ME624_1 Teaching mathematical thinking at Key Stage 3

Experiences of learning mathematics

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This free course provides a sample of level 2 study in Mathematics: www.open.ac.uk/courses/find/mathematics.

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Contents

- Introduction

- Learning outcomes

- 1 Overview

- 1.1 Experiences of learning mathematics

- 1.2 Mathematics: to know or to do?

- 1.3 Designing alternative programmes and curricula

- 1.5 Studying the Möbius band

- 1.6 Do, talk and record triad

- Conclusion

- Keep on learning

- References

- Acknowledgements

Introduction

This course is aimed at teachers who wish to review how they go about the practice of teaching maths, those who are considering
becoming maths teachers, or those who are studying maths courses and would like to understand more about the teaching process.

This OpenLearn course provides a sample of level 2 study in Mathematics.

Learning outcomes

After studying this course, you should be able to:

- understand some current issues in mathematics education, such as the relationship of mathematics content to mathematics processes

- understand a variety of approaches to the teaching of mathematics such as 'do-talk-record'

- approach mathematical problems and tasks in a flexible way.

1 Overview

1.1 Experiences of learning mathematics

You will come to this course with many memories of mathematics, both as a teacher and a learner. It may help if you start
by recalling memories of learning mathematics and making a record of them in your notebook.

When you work on a task, get into the habit of having your notebook to hand to record your thinking. Use the notebook in any
way that helps you to think about the work you have done. Some people find it helpful to divide a page into two columns using
the left-hand side to record memories and descriptions of incidents, and the right-hand side for reflection and commentary.

Task 1 Memories of learning

Review your most vivid memories concerned with learning mathematics. In particular, try to recall the following.

- Your own learning of mathematics as a pupil in school - what were your perceptions and emotions as you participated in those
lessons? Does any specific classroom incident come to mind?

- Your experiences of learning mathematics that happened outside the classroom.

- Your observation of children learning mathematics - try to recall a moment when you were aware of a particular learner and
their engagement, or lack of engagement, with the mathematics.

Make brief notes in your notebook that will help you recall any of the incidents invoked above

View discussion - Task 1 Memories of learning

Having thought about your personal experiences of learning mathematics the next task will provide you with an opportunity
to think about the way you view the learning of mathematics.

Task 2 Describing learning

Complete one or more of the following prompts.

- Learning mathematics is like …

- I enjoy learning mathematics when …

- I dislike learning mathematics when …

View discussion - Task 2 Describing learning

Clarifying what has worked well or badly for you in helping pupils to learn mathematics successfully can all be a useful starting
point in planning effective tasks that you offer learners. Meanwhile it could be very useful to your planning and teaching
to ask your pupils (or friends or colleagues) what their responses are to the prompts about learning mathematics given in Task 2.

Task 3 Describing teaching

Now try to write down some of your beliefs about teaching mathematics.

Complete the following prompts.

- Teaching mathematics is like …

- What I like most about teaching mathematics is …

- What I like least about teaching mathematics is …

View discussion - Task 3 Describing teaching

It is worth remembering that much of what teachers do is adapted, consciously or unconsciously, from what they have seen other
teachers do. Your sense of yourself as a teacher may be coloured by how closely you come to achieving what you have admired
or responded to in others—or it may depend on the extent to which you have found the ‘better way’ you were sure must exist.

You have now examined some of your beliefs about learning and teaching mathematics. But what are your beliefs about the nature
of mathematics itself? One of the key ways in which your perception of the nature of mathematics has developed is through
your own experience of learning and doing mathematics. This experience will also have a bearing on your notions of how mathematics
is learned and on your perceptions of the roles of teachers and pupils in mathematics classrooms. You may also experience
strong feelings and emotions relating to your own work on mathematical tasks. Don't be afraid to consider your feelings as
you reflect on the nature of mathematics and its role in schools.

Task 4 The nature of mathematics

Complete the following prompts.

- Mathematical ideas come from …

- Mathematics is important in schools because …

View discussion - Task 4 The nature of mathematics

When you considered your learning of mathematics, did you see it predominantly as a collection of topics (mathematical content)
or as a way of thinking (mathematical process)?

1.2 Mathematics: to know or to do?

1.2.1 To know or to do?

The so-called ‘content/process’ debate in mathematics involves discussion of the relative importance of content and process
in mathematics. It originated as part of a discussion about the nature of mathematics, particularly of school mathematics,
and of the purposes for which mathematics is learned. Identifying content and process in mathematics draws attention to the
idea that mathematics is a human activity.

