PGM310

PGM310 – Managing Professional Change and Development U13026864,563 words

Contents

An introduction – identify the problem -------------------------------------------- 2

Description – gather data ------------------------------------------------------------ 4

Critical Review – interpret data ----------------------------------------------------- 8

Reflective Discussion – act on evidence -----------------------------------------12

Conclusion – evaluate results -------------------------------------------------------16

References -------------------------------------------------------------------------------19

Appendix A – Research Outline -----------------------------------------------------20

Appendix B – Consent Form Adult Participant ----------------------------------21

Appendix C – Consent Form Young Person Participant ----------------------22

Appendix D – Intervention Action Plan --------------------------------------------23

Appendix E – Interview Schedule ---------------------------------------------------24

Appendix F – Questions ---------------------------------------------------------------25

Appendix G – Answers from year 6 student --------------------------------------26

Appendix H – Answers from mathematics specialist ---------------------------32

Appendix I – Observed Lessons -----------------------------------------------------37

Appendix J – Video lesson by W Fortescue-Hubbard --------------------------40

PWR – Mathematics 4,563 Words 1


An Introduction – identify the problem

How does the teaching of fractions in primary schools impact pupil progress in this area?

The research outline (appendix A) together with the interview schedule (appendix E) will consider whether the way fractions is taught affects children’s understanding of fractions.

It will try to ‘’recognise common pupil errors and misconceptions in fractions, and to understand how these arise, how they can be prevented, and how to remedy them’’ (DfEE, 1997, p. 36)

It will seek to find other factors which may have an impact on their understanding of fractions.

The education secretary Michael Grove is keen to see younger children studying fractions. As mentioned in the Guardian (Monday 8 July 2013) ‘’Five-year-olds to be taught fractions for the first time, for a solid grounding at an early age in preparation for algebra and more complex arithmetic.’’

Fractions has its own subheading in the new curriculum for each year group which clearly means it is critical to children’s understanding. In years 1, 2 and 5 the subheading is fractions, in years 3 and 4 the subheadings is fractions and decimals and in year 6 the subheading is fractions, decimals and percentages.

An overview of the Primary Curriculum in Mathematics states:In year 1, children must be able to recognise & use 1⁄2 & ¼.

By year 2, children must be able to find and write simple fractions and understand equivalence of e.g. 2/4 = ½.

At the end of year 3, children must be able to use & count in tenths, recognise, find & write fractions, recognise some equivalent fractions, add/subtract fractions up to <1 and order fractions with commonDenominator 2/4 = ½.

For year 4, children must be able to recognise tenths & hundredths, identify equivalent fractions, add & subtract fractions with common denominators, recognise common equivalents, round decimals to whole numbers and solve money problems.

In year 5, Compare & order fractions, add & subtract fractions with common denominators, with mixed numbers, multiply fractions by units, write decimals as fractions, order & round decimal numbers, link percentages to fractions & decimals.



For year 6, children need to be able to compare & simplify fractions, use equivalents to add fractions, multiply simple fractions, divide fractions by whole numbers, solve problems using decimals & percentages, use written division up to 2dp and introduce ratio & proportion (Based on National Curriculum published in September 2013)



Description – Gather data

For my research approach, I used a variety of data collection techniques as I felt this was necessary to get a good idea of what was happening in my research area. For my primary research, I used informal interviews, questions as well as lesson observations. For my secondary research, I used newspaper article from the Guardian, National Curriculum overview for year 1 to 6, circular on misconceptions and a text book called Teaching Mathematics with Insight by Ann D. Cockburn.

I tried to take account of all ethical considerations. I asked permission of the people who participated in my research (appendix b and appendix C). I tried to be as objective as possible. I have tried to accurately represent what I observed or what I was told. Finally I have tried not to take interview responses out of context.

I concentrated on the best students in mathematics, specialist mathematics teacher and an excellent performing class teacher who was very highly regarded by the head teacher as I had been a little disappointed in the teaching of mathematics at all my school placements when compared with the teaching of English. My own teaching showed that I too found English easier and more inspiring to teach than Mathematics. I decided to explore which topic of mathematics was causing the biggest challenge for both children and teachers.

By conducting informal interviews, I got a feel for what students and practitioners were facing on a daily basis.

Informal interview with a year six student from the higher mathematics set:

1. What area of maths do you find hard?

Problem solving which involves 2 steps or more as I do not know which operation to use.

