issues with language and learning: mathematics as a · pdf fileissues with language and...

Robert Beckett (BEC14128094)

Issues with Language and Learning: Mathematics as a Language

Mathematics Contexts and the Wider Curriculum

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Issues with Language and Learning: Mathematics as a Language Robert Beckett (BEC14128094)


University College London, Institute of Education

03/06/2015

Introduction

The language in which we are taught can sometimes create barriers to learning, especially within

mathematics (Pimm 1987). This assignment considers some of the struggles faced by pupils with

understanding and using subject specific language in the classroom. Within mathematics using specific

language is important in gaining basic and deeper understanding of concepts. After a brief review of

the broader picture, I will focus on difficulties found when using mathematical language, and the

development of understanding and use of the ‘mathematical register’ (Halliday 1978; Pimm 1987)

which is claimed to be the ‘language’ of mathematics.

Issues with language and learning

Much research has been undertaken in the field of language within education, and its relationships

to thought and learning, with Vygotsky and Piaget providing some of the most well established theories

(Austin and Howson 1979; Orton 1993).

Sapir said that “the feeling entertained by many that they think, or even reason, without language

is an illusion” (see Austin and Howson 1979, p.167) and in fact thought and language are intimately

connected. This links to Vygotsky’s ideas that language occupies an important and integral part of

thought (Austin and Howson 1979). He believed that egocentric speech was a transitional stage between

vocal and inner speech, defined as “to a large extent thinking in pure meaning” (Austin and Howson

1979, p. 166). In contrast Piaget felt that egocentric speech disappears as thought processes mature.

Barnes remarks that “verbalisation is important because ideally it makes our thought-processes open to

conscious inspection and modification” (see Orton 1993, p.140), which suggests that inner speech might

not always be sufficient.

These theories led to the development of social constructivism (Vygotsky) and cognitive

development (Piaget). In each there is consideration of knowledge, learning, motivation and instruction.

In cognitive development there is an emphasis on construction of knowledge from existing ideas and

concept development before speech, and so it is based around egocentric learning. Social

constructivism, however, is heavily focused on learning and knowledge being formulated through social

interaction such as discussion in order to develop concepts. Consideration of both theories can help to

develop the use of language to reduce the effect of learning obstruction in secondary schools.

If we consider communication and formation of concepts, Skemp (1987) suggests this is sped up

through language and thinking of examples and counter examples. He discusses how we could explain

a well-known concept, the colour red, to a man given sight after being blind from birth. Skemp

concludes that the most useful definition would be to point to a selection of red objects so the man can

form an understanding of the colour red. However, if then asked ‘What does colour mean?’ we cannot

use examples to explain colour as a concept. We need to have base knowledge before we can understand

higher order concepts (Skemp 1987) and the language we use can create barriers when learning

particular concepts and ideas. A similar theory is discussed by Vygotsky (1962) where we find it

difficult to separate the name of an object from its attributes once we have associated them.

2 Robert Beckett (BEC14128094)



People can generally interpret meaning from the context in which something is said (Orton 1993).

If you asked a child to put the knives and forks on the table you would not expect to find all the knives

and forks on the table, but enough for each person who would be eating. There is an expectation that

the child will know what we mean, and communicating meaning and the interpretation of language,

certainly at a young age, can be an issue with learning. Vygotsky (1962, p. 222) suggests that “behind

words there is the independent grammar of thoughts” and the fact that children can interpret what we

say in a different way is part of the relationship between language and learning. The importance of this

interpretation is discussed later within the context of mathematics.

From my own teaching experience it is apparent that pupils tend to separate subjects, not

connecting their relationships but preferring to keep them apart. This could be true for English and

French, for example, or in my own experience mathematics and science. When covering compound

measures, in particular speed and density, pupils asked why we were doing ‘science’ in a mathematics

lesson. The learners have taken both a thematic and conceptual meaning from the language of

compound measures. By mentioning density, the children thought we were doing science, however

compound measures is used in both subjects. The aim in science is to understand the physical situation;

whilst in mathematics the physical situation is a context, but the aim is to perceive the abstract

mathematical properties, not to understand the physical.

