1. THE UNIVERSITY OF NOTTINGHAM
XX4941: Practice Based Inquiry
How can I teach fractions in a way that addresses students’
misconceptions and provides opportunity for greater depth of
understanding?
Hayley Jones
Course: MA Education
Student ID: 4215307
Tutor: Mary Biddulph
Word Count: 6019

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Contents
Introduction............................................................................................................................. 2
What is Action Research?.....................................................................................................3
Finding a Focus....................................................................................................................... 8
Background Information..................................................................................................... 10
Methodology.......................................................................................................................... 14
Findings and Next Steps..................................................................................................... 18
Reflection and Conclusion .................................................................................................. 22
References ............................................................................................................................. 24
Appendices............................................................................................................................. 26
Appendix A – Elliott’s Action Research Cycle............................................................. 26
Appendix B – Letter of Consent.................................................................................... 28
Appendix C - Transcript of focus group....................................................................... 30

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Introduction
I am a NewlyQualifiedTeacher(NQT) of Mathematicsata Nottinghamshiresecondaryschool andas
part of the Practice Based Inquirymoduleforthe Universityof Nottingham MastersinEducation,Iam
undertakingapiece of actionresearch. The aimof thisassignmentistogainanunderstandingof what
actionresearchisandhowtoconductanethicalinquiry.ItisnotexpectedthatIshouldreachadefinite
‘answer’to my chosenproblem,butthat I should,throughreflection,reacha deeperunderstanding
of my researchfocus.The followingdescribesnotonlyhow my knowledge andunderstandingof the
nature of actionresearchhas developedbutalsohow I wentaboutresearchingmychosenfocusand
how the literature Ihave readaboutconductingapractice basedinquiryinformedthe choicesImade.
It seems appropriate to first begin with an account of what action research is, its origins and the
strengths and weaknesses of this approach to research.

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What is Action Research?
When undertaking the planning and implementation of a practice based inquiry, it seems
essential to have a secure understanding of what action research actually is and to become
familiar with the underpinning principles.
Concisely, action research is defined as
…a form of self-reflective problem solving which enables practitioners to
better understand and solve pressing problems in social settings.
(McKernan, 1991: 6)
There are several key aspects to be considered here, including the nature of action research
as well as the advantages and disadvantages of undertaking research in this way, all of
which will be addressed in this introduction.
The origins of action research are often credited to Kurt Lewin (Anderson, Herr and Nihlen,
2007; Townsend, 2010). Lewin believed that if research is focussed on practice, then it
should also be framed around actions (Townsend, 2010).
The research needed for social practice… is a type of action-research, a
comparative research on the conditions and effects of various forms of
social actions, and research leading to social action.
(Lewin in Townsend, 2010: 131)
It is Lewin who is also credited with giving action research a cyclical structure based around
planning, acting, observing and reflecting (Townsend, 2010). Elliott’s (1991) version
(Appendix A) of the action research cycle is more complex, with an emphasis on constant
reflection on the progress being made towards the research aims (Townsend, 2010). This
structure goes some way towards ensuring that the aims of the research are not lost in
the implementation of strategies, as suggested by Watkins (Anderson, Herr and Nihlen,
2007). Despite criticisms that such cycles and prescribed processes have a negative impact

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on the practitioner by restricting creative thought and the ability to react to different
circumstances (McTaggart, 1996), I personally feel I would benefit from having this
structure to follow when conducting my own practice based inquiry.
When reading about this cycle I was immediately struck by how it appears to be a
formalised, more rigorous version of how many teachers, and certainly myself, would
describe their practice. This was confirmed by Anderson, Herr and Nihlen (2007: 20), who
said that
All competent practitioners engage informally within these cycles… but
action research makes such reflection more intentional and systematic.
Whilst I believe that many practitioners are reflective, it is reflexivity that defines action
research. This is the process by which practitioners consider their own beliefs and
perceptions and how this impacts their practice and they then use this understanding to
bring about change (Townsend, 2010). However, this does mean that action research is
reliant upon the practitioner to accurately consider their positionalit y and to then be able
to provide objective, reliable research (Rust and Myers, 2006). Everything from the
research methods chosen and the way they are then analysed is affected by the
researcher’s positionality and it is only by being aware of the impact t hat this can have
and being transparent about where and why bias might occur that action research can be
considered trustworthy (Anderson, Herr and Nihlen, 2007). The impact this can have on
validity and the way that action research is regarded will be discussed later in this chapter.
Action research centres on insider knowledge and has high regard for the expertise and
experience teachers have. It is for this reason that teachers are often considered to be well
placed to be the research practitioners in schools, (Holly, 1989), despite the issues
described above. Corey, who first promoted action research in the field of education (1949,
1953, and 1954), believed that teachers would value the work of other teachers over that
of outsiders for the reasons stated above and that the conviction this instilled would mean
research would be more likely to result in changes. It is also thought that research as a

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form of continuous professional development for teachers would help avoid the deskilling
of teachers and reprofessionalise teaching (Clayton et al., 2008). The contrasting argument
is that often researchers struggle to make their work generalizable (Anderson, Herr and
Nihlen, 2007) but whilst this might be true for many action research projects, Lincoln and
Guba (1985) argue that the findings, though not generalizable, can sometimes be
transferred from one context to another, and that the burden of proof should lie with the
person trying to use the research in a new context.
Whilst action research initially sounds like an excellent way to develop one’s practice
further, it is important to consider all aspects of action research as it is not without its
criticisms. Most of the disagreement about the usefulness stems from the many differences
in perception between action research and traditional social science research. Initially, as
someone with a background in Mathematics and Science, I found these differences to be
disconcerting. However, further examination of the nature and background of ac tion
research allowed me to see the value in such methods and the way in which my thoughts
on this developed will be discussed later.
Argyris and Schön (1991) raised concerns about the conflict between ‘action’ and
‘research’. This conflict arises because action research inherently requires some form of
intervention, which is frowned upon by traditional social science researchers, who feel that
the research setting should not be interfered with. The advantages and disadvantages of
this conflict have been widely debated and the value of the outcomes of action research
often ends up at the centre of this debate (Anderson, Herr and Nihlen, 2007).
Another difference commonly discussed by commentators on action research (Townsend,
2010; Anderson, Herr and Nihlen, 2007; and Clayton et al, 2008) is the value of qualitative
research against more traditional quantitative research methods. Anderson, Herr and
Nihlen (2007) are quick to point out that the limited use of quantitative or ‘traditional’
research methods does not detract from the value of action research methods. Action
research, being mainly concerned with practices in social situations (Townsend, 2010)

