# 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

#### 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.’

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 …

#### 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

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 …

#### 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

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 …

#### 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

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)?