The two phrases ‘Would you like a cup of tea?’ and ‘Shall I put the kettle on?’ (Skemp 1987) both

imply having a hot beverage. This is Skemp’s (1987) idea about ‘deep’ and ‘surface’ structures within

language and how certain phrases can share similar surface (or deep) structures but hold different deep

(or surface) structures. This aspect of human language and interaction can create great difficulties for

new language learners and is discussed later within a mathematical context.

The mathematical register

Mathematical language is a widely explored topic and it has been suggested that the way we use

and express language in a mathematics classroom can become a barrier to learning, rather than assisting

pupils with their understanding (Pimm 1987). Linguists, such as Halliday (1978), have explored the

idea of a ‘mathematics register’ as a way of describing the differences of using mainstream languages

in a mathematical context. The term ‘register’ is a technical linguistic term describing a set of meanings

appropriate to a particular function of language, together with the words and structures which express

those meanings (Pimm 1987, p.75). Halliday (1978) claims that we should not think of a mathematical

register merely as terminology, but as meanings and the modes of argument used to express them. It is

clear that what makes up the mathematics register could be considered as the language of mathematics.

Registers have been developed in other disciplines (Lee 2006) and the notion corresponds to

Skemp’s (1987) idea of a ‘schema’ (a conceptual structure stored in memory) matched to a particular

structure. Skemp suggests that we all have a number of these ‘tuned structures’ corresponding to our

available schemas and we interpret words in terms of whichever one resonates with what is coming in.

Different structures could be resonated by the same input at different times by different people resulting

in different interpretations. For example in music we have ‘notes’, ‘keys’ and ‘bars’ which mean

something completely different in our day to day lives. This means that pupils, who are developing a

number of different registers through different subjects, could interpret sentences in different ways

depending upon which of their tuned structures resonate.

It has been identified that the mathematical register has a specific vocabulary (Lee 2006) using

words from three categories:




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1. Words that have the same meaning in everyday language as they do in ordinary English –

the words that are used to set mathematics in context

2. Words that have a meaning only in mathematical language – hypotenuse, isosceles,

coefficient, graph, take moments

3. Words that have different meanings in mathematical language and natural language –

difference, odd, mean, volume, value, integrate.

(Lee 2006, p.15)

Halliday (1978) suggested that vocabulary is open-ended and there are many ways in which a

language can add new meanings, including by inventing new words or by new meanings being attached

to words in response to societal change.

Communicating mathematics

Confusion can often arise during the communication of mathematics and acquisition of register

control. One key question is ‘Are pupils sufficiently aware of the existence of a mathematics register

which is employed within the school?’ (Pimm 1987, p.77). It is clear there is a change in language style

and behaviour, but how aware are they that they should now be communicating mathematically,

learning mathematics and learning how to speak like a mathematician.

An example of this confusion and misinterpretation of terms is described by Orton (1993, p.128)

when a child struggles to answer, “What is the difference between 47 and 23?” The answers, “One of

the numbers is bigger than the other” and “One number contains a 4 and a 7 but the other number

doesn’t” were true for the child’s notion of the word difference, however in the mathematics register

this term has an alternative meaning.

As an extension to the mathematics register, using mathematical symbolism as ‘shorthand’ can

cause further problems. These symbols contain an idea and “without an idea attached, a symbol is empty

[and] meaningless” (Skemp 1987, p.46). Skemp (1987) discusses his idea of deep and surface structures

in a mathematical context, also raised by Orton (1993), and considers how our mathematical symbolism

generally only gives us the surface structure.

A confusing example of symbolism occurs with vectors. When a vector 𝒂 is in typeface it is written

in bold, whereas when handwritten we underline the letter. When teaching this to year 10 students there

was confusion between the typeface notation used on the presentation and the underlined notation used

when I wrote out a vector. They could not understand the relevance of underlining the vector and I had

to explain that we need to differentiate between scalar values and vector values.