As a teacher of mathematics in the UK, you are faced with a national curriculum and, at the school level, a scheme of work
with short, medium and long-term plans. Your aim may be to help pupils use mathematics to ‘make sense of their world’ and,
to this end, you may wish to see them equipped with mathematical skills. But are these content skills (what mathematics they
should know), or process skills (being able to tackle and solve problems), or both?

Task 6 Content or process

Think about the distinction between the content skills and the process skills involved in learning mathematics.

Make two lists of mathematical skills: one headed ‘mathematical content’ and the other ‘mathematical process’.

What skills might each list contain?

Is it easier to write down the content skills or the process skills?

View discussion - Task 6 Content or process

1.2.2 Content

School mathematics curricula often focus on lists of content objectives in areas like number, arithmetic, statistics, measurement,
geometry, trigonometry, and algebra. A typical list of content objectives might contain over one hundred objectives to be
introduced or revisited and learned each year. These can be seen as hierarchical in nature but many textbooks do not attempt
to organise the objectives in ways that enable the bigger underpinning ideas to become apparent to the pupils. In addition,
the order of the listing has often resulted in applications of concepts being viewed as end-of-topic activities rather than being valued as providing meaningful and motivating contexts
for learning.

1.2.3 Process

Mathematical processes are different from content in that they overarch the subject and are not thought of as hierarchical.
A list of processes could contain:

- problem-solving (including investigating);

- mathematical modelling;

- reasoning;

- communicating;

- making connections (including applying mathematics); and

- using tools.

Each of the six processes listed here represents a wide range of component skills that usefully contribute to a pupil's mathematical
thinking as well as to their general thinking skills. In the task that follows, you are invited first to spend some time thinking
of examples of the different processes. You will then be able to consider in more detail what the component skills for each
process might be.

Task 6 Putting processes under the microscope

1. For each of the six processes listed above, write down two or three examples.

2. Click on the 'View document' link below to open ‘Processes uncovered’ and read a detailed listing of the possible components
for these processes provided by Andy Begg (Begg, 1994).

Click on the link below to open 'Processes uncovered'. With thanks to Begg, A. (1994/1996) in Neyland, J. (ed) Mathematics
Education: A Handbook for Teachers, Volume 1, Masterton nz: Wairarapa Education Resource Centre / Reston va: National Council
of Teachers of Mathematics,(pp. 183–1920)

Processes uncovered

View discussion - Task 6 Putting processes under the microscope

There is no unique or universal set of processes in mathematics. The following five mathematical processes have been mentioned
in various reports (for example, NCTM, 1989). They partially overlap with the six you have just been thinking about in Task
6, and the general educational skills listed in the Key Stage 3 National Strategy framework document for England and Wales
(DfEE, 2001)

- information processing skills;

- enquiry skills;

- creative thinking skills;

- reasoning skills; and

- evaluation skills.

As you will see in the next section, these general skills can be cross-matched with the mathematical content normally taught
in school. Indeed, it should be possible, for each mathematical topic that you teach, and for each of the six processes considered
in Task 6, to identify an example of that process in which that mathematical topic is applied.

1.3 Designing alternative programmes and curricula

Assuming that both the content of mathematics and the processes need to be included in programmes and curricula, the problem
becomes one of how a suitable curriculum can be structured. One possibility is to construct a very specific curriculum with
clearly defined objectives for both content and processes separately, and possibly with suggested learning activities. However,
content and process are two complementary ways of viewing the subject.

An alternative is to see the curriculum in a two-dimensional array with content areas drawn against processes; to use a weaving
metaphor, one can be thought of as the weft and the other the warp. The five major content areas (number, algebra, measurement,
geometry, statistics) and the six key processes (problem-solving, mathematical modelling, reasoning, communicating, making
connections, using tools) would therefore form the thirty combinations indicated in the matrix below

Table 1 The Content/Process Matrix

Number

Algebra

Measures

Geometry

Statistics

Problem-solving

Modelling

Reasoning

Communicating

Connecting

Using tools

Task 7 Using a content/process matrix

What is your initial response to this matrix?

Think of a recent mathematics lesson. What was the content of that lesson? Can you identify any mathematical processes that
the pupils used during the lesson (you will need to look again at the Appendix to remind yourself of the detail of these processes)?

