PGM310

PGM310 – Managing Professional Change and Development U1302686
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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

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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 common
Denominator 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

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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)

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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

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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

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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

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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

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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!”

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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
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?

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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).
works, cannot apply in a
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?

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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.

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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
cannot apply this to a specific
Whilst modelling, make links to division when
splitting the group into equal parts and
multiplication when finding a number of the
parts.

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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.

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pocket money. Nowadays this is far less common due to children’s more restricted lives on safety grounds,
and due to the introduction of the plastic card. It is no longer wise to assume, therefore, that 6- and 7-year-
olds are experienced money users: indeed, in the future, notes and coins may be a thing of the past.
More generally mathematical errors and misconceptions may occur when teachers make unwarranted
assumptions about their pupils’ experience. In the case of money, lack of experience may be one of the
reasons that young children who have been used to counting cubes, consider all coins —regardless of
denomination—as equal in value.
Child: Expertise
Although some may not like the concept, on entering school children have to learn to ‘play the game’. I
have written about this in some detail elsewhere (Cockburn, 1995) but, to take an example from Dickson,
Brown and Gibson (1984, p. 331), Percy was shown a picture of twelve children with the following problem
written beneath them:
‘I have 24 lollies and I want each child to have the same number of lollies. How many lollies will
I give each child?’
Percy’s response was,
‘I would give each child one lolly and keep 12 for myself.’
Percy it seems, was 12-years-old but, despite his age, I would suggest that either he did not possess, or
chose not to use, experience in ‘playing the game’.
FIG 1.1 Some of the commonest sources of mathematical errors
MAKING MATHEMATICAL ERRORS 5
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.

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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
all
the 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
improper fraction to a
Divide the numerator by the denominator to get the
whole

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mixed number 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.

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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".

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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

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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

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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
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
Research Question(s):
How does the teaching of fractions in primary schools impact pupil progress in this area?

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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
Research Question(s)
Purpose Statement

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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 ____________

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

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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 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:
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
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

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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.

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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

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Appendix H – Answers from mathematics specialist

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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

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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

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11/4= 2 ¾ 11/5= 2 1/5

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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

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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.

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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?

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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.

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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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