2. What is 1/7 of £49?I don’t know. I am not sure what 1/7th means.

3. Draw me a circle, can you show me 1/7 in this circle?He draws 7 lines going from the centre to the circumference and shades one of the sectors.He understands that £49 will be shared amongst 7 children. The operation needed is £49/7 = £7

4. What is 1/7 as a percentage?I don’t know what percentage means



5. Do you know what a percentage is out of?100 so is it 1/7 x 100= 100/7 = 14 2/7 ?

Informal conversation with the maths specialist:

He has been especially employed by the school to improve the children’s SATS result.

Which area of mathematics do the children find most challenging?

Without a doubt, all the children struggle with fractions, in particular multiplication and division of fractions.

I asked the same year 6 boy whose level is 4c (appendix G) and the specialist maths teacher (appendix H) to complete a set of questions. They both had the same set of questions (appendix F).

A small extract of some of their answers is shown below:

No Answer year 6 student Answer Maths Specialist

1 Use the fraction button on the calculator. It will provide the simplest form.

I know I have the simplest form when I cannot find any more common factors.

2 2/3 =12/18 A fraction which is equivalent to 2/3 are many. By multiplying top and bottom by the same number, all these will be equivalents.

I needed to see 3 x what =18?

Using my knowledge of 3 times table, I knew 3x6=18

As I multiplied the bottom by 6, I had to multiply the top by 6 so 2x6=12, hence my equivalent fraction was 12/18



3 2/5 x X = 32

Use algebra: 2X = 32 x 5

X = 16 x 5 = 80

I can calculate any fraction of a total as I understand the algebraic form I need to put this in.

a/b x X = Y is my general term

A further six children from my lunch time maths club were given the same set of questions. Their levels ranged from 3a to 4b.

Finally a year 7 student whose level is 6b was given the questions.

Since these seven children only partially completed the questions, their answers have not been attached but they have been discussed.

I observed four lessons linked to fractions (appendix I), they show how fractions are taught at school. These lessons are based on upper KS2. I wanted to find out if the way a topic was taught had any effect on learning.

Below is an extract of one of the lessons:

How would you order this set of fractions?

½ 2/5 7/10

Step 1: write our tables until we find a number which appears in all the tables.

2 times table: 2,4,6,8,10,12,14

5 times table: 5,10,15,20,25,30

10 times table: 10,20,30

Step 2: convert all the fractions until the denominator is the same, remember to apply the golden rule: what you do to the denominator, do the same to the numerator

½ = 5/10

2/5 = 4/10

7/10



Step 3: order the fractions

4/10 5/10 7/10

Step 4: convert back to the original fraction

2/5 ½ 7/10



Critical review – Interpret data

The informal interview was a verbal communication process. I asked some questions which led to further questions and I noted down their response.

The year six student said that he was not sure which operation (addition, subtraction, multiplication or division) to use in multi step problems. Quinlan (2014) makes a good case for the memorization of certain kinds of knowledge, such as the multiplication tables, in order to have at our disposal information that can help us solve problems, rather than relying on search engines to help us solve very specific problems.

He did not understand what 1/7th meant. He seemed to understand the questions better when I applied a pictorial emphasis, for example, when I asked him to imagine a cake which has seven parts and he can have one part of it, he had no problem understanding what I meant and he came up with his own version of the answer.

The national curriculum subheadings for teaching years 1 to 6, as noted earlier, show a clear link between fractions, decimals and percentages. With this in mind, when I asked him to express 1/7th as a percentage, I had hoped he would see the link but there was a blank look. Again, I had to reword my question to get him thinking. I was left thinking ‘’are the concepts too abstract for the children or have they not understood the teaching?’’

The mathematics specialist was very clear that fractions was causing a real problem for the children. He was not able to explain why there was a problem, only that there was. He stated that in his twenty years of teaching, fractions seemed too hard for the children to grasp. I asked if the way fractions is taught could be a problem, he felt that yes that could be a contributing factor. He also felt there were misconceptions which may not have been addressed early enough. I had formed my questions using a circular called ‘’Misconceptions with the key objectives’’ which had been put together by a group of primary school teachers. “The aim of this working group was to produce research material and guidance for teachers to support the planning for misconceptions.”

A quote from Nuffield (1991):

“It has been said that ‘fractions’ have been responsible for putting more people off mathematics than any other single topic. In fact the very word fraction has been known to make strong men wince!”

The same year 6 student (Level 4b) who is in the higher maths group struggled to give clear explanations on some of the questions. His answer to



question one shows a reliability on calculators rather than an understanding of fractions. He did not attempt the explanation part of the questions. As I studied his answers, I began to wonder if the set of questions I had prepared based on what is taught in primary school for that year group, by looking at the national curriculum and studying misconceptions, was too hard or simply not written in an user friendly way.