Click on the link below to open 'Processes uncovered'. With thanks to Begg, A. (1994/1996) in Neyland, J. (ed) Mathematics
Education: A Handbook for Teachers, Volume 1, Masterton nz: Wairarapa Education Resource Centre / Reston va: National Council
of Teachers of Mathematics,(pp. 183–1920)

Processes uncovered

View discussion - Task 7 Using a content/process matrix

It may help you to look back at the Appendix when you are working on later tasks in this course. Try to be aware of the mathematical
processes you are using as well as the more obvious areas of mathematical content involved.

Task 8 Reading

Read the article Paul Bunyan versus the Conveyor Belt by W. H. Upson. While you are reading, note down your reactions to it.

Click on the link below to read Paul Bunyan versus the Conveyor Belt. With thanks fo W. H. Upson

Paul Bunyan versus the Conveyor Belt

View discussion - Task 8 Reading

When you were reading the article you might have also been aware of how you could use it with your pupils. If you felt it
inappropriate to read the article in full to your pupils you could summarise the story. Using articles and stories in this
way can provide useful examples of the links between everyday life and mathematics.

Task 9 Process

Look back at the article and work out what processes Bunyan used when working on the conveyor belt problem. List them in your
notebook; you will need these later.

So far you have read an article - that is, you have done something. You have also recorded some notes about what you read. You may also have talked to yourself in order to understand
the article more clearly. As you will see in the final section of the course, these three elements form a useful framework
for helping you to think about pupil learning. It will be referred to as the do-talk-record triad, or simply, DTR.

Simply reading words on a page does not mean that you have necessarily engaged with the mathematical ideas or done any mathematical
thinking. In order to help you work on the ideas in a ‘hands-on’ way, you are asked to model the conveyor belt problem using
a physical model.

1.5 Studying the Möbius band

Task 10 The Möbius band

Take a long thin strip of paper (preferably squared or graph paper) about 30 cm by 3 cm. Give one end a half twist and then
tape it together. This is a Möbius band as shown in Figure 1.

Figure 1

View description - Figure 1

Check that the Möbius band has just one face by using a pencil to mark down the centre of the strip—it meets up with itself!

Look at the band and imagine cutting down the middle of it, along the pencil line. Can you see what is going to happen? Make
the cut.

Repeat this process for a second cut down the (new) centre of the strip.

View discussion - Task 10 The Möbius band

You have now tried out the basic idea of a Möbius strip but it can be expanded further and this involves more cutting and
sticking (doing), and more talking and recording … but not always in that order.

The Möbius band was discovered, in the nineteenth century, by the German mathematician and astronomer Augustus Ferdinand Möbius
(1790–1868). The Möbius band is a standard problem type in an area of mathematics called topology-a branch of geometry concerned
with the properties of a figure that remain unaffected when a shape is distorted in some way (perhaps when stretched or knotted).
Topology has applications in contexts that involv surfaces and this includes crystallography, biochemistry (for example, in work with DNA), and electronics.

As you work on the task, make notes about how you work on it and the discoveries you make. Note down any predictions or conjectures
you make and remember to record your findings. Also think about the processes you have used and how you might present the
task to someone else.

Task 11 Return of the Möbius band

Make some more Möbius bands but this time make one with 2 half twists, one with 3 half twists and one with 4 half twists.
It is a good idea to label them at this stage or perhaps use strips of different colour.

Note down how many faces each band has and then cut each one down the centre line.

Before you cut, remember to predict what you think will happen.

While you work on the task, explain to yourself or to someone else what you are doing and what you are thinking.

Record your predictions and results

How did you feel when you were doing this task? Both adults and children often report being very excited by the results but
also pretty baffled. It is sometimes difficult to see what is happening with some mathematical problems, and even trickier
to predict what is going to happen. Many people record their findings and conjectures in a fairly haphazard way but even so
they can usually retrace what they did. When working on an investigation it is common for people to make jottings rather than
organised notes. The advantage of this form of note taking is that it does not slow down the investigation.

However, if you wanted to explain your findings to someone else you might reorganise your notes in a way that someone else
could follow. Too often, pupils can get caught up with the presentation of the work rather than the exploration of the mathematics.
This can result in a loss of creativity and of a sense of purpose and enjoyment. It also is easier for some pupils to explain
their findings verbally rather than having to write them down, while others find it easier to do annotated drawings.

Task 12 Recording

Look back at the way you recorded your findings. Was it ordered, apparently haphazard, neat and tidy? What was the purpose
of your recording?

