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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ContentsIntroduction.......................................................................................................................................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......................................................................................................................................27

Appendix A – Elliott’s Action Research Cycle..................................................................27

Appendix B – Letter of Consent............................................................................................28

Appendix C - Transcript of focus group.............................................................................30

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Introduction

I am a Newly Qualified Teacher (NQT) of Mathematics at a Nottinghamshire secondary school and as

part of the Practice Based Inquiry module for the University of Nottingham Masters in Education, I

am undertaking a piece of action research. The aim of this assignment is to gain an understanding of

what action research is and how to conduct an ethical inquiry. It is not expected that I should reach a

definite ‘answer’ to my chosen problem, but that I should, through reflection, reach a deeper

understanding of my research focus. The following describes not only how my knowledge and

understanding of the nature of action research has developed but also how I went about researching

my chosen focus and how the literature I have read about conducting a practice based inquiry

informed the choices I made. 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

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negative impact 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 positionality 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 that 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

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research as a 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 action 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) benefits from the use of qualitative and narrative methods that are

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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 there 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 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

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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).

After some reflection, I have chosen to consider the pedagogy surrounding an aspect of the

curriculum 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 particularly true of some of the lower attaining groups I have worked with. It was the first time I

had taught the material, however I struggled to teach it in ways I believe to be appropriate for

mathematics teaching. As a result I resorted to teaching the pupils the rules, or simply the process,

knowing that they had gained little conceptual understanding. Upon returning to the topic a couple

of weeks later, 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 accurately recall the rules and that because they had no understanding of where the

rules had come from despite my best efforts, the students were no better off than they had been at

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’ misconceptions and

provides opportunity for greater depth of understanding?

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Reading literature that explained how to choose a good focus for practitioner inquiry went a long

way 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) and from observation and reflections concerning my own practice (Hubbard and Power, 1993).

My focus is centred on “content knowledge” rather than the context of my teaching for the reasons

stated above. Dana and Yendol-Hoppey, (2009), describe eight passions that are good starting points

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, that I will find more value in conducting the action research and hope to take my findings

to implement change in my own practice (Anderson, Herr and Nihlen, 2007). I feel that my chosen

focus is well suited to practitioner inquiry since it will help improve my understanding of students’

misconceptions of fractions and so be developing my content knowledge as well as helping me to

move 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

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is, they confuse this with finding a half of four and so incorrectly give the answer as being

two. This may 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 612

=39

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 34

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 49−14=35

,

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:

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Figure 3.1: List of different interpretations of fractions as given

by Hansen (2011: 33).

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 mathematical

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.

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Critchley (2002) also places merit on letting children solve a real problem involving

fractions in order to support children’s understanding.

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 collecting

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 epistemological 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

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signed and informed consent for my participants in line with the University of

Nottingham’s guidelines (Appendix B).

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

epistemological 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 participate could not

because she simply could not fit it into her timetable before the end of the year.

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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 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

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relationship with the 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

extremely 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 to my positional bias but felt that this was more suitable than a general ‘what

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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 teaching fractions, it is clear that I begin at the iconic stage, using diagrams and

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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 experience 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

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and what their experiences 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

consistent 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) Images of fractions as processes and images of fractions in processes. In M.

J. Høines and A. B. Fuglestad (eds), Proceedings of the 28th PME international conference,

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 practice (special issue: teacher knowledge construction in collaborative settings) 12(7): 69-86.

Ryan, J., and Williams, J., (2007) Children’s Mathematics 4-15. Open University Press.

Sim, J., (1998) Collecting and analysing qualitative data: issues raised by the focus group. Journal of Advanced Nursing, 28: 345-352.

Somekh, B., (1994) Inhabiting each other’s castles: towards knowledge and mutual growth through collaboration. Educational action research, 2(3): 357-381.

Swan, M., (2001) Dealing with misconceptions. In: Issues in mathematics teaching. London: RoutledgeFalmer.

Townsend, A., (2010) Action Research. In: Hartas, D. (Ed.) Education Research and Inquiry: Qualitative and Quantative Approaches: 131-145. London: Continuum.

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Trochim, W. M K., (2006) Qualitative Validity. Research Methods Knowledge Base [online]. Available at http://www.socialresearchmethods.net/kb/qualval.php [Accessed 19/09/2014].