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benefits from the use of qualitative and narrative methods that are appropriated from areas
such as anthropology and sociology (Anderson, Herr and Nihlen, 2007).
The above means that the validity of action research is often called into question
(Anderson, Herr and Nihlen, 2007). It is for this reason that collaboration is commonly
encouraged when undertaking action research as a way of working towards counteracting
some of the bias that is inherent within it. Triangulation is also an important way of
increasing the validity and trustworthiness of action research. The idea is that different
perspectives demonstrate that positionality has not impacted upon the research (ibid.).
Whilst this must be effective to some extent, it can be argued that when collaboration
occurs within the same community, such as a school, then t here is the likelihood that all
participants and collaborators share the same inherent biases.
Perhaps more importantly is the political aspect of conducting an action research inquiry.
Anderson, Herr and Nihlen (2007) argue that despite the small scale nature of individual
pieces of action research, and the fact that qualitative research methods tend to be the
main approach used, which casts doubt in the minds of some on the generalizability of the
results of action research, the work could be used to bring about educational change on a
national level. The political implications occur from the nature of action research itself. As
a collaborative, democratic process that allows for discussion and debate, action research
often challenges the status quo of an establishment. It is the emphasis on collaboration
(Townsend, 2010; Anderson, Herr and Nihlen, 2007) that creates a feeling of commitment
to the cause within a community and encourages practitioners to push to make changes
to their own practice or even to the rules and codes of practice of an establishment
(Anderson, Herr and Nihlen, 2007)..
Finally, the barriers to this kind of research must be considered. One of the highlighted
issues for teachers undertaking action research is the time available to commit to such in
depth study. I would argue that this is the main limitation for most teachers when deciding
whether to conduct such a project. I chose to complete this project as a Newly Qualified

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Teacher (NQT) as I appreciated the chance to improve my practice early in my teaching
career at a time before extra responsibilities might impact upon my ability to do so. For
this reason I am also grateful that undertaking action research is suitable for a professional
at any point in their career (Dana and Yendol-Hoppey, 2009) and I look forward to taking
my new skills and understanding into the rest of my teaching career.

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Finding a Focus
Deciding on an area of focus for my action research was something I found particularly
challenging, although I was reassured that this was normal when searching for an action
research focus (Dana and Yendol-Hoppey, 2009). As an NQT, I was worried about the
extent to which any work I completed might have an impact, and be acted upon, within
my establishment. This meant that I avoided any system based concerns that I had and
focused on concerns about my practice, where I was more confident of being able to
implement any changes as a result of my findings (Nixon, 1981).
Aftersome reflection,Ihave chosentoconsiderthepedagogysurroundinganaspectof thecurriculum
that I have struggled with this year. Teaching mathematics in a way that encourages deeper
understanding of the material, without just giving rules to follow, is something that is extremely
important to me when considering my personal philosophy of teaching. However, when trying to
teach fractions this year, I have not been able to approach my teaching in this manner. This is
particularlytrue of some of the lower attaininggroupsI have workedwith.It was the first time I had
taughtthe material,howeverIstruggledtoteachitinwaysIbelievetobe appropriateformathematics
teaching. As a result I resorted to teaching the pupils the rules, or simply the process, knowing that
theyhad gainedlittle conceptual understanding. Uponreturningtothe topicacouple of weekslater,
the students performed poorly and had little recollection of the rules they had been taught. My
conclusions, when reflecting on the follow up lesson, were that the students had not been able to
accuratelyrecall the rulesandthat because theyhadno understandingof where the ruleshadcome
from despite mybestefforts,the studentswere no betteroff than theyhad beenat the start of the
topic. These wanderings and the desire to improve this aspect of my teaching led to the following
question for my practice based inquiry:
How can I teach fractions in a way that addresses students’ misconceptionsand
provides opportunity for greater depth of understanding?

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Readingliterature thatexplainedhowtochoose agood focusforpractitionerinquirywentalongway
in reassuring me that I had selected a worthy focus. Certainly I felt secure in knowing that I had
followed advice in choosing a focus that had arisen from a felt difficulty (Dana and Yendol-Hoppey,
2009) andfrom observationandreflectionsconcerningmyownpractice (Hubbard and Power, 1993).
My focus is centred on “content knowledge” rather than the context of my teachingfor the reasons
statedabove.Dana and Yendol-Hoppey,(2009), describe eightpassionsthatare good startingpoints
for finding a focus. Whilst they have provided eight individual themes it makes sense to me that an
area for focus might touch on more than one of these. For example, I feel that my own question
touches on the following passions:
 Curriculum development
 Developing content knowledge
 Developing teaching strategies
I hoped that having chosen something that I am passionate about and that has emerged from a
dilemma,thatIwill findmore valueinconductingthe actionresearchandhope totake myfindingsto
implementchange inmyown practice (Anderson,HerrandNihlen,2007).I feel thatmychosenfocus
is well suited to practitioner inquiry since it will help improve my understanding of students’
misconceptionsof fractionsandsobe developingmycontentknowledge aswellashelpingme tomove
forward towards better practice by developing my teaching strategies (Hubbard and Power, 1993;
Dana and Yendol-Hoppey, 2007; Elliott, 1995).

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Background Information
Throughout my initial teacher training programme and during my NQT year, I have always
been advised of the importance of understanding and addressing children’s misconceptions
in mathematics. It was during the 1980s that researchers’ interest in the nature of
students’ mistakes in mathematics began to increase in popularity (Swan, 2001). There
has been a lot of research into the children’s understanding of mathematics and it seemed
essential to read current literature about the misconceptions students have about fractions
specifically before beginning my search for examples of good practice.
“Traditional instruction in fractions does not encourage meaningful
performance.”
(Lamon, 2001: 146)
Research by Lamon (2001) suggests that by teaching fractions using traditional methods
we do not provide students with the understanding of the material. Her research goes on
to show that when students are taught for understanding, they are able to solve more
complex problems successfully (ibid.)
When teaching fractions for understanding, there are several common misconceptions that
should be addressed (Hansen, 2011). The main types are as follows and will be detailed
briefly:
 Modelling
 Overgeneralisation
 Objects and process
 Incorrect intuitions
Mathematical models identify relationships between the different variables and parameters
of a problem. Many students struggle with being able to correctly model a problem (Ryan
and Williams, 2007). For example, when asked what four divided by a half is, they confuse
this with finding a half of four and so incorrectly give the answer as being two. This may

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also occur because the latter is a question they are more likely to encounter in real life
(ibid.).
Overgeneralisations occur when students take a rule that they have learnt in one particular
aspect of mathematics and believe it can also be used and applied in another aspect (Ryan
and Williams, 2007). Certainly in mathematics, there is commonly an overuse of the
additive strategy rather than the multiplicative strategy, exhibited often when students are
working with equivalent fractions (Hart, 1981). When trying to find equivalent fractions,
students might state that
6
12
=
3
9
because they have added or subtracted the same amount
to the numerator and denominator rather than multiplying or dividing.
The misconception concerning object and process occurs when students cannot
comprehend that a fraction is an object that is the result of the process of division (Herman
et al., 2004). This means that students struggle with the concepts of
3
4
being the answer
to the process of 3 ÷ 4.
Students often have incorrect intuitions in mathematics. This is when a pupil forms a false
understanding of a problem (Nickson, 2004). When considering fractions, students find it
difficult to accept fractions as a single entity and instead view and treat the numerator and
denominator as two distinct numbers (ibid.). This results in errors such as
4
9
−
1
4
=
3
5
, where
students have performed the subtraction on the numerator and then the denominator
separately.
All of these misconceptions are compounded by the fact that there are various
interpretations of fractions (Hansen, 2011). The relative conceptual difficulties of these
interpretations are commonly accepted by mathematical educators (Charalambous and
Pitta-Pantazi, 2005). The different possible interpretations are given below:
Figure 3.1: List of different interpretations of fractions as given by
Hansen (2011: 33).