A bright grammar school year seven student (Level 6b) gave her answers to the first two questions.

I know I have the simplest form of a fraction when the numerator and denominator can no longer be divided by the same number.

12/18 is equivalent to 2/3 and has a denominator of 18. I did this because as the original denominator was 3 and the new denominator I was given was 18, I had to divide them to find 6. After this I did 6x2 to find 12.

She was able to give logical and clear explanations but she could not understand some of the questions.

Below is an extract from the circular, it is bewildering that a year 7 student who is considered well above average in mathematics could not make sense of the questions which was taken directly from the circular.

Objective Misconception Key Questions

Reduce a fraction to its simplest form by cancelling common factors.

Lack of understanding that fractions can be ‘equivalent’ (i.e. same size but split into different number of equal parts). Therefore children struggle with concept of reducing to a simpler equivalent.

Some children may remember the ‘more abstract’ rule ‘ whatever you do to the bottom, do to the top’ (and vice versa) but due to lack of understanding why this works, cannot apply in a

What clues did you look for to cancel these fractions to their simplest form?

How do you know when you have the simplest form of a fraction?

Give me a fraction that is equivalent to 2/3, but has a denominator of 18. How did you do it?

2/5 of a total is 32. What other fractions of the total can you calculate?



Use a fraction as an operator to find fractions of numbers or quantities (e.g.5/8 of 32, 7/10 of 40, 8/100 of 400 centimetres).

context.

Children only see a fraction as a part of a whole ‘one’ (a strip or a circle)– Do not understand can be applied to a group of objects, a number or a measurement greater than 1.

Some children may remember the divide by the denominator and times by the numerator’ but do not understand why and hence cannot apply this to a specific context.

Lack of understanding as to what a fraction actually is:-

Children haven’t made the link with fractions and so struggle to find 50% (1/2), 25%(1/4), 75%(3/4), 40% (4/10) etc.

Children realise a link with fractions but use the value of the percentage as the denominator and subsequently they divide by value. E.g. They think 24% is equal to 1/24 and so to find 24% of 300 they would simply divide by 24.

Using a set of fraction cards (e.g. 3/5, 7/8, 5/8, ¾, 7/10 etc.) and a set of two-digit number cards, ask how the fractions and numbers might be paired to form a question with a whole-number answer. What clues did you use?



The six students who had been given the questions managed to answer a few questions and did not attempt any of the explanation ones.

With the taught lessons, in lesson 1, I observed student A, she lost her concentration very quickly, she started using wrong fraction. She did not know which tables to write as she told me this had not been explained to her but observation of student T in lesson 2 showed she has got all the correct answers. She gets a star to put in her sticker book.

The class teacher was very highly regarded by the head teacher and although she did not like mathematics the way she liked English, she was an articulate and enthusiastic teacher. All the observed lessons in this research are from her lessons. In a fifty minute lesson, fifteen minutes was spent explaining the topic and using the assessment for learning (AFL) strategies such as whiteboards and show of hands to ensure the children understood the basics, before moving on to attempting the questions.

She taught the same children literacy as well, and I wondered if her enthusiasm for one subject over the other affected the amount of learning in her classes. I was also left thinking that perhaps she does not understand the needs of the class.

Quinlan (2014) states that:

“There is no guarantee that all of the learners in a class are going to be engaged by the same topic.”

True, although I maintain that a good teacher in the right conditions can make any topic engaging for any pupil.



Reflective Discussion – Act on evidence

The intervention action plan (appendix D) tries to address findings by looking at relevant literature to come up with planned changes which are likely to benefit practice as can be seen below.

Research Finding Implication from Literature

Intended change / action

(i.e. What did you discover from your data?)

Basic principles had not been understood

(i.e. How does literature relate to your research findings?)

In their analysis of the 1998 Key Stage 2 tests, QCA highlighted that questions relating to fractions, posed difficulties.

(i.e. What planned changes or actions are likely to benefit practice?)

We need to have children solve lots of problems using either visual models or fraction manipulatives. Another way is to ask them to DRAW fraction pictures for the problems. That way the students will form a mental visual model and can think through the pictures.

Gaps in understanding

Children only see a fraction as a part of a whole ‘one’ (a strip or a circle)– Do not understand can be applied to a group of objects

Finding a fraction means finding a part of that ‘whole’ group (revisit concept of equal parts). Physically show the moving of the objects to split into groups (the denominator). Find one ‘part’, then look at the numerator and determine how many of those parts are ‘needed’.