The mathematical register uses metaphor to help convey meaning (Halliday 1978; Pimm 1987;

Lee 2006). This occurs at every level of mathematics and well known examples are that a function is a

machine and an equation is a balance. Recognising the use and existence of these metaphors can be

difficult owing to how the mathematical register is used. As in ordinary English, stretching these

metaphors can result in failings such as the inclusion of negative numbers in an equation balance (Lee

2006). This use of metaphor over simile can cause misconceptions and confusion for pupils, who might

give up if they fail to decipher the literal sense of a statement (Pimm 1987).

Issues with using mathematical language

“Who has the right to say mathematics is a language, or mathematics is not a language? Language

belongs to all the people who use it” (Wagner 2009, p. 454). When considering the vocabulary used in




the mathematics register, we can see that words such as ‘square’ and ‘base’ have different meanings in

different mathematical contexts (Lee 2006). It is therefore important to define context in order to

prescribe definite meaning to the words used. An example would be: ‘What is the meaning of the term

negative?’ Unless we have the context for this definition we cannot respond. In school, children are

often provided with the opportunity to learn definitions and prescribed meanings, however this is very

different to the way children learn fluency in language (Wagner 2009). Fluency in a language is attained

when someone is able to think in that language (Lee 2006), though Pimm (1987) suggests that fluency

in mathematics is when one is knowledgeable or has control over mathematical concepts. So how can

we encourage this fluency in mathematics?

By developing ‘mathematical literacy’ we can help to encourage this fluency by improving

mathematical language acquisition and understanding. A definition of ‘mathematical literacy’ was

given for the PISA (Programme for International Student Assessment) 2012 cycle which reads:

‘Mathematical literacy is an individual’s capacity to formulate, employ, and interpret mathematics

in a variety of contexts. It includes reasoning mathematically and using mathematical concepts,

procedures, facts, and tools to describe, explain, and predict phenomena. It assists individuals to

recognise the role that mathematics plays in the world and to make the well-founded judgments

and decisions needed by constructive, engaged and reflective citizens.’ (OECD 2010)

So by improving mathematical literacy, the ability to reason and use concepts, solve problems and

explain why, should in turn improve mathematical language. On the other hand, improvement of

mathematical language could in turn improve mathematical literacy and so it seems there is a very close

link between the two.

Lee (2006) briefly explores the idea of teaching the mathematical language as an additional

language, since it holds many of the features of a natural language. She suggests that teaching

mathematics as a foreign language might overcome issues and barriers that occur when pupils are

required to use the mathematical register. Pupils might be expected to master words, grammar and

syntax of mathematics, along with cultural aspects of people who use it, in order to fully grasp certain

ideas. Lee suggested that pupils need to divulge and become engrossed with the social and cultural

aspects of mathematics before they can express mathematical ideas and concepts through efficient use

of the mathematics register.

There is a very common issue in learning mathematics where pupils worry about ‘doing it right’

(Lee 2006). There are two common responses from those who are unsure. The first is where the pupil

prefers to sit and do nothing waiting for an answer just in case they get it wrong. Lee (2006) suggests a

struggle in this situation where the teacher uses a lot of energy trying to overcome a pupil’s deep

conviction that there is only one way to solve a problem. These pupils also would rather copy out a

method they don’t understand rather than fix their own small mistakes. The second response is seeking

teacher attention to check every step of the solution. I have experienced both responses several times

during my teaching experience. For example, one bright student who always tries hard often has his

hand up for me to check his workings. I try to avoid providing an answer but instead ask him to explain

his work and reasoning, after all what value is there in knowing how to do something if you don’t

understand why it is done?

Investigating issues with language and learning in a London academy

The academy where this investigation was conducted splits the cohort of pupils into three miniature

schools each housing approximately four-hundred-and-fifty key stage three and four (KS3/4) pupils,




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with a further miniature school housing a similar number of sixth form students. When referring to a

‘school’ I will be talking about the miniature school I was based in (which had an accelerated

curriculum), and reference to the ‘academy’ will cover all of the miniature schools.

Within the school the English Department try to address as many of the language issues and

barriers as possible before they occur. Pupils are provided with a ‘cover sheet’ at the front of their books

which provides definitions of key glossary terms and guides to using punctuation and connective words.