Wood, D., (1988) How children think and learn. Blackwell: Oxford.

Appendices

Appendix A – Elliott’s Action Research Cycle

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 JonesTeacher of Mathematics

Participant Consent Form – Return to Miss H Jones by the 9th of July, 2014.

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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 thinking about fractions and any ideas that you do already that you think 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 found at the start that it’s pretty easy to draw a circle and split it up and say ‘that’s a half’ or ‘that’s a quarter’… although I am still not sure that they are entirely convinced that that is what a fraction is and that’s not always what a quarter is in all circumstances. I suppose its addressing some of those things and how you get round those, including the harder operations… I can draw some pizzas to show addition and that kind of thing, but when it’s multiplication and division…

Teacher A: I think that one of the difficulties I find, is kids have got to be… it’s almost like the kids have got to experience fractions before you can teach them. Because a fraction can be many things can’t it, can be a fraction of a whole, a fraction of a group, you can have the concept of one number divided by another. It can be an operator, so there are many things it can be, it comes in different guises. And it depends on the youngster, for some youngsters you need to go back to… almost like a playfulness of understanding what a fraction really means. If I say ‘half of a group’, what does that really mean? Or if I say ‘three sevenths of a group’, what does that mean? Without that sense of what a fraction is you do end up, like I ended up, in the worst case scenario just having to teach rules, which is not very satisfactory really because you don’t feel like your youngsters are moving on any further 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 extremely difficult to find a way to teach which really goes to the heart of what a fraction is. One of the top flight mathematicians… one in particular that I’m thinking of… when it comes to describing what a fraction is, he says ‘a fraction could be this, could be 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 talking about it as a part of a group or one number divided by another. It’s hard isn’t it?

Teacher B: I thoroughly agree with you [directed at Teacher A]. I’ve been doing it for forty years now and I haven’t found a method yet that I feel really works. I would say that just listening to you talking about fractions in a group, I would say we don’t do enough of herding a 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 into eighths or whatever… and then tell me what five eighths is. And you could

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either do that with people or you could, with the whiteboards and so forth, you could have those of little squares drawn on the board couldn’t you, and then they could just move them into four areas or three areas, depending. So you could get them to experience a little bit more and then sort of say, ‘ok now split them up in eighths… so what’s an eighth of 24 and then what are three eighths of 24’…. you know, you could do stuff like that especially with year seven.!

Teacher C: It’s interesting because I find with top ability year eight or year seven, that if you went back to basics of what fraction really is, I think they’re pretty secure on what a fraction is. 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 them to the understanding of ‘of’ and ‘times’ meaning the same. So a quarter times two thirds is the same as a quarter of two thirds, and then you can get at that. Teacher A talking about the different meanings of fraction, I was trying to think of terms of things that the students are familiar with like 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 clock faces which are nought to twenty-four for a full day. And you could get them to use that pie chart to calculate a quarter of a minute, how many seconds is it… so they have to work a quarter of sixty, they have to understand that they have to divide sixty by four, but they could also then shade it…they could actually split the sixty seconds into four pie chart pieces as well. So you’ve sort of got a picture of quarters, got an idea of quantity that is associated with them. You’re finding a quarter of something fairly tangible, rather than an abstract idea of a quarter. We often give them rectangles to shade, why that size of rectangle when we give them different sizes of rectangles depending on what fraction we want to do. So they’re not really working out of anything very concrete, but the sixty seconds in an hour, there’s lots of fractions that you can do with that…. that work nicely and fractions of a day, in terms of twenty-four hours, there is also a lot of fractions you could do with that . Right, ok, so, I was thinking of trying to get the pictures so 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. And it’s automatically bringing in a little bit of the idea that three eights is divide by eight 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 ‘find a third times 60’. Although again it gives an opportunity to talk about of and times meaning the same thing. [others murmer agreement] And it was a question of, when they’re adding fractions, you know, instead of saying ‘what is a quarter plus three eighths?’ whether you could say ‘what’s a 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 them build up various charts ok so a quarter of the day add three eighths of the day is

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so many hours and looking at that, that’s equivalent to that fraction so you can actually see 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 carefully so that it’s you know a quarter plus three eighths is five eighths and you can get them to sort of think about why that is. And try to develop the rules that they work with.