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 Part of a whole – here an object is ‘split’ into two or more equal
parts.
 Part of a set of objects – what part of a set of objects has a
particular characteristic?
 Numbers on a number line – numbers which are represented
between whole numbers.
 Operator – the result of division.
 Ratio – Comparing relative size of two objects or sets of objects.
Hansen (2011) states that teachers should be fully aware of these interpretations when
introducing fractions to children in order to do so in a meaningful way. It is this ‘meaningful
way’ that I am interested in researching in order to improve my classroom practice. Whilst
there is a lot of literature about the misconceptions children have concerning fractions, it
has been difficult to find literature containing many practical tips and advice for the
planning of successful lessons on this topic.
From previous work on misconceptions, I know that in order to develop understanding I
would like my lessons to be dialogic since classrooms should be a place where discussions
take place in order to share approaches and to openly address the difficulties presented
above (Swan, 2001).
The only way to avoid the formations of entrenched misconceptions is
through discussion and interaction. A trouble shared, in mathematic al
discourse, may become a problem solved.
(Wood, 1988: 210)
Group discussions enable students to vocalise their thoughts and discuss them with peers,
which in turn allows for cognitive conflict through contradiction of opinions (Swan, 2001).
One way of achieving this atmosphere is through problem solving style activities. Critchley
(2002) also places merit on letting children solve a real problem involving fractions in
order to support children’s understanding.

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These practical aspects will be the beginnings of any lessons I plan as a result of my
research and I now hope to find examples of good practice specifically related to teaching
fractions.

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Methodology
As someone with a background in mathematics and science, when considering research
methodology I instinctually lean towards the more scientific model of research:
formulating a hypothesis, planning and conducting experiments as a way of collec ting data
and then analysing this data in order to decide whether the hypothesis can be accepted or
rejected. It did not take long to realise that this very positivist outlook on research simply
would not work on a social study such as my own focus for practitioner enquiry. The helical
process of action research explained earlier in this essay is the research structure I intend
to follow since it better meets the aims of my practitioner enquiry. As discussed earlier,
this was something that I found conflicting but further examination of my epistemologic al
views allowed me to reconcile these feelings. I realised that since beginning my studies in
education I have always felt most strongly persuaded by the social constructivist
epistemological view, using theories from Vygotsky, Bruner and Wood to inform my
teaching. And so, when considering educational research, I followed the same theories I
do when teaching. It is worth noting that not only will this affect the methods I choose but
also how I analyse and interpret the results (Crotty, 1998).
Since our session on ethics I had begun to realise what a complex role they would play in
forming my research, although the BERA Ethical Guidelines (2011) seemed to me to
overcomplicate the principles and come across very much as a legal document designed
to protect the researcher as much as the participants. I cannot help but feel that
essentially, the ethical principles of research are very simple to follow from a moral
perspective and would be followed from being humane rather than simply adhering to a
set of rules. That being said, I understand their importance in protecting participants from
unethical research and can see the value in everyone following a set code of practice,
especially as someone relatively new to research. In following the guideline, I ensured that
I had completed the ethical statement form and produced a letter for signed and informed
consent for my participants in line with the University of Nottingham’s guidelines
(Appendix B).

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Ethics is not only about the participants but the very nature of the research that is being
undertaken. In order for an enquiry to be considered ethical, work should be open to peer
scrutiny and be thought as trustworthy as possible (Trochim, 2006). I consider this to be
especially true of action research where the data examined is primarily qualitative and
therefore more easily impacted upon by any inherent bias that may exist (Hulme et al.,
2011). The above seemed particularly applicable to my own research, where I will be
asking staff for opinions on pedagogy, which are subject to an individual’s epistemologic al
outlook and hence naturally subject to bias. It is for this reason that I am pleased that my
project has become so collaborative, as will be explained below.
I had decided that observation would be a good initial step to beginning to find out what
is happening in other teacher’s classrooms since Robson (1993: 192) stated that
observation ‘is commonly used in the exploratory phase, typically in an unstructured form,
to seek to find out what is going on in a situation.’ I felt that this would be well suited to
the reconnaissance stage of the action research cycle and had planned to observe more
experienced members of the department teaching fractions to low attaining Key Stage 3
groups so that I could look for approaches that tackle pupils’ misconceptions more overtly
and include explanations or activities that promote conceptual understanding. However, I
encountered a few problems with this method during the initial planning stages.
The main problem was that members of the department were unwilling to participate in
the observational aspect of the research, despite assurances of anonymity and other
ethical procedures (BERA, 2011) as well as reiterating that the focus was for me to improve
my own practice, not judge theirs. This problem was compounded by the strain such
participating in such a project would place on teacher time. This, along with the time of
year, meant that one member of staff who did wish to partic ipate could not because she
simply could not fit it into her timetable before the end of the year.
The discussions I had with teachers when trying to find participants highlighted to me that
most members of staff felt uncomfortable or unconfident when teaching fractions and it

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was this that meant most people were unwilling to participate. Staff were open about this
being the reason they did not wish to take part.
In depth discussion with a few members of the department revealed that they would be
happy to share their thoughts and best practice through other methods. Since we wanted
the project to be of benefit to the department, we decided our aim would be to work
together to plan a sequence of lessons, which could then be taught, observed, reflected
upon and improved as part of the action research cycle until we had created a set of
lessons that we felt were more focussed on understanding.
We decided that a good place to start would be to organise a very small focus group where
we could discuss and begin to understand more clearly what some of the obstacles are
when teaching fractions in a secondary school, as well as any initial ideas that would help
add an element of conceptual understanding to our lessons. Although the focus group had
only a small number of participants, I felt that the principles of running a focus group
would help guide the way we should interact. I felt the benefits of this method would
outweigh any disadvantages, especially considering the type of information I was trying
to gather. Kreuger (1994) said that for participants to be fully engaged with the research
process and to openly discuss opinions, the environment must be permissive and non-
judgemental. The freedoms given by a focus group would hopefully encourage an
atmosphere where participants would feel confident in sharing and clarifying ideas (ibid.).
Somekh (1994: 360) states that ‘members in collaborative projects start from the
assumption that there is a status differential.’ This can be due to factors such as
experience, salary and role within the establishment. It is for these reasons that I carefully
considered the participants involved and did not include the head of department and the
head of school, who is also a maths teacher, so that participants felt they could speak
openly without judgement. I also had to consider how my own position might affect the
honesty of the group but on reflection decided that my good working relationship with the

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participants, along with ethical assurances, would mean that I could deem comments from
the focus group to be trustworthy.
I had given myself the role of moderator, or facilitator, and although I was prepared for
how difficult a position it would be (Robson, 1993) I was surprised by how hard I found it.
Sim (1998: 347) states that:
The moderator has to generate interest in and discussion about a
particular topic, which is close to his or her professional or academic
interest, without at the same time leading the group to reinforce existing
expectations or confirm a prior hypothesis.
In order to generate interest and discussion, I used the guide by Robson (1993) for starting
an interview and this clear start meant that the discussion was focussed. What I found
particularly hard to manage was that the nature and experience of the participants meant
that they remained focussed and that the discussion was well balanced between
participants and as such I struggled to contribute to the discussion as a moderator. Whilst
it can be detrimental to a focus group if the moderator does not direct the discussion, there
was no negative impact from my lack of input and by not interrupting the discussion I have
avoided the possibility of overly influencing what is discussed by my own comments and
causing bias.