Common errors

“Recognise common pupil errors and misconceptions in mathematics, and to understand how these arise, how they can be prevented, and how to remedy them” (DfEE, 1997, p. 36)

Explore some of the most common reasons for children making mathematical errors in school by using Cockburn’s model “Some of the commonest sources of mathematical errors” include them in the lesson plan and discuss them with the children so they can been corrected.

Struggled with which operation to use

Some children may remember the divide by the denominator and times by the numerator’ but do not understand why and hence

Whilst modelling, make links to division when splitting the group into equal parts and multiplication when finding a number of the parts.



cannot apply this to a specific context

Could not grasp the link between fraction, decimal and percentags

Children regard fractions, decimals and percentages as three abstract ideas. Unable to make links between the three.

Draw 3 number lines underneath each other– same length – 0 to 1 (100%). One is the ‘fraction line’, one the ‘decimal line’, one the ‘percentage line’ – All representing the same amount. Mark on a fraction such as ¾. Show equivalent decimal 0.75 on relevant line. Show equivalent percentage 75% on relevant line.

Fractions have often been considered as one of the least popular areas of mathematics. Many children consider the concept of fractions as ‘difficult’ and too often children have had difficulty understanding why they are carrying out a particular procedure to solve a calculation involving fractions.

This lack of understanding is the main reason why there are errors in fractions. Of course there are a variety of other reasons as well. They may be due to the pace of work, the slip of a pen, slight lapse of attention, lack of knowledge, misunderstanding, or teachers lack of confidence in how best to teach this area.

Some of these errors could be predicted prior to a lesson and tackled at the planning stage to reduce or make any possible misconceptions negligible. In order to do this, the teacher needs to have the knowledge of what the misconception might be, why these errors may have occurred and how to unravel the difficulties for the child to continue learning.

Cockburn (1999) suggests the following model to explain some of the commonest sources of mathematical errors.



Cockburn (1999) summarises the link between the child, the task and the teacher very well. She starts with the child and asks key questions:

Does the child know which procedures to apply?

Does the child know how to use the procedures correctly?

Does the child understand the task both in terms of the language used and the mathematical implications?

Baroody (1993) said, ‘’For children, mathematics is essentially a second or foreign language’’ (p.2-99). If this is the case, then teaching fractions in a way which they can understand is so important as it is a building block for other key topics in mathematics, just as learning and recognising the alphabet is key to reading.

Teachers are very busy and perhaps not much thought goes on in the planning stage as to how a pupils’ imagination or creativity can be a contributing factor in their understanding of fractions. The contrast between a really creative and imaginative lesson such as English and an unimaginative lesson such as mathematics is the reason why children are struggling.

The relationship between the child and the teacher plays an important part in how well the child does in class. If an able child does not get along with the teacher, then he or she may not do well in the subject. If a teacher prejudges that the child is not very good in the subject, the child may not try their best.

As in all cases, mood does affect outcome. It may be hard to focus on the lesson if the child has not slept well or is worried about something.



Children are becoming confused as calculations involving fraction are introduced too early, when certain children still require more experience with the visual and practical aspect of creating simple fractions of shapes in order to gain a more secure understanding of what a fraction actually is.

Children need to have a firm understanding of what the denominator represents and the numerator represents through the use of visual (and kinaesthetic) resources.

“The headlong rush into computation with fractions, using such mumbo-jumbo as ‘add the tops but not the bottoms’ or ‘turn it upside down and multiply’, has often been attempted before the idea of a fraction or fractional notation has

been fully understood.”

(Nuffield Maths 3 Teachers’ Handbook: Longman 1991)

It is essential that children have this pre-requisite knowledge of fractions in order to use and apply their knowledge within a range of different contexts.

As many teachers and parents know, learning the various fraction operations can be difficult for many children.

And the simple reason why learning the various fraction operations proves difficult for many students is the way they are typically taught. Just look at the amount of rules there are to learn about fractions!

1. Fraction addition - common denominators

Add the numerators, and use the common denominator

2. Fraction addition - different denominators

First find a common denominator by taking the least common multiple of the denominators. Then convert allthe addends to have this common denominator. Then add using the rule above.

3. Finding equivalent fractions

Multiply both the numerator and denominator by a same number.

4. Convert a mixed number to a fraction

Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the common denominator as in the fractional part of the mixed number.

5. Convert an Divide the numerator by the denominator to get the



improper fraction to a mixed number

whole number part. The remainder will be the numerator of the fractional part. Denominator is the same.