Teacher A: There’s also that element of sharing, isn’t there? Because if I say five eighths of something

Teacher B: Yeah

Teacher A: I’m splitting into eight equal parts so I’m sharing equally amongst eight and taking five of those parts. And again that’s another deeper understanding of fractions and that fits in with what you were saying.

Teacher B: And when they’re doing the addition of fractions they could have the clocks all around so 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.

Teacher A: And also, if there was a magic bullet professionally it would be out there, wouldn’t it?

[Others agree]

Teacher C: And I think, what I like about what you’re saying was that you’re actually then bringing in a lot of different fractions stuff continuously and I wonder sometimes whether because of the way schemes of work are, we kind of do sections and we link a little bit but actually we mainly do fractions in sections and actually if we did fractions bringing it all in would that help make those links, like we were saying.

Teacher A: I think we certainly become abstract too quickly, too early. You know, we abstract it into five eighths times three sevenths , that’s an abstraction. Meaningless in terms of … and what your idea then of bringing things together where you’ve got a real physical context, which interchanges between a quantity and a fraction, because fractions represent quantities as well, so you’ve got interchange going on. You can kind of think of it, I don’t know, I don’t want to bring decimals into it, but obviously you’ve got that connection there.

Teacher B: I think they just need more experience of fractions where they’re actually using them to calculate something that is at least partly meaningful in some way.

Teacher A: And it’s that experience in a way sometimes I feel like we’re trying to make up for a lost experience or a lost investigation in childhood, real childhood, when they’re playing. This idea at the foundation stage of play being the means is so important and people

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don’t realise when kids are playing in reception class and the foundations stage play should be the element of what they’re doing because by doing that they’re actually building some concrete experiences.

[others agree]

Teacher A: So the other ethos is that they’re sitting down at a desk and they’re trying to formalise things too early, I think and we’re trying to pick it up afterwards. Now the bright ones are cognitively aware, top sets in year 7 and 8, they’ll comfortably go in 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 understand it? Because I was always in a top set and actually when I think about timesing two fractions together, I’m a flipping maths teacher, and dividing two fractions, I always think to myself, do I actually understand it well enough to teach it?

Teacher A: I think I understand in the case of certain fractions like if I say a half of a half that makes sense and if I say 1 divided by a third that makes sense, but once I go beyond and you get more complicated numbers it becomes an abstraction to me, it’s just an operation and …

All: you follow the rule.

Teacher A: so ultimately we follow the rule. You can’t constantly reflect on what it means but you’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.

Teacher B: I think for some quite bright kids, A they’re good at just remembering the rules and they work with them but somehow or other they think yeah that sort of makes sense, even if 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 down I’d have to really think about the deeper meaning to it. The other… I was trying to think about different representations of fractions as well, and it was because I was just trying to play devil’s advocate as to how difficult adding fractions is and I thought, if you presented them, as they do in university, as an abstract operation on an ordered pair, where a b circle c d equals… and then you give that mathematical formulae you’d just

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say ‘What?! You must be joking! What’s all that about? Alright I 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.

Teacher B: That’s what I mean. And it’s thinking about ordered pairs, whether as an aide for them to see that two fractions are equivalent, if they have a two dimensional grid and they … 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 back to front really thinking about it, but equivalent fractions will have the same gradient. So if they’ve got this grid with the fractions on, if you say can you find two fractions which are the same, or three fractions, they all lie on a straight line. And what do you notice about the straight line? It goes through the origin. And again you can actually relate that to gradient. They’re the same steepness so ½, 2/4, 3/6, 4/8, 5/10, they’re all lying on that same line, they give the same steepness of line. you can actually find equivalent fractions by finding ones which line up. It’s just another experience, another way of thinking of equivalent fractions rather than just ‘it means you can find a number that goes into top 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.

Teacher A: the other thing, they used to have things like cubes and air rods, which we don’t see much of. I think primaries have them sometimes, where you’ve got the rods of different colours. And you start off with a rod this long and 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 build and play and things like that need to be happening in the early years before they come 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.

Teacher C: There has been a number of times where I use a similar thing called the fraction grid and it’s got all the different fractions and then they colour and see which ones are 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.