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Findings and Next Steps
Convention and ethics dictate that data collected from any type of interview should be
transcribed (Robson, 1993). The simplest way to do this was to take an audio recording
of the focus group and transcribe the information at a later date so that my attention was
focussed on the group. There is however no strict convention for the analysis of qualitative
data, unlike that which can be observed with quantitative data (ibid.). Whilst some argue
that there is no need for such developments for qualitative data since this would go against
the very nature of this type of research methodology, others wish for a ‘scientific’ approach
(ibid.).
Tesch (in Robson, 1993: 457) reduces and categorises the various approaches to analysing
qualitative data into four basic groups.
1. The characteristics of language;
2. The discovery of regularities;
3. The comprehension of the meaning of text or action; and
4. Reflection.
Robson (1993) states that whatever approach one takes to analysing the data, it must be
explicitly explained in order to preserve the transparency and hence the trustworthiness
of one’s research. I have used method four from Tesch’s types of analysis listed above; a
method that is least structured and most interpretive. I struggled to accept this method
and initially sought more structure and so tried coding the transcript. I found that whilst
there were distinct themes emerging from the text, these themes did not appear as single
words or phrases and so I decided a more interpretive method was appropriate. My
research was aimed at gathering ideas to help improve my own practice and whilst I feel
that it is of high value to me and perhaps others in the department, it is also extremel y
individualised. For this reason I felt that the analysis did not require a strict system of
coding to look for patterns and so I chose to read the transcript and reflect on the
discussions that had taken place. I am aware that this makes the results highly subjective

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to my positional bias but felt that this was more suitable than a general ‘what works’
approach. Townsend (2010) argues that an action researcher should never be detached from
the research and by analysing the transcript in this way I was able to pick out the suggestions
that I felt were most suited to my own style of teaching. This meant that I have been able to
reflect upon my practice in relation to what was discussed. Elliot (1995: 1) believes that any
educational action research that does not improve the participants practice is “very dubious”
and so I am pleased that I have already begun to make changes to the way that I teach.
There were two main, closely linked themes that emerged from my focus group that I want to
take forward:
 Teacher A’s belief in the value of experience.
 Teacher B’s ideas about how to integrate something concrete into the lessons.
Both of these themes are centred on the idea of turning what is essentially a very abstract
concept for students to understand in to something more concrete.
Teacher A’s main argument was that we ‘become abstract too quickly’ (Appendix C) and
this led all the participants to the conclusion that pupils needed a more concrete experience
of fractions. This is something that rang true and I think this is because I could relate it to
Learning Theories that I have studied, particularly Bruner’s stages of representation, a
social constructivist school of thought. Bruner’s three stages of representation are very
closely linked to the ways in which someone “knows” something (Bruner et al., 1966:6)
and are labelled as follows:
1. Enactive – learning through doing;
2. Iconic – learning through pictures or images; and
3. Symbolic – learning through symbols such as language.
Throughout Studies in Cognitive Growth (Bruner et al., 1966) these stages are referred to
as linear stages of child development but I would argue that we go through a cycle of
these stages every time we learn something new. Reflecting upon my current approach to

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teaching fractions, it is clear that I begin at the iconic stage, using diagrams and pictures
to help illustrate my point before quickly moving on to the symbolic stage, missing out the
enactive stage entirely. This is something I would like to address in my future teaching of
fractions and I will therefore aim to incorporate some of the more practical aspects
suggested by Teacher B (Appendix C). Introducing this element of hands on experienc e
would move my teaching away from the ‘traditional instruction’ described by Lamon (2001:
146) and hopefully begin to encourage ‘meaningful performance’ from the students (ibid.).
It is my hope that by going back to this enactive stage I can enable students to engage
with the material and create the cognitive conflict and the disequilibrium required for
addressing the misconceptions described by Hansen (2011) and allow for cognitive growth
(Swan, 2001 and Bruner et al., 1966). I feel that giving students concrete examples and
experiences to refer back to will be particularly helpful in addressing the misconceptions
arising from poor modelling as described by Ryan and Williams (2007) and pupils’ incorrect
intuitions (Nickson, 2004). All of this goes hand in hand with Critchley’s belief that allowing
children to solve real problems will support their understanding (2002).
Before moving on to the action stage of Elliott’s action research cycle, I would like to
conduct further research as part of the reconnaissance phase. I had thought that a good
way to get an idea of how to implement some of the more practical ideas and to further
my understanding of what children’s experiences of fractions are when they come to
secondary school, would be to visit some of the feeder primary schools and observe some
lessons. There are several problems with this, though the main would be how I could get
an accurate representation of what is happening since my presence in the classroom, along
with prior warning of what I was looking for, would likely cause the teacher to act and to
plan their lesson differently to how they normally would. This means I would not be
guaranteed a true reflection of how fractions were normally taught to the students
(Robson, 1993). Due to the limitations of time and resources, it is unlikely I would be able
to counteract this. In order to triangulate my findings, I could also interview some students
entering year seven to find out how they feel about fractions and what their experiences

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of fractions are up to that point. By collaborating all this information together I feel I would
get a better understanding of the situation and from there be able to consider the
necessary action points and through reflection decide how to proceed.

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Reflection and Conclusion
As discussed in the introduction to this assignment, the aim was not to complete all phases
of an action research cycle but to get an experience of conducting a practitioner enquiry.
I feel that my work on this research has helped me develop an understanding of what it
means to undertake a true piece of ethical research in an educational setting.
I feel proud of the work I have completed and I am satisfied that my understanding of my
initial problem has vastly improved. I believe that my research fulfils the criteria for validity
and trustworthiness as set out by Anderson, Herr and Nihlen (2007). The criteria are as
follows:
 Outcome Validity/Trustworthiness;
 Process Validity/Trustworthiness;
 Democratic Validity/Trustworthiness;
 Catalytic Validity/Trustworthiness; and
 Dialogic Validity/Trustworthiness.
In terms of the research I have undertaken, meeting this criteria means that I have gained
a deeper understanding of my research problem; my research can continue on to the next
steps in order to gain further understanding; the involvement of other members of the
department means that the process has been collaborative and democratic and c onsistent
communication with a critical friend means that the process has also been dialogic; and
finally the participants, as well as myself are eager to find out more in order to continue
making changes to our practice and therefore the research has acted as a catalyst for
change.
Although the findings of my research can be considered trustworthy since they meet the
guidance above, I still feel it would be prudent to conduct the next steps of the research
in order to triangulate the results and as such increase the trustworthiness of my research
(Anderson, Herr and Nihlen, 2007).

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I have been more surprised by how conducting this research has changed the way I
perceive myself as a professional. I have learnt, through this module and wider reading,
that there is no way of avoiding politics in an educational setting (Kelly, 2009; Anderson,
Herr and Nihlen, 2007) and I am aware that we as professionals have little opportunity to
exercise judgement on how and what we teach (Elliott, 2005). Several of the readings
discuss reprofessionalising teachers and placing more value on their specialist knowledge
through action research as CPD (Elliott, 2005; Anderson, Herr and Nihlen, 2007 and
Clayton et al., 2008). Whilst I am sceptical of the impact action research can have on
school and national systems when undertaken by individuals, I am pleased with the power
this project has given me, not only to improve my day to day practice but also to consider
more carefully the educational policy that is presented to me at a time when educational
provision is becoming increasingly marketised (Elliott, 2005).
I am optimistic that over time, action research will play a bigger role within the educational
setting as more teachers become empowered and strive for social and educational change.
In my school in particular, I would like to believe that a collaborative piece of action
research driven by the search for social justice might have the power to make changes to
whole school systems where previously this has not been possible.