This cover sheet differs depending on the year group, but remains at the front of their books for the

duration of that academic year, with teachers referring to it when necessary. Pupils can use this sheet

for their independent learning, as well as for part of the success criteria when self or peer assessing

grammatical or literacy based work. In every subject five blank pages are left at the front of exercise

books so pupils can form a glossary of key terms each week throughout the year. This ensures that

pupils have a reference point for understanding definitions of key terms introduced during the year.

This could be linked to Piaget’s cognitive development theory, since students are expected to construct

their own definitions, building on knowledge they have gained. The Social and Religious Studies (SRS)

and English Departments regularly remind students to update this glossary; however the Humanities

and Mathematics Departments admitted that time constraints often prevent this, in particular with KS4

classes.

Teachers of English and humanities say they do not usually see a language barrier for pupils, with

most grasping terminology quite well, especially at KS3. Terminology, in particular related to relevant

topics, is taught along the way (e.g. ‘trenches’ when covering World War topics), and keywords are

displayed at the beginning of lessons to allow visualisation. Within history there is a problem with

students over abbreviating words (e.g. World War 1 – WW1), but addressing issues like this is difficult

owing to the large amount of content to cover. Understanding technical terms within English, such as

‘onomatopoeia’, is done through assessment for learning strategies, such as cards sorts, and students

might be asked to construct a sentence including use of a simile or a metaphor. These ideas build upon

the social constructivist approach to learning with more significant emphasis upon interactive and

collaborative learning. The SRS Department uses extended writing as the main form of assessment at

KS3, with relatively simple phrasing to ensure there is no issue decoding the question. They also employ

success criteria in which the pupils are encouraged to use religious terminology to reach higher levels.

The English Department use a similar strategy across the curriculum as a way of ensuring that students

are aware that they need to understand technical terms, and use them to reach a higher level or grade.

For SRS at KS4 there is a large literacy issue with the phrasing of GCSE questions being complex or

open to interpretation, which in turn creates problems with decoding them. At this level teachers

struggle more with generic literacy terms such as ‘controversial’ or ‘persists’ rather than key

terminology. As a result the SRS Department across the academy have begun to use more generic

literacy marking as a strategy to try and address this. This decoding issue is also apparent across key

stages in science where the question phrasing causes problems particularly within application questions

where pupils are indirectly expected to apply knowledge they have been taught to a new situation.

It is clear that there are issues with language and learning across the academy, but each department

is not only aware of these, but also has ideas and strategies in place to deal with them.

How ‘mathematical language’ is encouraged in relevant subjects in a London academy

The academy encourages literacy skills within all lessons and this is emphasised strongly within

the school. In the Mathematics Department children are encouraged to provide their answers in full

sentences and to provide justification or reasoning for their answer, both orally and in written




communication. Generally this is expected for articulating reasoning of problem solving to deepen and

reinforce understandings, and allowing students to talk in a professional composed and confident

manner. Each Mathematics Department uses this policy to encourage the learning of key words and

definitions. Key mathematical terminology is introduced to pupils regularly and they are encouraged to

note these words in their books (though not always in the glossary section mentioned earlier).

Sometimes pupils find the introduction of new vocabulary confusing and unnecessary (e.g. using

product instead of multiply) which becomes a barrier to learning. The department also adopts the

school’s marking policy whereby each pupil is given at least one subject and one literacy target for

every lesson. The literacy targets can be generic or subject specific and examples such as writing out

misspelt words three times or providing full definitions of key terminology are regularly evident in

teachers marking.

Whilst there will always be barriers within written work, it is possible for pupils to verbalise their

ideas. They are expected to make use of terminology when providing answers in full sentences. This

relates to the social constructivist model in which learning can only happen through interaction such as

discussion and sharing ideas. All Mathematics Departments in the academy are also aware of the need

to acquire key vocabulary and to gain understanding of the deep structures of concepts as discussed by

Skemp (1987).

Contrastingly, confidence in mathematical skills within science lessons seems to be a major barrier.