6. Simplifying fractions Find the (greatest) common divisor of the numerator and denominator, and divide both by it.

7. Fraction multiplication

Multiply the numerators and the denominators.

8. Fraction division Find the reciprocal of the divisor, and multiply by it.9. Comparing fractions Convert the fractions so they have a common

denominator. Then compare the numerators.

10. Convert fractions to decimals

Divide using long division or a calculator.

Conclusion – Evaluate results

I used various types of data to see if I could work out what it is about fractions which is causing children and teachers so much problems.

My primary data indicated real issues with understanding fractions and making links with other operations.

I used a lot of secondary data to see if what I had found was similar to what had been written about problems associated with fractions.

I started with the informal interview which showed this to be the area to focus on.

I based a set of questions around a circular which discussed some of the misconceptions children have when looking at fractions. After analyzing the answers, it became obvious that there were gaps in their understanding. Where did this gap stem from? To get an answer to this, I observed several lessons.

I deliberately chose the best teacher. I found that not all the children were engaged. It could be because of the class size (30), the ability range (classes split into 3 ability groups), topic was too abstract (a significant number of children could not make links with multiplication or division) or the teacher was not as interested in Maths as she was in English.

My research involved upper key stage 2 children. Perhaps the problem starts at a much younger age.



I went back to the national curriculum. I noticed the depth of knowledge children are expected to acquire by the time they leave primary school. This has been highlighted in detail during the introduction stage of this research.

Fractions should be taught in a similar way to teaching a foreign language. Teachers should not make any assumptions about how much the children already know. Year 6 children should have acquired the pre-requisite knowledge of fractions by the end of year 5. Teachers need to look at what children already know and teach them accordingly.

Cockburn (1999) said it perfectly:

“If teachers can identify common misconceptions relating to fractions and identify ways to address these misconceptions through the teaching of appropriate pre-requisite skills this may be a good starting point to addressing the problems associated with teaching fractions.”

So instead of merely presenting a rule as many books do, a better way is to use visual models or manipulatives during the study of fraction arithmetic, that way fractions become something real and concrete to the student, and not just a number on top of another without a meaning. The student will be able to estimate the answer before calculating, evaluate the reasonableness of the final answer, and perform many of the simplest operations mentally without knowingly applying any "rule."

Of course textbooks DO show visual models for fractions, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! We need to have children solve lots of problems using either visual models or fraction manipulatives. Another way is to ask them to DRAW fraction pictures for the problems. That way the students will form a mental visual model and can think through the pictures.

If children think through pictures, they will easily see the need for multiplying or dividing both the numerator and denominator by the same number. But before voicing that rule, it is better that children get lots of 'hands-on' experiences with fraction pictures they draw themselves. They can even have fun splitting the pieces further or conversely merging pieces together. They may find the rule themselves even - and it will make sense. If they forget the rule later, they can always fall back to thinking about splitting the pieces and re-discover it.

They cannot get through algebra without knowing the actual rules for fraction operations. But by using visual models extensively in the beginning stages, the rules will make more sense, and if 10 years later the student has forgotten the rules, he should still able to "do the math" with the pictures in his mind, and not consider fractions as something he just "cannot do".



Finally an excellent video lesson by W Fortescue-Hubbard (appendix J) shows an amazing way to introduce fractions to young children. Perhaps this is the way forward. A small extract from the lesson is shown below:

½ ½ 1/2

The above are all halves, how can that be?

I expect you always thought that when you had a half they had to be 2 identical sizes.

Ah, but

This is a ½ of a large bar of chocolate

This is a ½ of a medium size bar of chocolate

And this is a half of a small size bar of chocolate



When you are talking about fractions, you are talking about a part of a whole which means when we define our whole, our whole could be quite big or our whole could be quite small.

References

Cockburn, A.J. (1999) Teaching Mathematics With Insight, 1st edn. London and New York: Taylor & Francis.

BAROODY, A.J. (1993) Problem Solving, Reasoning and Communicating, New York: Macmillan.

Tidd, M. (2013) Overview of the Primary Curriculum Based on National Curriculum published in September 2013. Available at: www.primarycurriculum.me.uk (Accessed: 17 April 2014).

NCETM Misconceptions with the key objectives2. Available at: https://www.ncetm.org.uk (Accessed: 17 April 2014).