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References
Anderson, G.L., Herr, K., and Nihlen, A.S., (2007) Studying your own School: An
educator’s Guide to Practitioner Action Research. Thousand Oaks, California:
Corwin Press.
Argyris, C., and Schön, D., (1991) Participatory action research and action science
compared: a commentary. In W. R Whyte (Ed.) Participatory action research:
85-96. Newbury Park, CA: Sage.
BERA, (2011) Ethical Guidelines for Educational Research. London.
Bruner, J.S., Greenfield, P., and Oliver, R., (1966) Studies in cognitive growth. Cambridge,
MA: Harvard University Press.
Charalambous, C. Y., and Pitta-Pantazi, D., (2005) Revisiting a theoretical model on
fractions: implications for teaching and research [Online]. Available at
http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol2CharalambousEtAl.
pdf [Accessed 26/07/2014].
Clayton, S. et al., (2008) ‘I know it’s not proper research, but…’: how professionals’
understandings of research can frustrate its potential for CPD. Educational Action
Research, 16(1): 73-84.
Corey, S. M., (1949) Action research, fundamental research and educational practices.
The teachers college record, 50(8): 509-514.
Corey, S. M., (1953) Action research to improve school practices. New York: Bureau of
Publications, Teachers College, Columbia University.
Corey, S. M., (1954) Action research in education. The journal of educational research,
47(5): 375-380.
Critchley, P., (2002) Chocolate Fractions. Times Educational Supplement, 19 January.
Crotty, M., (1998) The Foundations of Social Research. London: Sage
Dana, N. F. and Yendol-Hoppey, D., (2009) The Reflective Educator’s guide to classroom
research. Thousand Oaks, California: Corwin Press.
Elliott, J., (1991) Action research for educational change. Buckingham: Open University
Press.
Elliott, J., (1995) What is good action research? Some Criteria. Action Researcher, no 2.
Poole: Hyde Publications.
Elliott J., (2005) Becoming critical: the failure to connect. Educational action research,
13(3): 359-373.
Hansen, A., (2011) Children’s errors in mathematics, second edition. Exeter: Learning
Matters Ltd.
Hart, K., (1981) Fractions in K. Hart (Ed.) Children’s understanding of mathematics: 11-
16: 66-81. London: John Murray.

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Herman,J.,et al., (2004) Imagesof fractionsas processesandimagesof fractionsinprocesses.InM.
J. HøinesandA.B. Fuglestad(eds), Proceedingsof the28th
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4, 249-256.
Holly, M. L., (1989) Reflective Writing and the Spirit of Inquiry. Cambridge Journal of
Education, 19(17):1-80.
Hubbard, R. S., and Power, B. M., (1993) The art of classroom inquiry: a handbook for
teacher-researchers. Portsmouth: Heinemann.
Hulme, M., et al., (2011) A guide to practitioner research in education. London: Sage
Publications Ltd.
Kelly, A. V., (2009) The curriculum theory and practice. London: Sage Publications.
Kreuger, R. A., (1994) Focus Groups: A practical guide for applied research 3rd ed.
Thousand Oaks, CA: Sage Publications.
Lamon, S. L., (2001) Presenting and representing: From fractions to rational numbers. In
A. Cuoco and F. Curcio (eds), The roles of representation in school mathematics-
2001 Yearbook: 146-165. Reston: NCTM.
Lincoln, Y. S., and Guba, E. G., (1985) Naturalistic Inquiry. Thousand Oaks, CA: Sage.
McKernan, J., (1991) Curriculum action research: A handbook of methods and resources
for the reflective practitioner. London: Kogan Page.
McTaggart, R., (1996) Issues for participatory action researchers, in O. Zuber-skerritt
(ed.) New directions in action research. London: Falmer Press.
Nixon, J., (1981) A teacher’s guide to action research. London: Grant-McIntyre.
Nickson, M., (2004) Teaching and learning mathematics second edition: A guide to
recent research and its applications. 2nd ed. London: Continuum.
Robson, C., (1993) Real world research. Oxford: Blackwell.
Rust, F., and Myers, E., (2006) The bright side: Teacher research in the context of
educational reform and policy-making. Teachers and teaching: theory and
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12(7): 69-86.
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London: RoutledgeFalmer.

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Townsend, A., (2010) Action Research. In: Hartas, D. (Ed.) Education Research and
Inquiry: Qualitative and Quantative Approaches: 131-145. London: Continuum.
Trochim, W. M K., (2006) Qualitative Validity. Research Methods Knowledge Base [online].
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Wood, D., (1988) How children think and learn. Blackwell: Oxford.
Appendices
Appendix A – Elliott’s Action Research Cycle

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Elliott’s action research cycle (Elliott, 1991).

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Appendix B – Letter of Consent
Monday 7th July, 2014.
Dear
RE: Masters Research participant information and consent form
I am currently working on my Masters in Education at the University of Nottingham. I
am undertaking a research project on my teaching of fractions and how I might improve
my teaching methods in order to encourage understanding rather than wrote learning. I
am hoping conduct a group interview focussing on how experienced teachers currently
approach and any ideas they have for improving practise. My focus will be on what the
teacher does to encourage understanding, through questioning, different activities or any
other approach. All information gained from this interview will be treated anonymously,
with the utmost confidentiality and no individual will be identifiable as a result of the
research. After the interview, you will be able to read a transcript of the discussion to
check the accuracy of the recording and you are also entitled to withdraw form the
process at any time.
I am writing to ask if you are willing to give your consent to participate in this process. If
so then please complete the attached participant consent form and return to me at
school by the 9th of July.
If you have any questions or wish to discuss this with me further then please contact me
by phone on: 01949 87555 or email at: hjones@toothillschool.co.uk .
Yours sincerely,
Miss H Jones
Teacher of Mathematics

30. XX4941 H Jones:4215307
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Participant Consent Form – Return to Miss H Jones by the 9th of July, 2014.
Project title: How can I teach fractions in a way that addresses pupils’ misconceptions
and does not rely on a set of rules?
Researcher’s name: Hayley Jones
Supervisor’s name: Mary Biddulph
 I have read the Participant Information Sheet and the nature and purpose of the
research project has been explained to me. I understand this information and
agree to take part.
 I understand the purpose of the research project and my involvement in it.
 I understand that I may withdraw from the research project at any stage and that
this will not affect my status now or in the future.
 I understand that while information gained during the study may be published, I
will not be identified.
 I understand that I will be videotaped during the interview.
 I understand that an electronic copy of both the transcript and audio recording
will be stored on Miss Jones’ private laptop, and only she will have access to this.
A hard copy of the anonymous transcript will be submitted to the research
supervisor when the assignment is due.
 I understand that I may contact the researcher or supervisor if I require further
information about the research.
Signed …………………………………………… (Research participant)
Print name ……………………………………… Date ……………………
Contact details
Researcher: hjones@toothillschool.co.uk or 01949 87555
Supervisor: mary.biddulph@nottingham.ac.uk
School of Education Research Ethics Coordinator:
educationresearchethics@nottingham.ac.uk