Some pupils assume they have poor mathematical ability and therefore often fail to attempt questions,

whilst there are also pupils who are very good at mathematics who struggle to transfer their skills to

science lessons owing to the more covert and complex manner in which information is arranged. This

lack of ability to transfer mathematical skills to science causes re-teaching of core skills, wasting time

and frustrating pupils and teachers. In terms of strategies to address these issues, across the academy

there is training to help improve numeracy teaching within science lessons at least twice a year and the

miniature schools share their resources and approaches as a way of improving. Teachers find this

training useful, however feel that further professional development in mathematics specific subject

aspects (e.g. solving equations) would have a greater and more direct impact on the pupils.

There is a strong emphasis in developing pupils’ mathematical literacy (OECD 2010) within the

school in mathematics and science. In mathematics, pupils are often provided with DfES (2005)

Improving Learning in Mathematics activities or ‘rich’ tasks from the website NRich (1997-2015),

helping to develop their mathematical reasoning and understanding of concepts. This provides

opportunities for pupils to talk about their ideas and knowledge before coming to conclusions, again

relating to the social constructivist model and similar to Vygotsky’s suggestion that we seek

understanding before we are willing to internalise the information.

The academy is encouraging the development of literacy, mathematical language and mathematical

literacy during lessons. Currently strategies and resources within each miniature school stretch only as

far as making use of the individual school policies. For my placement school this is making use of the

exercise book glossary and marking policy, and creating or finding resources or strategies to aid and

improve this mathematical language acquisition is a development area for the school.

Developing understanding and fluency in use of the mathematical register

In order to develop pupils’ fluency of the mathematical register, I have continued the school’s

emphasis on developing pupils’ mathematical literacy (OECD 2010) through the use of activities such

as open ended discussions, tasks and experiments. I have, in particular, made use of the DfES (2005)

Improving Learning in Mathematics tasks which have a strong element of Vygotsky’s social




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constructivist approach with collaborative learning. These tasks open up the opportunity for pupils to

discuss their ideas and develop their mathematical speech whilst solving mathematical problems.

There is scope to develop their use of the mathematical register further. I wanted to design an

activity that would deepen the development of understanding and fluency in using terminology and in

particular using the mathematical register.

Vygotsky’s social constructivist approach is one that I now see as vital for pupils to develop their

understanding of the mathematical register (Halliday 1978; Pimm 1987) and use it in a fluent and

appropriate manner during mathematics lessons. To develop this, in addition to competency in

mathematical literacy, it is key for pupils to discuss ideas in an open environment that enables mistakes

to be built upon. This allows all suggestions to be placed out in the open before clarifying and modifying

their own ideas to end up “with a stronger and more cohesive structure than before” (Skemp 1987,

p.43).

With this approach in mind, the activity I designed focussed on developing understanding of certain

key terms. Within mathematics the school’s policy for keeping a glossary, whilst useful, can lead to

some ambiguity. Although the necessity to explain the context in which a term is being used is not

normally apparent, within mathematics there are a number of terms which could be used in different

areas or aspects of the subject and even terms which could have multiple definitions. Two examples are

‘triangle’ and our earlier example ‘negative’. If providing a definition in your glossary page for triangle

do you write ‘has three sides’, ‘has three angles’, ‘angles add up to 180°’ or even something completely

different? Could a pupil even write about spherical triangles (Pimm 1987)? If being asked to define

negative, although you might be able to in that context, what happens when the term appears again in a

new aspect; is it redefined? Which definition is correct?

When considering the notion of a triangle Pimm (1987, p.102) schematises the situation with

branches from the word to separate two ideas, a straight-sided triangle and a spherical triangle. However

considering some of the other issues with this term I felt that this idea could be extended so that terms

with various definitions, ideas or contexts could be displayed with clarity for pupils in the form of a

spider diagram. This would allow space for all of the key aspects of a triangle to be defined in one space

(see Appendix A), and would also allow each particular context of the word negative to be given around

the word (see Appendix B).

In order to implement this activity I needed to ensure that it held an element of social interaction

forming the basis of higher psychological functioning. Knowledge and understanding of terms used in

the mathematical register is acquired through collaborative and interactive learning, and through open

discussions to share and formulate ideas about concepts. The activity would hold elements of student-

student scaffolding (van Oers 2014), as well as opportunities for teacher interaction if students got stuck

with their discussion. This would help to develop pupils’ knowledge, skills and confidence in the task

as well as the concept.