View discussion - Task 12 Recording

As a teacher it is important not only to reflect on why you are offering a particular task to pupils but also how you are
asking them to record their findings. The Content/Process matrix may help you do this.

Task 13 Content and process

Look back at the Content/Process matrix. If you were using the Möbius band task with pupils which cells do you think you could
fill in? You may find the list of mathematical processes useful. You may also wish to refer to the list of processes you identified
in Task 9.

Now try the Möbius band task with someone else; it does not have to be a pupil. Watch what the person does when they are working
on the task and ask them to explain their thinking as they work. Pay particular attention to the process skills that they
draw on.

Look back at the matrix and see if you can fill in any more of the cells.

View discussion - Task 13 Content and process

When you consider using a task with pupils it may help you to consider whether it has a limited number of purposes or it could
be used in a wide variety of ways. A task that has a variety of purposes was described by Ahmed (1987) as a ‘rich mathematical
task’ being one which:

- must be accessible to everyone at the start;

- needs to allow further challenges and be extendible;

- should invite children to make decisions;

- should involve children in speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting;

- should not restrict pupils from searching in other directions;

- should promote discussion and communication;

- should encourage originality/invention;

- should encourage ‘what if’ and ‘what if not’ questions;

- should have an element of surprise;

- should be enjoyable.

You may find this to be a useful checklist when you are considering using a new task with pupils. The Mobius strip task can
be used in a way that fulfils many of these criteria. You are now asked to explore the second criterion in Ahmed's list by
extending the initial task and making it more challenging

Task 14 Stretching the Möbius band

1. Before reading on, think about how the Mobius band activity might be extended.

2. Look at your results so far from Tasks 10 and 11. Construct a table 1.1 in which you can put your results from Tasks 10 and
11. Using these results, can you predict what will happen with 5 half twists, and so on?

Table 1.1

Number of half twists

Expected result

Actual result

0

2 strips, same length as original, half the width

1

1 strip, twice the length of original, half the width

2

2 linked strips, same length as original, half the width

3

1 strip, twice the length of original, half the width

4

2 linked strips, same length as original, half the width

View discussion - Task 14 Stretching the Möbius band

The layout of the table can sometimes help you to see the pattern, or the general case. From the comments on this table it
looks like the strips with an odd number of twists behave in a different way from those with an even number.

Task 15 The revenge of the Möbius band

Another extension of the Mobius band task might be to make the cut along a different parallel other than down the middle of
the strip. For example, you could investigate what happens when you make the cut one-third of the way along each band.

Before you make each cut, take time to predict what will happen. Make your conjecture and record it. Also record any surprises
or questions that you have.

Can you predict what will happen with 10 half twists or 37 half twists?

View discussion - Task 15 The revenge of the Möbius band

While you were working on these tasks, you were asked to do some activity, talk about it and record your conjectures and findings.
This triad of do–talk– record is a useful device for seeing what is going on in mathematics classrooms and is the subject of the final section of this
course.

1.6 Do, talk and record triad

The do–talk–record triad (DTR) is a description of what is likely to take place in collaborative mathematics classrooms. It is concerned with
observable events, and with the learner rather than the teacher, though many teaching insights flow from it. Although the
order of the triad suggests that it should be followed in a particular sequence, this is not necessarily the case. Sometimes
talking comes before doing or recording before talking. It also takes time for a learner to move from doing and talking to
recording and back to doing again. At each stage the learner needs time to think and reflect on their work and results.

In a classroom where DTR is in use, you are likely to see learners who are prepared:

- to think for themselves;

- not to be afraid to communicate their thinking for fear it may be wrong;

- to accept that wrong answers can be helpful;

- to listen to their peers for comments in their own words;

- to question their peers' ideas asking for justification, examples, or proof

In the classroom, teaching with an emphasis on DTR would look like the framework in Figure 2.

Figure 2

View description - Figure 2

This framework links with ideas of going from the concrete to the abstract, and with notions of the importance of practical,
kinaesthetic, or enactive experience on which to build understanding. It emphasises too the importance of language in learning,
both the pupils' own language and the language of mathematics.

The ideas implicit in this do–talk–record framework are set out in Table 2.

Table 2

Form of activity

Pupils should …

Do

Action and concrete experience in multiple embodiments

… work with particular examples of a more general idea.

Talk

Language patterns injected,explored, listened to, developed

… talk about their work with these particular examples in their own terms.

Record

Stories written in pictures and words, successive ‘short-handing’, ultimately leading to standard notations

… be encouraged to make their own written record of such activities. Initial records might well be in the form of pictures
or words.