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Appendix C - Transcript of focus group
Researcher:So,we are thinkingaboutfractionsandanyideasthatyoudo alreadythatyouthink might
work in a way that addresses misconceptions and gives a deeper understanding than
some of the written and more structured methods that I resort to.
Teacher C: Are you talking about anything specific in terms of fractions?
Researcher: I foundat the start that it’s prettyeasy to draw a circle and splitit up and say ‘that’s a
half’ or ‘that’s a quarter’… although I am still not sure that they are entirely convinced
that thatis whata fractionisand that’snotalwayswhat a quarteris in all circumstances.
I suppose itsaddressingsome of thosethingsandhow youget roundthose,includingthe
harder operations…I can draw some pizzasto show additionand that kindof thing,but
when it’s multiplication and division…
TeacherA: Ithinkthatone of the difficultiesIfind,iskidshave gottobe…it’salmostlikethe kidshave
got to experience fractionsbefore youcan teach them. Because a fraction can be many
thingscan’tit,canbe afractionof awhole,afractionof agroup,you canhave theconcept
of one numberdividedbyanother.Itcanbe an operator,so there are manythingsitcan
be, it comes in different guises. And it depends on the youngster, for some youngsters
youneedtogo backto… almostlike aplayfulnessof understandingwhatafractionreally
means.If I say‘half of a group’,whatdoes that reallymean?Orif Isay ‘three seventhsof
a group’,whatdoes that mean? Withoutthat sense of what a fraction isyou do endup,
like I ended up, in the worst case scenario just having to teach rules, which is not very
satisfactoryreally because youdon’tfeel like youryoungstersare movingonanyfurther
into their understanding. It’s almost like we have to go back and play. I think a certain
argument for younger children, and that is… baking, [others agree] or doing something
that is working out the proportions of things, the fractions of things, you’re weighing,
you’re looking at weights and measures which are all related to fractions of quantities,
isn’t it. Without that kind of underpinning understanding, my experience of it, I find
extremelydifficulttofindawaytoteachwhichreallygoestothe heartof whatafraction
is.One of the top flightmathematicians…one inparticularthatI’m thinkingof…when it
comesto describingwhatafractionis,he says‘afractioncould be this,couldbe that,’all
the different guises that I was talking about. So when you talk about fractions and
teaching a fraction you’ve got to start also breaking it down into what part of that area
of fraction you’re talking about. Are you talking about it just as a part of something,or
are you talkingaboutit as a part of a groupor one numberdividedbyanother.It’shard
isn’t it?
TeacherB: I thoroughlyagree withyou[directed atTeacherA].I’ve beendoingit forforty yearsnow
andI haven’tfoundamethodyetthatIfeel reallyworks.Iwouldsaythatjust listeningto
you talkingaboutfractionsin a group,I wouldsaywe don’tdo enoughof herdinga few
children together in a space and asking one of the people that’s not in the herd to split
the group into quarters or ask them what a quarter of the group is or to split the group
intoeighthsorwhatever…andthen tell me whatfive eighthsis. Andyoucouldeitherdo
that withpeople or you could,withthe whiteboardsandso forth,you could have those

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of little squares drawn on the board couldn’t you, and then they couldjust move them
intofourareasor three areas,depending.Soyou couldgetthemtoexperiencealittlebit
more and then sort of say, ‘ok now split them up in eighths… so what’s an eighthof 24
andthenwhatare three eighthsof 24’…. youknow,youcoulddostuff likethatespecially
with year seven.!
Teacher C: It’s interesting because I find with top ability year eight or year seven, that if you went
back tobasicsof whatfractionreallyis,Ithinkthey’reprettysecureonwhatafractionis.
But actually, when you then put it into the terms of multiplying fractions, dividing
fractions, it’s like that understanding doesn’t help. I don’t know whether that’s us as
teacher…
Teacher B: I think multiplication and division is a completely other area of work justifying why that
works. And you’ve got this whole business of leading themto the understanding of ‘of’
and ‘times’ meaningthe same.Soa quartertimestwo thirdsis the same as a quarterof
two thirds,andthenyou can get at that. TeacherA talkingaboutthe different meanings
of fraction,I wastryingto thinkof termsof thingsthatthe studentsare familiarwithlike
time; minutes in an hour, hours in a day, and you could have clock faces like nought to
sixty for every minute in an hour or for every second in a minute.You could have other
clockfaceswhichare noughtto twenty-fourforafull day.Andyoucouldgetthemtouse
that pie chart to calculate a quarter of a minute, how many seconds is it… so they have
to worka quarterof sixty,theyhave to understandthattheyhave todividesixtybyfour,
but theycouldalsothenshade it…theycouldactuallysplitthe sixtysecondsintofourpie
chart pieces as well. So you’ve sort of got a picture of quarters, got an idea of quantity
that isassociatedwiththem.You’re findingaquarterof somethingfairlytangible,rather
thanan abstract ideaof a quarter.We oftengive themrectanglestoshade,whythatsize
of rectangle whenwe give themdifferentsizesof rectanglesdependingonwhatfraction
we want to do.So they’re notreallyworkingoutof anythingveryconcrete,butthe sixty
secondsinan hour, there’slotsof fractionsthat youcan dowiththat…. that work nicely
and fractions of a day, in terms of twenty-four hours, there is also a lot of fractionsyou
coulddo withthat. Right,ok, so, I was thinkingof trying to get the picturesso that they
could do a quarter of a minute but have it quantified as 15 seconds and other fractions
of a minute and you can have an eighth of a day in hours and again you can quantify it.
Andit’sautomaticallybringinginalittle bitof the ideathatthree eightsisdivide byeight
and multiply by three and so forth, so it brings a little bit of that into it. Because they
quite often have trouble dealing with the ‘finda third times 60’. Although againit gives
an opportunity to talk about of and times meaning the same thing. [others murmer
agreement] And it was a question of, whenthey’re adding fractions,you know, instead
of saying‘whatisa quarterplusthree eighths?’whetheryoucouldsay‘what’sa quarter
of a day plus three eighths of a day?.’
Teacher C: hmm. Put it more in context and help them use previous knowledge.
Teacher B: And then, you’ve got a chart of fractions of a day as well you know that you’ve made
thembuildupvariouschartsok so a quarterof the dayadd three eighthsof the dayisso
manyhours andlookingatthat, that’sequivalenttothatfractionso youcan actuallysee

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that adding fractions gives you a fraction. So adding two fractions of a day gives you a
fraction of a day and then maybe you start to explore, you know, you’ll set things up
carefullysothatit’syouknow a quarterplusthree eighthsisfive eighthsandyoucanget
themto sortof thinkaboutwhythatis.Andtry to developthe rulesthattheyworkwith.
TeacherA:There’salsothatelementofsharing,isn’tthere?Becauseif Isayfive eighthsof something
Teacher B: Yeah
TeacherA: I’msplittingintoeightequalpartssoI’msharingequallyamongsteightandtakingfiveof
those parts. And again that’s another deeper understanding of fractions and that fits in
with what you were saying.
TeacherB: Andwhenthey’re doingthe additionof fractionstheycouldhavethe clocksall aroundso
they’re shading the amount.
Teacher A: A sort of investigative project
Teacher B: it’s a lot of diagrams to put together sheet wise or however you do it there’s a lot of
reprographics involved in that and it’s just that I know I’ve not been successful in forty
years of really teaching fractions well you know so it’s trying something different.
TeacherA: Andalso,if there was a magic bullet professionally it would be out there, wouldn’t it?
[Others agree]
TeacherC: AndI think,whatIlike aboutwhatyou’re sayingwasthatyou’re actuallythenbringingin
a lot of differentfractionsstuff continuouslyandI wondersometimeswhetherbecause
of the wayschemesof workare,we kindof dosectionsandwe linkalittlebitbutactually
we mainlydo fractionsinsectionsandactuallyif we didfractionsbringingitall inwould
that help make those links, like we were saying.
TeacherA: I thinkwe certainlybecome abstracttooquickly,tooearly.Youknow,we abstractitinto
five eighthstimes three sevenths , that’s an abstraction. Meaningless interms of … and
whatyour ideathenof bringingthingstogetherwhereyou’vegotareal physical context,
which interchanges between a quantity and a fraction, because fractions represent
quantitiesaswell,soyou’ve gotinterchange goingon.Youcan kindof thinkof it, I don’t
know, I don’t want to bring decimals into it, but obviously you’ve got that connection
there.
TeacherB: I thinktheyjustneedmore experience of fractionswhere they’re actuallyusingthemto
calculate something that is at least partly meaningful in some way.
TeacherA: And it’sthat experience inaway sometimesIfeel likewe’re tryingtomake upfor a lost
experience oralostinvestigationinchildhood,real childhood,whenthey’replaying.This
idea at the foundation stage of play being the means is so important and people don’t
realise whenkidsare playinginreceptionclassandthe foundationsstage playshouldbe
the elementof what they’re doingbecause bydoingthat they’re actuallybuildingsome
concrete experiences.