After several lessons on surds with year 10 I decided to try and implement my activity as a way of

expressing key ideas about surds in one place. This could later become a useful revision tool. I invited

the pupils to discuss their ideas about the word surd as well considering concepts discussed in lessons

in order to formulate a spider diagram linking everything together.

During the discussions I heard some pupils say, “Well isn’t a surd a square root which doesn’t have

a root? Like a decimal or something?” Often their peer was able to respond with the comment that a

surd is an infinite decimal number, an irrational number. When unable to reach this conclusion I tried




to prompt discussion about whether or not √4 was in fact a surd. Pupils reached the conclusion that this

was not a surd since it has a solution, whereas something like √2 is an everlasting decimal number. One

small group of pupils even made the statement that “You can’t round a surd to a decimal place otherwise

it wouldn’t be a surd.” Pupils had used some of the resources provided in the DfES (2005) Improving

Learning in Mathematics task ‘N11 Manipulating surds’ in their most recent lesson and as a result were

generally familiar with the rules used when performing algebraic operations on surds. Pupils briefly

discussed and agreed that they could include all of the rules (including the incorrect ones) as part of

their spider diagram. Whilst not necessarily a definition, it is an aspect of surds which will be useful to

remember, and including those rules which don’t work may help to remind them of misconceptions that

could arise. Students also discussed the potential inclusion of examples of these rules as well as

examples of how to simplify surd expressions.

It was excellent to see that pupils had taken on board ideas surrounding the concept of surds during

their lessons, and were also then able to apply this in a discussion to assist other learners who may have

missed content, or struggled with concepts, to formulate the key information about surds. Although the

class managed to express their ideas through an interactive discussion, I hadn’t thought through my

introduction of the activity and it did not provide the full results I had hoped for. I think there may have

been an element of reluctance from pupils as they wanted to ‘do it right’ (Lee 2006) and so held

discussions developing their ability to use the relevant part of the mathematical register, however they

refrained from putting these to paper. One student produced a diagram of ideas seen in lessons (see

Appendix C), whilst I fear that for many of her peers who did not record this information the knowledge

and understanding may be lost.

Following this, I tested the activity with my year 7 class having spent a few lessons on the topic of

exploring shapes The definition of shape is certainly vaguer than the term surd so I hoped for more

success with this implementation. The pupils had recently spent a lesson completing the Improving

Learning in Mathematics task ‘SS1 Classifying shapes’ (DfES 2005) and considering different

properties shapes might have. I explained that I wanted them to discuss their ideas about what the word

‘shape’ could mean and then to produce a spider diagram linking these suggestions together. I suggested

that they could also provide ideas from prior knowledge. To prevent the element of reluctance seen with

the year 10 class, I made it clear that mistakes might be made. Following the results of the year 10

lesson I decided to further my scaffolding (van Oers 2014) slightly through an example (see figure 1)

in order to provide more clarity of expectations. This scaffolding also helped to readdress a

misconception that any shape is a polygon, which some pupils ran into problems with during their starter

activity that same lesson.

Pupils were active in their discussions during the activity, with most immediately discussing

possible examples of a polygon and also examples of the opposite, not a polygon. Others considered

Figure 1




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some of the potential properties these shapes might have, using ideas from the lesson as well as some

others. Many suggested that all (two-dimensional) shapes had area and perimeter and agree to include

these in their diagrams, and some discussed the relevance of parallel line and symmetry within shapes

(all classifications seen in the DfES (2005) activity). Some pupils even extended this to include

‘definitions’ of the terms, for example “parallel lines – never meet or touch together.” Pupils again

discussed a question raised in the starter activity about whether or not a circle was a polygon with most

agreeing that due to it containing no angle (formed by two straight lines) it was not a polygon. Although

not every pupil had fully grasped the use of a spider diagram (to link connecting ideas), all pupils were

able to create a diagram containing key aspects relating to shapes.