In many mathematics classrooms, it is common for doing and recording to take priority over talking. But it is the act of verbalising
a problem that can often be of most help. Teachers can help with this verbalisation by the use of prompts such as the following.

- Explain the question to me.

- What do you know?

- What are you trying to find out?

- What have you done so far?

Often it is the act of verbalisation and the hearing of the words that provides a key to moving forward. It may be the case
that hearing the words in your head is not enough, in which case say them out loud, even if you are not talking to anyone.

You have worked on the Möbius band problems yourself so now it is time to work with pupils. The way you approached the task
may be different from that of your pupils. Remember to note down any differences or similarities.

Task 16 The Mobius band meets the pupils

- Think about how you might introduce the Möbius band to a group of pupils. What would you like the pupils to do in terms of
doing, talking and recording? Think about what you hope the pupils will get out of working on this task?

- Try out your ideas by introducing the Möbius band to a group of pupils. Look for evidence of the pupils doing, talking and
recording

Before you proceed further think about the task you have just completed, the mathematics involved, and the strategies you
used. Were you surprised by some of the results? Jot down any further reflections in your notebook.

Conclusion

In this course you have been introduced to the difference between mathematical content and processes. You have worked on the
do–talk–record (DTR) framework for learning mathematics.

Keep on learning

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References

Ahmed, A. (1987) Better Mathematics, London, HMSO.

DfEE (2001) Key Stage 3 National Strategy: Framework for Teaching Mathematics: Years 7, 8 and 9, London, DfEE.

NCTM (1989) Curriculum and Evaluation Standards for School Mathematics Reston VA, National Council of Teachers of Mathematics.

Acknowledgements

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Task 1 Memories of learning

Discussion

One teacher recalled how things frequently practised in early childhood could become automated without one realising it. She
wrote:

‘As a small child I used to count out loud the stairs in our house when I went up and down. There were fourteen every time.
This habit has stuck with me and I do it subconsciously now. Most houses (so I've found) have fourteen stairs. When we moved
to our current house I didn't realise I was counting the stairs but the very first time I climbed them I said, ‘My goodness,
how odd, there are fifteen stairs’. I checked. There were! I still count them now.’

Back

Task 2 Describing learning

Discussion

Comment

Many teachers offer their personal metaphors and images such as:

Learning mathematics is like …

- using common sense

- learning the rules of the game

- opening curtains

- a voyage of discovery.

Some have commented that it seems like different things at different times and that their responses vary. The changes might
depend on the kind of mathematics being worked on or on how well things were going. So, for one teacher learning mathematics
was like ‘climbing a wall: sometimes easy, sometimes very hard’.

Personal images are also invoked when teachers try to say what they like or dislike about learning mathematics. Here are some
examples.

I enjoy learning mathematics when …

- I reach the light at the end of the tunnel.

- I seem to be discovering the truth, unravelling a mystery

- I get completely absorbed and forget about the time

- somebody says something about a topic I thought I knew and it gives me a new way of looking at it.

Responses to ‘I dislike learning mathematics when …’ are sometimes related to the mathematics itself:

- I can't see why anybody might be interested in ‘the answer’;

sometimes to the social context of learning:

- I'm in a group and everyone around seems to be better at it than I am;

and sometimes to both:

- I get completely stuck and there is no one around to ask.

Often memories of a change from a positive state to a negative one–or negative to positive–are reported, and some learners
have found that this change of state can happen several times in a lesson or study session.

One minute you are jogging along happily thinking you can see just what is going on and then you grind to a halt and decide
that you must be really stupid. When you are on a high it's very difficult to remember what it's like to be low, and vice
versa. I am just beginning to realise that this happens to almost everyone–and that must include the children

Back

Task 3 Describing teaching

Discussion

As with learning, metaphors for teaching may be very personal and may vary, even for an individual at different times. Some
of these that have been reported include the following.

Teaching mathematics is like …

- pushing water uphill (sometimes downhill).

- learning something new every day.

- banging one's head against a wall

- juggling balls in the air.

- opening up new vistas for pupils

- a roller coaster.

- helping pupils to climb a hill–sometimes a steep one.