Quinlan, O. (2014) The Thinking Teacher, 1st edn. Wales: Independent Thinking Press

Nuffiled Maths 3 Teachers’ Handbook (1991): Longman

Fortescue-Hubbard, W. (2010) Starting off with fractions http://archive.teachfind.com/ttv/www.teachers.tv/videos/starting-off-with-fractions.html


https://www.ncetm.org.uk/

http://www.primarycurriculum.me.uk/


Appendix A – Research Outline

Research Outline

Student Number: U1302686

Primary With Route (PWR): Mathematics

Overall aim for the research:

This research question will consider whether the way fractions is taught affects childrens understanding of fractions.

It will look at how children feel about fractions, what is their understanding of this term.

It will seek to find other factors, eg knowledge of multiplication, which may have an impact on understanding fractions.

Rationale for the chosen focus:

After discussion with the maths specialist on the one area of mathematics which the children are finding challenging, I was told fractions is proving difficult for them to grasp particularly multiplication and division of fractions.

Research Question(s):


Data Collection Method(s):

Informal Interview 1 child and 1 adult (maths specialist)

Formal interview 1 child and I adult

Formal interview of 6 children in maths club



Formal interview of 1 child in year 7

4 observed fraction lessons

Relevant Literature:

National Curriculum, Curriculum Overview for Years 1-6

National Centre for Excellence in the Teaching of Mathematics

Teaching Mathematic with Insight by A.D Cockburn

Misconceptions with the Key Objectives2

The Thinking Teacher by O Quinlan

Starting off with fractions (Video) by W Fortescue-Hubbard

Ethical Considerations:

All names will be deleted

Appendix B – Consent Form Adult Participant

Consent Form

UNIVERSITY OF EAST LONDON

Consent to Participate

(PWR MATHEMATICS)

Research Problem

After discussion with the maths specialist on the one area of maths which the children are finding challenging, I was told fractions is proving difficult for them to grasp particularly multiplication and division of fractions.

Research Question(s)


Purpose Statement



It will seek to find other factors, eg knowledge of multiplication, which may have an impact on understanding fractions



Investigator’s Name (BLOCK CAPITALS)FARHANA IMRAN………………………………………………………..

Investigator’s Signature…………………………………………………………………………………

Date: 14th March 2014………………………….

Appendix C – Consent Form Young Person Participant

Consent Form

UNIVERSITY OF EAST LONDON

Consent to Participate

Part 2 (to be completed by the parent of guardian):


I have been fully informed about the nature and purpose of the study on [insert project subject or title].

I have had the opportunity to discuss details and ask questions about the study.

I understand what is being proposed and the procedures in which I will be involved.

I freely and fully consent to participate in the study, and understand that I have the right to withdraw from the study at any time without being obliged to give a reason.

Name ______________________________________________________

Signature __________________________ Date ____________

I do agree to take part in the study on FRACTIONS

I agree to take part in an individual interview with my teacher

I know what the study is about and the part I will be involved in.

I know that I do not have to answer all of the questions and that I can decide not to continue at any time.

Name _____________________________________________________

Signature __________________________ Age____________

I have been fully informed about the nature of the study.

I give permission for the child to be included.

Name _____________________________________________________

Relationship to Child __________________________________________

Signature __________________________ Date ____________


Investigator’s Name (BLOCK CAPITALS)……FARHANA IMRAN………………………………………………………..

Investigator’s Signature…………………………………………………………………………………

Date: …14th March 2014……………………….

Appendix D – Intervention Action Plan

Student Number: u1302686

Research Finding Implication from Literature

Intended change / action

(i.e. What did you discover from your data?)

Basic principles had not been understood

(i.e. How does literature relate to your research findings?)

In their analysis of the 1998 Key Stage 2 tests, QCA highlighted that questions relating to fractions, posed difficulties.

(i.e. What planned changes or actions are likely to benefit practice?)

We need to have children solve lots of problems using either visual models or fraction manipulatives. Another way is to ask them to DRAW fraction pictures for the problems. That way the students will form a mental visual model and can think through the pictures.

Gaps in understanding

The circular ‘Misconceptions with the Key Objectives2’ - Children only see a fraction as a part of a whole ‘one’ (a strip or a circle)– Do not understand can be applied to

a group of objects

Finding a fraction means finding a part of that ‘whole’ group (revisit concept of equal parts). Physically show the moving of the objects to split into groups (the denominator). Find one ‘part’, then look at the numerator and determine how many of those parts are ‘needed’.

Common errors

“Recognise common pupil errors and misconceptions in mathematics, and to understand how these arise, how they can be prevented, and how to remedy them” (DfEE, 1997, p. 36)

Explore some of the most common reasons for children making mathematical errors in school by using Cockburn’s model “Some of the commonest sources of mathematical errors” include them in the lesson plan and discuss them with the children so they can been corrected.

