Working mathematically

Introduction

In this course you will explore a number of major issues in mathematics education. For example, when you think about mathematics, do you see it mainly as a collection of topics (mathematical content) or as a way of thinking (mathematical process)? Later in the course you will be asked to consider further the distinction between content and process in mathematics. You will also be asked to consider a particular organising framework for thinking about your teaching. It is referred to as the do–talk–record (DTR) framework. You will also be asked to work on some mathematics yourself and then reflect on what you have done. The mathematical context used in this course is Möbius strips. This context has been chosen to give you the opportunity to work on some mathematics at your own level. After you have completed several mathematical activities relating to Möbius strips, you will be expected to select parts of these activities, adapting them as necessary, to try out with your learners (or others). This work with learners will give you the opportunity to apply the DTR framework to your own teaching.

But first, take a little time to reflect on your own feelings about your aims for this course. What do you want to work on with regard to the learning and teaching of mathematics and what is of most concern to you? Activity 1 will help you engage with these and related questions.

This OpenLearn course is an adapted extract from the Open University course ME620 Mathematical thinking in schools.

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 the use of pedagogical frameworks used in teaching mathematics such as do–talk–record

• approach mathematical problems and activities in a flexible way

• recognise a personal way of working on mathematics activities.

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. Studying the course will add to your personal store of material, either through undertaking mathematical activity yourself or by introducing mathematical activities to learners.

When you work on an activity, 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.

1.1  Memories of learning

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

1.2 Describing learning

Clarifying what has worked well or badly for you in helping learners to learn mathematics successfully can all be a useful starting point in planning effective activities that you offer learners. Meanwhile it could be very useful to your planning and teaching to ask your learners (or friends or colleagues) what their responses are to the prompts about learning mathematics given in Activity 3.

1.3  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 learners in mathematics classrooms. You may also experience strong feelings and emotions relating to your own work on mathematical activities. Don’t be afraid to consider your feelings as you reflect on the nature of mathematics and its role in schools.

1.4  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)?

2  Mathematics: 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, 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 learners 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?

2.1  Content

School mathematics curricula have traditionally focused 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 learners. 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.

2.2  Process

Mathematical processes are different from content in that they overarch the subject and are not thought of as hierarchical. You may find that once you have identified and clarified in your mind the key mathematical processes, you are then ready to make the shift from mathematical content to mathematical thinking. A list of processes could contain:

• problem solving (including investigating)
• mathematical modelling
• reasoning
• communicating
• making connections (including applying mathematics)
• using tools.

Each of the six processes listed here represents a wide range of component skills that usefully contribute to a learner’s mathematical thinking as well as to their general thinking skills. In the activity 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.

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 Activity 7, and are similar to the key processes listed in the National Curriculum in England (Qualifications and Curriculum Authority, 2007):

• information processing skills
• enquiry skills
• creative thinking skills
• reasoning skills
• 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 and this is something that you are asked to do in Section 4.

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 in Figure 1.

Figure 1 The content/process matrix

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The diagram is in the form of a two way table representing the matrix that connects content with process. The labels in the vertical left-hand column are Problem solving, Modelling, Reasoning, Communicating, Connecting and Using tools. The horizontal labels are Number, Algebra, Measures, Geometry and Statistics

Figure 1 The content/process matrix

It may help you to look back at the Appendix when you are working on later activities 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.

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

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 or with a colleague in order to understand the article more clearly. As you will see in the final section of this course, these three elements form a useful framework for helping you to think about learner learning. It will be referred to as the DTR triad.

3.1  The Möbius band

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.

Activity 11  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 2.

Figure 2 The Möbius band

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The diagram shows a picture of a Möbius band with one twist as described in the text.

Figure 2 The Möbius band

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.

Comment

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.

A useful general strategy for stimulating pupils’ curiosity and channelling the energy that results is to invite them to pose ‘what if?’ questions. As an illustration, when you worked on the Möbius band in Activity 11, some suggestions were provided in these activities for things to try. A more open-ended (and in many ways more satisfactory) version of this activity would be to ask you to pose a series of ‘what if…’ questions for yourself. For example, what if I used two half twists/three half twists, and so on. Would you have welcomed more openness in this activity? For example, were you simply following the instructions or did you start to feel that there was evidence of a ‘what if?’ energy taking hold within you? If this is the sort of energy that you wish to encourage in your classroom, it is worth considering how it might be fostered?

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 involve surfaces and this includes crystallography, biochemistry (for example, in work with DNA), and electronics.

As you work on the activity, 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 activity to someone else.

How did you feel when you were doing this activity? 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, learners 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 learners to explain their findings verbally rather than having to write them down, while others find it easier to do annotated drawings.

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

When you consider using an activity with learners 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. An activity that has a variety of purposes was described by Ahmed (1987) as a ‘rich mathematical activity’ 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 learners 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.

(Ahmed, 1987, p. 20)

You may find this to be a useful checklist when you are considering using a new activity with learners. The Möbius strip activity 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 activity and making it more challenging.

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.

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

4  Do–talk–record triad

The DTR triad 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 3.

Figure 3 Do–talk–record framework

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The do–talk–record framework is a series of five boxes with text in. The first box text is ‘action and concrete experience of the process’, an arrow leads from this box to the second box. The second box text is ‘linked through language development and much discussion to’, an arrow leads from this box to the third box. The third box text is ‘pictures of that action and words which describe the process’, an arrow leads from this box to the fourth box. The fourth box text is ‘linked as a result of frequent use and abbreviation to’, an arrow leads from this box to the fifth, and final, box. The fifth box text is ‘standard notation and layout of an algorithm’.

Figure 3 Do–talk–record framework

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 learners’ own language and the language of mathematics.

The ideas implicit in this DTR framework are set out in Table 2.

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 learners. The way you approached the activity may be different from that of your learners. Remember to note down any differences or similarities.

Before you proceed further think about the activity 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 thought about the difference between mathematical content and processes. Finally, you have worked on the DTR framework.

Appendix: processes uncovered

The following listing provides possible components for the six processes (Source: Begg, 1994, pp. 183–92).

Appendix: Paul Bunyan versus the conveyor belt

Here’s a story about the legendary Paul Bunyan that shows how the Möbius strip has some ‘practical’ applications.

© Hazlett W. Upson (1960) ‘Paul Bunyan versus the conveyor belt’, John Murray Publishers

References

Acknowledgements

This course was written by Barbara Allen and Alan Graham.

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence.

The material acknowledged below is Proprietary and used under licence (not subject to Creative Commons Licence). Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:

Course image: Rob Deutscher in Flickr made available under Creative Commons Attribution 2.0 Licence.

Extract from Upson, W.H. (1960) ‘Bunyan Versus the conveyor belt’, John Murray Publishers.

Every effort has been made to contact copyright owners. If any have been inadvertently overlooked, the publishers will be pleased to make the necessary arrangements at the first opportunity.

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