Ways of completing the prompt ‘What I like most about teaching mathematics is …’ often mention enjoyment and achievement,
the latter especially after a struggle. Thus responses from teachers include

- the ‘aha’ factor;

- a sense of achievement

- trying to make the subject enjoyable

- watching the penny drop and seeing pupils' pleasure at mastering a new skill;

- when someone has been struggling and they suddenly discover they can understand and they can do it;

- when I've taught a difficult subject and pupils think it is easy

- seeing pupils enjoy themselves;

- I've always enjoyed doing mathematics;

- the sense of fulfilment when a problem clicks for a pupil.

Dislikes with respect to mathematics teaching are frequently related to pressures: sometimes from external sources, and sometimes
intrinsic to any mathematics teaching situation. Thus:

What I like least about teaching mathematics is …

- having to deal with large classes that lack motivation

- lack of time needed to present the subject in such a way that even the weakest pupils can appreciate some of the fun, pattern
and power.

- heavy marking load

- record-keeping

- having to teach pupils who spoil, or attempt to spoil, the learning experience for others in the class.

- when too much pressure is put on pupils (often from parents) to excel, and as a result they achieve less

Back

Task 4 The nature of mathematics

Discussion

The first prompt is often answered in terms of where mathematical ideas in the classroom come from:

- books (used in classroom), magazines, colleagues, my brain;

- teachers, parents, TV, environment

- surroundings, books, everyday life

- human activity, the world around us.

Other answers are more global, such as

- within;

- people;

- experience;

- life, anywhere and everywhere.

Reasons given for the intrinsic importance of mathematics often betray a more personal view of what mathematics is ‘about’.
They include:

- its beauty and the support it gives to other disciplines;

- it gives a different perspective on things;

- it is cross-curricular and, in general, pupils will need some mathematics in the world of work;

- it promotes logical thought and approaches;

- it is involved in so many areas of life;

- it is a type of thinking not experienced in many other subjects, as well as a tool for some;

- it is a language used across the curriculum and it trains disciplined thinking;

Other reasons are more instrumental. Mathematics is important in schools because:

- someone said so and it could be taught via every other subject-but that would do someone out of a job

- it is seen as a measure of academic ability

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Task 6 Content or process

Discussion

Many teachers find it easier to write down the content list because they are used to working with documents that relate to
a national, regional or school curriculum.

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Task 6 Putting processes under the microscope

Discussion

One teacher found part 1 of this task difficult to do (he was particularly stuck on the ‘modelling’ process) until it was
suggested that he might find it helpful to think about classroom situations where his pupils were engaged in modelling. He
was then able to come up with several graphical examples of modelling based on a recent data handling investigation that his
pupils had carried out.

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Task 7 Using a content/process matrix

Discussion

Some teachers (and pupils) find it difficult to identify mathematical processes, particularly modelling and connecting. This
course is designed to help you become aware of a range of mathematical processes and to consider how they can be used to encourage
learning

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Task 8 Reading

Discussion

The article shows a clear link between the way mathematics can be used to solve an everyday problem. But it is this link between
school mathematics and everyday life that many pupils find problematic. Articles like this one can be a useful catalyst for
motivating pupils and encouraging classroom discussion and investigation.

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Task 10 The Möbius band

Discussion

Since you have already read the article, it may have been obvious to you what would happen. But many adults and children actually
need to see it happen in order to believe it. After the first cut, you should have a band that was twice as long as the original
and half the width. The second cut produces two connected bands of the same length but half the width–a less obvious result.

As you worked on this task, were you simply following the instructions or was there evidence of a ‘what if?’ energy taking
hold? If this is the sort of energy that you wish to encourage in your classroom, how might it be fostered?

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Task 12 Recording

Discussion

The purpose of your recording was at least threefold. The first was that you had been asked to do it and this recording could
form part of a TMA. The second was that recording may have helped you to keep track of what you were doing. The third possibility
is that the notes you made, and the way you recorded them, may have helped refine your predictions and conjectures.

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Task 13 Content and process

Discussion

The processes involved in working on the Möbius band included problem solving, modelling, reasoning, communicating, connecting
and using tools. It is a rich mathematical task because it involves a variety of processes

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Task 14 Stretching the Möbius band

Discussion

1. One possible extension of the Möbius band is to consider what might happen with 5 half twists, 10 half twists, 37 half twists?
However, there are clearly problems that might emerge in continuing to do this as a practical activity.

2. You need to be able to predict what will happen rather than continuing to make and cut the bands. In general, setting out
results in the form of a table is a useful way of investigating patterns in your results.

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Task 15 The revenge of the Möbius band

Discussion

A table of results has not been included this time because you may want to use the work you have done as part of your research.

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Figure 1

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Figure 2

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