Teacher A: Useful, but again if you’ve got a class of 30 you can’t, it doesn’t work. The other thing is when I was on the PGCE I did a ... it was to do with fractions and teaching kids adding

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fractions and equivalent fractions and it was just having simple bars and if I wanted to add half and a quarter, it was like using air rods , it was a power point presentation, then effectively I say well… before I can add those two, we need to be adding the same physical size of thing so I need to split the (points to the half) so I’ve got three quarters .

Teacher B: I think in your diagram it makes more sense to say you’ve got three eighths personally

[all laugh]

Teacher B: you can see why the kids get confused.

Teacher C: and this kind of understanding can 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.

Teacher C: What I want to know is multiply and division. That’s the one I always struggle 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 heuristically and 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

Teacher B: well even with top set I’d say what does 42 divided by 7 equal 6 mean? And you’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, they draw the lines don’t they? D actually you know, they would draw a picture for this wouldn’t they? Well that’s a third, that’s a third, that’s a third, that makes up 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 really is, is where kids are weak in the first place. [others agree] so how may thirds are there in a whole one, well yeah there are three but that’s … but here is where you’ve got the problem because here is where thirds change their nature. If you do 2 divided by a third, how many thirds are there in two? Well you can either say 6, or there’s always three thirds make a whole one, whatever it is, you know.

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[laughing]

Teacher C: Yeah! No wonder the kids get confused!

Teacher B: If you’ve got 24 kids and split into quarters that means you split them into 4 equal bits so there are 4 quarters in 24. You’ve just split 24 into 4 equal quarters haven’t you? But 24 divided by a quarter means something different, doesn’t it?

Teacher A: so where I’d go from there, if we get to that bit, then are we happy if we write 2 over one is the same as 2?

Teacher C: Yeah

Teacher A: and this is heuristic. It’s saying… it’s a plausibility argument! If I do that, flip that up, 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 …

Teacher B: It’s a plausibility argument and two negatives make a positive, two opposites cancel each other out…

Teacher A: and again it’s just to try and give a cognitive hook. Just something to say ‘ yeah I 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…

Teacher A: And again yeah it’s 2 divided by 1. I mean even the division symbol is a fraction it 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]

Teacher B: You can get your calculator out as well and instead of saying what’s two divided by a half, or you can start off a little further back, you can say ‘Everyone use your calculator. How many sevens in 42?’ and they’ve got to reinterpret that as 42 divided by seven. How many threes are there in some big number so they can’t work it out in their head and they start to get that how many of this are there in that is a division so you sort of build the feeling of what a division calculation is. That to them is not actually obvious anyway, 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

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how many halves are there in 20? And they’ll know that’s 20 divided by ½ and well the answers 40… 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.

Teacher A: The other thing as well is I feel that you kind of hit, with any understanding, a brick wall, if like you [Teacher B] said, there’s no concept of division. If you don’t know how to divide two numbers then the fraction becomes meaningless because you can’t look at a fraction 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.

Teacher A: I still think it starts before we get here. It starts in the primary with playing with quantities 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 then becomes a confidence issue that they don’t understand it, 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 explored it through language, you can draw pictures and you start to abstract about it so that’s you’re starting to get into abstraction. When you get to being in to symbols. So we are coming in here (points to symbols) in year seven and eight. They’re weak on that, they’re weak on that (pointing to E and L). They’ve seen some pictures but with no real understanding. So, it’s in my mind that (pointing to E and L again) has to happen before you can build these properly. So what we do is we apply sticking tape in a sense because we give them a set of rules which allows them to work with symbols at the symbol level but without any cognitive understanding of what they’re doing. This gets them through their GCSE, but doesn’t give them any understanding of fractions.

Teacher C: It puts it into perspective when you look at it like that, doesn’t it? There’s the 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 important for kids to talk to each other about their ideas because that is formulating

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that, isn’t it? If you’re not expressing your ideas you’re kind of living in a vacuum and you kind 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 actually work with imagery without a language. But I think there won’t be many like that.

Teacher A: But I think you can only… your thinking is only as powerful as your language.

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 things that you can’t reason about with English. Well, you can’t reason easily with English. So without that language, and that’s when you move into the abstract, and you move into an abstract language you are communicating and thinking in that language and 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. So to my mind, we’re kind of, we’re looking for solutions here (pointing to S) when I suspect the solution is back here (pointing to E and L).

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