Struggled with which operation to use

The circular ‘Misconceptions with the Key Objectives2’ - Some children may remember the divide by the denominator and times by the numerator’ but do not understand why and hence cannot apply this to a

Whilst modelling, make links to division when splitting the group into equal parts and multiplication when finding a number of the parts.



specific context

Could not grasp the link between fraction, decimal and percentags

The circular ‘Misconceptions with the Key Objectives2’ - Children regard fractions,decimals and percentages as three abstract ideas. Unable to make links between the three.

Draw 3 numberlines – underneath each other– same length – 0 to 1 (100%). One is the ‘fraction line’, one the ‘decimal line’ , one the ‘percentage line’ – All representing the same amount. Mark on a fraction such as ¾. Show equivalent decimal 0.75 on relevant line. Show equivalent percentage 75% on relevant line.

Appendix E – Interview Schedule

Research Outline

Student Number: U1302686

Primary With Route (PWR):Mathematics

Overall aim for the research:



It will seek to find other factors, eg knowledge of multiplication, which may have an impact on understanding fractions.

Rationale for the chosen focus:

After discussion with the maths specialist on the one area of mathematics which the children are finding challenging, I was told fractions is proving difficult for them to grasp particularly multiplication and division of fractions.

Research Question(s):


Data Collection Method(s):

Informal Interview 1 child and 1 adult (maths specialist)

Formal interview 1 child and I adult

Formal interview of 6 children in maths club

Formal interview of 1 child in year 7

4 observed fraction lessons

Relevant Literature:



National Curriculum Overview for Years 1-6

National Centre for Excellence in the Teaching of Mathematics

Teaching Mathematic with Insight by A.D Cockburn

Misconceptions with the Key Objectives2

Ethical Considerations:

All names will be deleted

Appendix F – Questions

Key Questions

1 How do you know when you have the simplest form of a fraction?

2 Give me a fraction that is equivalent to 2/3, but has a denominator of 18. How did you do it?

3 2/5 of a total is 32. What other fractions of the total can you calculate?

4

i

ii

iii

Using a set of fraction cards (3/5, 7/8, 5/8, ¾, 7/10) and a set of two-digit number cards(56,12,15,70,40), ask how the fractions and numbers might be paired to form a question with a whole-number answer. What clues did you use?

What did you look for first?

Which part of each number did you look at to help you?

Which numbers did you think were the hardest to put in order? Why?

5 Give me a number somewhere between 3.12 and 3.17. Which of the two numbers is it closer to? How do you know?

6 Tell me some fractions of numbers that are equal to 2, 5, 10, 15, etc. How did you go about working this out? How do these relate to the division questions?

7 Tell me two fractions that are the same as 0.2. Are there any other decimals that have fractions that are both fifths and tenths? How many hundredths are the same as 0.2?

8 Tell me some fractions that are equivalent to ½. How do you know? Are there others? Repeat for fractions like ¼ and ¾, 1/3 and 2/3.



9 Tell me some fractions that are greater than ½. How do you know? What about fractions that are greater than 1?

10 Which would you rather have 1/3 of £30 or ¼ of £60? Why?

11 What numbers/shapes are easy to find a third/quarter/fifth/tenth of? Why?

Appendix G – Answers from year 6 student















Appendix H – Answers from mathematics specialist











Appendix I – Observed Lessons

Lesson 1:

A KS2 teacher asks her class what have we been doing with fractions? She explains that we have covered equivalents, order fractions and common denominator. She models how to order fractions.

How would you order this set of fractions?

½ 2/5 7/10

Step 1: write our tables until we find a number which appears in all the tables.

2 times table: 2,4,6,8,10,12,14

5 times table: 5,10,15,20,25,30

10 times table: 10,20,30

Step 2: convert all the fractions until the denominator is the same, remember to apply the golden rule: what you do to the denominator, do the same to the numerator

½ = 5/10

2/5 = 4/10

7/10

Step 3: order the fractions

4/10 5/10 7/10

Step 4: convert back to the original fraction

2/5 ½ 7/10

Lesson 2:

Choose any 5 of these fractions.

½ 1/3 ¼ 1/5 1/6 1/8 1/10 2/5 2/3 ¾ 3/10 1



We need to find 5 equivalent fractions.

What is the equivalent fraction?

8/16 = ½

6/30 = 1/5

6/6 = 1

4/12 = 1/3

8/80 = 1/10

Lesson 3:

LO: To add fractions

There are 3 simple steps to fractions

Step 1: same denominator

Step 2: Add numerator

Step 3: Simplify the fraction if you need to

How would I simplify 3/6? This is ½.