What Skemp (1987) suggested about creating stronger, structured ideas following an interactive

and collaborative discussion was really evident during the activity with the year 7 class and was far

more successful than the previous attempt. The pupils had clearer expectations of the activity,

addressing the discrepancies which featured during the year 10 lesson. Pupils held strong and productive

discussions, considering ideas from lessons and from their general knowledge to create some detailed

diagrams (see Appendix D) with different aspects which could be applied to the word shape. Whilst

they didn’t necessarily come up with a specific definition, they were able to provide examples of

different types of shape and their properties. The students were then able to easily connect these ideas

and condense the information into one space.

Following the work on each topic it was clear that both year 10 and year 7 had gained confidence

in the terminology sufficient to discuss it in an open environment and collaborate on their ideas. The

year 7 class yielded more success as I had provided clearer instruction. The activity could be improved

even further by inviting pupils to provide more detailed diagrams containing more of their discussions

and ‘definitions’, rather than just related terminology (which might also require definitions).

Conclusion

There has been considerable research into issues between language and learning and the most

extensive academic research in using mathematics as language has been conducted by Pimm (1987).

There has been a focus on the issue of ensuring fluency in using what was termed the ‘mathematics

register’ by Halliday (1978) by a number of other researchers. One of the key aspects expressed in the

research is to encourage exploration of mathematical ideas and language through social interaction such

as group work activities. For me this began by continuing emphasis created within a London Academy

on the development of mathematical literacy (OECD 2010) by using DfES (2005) activities, as well as

others, to encourage the development of problems solving through the means of discussion. Whilst

considering the work conducted by Pimm, I was able to design an activity, using spider diagrams, which

built upon the ideas around keeping a glossary of key terminology and definitions to develop pupils

understanding. Through a social constructivist approach, pupils developed their ideas through

discussion before coming to conclusions. This activity addressed some of the concerns surrounding

ambiguity of mathematically terminology and helped pupils to become more confident in expressing

their ideas before forming a strong and solid structure about a mathematical concept, which in turn

helped develop their use of the mathematical register.

References

Austin, J. L. and Howson, A. G. (1979) Language and mathematical education. Educational Studies in

Mathematics 10, pp. 161-97.




DfES. (2005). Improving Learning in Mathematics. London: Standards Unit, Teaching and Learning

Division.

Halliday, M. A. K. (1978). Language as social semiotic (pp. 195-204). London: Edward Arnold.

Lee, C. (2006). Language for learning mathematics. Maidenhead, England: Open University Press.

NRich. (1997-2015). NRich enriching mathematics website. University of Cambridge. Retrieved 2 May

2015 from http://nrich.maths.org/frontpage .

OECD. (2010). PISA 2012 mathematics framework. To OECD, November 30, 2010. Draft subject to

possible revision after the field trial. Retrieved 20 April 2015 from:

www.oecd.org/pisa/pisaproducts/46961598.pdf .

Orton, A. (1993). Learning mathematics: Issues, Theory and Classroom Practice. London: Continuum.

Pimm, D. (1987). Speaking mathematically. London: Routledge & Kegan Paul.

Skemp, R. (1987). The Psychology of Learning Mathematics. L. Erlbaum Associates.

Wagner, D. (2009). If mathematics is a language, how do you swear in it? The Montana Mathematics

Enthusiast, 6(3), pp. 449-458. Retrieved 18 February 2015 from:

http://www.math.umt.edu/tmme/vol6no3/Wagner_article10_vol6no3_pp449_458.pdf .

van Oers, B. (2014). Scaffolding in Mathematics Education. In S. Lerman (Eds), Encyclopedia of

Mathematics Education. Springer.

Vygotsky, L. S. (1962). Thought and Language. New York: MIT Press/Wiley.

http://nrich.maths.org/frontpage

http://www.oecd.org/pisa/pisaproducts/46961598.pdf

http://www.math.umt.edu/tmme/vol6no3/Wagner_article10_vol6no3_pp449_458.pdf




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Appendix A – Example higher level spider diagram for the term triangle

Appendix B – Example higher level spider diagram for the term negative




Appendix C – Year 10 surds spider diagram

Appendix D – Year 7 shape spider diagrams




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