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[others agree]
Teacher A: So the other ethos is that they’re sitting down at a desk and they’re trying to formalise
thingstooearly,Ithinkandwe’re tryingtopickitupafterwards.Now the brightonesare
cognitivelyaware,topsetsin year 7 and 8, they’ll comfortablygoin and you think,well
what’s the difference? Have they had the same experience as others? Well they’ve
probably just taken more from the experiences they’ve had, haven’t they?
Teacher C: It’s interesting though, is just because they can do it, does that mean that they can
understandit?Because IwasalwaysinatopsetandactuallywhenIthinkabouttimesing
two fractionstogether,I’ma flippingmathsteacher,anddividingtwofractions,Ialways
think to myself, do I actually understand it well enough to teach it?
TeacherA: I thinkI understandinthe case of certainfractionslike if Isay a half of a half that makes
sense andif Isay1 dividedbyathirdthatmakessense,butonce Igobeyondandyouget
more complicatednumbers itbecomesan abstractionto me, it’sjust an operationand
…
All: you follow the rule.
TeacherA:so ultimatelywe followthe rule.Youcan’tconstantlyreflectonwhatitmeansbutyou’ve
got some kind of cognitive hook that you can look at when you need to. And it’s the
cognitive hook we’re not providing, isn’t it?
Teacher C: Yeah.
Teacher A: it’s that kind of understanding…
Teacher C: So we’re jumping in with more complex stuff before actually they’ve…
Teacher A: and it’s turned up with GCSE kids who are just remembering to cross multiply, the fish
method, no idea what they’re doing. They might get two marks in the exam. Do they
know anything about maths? No, not at all.
TeacherB: I thinkforsome quite brightkids,A they’re goodatjust rememberingthe rulesandthey
workwiththembut somehow orother theythinkyeahthat sort of makessense, evenif
they don’t understand it at a very deep level, somehow it feels right. It’s that intuitive
sort of feel ‘yeah that feels right, I’m happy with that.’ And I think there’s a lot of
mathematics that I teach that is you know it feels alright to me and if you pinned me
downI’d have to reallythinkaboutthe deepermeaningtoit.The other… I was tryingto
think about different representations of fractionsas well,and it was because I was just
tryingto play devil’sadvocate astohow difficultaddingfractionsisandI thought,if you
presented them, as they do in university, as an abstract operation on an ordered pair,
where abcircle c d equals…andthenyougive thatmathematicalformulaeyou’djustsay
‘What?! You must be joking!What’sall that about?AlrightI can work a few out but I’ve
no idea what this is about.’
Teacher A: well perhaps the experience you have is what they have when they see it.

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TeacherB: That’swhatI mean.Andit’sthinkingaboutorderedpairs,whetherasanaide forthemto
see thattwofractionsare equivalent,if theyhaveatwodimensionalgridandthey…and
you know 5/8 would be, you’d have to decide which way round to do it. I think the x
would be 8 and the y would 5. But if you plot that fraction at (8, 5) therefore, it’s a bit
backto frontreallythinkingaboutit,butequivalentfractionswill havethesame gradient.
Soif they’ve gotthisgridwiththe fractionson,if yousaycanyoufindtwofractionswhich
are the same, or three fractions, they all lie on a straight line. And what do you notice
aboutthe straightline?Itgoesthroughthe origin.Andagainyou can actuallyrelate that
to gradient.They’rethe same steepnessso½,2/4,3/6, 4/8, 5/10, they’re all lyingonthat
same line,theygivethe same steepnessof line.youcanactuallyfindequivalentfractions
by finding ones which line up. It’s just another experience, another way of thinking of
equivalentfractionsratherthan just ‘itmeansyou can finda numberthat goes intotop
and bottom’, which is fair enough but they don’t actually…
Teacher A: get it.
Teacher B: Why does that make them the same? Because 15/24 looks very different to 5/8 to me!
But you could talk about generating… I don’t know, it was just another experience.
Teacher A: It’s interesting, I’ve never thought of equivalent fractions being on a straight line
Teacher C: no I’ve not.
TeacherA:the otherthing,theyusedtohave thingslikecubesandairrods,whichwedon’tsee much
of.Ithinkprimarieshavethemsometimes,whereyou’ve gottherodsof differentcolours.
And youstart off witha rod thislongand you’ve got two rods half the size that fit along
and then 4 and you can build… and again it’s a kinaesthetic thing we can buildand play
and things like that need to be happening in the early years before theycome through
but for some groups you kind of wonder…
Teacher C: did it ever happen?
Teacher B: and for our SEN groups we still need to be doing that.
TeacherC: There has beenanumberof timeswhere Iuse a similarthingcalledthe fractiongridand
it’sgotall the differentfractionsandthentheycolourandsee whichonesare equivalent.
Teacher B: But it’s actually pushing the rods around…
Teacher C: but I do like… I’ve got the rods as well and I think they’re really good.
TeacherA: Useful,but againif you’ve gota class of 30 you can’t,it doesn’twork.The otherthingis
when I was on the PGCE I did a ... it was to do with fractions and teaching kids adding
fractions and equivalent fractions and it was just having simple bars and if I wanted to
add half anda quarter,it waslike usingairrods, itwas a powerpointpresentation,then
effectivelyIsaywell…before Icanaddthose two,we needtobe addingthe samephysical
size of thing so I need to split the (points to the half) so I’ve got three quarters .
TeacherB: I thinkinyour diagram it makes more sense to say you’ve got three eighths personally

