

http://www.partnership.mmu.ac.uk/cme/Student_Writings/CDAE/JulieSutcliffe.html

http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol2Amato.pdf

http://www.people.ex.ac.uk/PErnest/pome14/rowe.htm

http://www.qca.org.uk/itl.aspx

#################Ideas Ideas################### Ideas Ideas####################

Where you can look for ideas

You will have been provided with an Assignment 2 Reader, which contains research on the five themes and a list of sources which might help you get started in your research – We strongly recommend you find the time to read all the articles in this reader as it is important for your teaching that you are aware of the common misconceptions pupils have in all these areas of the curriculum.

Your course text (Nickson, 2004) will give you a broad overview of the main themes in each of the areas and it is up-to-date. You will be able to get further references and many of these will cite some of the sources below. Apart from the two papers by Vlassis, the articles are from three edited books. There is some overlap between them and the first four chapters come from the landmark CSMS research conducted in the 1970s. This datedness does not make the work redundant as you will see that much of what is written is still very pertinent.

Hart, K. (1981) Fractions in K. Hart (Ed.) Children’s Understanding of Mathematics: 11-16. John Murray, London, pp. 66-81

Dickson L, Brown M and Gibson O (1984) Children Learning Mathematics: a teacher’s guide to recent research, The Alden Press, Oxford pp. 274-320

Behr, M. J., Harel, G., Post, T. and Lesh, R. (1992) Rational number, ratio and proportion.

Behr, M. J., Harel, G., Post, T. and Lesh, R. (1992) Rational number, ratio and proportion in Nickson, M. (2004) Teaching and Learning Mathematics 2nd Edition: a guide to recent research and its applications. London, Continuum. 28-46.

Booker, G. (1998) Children’s construction of initial fraction concepts. Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education. 2,128-35.

CSMS mathematics team, In PGCE mathematics 2005-6;Assignment 2-Reader; Nottingham

CSMS mathematics team, In Hart, K. (1981) Children's Understanding of Mathematics: 11-16, London: John Murray

Grouws, D. (ed) Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan Publishing Company. 296-333.

Hart, K. (1981) Children's Understanding of Mathematics: 11-16, London: John Murray

Keiren, T. (1988) Personal knowledge of rational numbers: its intuitive and formal development.

Nickson, M. (2004) Teaching and Learning Mathematics 2nd Edition: a guide to recent research and its applications. London, Continuum

You should also make use of professional journals (MT or Maths in Schools), Ofsted subject reports, GCSE Examiners’ Reports and QCA standards reports to inform your work

-------------- Has to do with piaget and vigotsky------------------

http://www.caaweb.co.uk/files/research/piaget_vygotsky.pdf