¼ + ¼ = 2/4 = ½

1/3 + 1/6 what do you do if the denominators are different?

1/3 = 2/6 make sure all denominators are the same.

2/6 + 1/6 = 3/6 = ½

Lesson 4:

What is a mixed fraction? It is a whole number and a fraction. Eg 1 1/3

What do I mean by converting? Change it from improper fraction to mixed fraction.

Step 1: Divide the numerator by the denominator

Step 2: Write down the whole number answer

Step 3: Then write down any remainder above the denominator

11/4 11/5



Step 1: 11/4 11/5

Step 2: 2 whole number 2 whole number

Step 3: 3 remainder 1 remainder

11/4= 2 ¾ 11/5= 2 1/5



Appendix J – Video lesson by W Fortescue-Hubbard

Fractions is one of the topics in mathematics which pupils find very difficult, this is quite often because when they first started learning about fractions they never really got the chance to rip up pieces of paper.

½ ½ 1/2

The above are all halves, how can that be?

I expect you always thought that when you had a half they had to be 2 identical sizes.

Ah, but

This is a ½ of a large bar of chocolate

This is a ½ of a medium size bar of chocolate



And this is a half of a small size bar of chocolate

When you are talking about fractions, you are talking about a part of a whole which means when we define our whole, our whole could be quite big or our whole could be quite small.

Let’s see how this works when we use paper. Here I’ve got one whole piece of paper that’s my whole.

I am going to take that piece of paper and I am going to divide it exactly in two. The two pieces need to match.



And what I’ve got now is one out of two pieces. I’ve got another out of 2.

What I’ve got is my one whole and I’ve divided it into two equal pieces.

My two parts make a whole.

Let’s take a different piece of coloured paper. Lets see how that works this time.

This time I’m going to take my one whole and I am going to divide it into three equal pieces. And they need to be exact. Each piece has to be identical to the other.

ok can we name each of these pieces?

How many pieces? 1,2,3

I’ve got one out of three

I’ve got another one out of three

And my last one out of three



One whole one divided into three equal pieces

A third

Three thirds make up one whole

So where can that take us?

Let’s have a look at what happened when we have more than three thirds.

So we have 1,2,3 pieces that’s three thirds

Three thirds are one whole one

1/3 + 1/3 + 1/3 3/3 = 1

And we can see that fits exactly on top.

So what have we learnt so far?

Well, we discovered that you record a fraction by the number of pieces that you have divided that whole shape into.

Here is some pieces.

Take them out to make the whole.

Check that is definitely one whole piece.

So what is the name of these fractions?

1,2,3,4,5 so the name is one chopped into five.



1/5 + 1/5 + 1/5 + 1/5 + 1/5 gives one whole

What happens if we have more than 5 bits?

1 2 3 4 5 6 7 8 9 10 11

I’ve got 11 fifths 11/5

In mathematics when we have the top number being larger than the bottom number, we have a name for that, it’s called a top heavy fraction.

Another way of writing 11/5 is to look at how many whole ones we have in this 11 fifths. So we need to have a look again at our whole pieces of paper.

If you remember for one whole number we need to have five fifths. 5/5 makes one whole. And can I make another one? Let’s see. Yes I can make another one. So what I’ve got one whole one, two whole one but I’ve got a fraction left over 1/5.

So here we’ve got two whole ones and 1/5. This in mathematics is called a mixed number.

We have the whole part with the fraction part. So you can see that we can write 11/5 as 2 1/5.

Let’s have a look at 3 2/5

First of all we need to have 1,2,3 whole ones

and we’ve got 2/5 at the end.

If you remember one whole one we have 5/5 which is going to be true for each of our whole numbers.

So how many fifths have we got altogether?



1=5/5 1=5/5 1=5/5 1/5 1/5

We’ve got 5/5 and we can write that down plus another 5/5 there, another 5/5 there and our 2/5 so altogether we’ve got 5 10 15 16 17

3 2/5 = 5/5 + 5/5 + 5/5 + 2/5 =17/5

So 3 2/5 = 17/5 as a top heavy fraction.

So where has this initial journey into fraction taken us?

First of all it has helped us to understand, using the chocolates, how important it is to define how big our whole is to start with.

Then we have looked at dividing a whole into equal size pieces and how we can take the number of the pieces to make back into a number of wholes. And through that journey I think we’ve learnt that the bigger the number at the bottom then the smaller that part of the whole is.

Clever isn’t it?


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