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[all laugh]
Teacher B: you can see why the kids get confused.
Teacher C: and this kind of understandingcan stem from the equivalent fractions side of things.If
they’re confident with their equivalent fractions that then can lead into this.
Teacher B: And of course at the back of all of this is their confidence with number.
TeacherC: What I want to knowis multiply anddivision.That’sthe one I alwaysstruggle with.How
would you explain multiplying and dividing fractions, even if it’s the simple ones.
Teacher A: The dividing one I always try a bit heuristicallyand say well if I’ve got 1 divided by one
third, everyone’s happy that’s 3. If I’ve got 2 divided by…
Teacher B: Sorry, why is it three? Justify.
Teacher A: OK. Because there’s three thirds in one whole. So it goes into that one three times.
Teacher B: yeah, you see I don’t think they understand that when…
Teacher A: this is top set
TeacherB: well evenwithtopsetI’dsaywhatdoes42dividedby7equal 6mean?Andyou’re saying
how many 7s can you make out of 42 whole ones? You can make 6.
Teacher A: well I’m saying how many thirds fit inside that one.
Teacher C: that to me… because actually if you say to lower ability kids how many 7s are in 42 or
whatever,theydrawthe linesdon’tthey?Dactuallyyouknow,theywoulddraw apicture
for this wouldn’tthey? Well that’sa third,that’s a third,that’s a third, that makesup a
whole one.
Teacher A: And again that depends on an understanding of sharing that into three equal pieces.
Teacher C: yeah it does
Teacher A: you’re always going back to the previous level, which may not be there.
Teacher B: Yeah. What division reallyis, is where kids are weak in the first place.[others agree] so
howmay thirdsare there ina whole one,well yeahthere are three butthat’s…but here
iswhere you’ve gotthe problembecause hereiswherethirdschange theirnature.If you
do 2 dividedbya third,how many thirdsare there intwo? Well you can eithersay 6, or
there’s always three thirds make a whole one, whatever it is, you know.
[laughing]
Teacher C: Yeah! No wonder the kids get confused!
TeacherB: If you’ve got24 kidsandsplitintoquartersthatmeansyousplittheminto4equal bitsso
there are 4 quartersin 24. You’ve justsplit24 into4 equal quartershaven’tyou? But 24
divided by a quarter means something different, doesn’t it?

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TeacherA:so where I’dgofromthere,if we gettothatbit,thenare we happyif we write 2overone
is the same as 2?
Teacher C: Yeah
TeacherA: andthisis heuristic.It’ssaying…it’saplausibilityargument!If Ido that,flipthatup,I get
4 over one, which is 4.
Teacher C: Right
Teacher A: and guess what, it works for all of them.
Teacher C: So this used kind of like, they can visualise it can’t they and then you go into a more
general …
TeacherB: It’sa plausibilityargumentandtwonegativesmake apositive,twooppositescancel each
other out…
TeacherA: and againit’s justto try and give a cognitive hook.Justsomethingtosay ‘ yeahI can see
how it works there and now I can leap into that extraction with faith that it’s working.
Teacher C: because I’ve come across, when you do the 2 over 1, they’re like ‘why, why is it 2?’
Because they’re thinking about ‘is it division, is it not?’ It starts to open up…
TeacherA: Andagain yeahit’s2 dividedby1.I meaneventhe divisionsymbolisafractionit means
that divided by that.
Teacher C: They find that really hard don’t they?
Teacher A: You’re interchanging between division as fractions now, another layer, isn’t there?
[others agree]
TeacherB: Youcan getyourcalculatorout aswell andinsteadof sayingwhat’stwodivided byahalf,
or youcan start off a little furtherback,you can say ‘Everyone use yourcalculator.How
manysevensin42?’andthey’vegottoreinterpretthatas42dividedbyseven.How many
threesare there insome bignumbersotheycan’tworkitoutin theirheadandtheystart
to getthat how manyof thisare there inthat is a divisionsoyousortof buildthe feeling
of what a divisioncalculationis.That to them is not actuallyobviousanyway,especially
to the lower ability. So loads of work and then you can say ok, you can actually, and of
course their calculators have the fraction button, so you can say how many halves are
there in 20? And they’ll know that’s20 dividedby½ and well the answers40… And just
those experiences… And realise that’s double it,, you know?
Teacher C: so using their calculator as well. We perhaps don’t do enough of that kind of thing.
Teacher B: so you could develop… There’s just so much, isn’t there? There’s no easy answer.
TeacherA: The otherthingas well isI feel thatyoukindof hit,withany understanding,abrickwall,
if like you[TeacherB] said,there’snoconceptof division.If youdon’tknowhow todivide

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twonumbersthenthe fractionbecomesmeaninglessbecause youcan’tlookatafraction
and have any sense of what it means unless you’ve got any sense of understanding of
division. I think I’m right in saying that.
Teacher C: No, I agree.
Teacher B: So we need a pre unit on division and getting them into sort of doing a bit of division.
TeacherA:Istill thinkitstartsbefore wegethere.Itstartsinthe primarywithplayingwithquantities
and splitting things up and sharing things out because you can’t get to division until
you’ve got the understanding of sharing. So you’re sharing things out and then you’re
starting to get to being able to formalise it into division, aren’t you?
Teacher C: Yeah. Because I think as well, if you ask a kid what’s the one thing they always need to
work on, the majority of the kids it’s fractions and I wonder if that’s because, it’s like
you’re saying, we jump into it too fast. Or even some primary schools jump in to it too
fast and it thenbecomesa confidence issue thattheydon’tunderstandit,not an ability
issue.
Teacher A: It’s that ELPS. Basically you’ve got to be able experience something first, once you’ve
experienced it, you talk about it through language. Once you’ve talked about it and
exploreditthroughlanguage,youcandraw picturesandyoustart to abstract aboutit so
that’syou’re startingtogetintoabstraction. Whenyougettobeingintosymbols.Sowe
are coming in here (points to symbols) in year seven and eight. They’re weak on that,
they’re weak on that (pointingto E and L). They’ve seen some pictures but withno real
understanding.So,it’sinmy mindthat (pointingtoE and L again) has to happenbefore
youcan buildthese properly.Sowhatwe doiswe applystickingtape ina sense because
we give thema set of ruleswhichallowsthemto work withsymbolsatthe symbol level
but without any cognitive understanding of what they’re doing. Thisgets them through
their GCSE, but doesn’t give them any understanding of fractions.
TeacherC: It puts it intoperspective whenyoulookatit like that, doesn’tit?There’sthe three bits
before you’ve even got the last bit.
Teacher A: And that, that’s intuitive as well, isn’t it? Because you know if I, I can really talk about
something I’ve experienced, through language isn’t it and then we all share out
experiences through language and then we can start to abstract. That’s why it’s so
importantforkidstotalktoeachotherabouttheirideasbecausethatisformulatingthat,
isn’tit?If you’re notexpressingyourideasyou’rekindof livinginavacuumand youkind
of don’t really get that next step perhaps.
Teacher B: I think there’ll be some who go straight into a picture format and actually manipulate
pictures and diagrams in their heads and they don’t actually do it with a language they
can actuallywork withimagerywithoutalanguage.But I thinkthere won’tbe many like
that.
Teacher A: But I think you can only… your thinking is only as powerful as your language.

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Teacher B: I know several doctors of philosophy who would argue otherwise anyway never mind
Teacher A: I would argue because you’ve got mathematics as a language that allows you to think
about thingsthatyou can’t reasonaboutwithEnglish.Well,youcan’treason easily with
English.So withoutthatlanguage,and that’swhenyou move into the abstract, and you
move intoanabstractlanguage youare communicatingandthinkinginthatlanguageand
that replaces or presents the abstraction, that allows you to move forward. When you
think about solving an equation you try and solve a simple equation, try and solve a
quadratic, for example, with just purely English, without any symbols and you run into
difficulties.Sotomy mind,we’re kindof,we’re lookingforsolutionshere (pointingtoS)
when I suspect the solution is back here (pointing to E and L).