In attempting to look more closely at mathematical activities in an attempt to explore their possibilities, I believe we must start from the words and ideas of maths educators such as David Kent, Caleb Gattegno and David Wheeler rather than from our pre-conceived judgements.

For example, what are our views on children’s mathematical powers?

Do we agree with Gattegno when he writes: “Children spontaneously stay with problems. And they stay for as long as is required. They consider abstraction (the simultaneous use of stressing and ignoring) naturally as their birthright. They give proof that they know many concepts but, more than that, that they know how to generate them in their awareness…. Moreover, they live close to their powers of transformation and their mental dynamics.” (Gattegno, 1981)

I believe that we have to take time to watch young children learning to make sense of the world to re-discover and appreciate the powers that children possess of their own. If we are sensitized by some of the questions posed by Gattegno in the above article, we may begin to share Wheeler’s “strong rational conviction that children have the necessary functionings to mathematize.” (Wheeler, 1975)

Gattegno feels strongly about the lessons to be learnt from such observations. “A method of educating for the future does exist – if we know how to acknowledge what is given us and already is in us, and with this, encounter what is but is not yet part of us… It happened that every one of us as a child did precisely this. For a while we did not talk, we did not speak, and after a while we did both. That is to say, we met what was and we managed to make it our own.” (Gattegno, 1970)

It is important to start with a belief in the powers of children and their ability to mathematize before the tackling of the issue of pupil-centered activities in mathematics, because if we do not recognize these powers we can never be in a position to set suitable tasks.

Wheeler (1975) provides an additional insight into the ways in which we have allowed mathematical activities to become constrained. “We need to consider how to avoid the danger that mathematical activity becomes a label for something too diffused and generalized, a way of learning in which almost anything goes. It may be another step… to substitute for the encouragement of mathematical activity an education which zeroes in on mathematization… the shift of emphasis can take us ever further away from an exclusive reliance on external criteria of quality derived from the mathematics of the past.

“Even though the aim of mathematical activity was designed to stress the importance of the ‘process’ over the ‘product,’ we have tended to reassure ourselves that what we were encouraging was actually mathematical activity by making sure that the product was recognizably familiar mathematics. So, in a way the nature of the product still dominates our judgements.”

I am certain that this is the major dilemma facing anyone attempting to allow pupil-centered mathematical activity to take place in their classroom. How can I justify the time spent? Is the work that they are doing obviously in the syllabus? Are they working reasonably quietly? Is it just a fun lesson? Do I really believe the activity is benefiting their learning?

If all the activity is only to make the lesson more fun, we can learn from John Trivett’s insights. “I began to see what I had been doing over the previous years: glamorize the mathematics, obscure it… to make it attractive and pleasing to the learners. I had dressed up the subject matter and the learning of it with the subtle implication that real mathematics is hard, is dull, is unattainable for the majority of boys and girls and that the best we teachers can therefore do is to sweeten the outward appearance, give extraneous rewards and indulge in entertainment to sweeten the bitter pill.” (Trivett, 1981)

These comments bring into question the whole role of the mathematics teacher during pupil-centered activities. Perhaps the key answer to this problem lies in a genuine belief in the following views expressed over a century ago:

“That if real success is to attend the effort to bring a man to a definite position, one must first of all take pains to find HIM where he is and begin there. …For to be a teacher does not mean simply to affirm that such a thing is so, or to deliver a lecture, etc. No, to be a teacher in the right sense is to be a learner, put yourself in his place so that you understand what he understands and in the way that he understands it…” (Kierkegaard, 1854)

I maintain that the only way in which we can attempt to fulfill what I believe are essential requirements of teaching is to remove ourselves from the center of the stage. To realistically do this, we need to design material that occupies children and this gives mathematical activities an added attraction.

David Wheeler describes his vision of the role of the teacher as follows:

“…the teacher must withdraw as much of himself as possible in the teaching situation… He must use every means he can to focus the attention of the children on the problem, and this means that he must efface himself from their attention…

“If we watch the teacher at work we see that:

–He sets the situation, giving essential information, but beyond that he tells the children nothing;

–He obtains as much information from the children as possible, by observing, asking questions, and asking for particular actions;

–He works with this feedback immediately;

–Except on rare occasions he does not indicate whether a response is right or not, though he often asks the children which it is;

–He accepts errors as important feedback telling him more than correct responses, and by directing children’s attention back to the problem he urges them to use what they know to correct themselves…” (Wheeler, l970)

The main problem that remains if we accept the value of this teaching model is to “set the situation.” Certainly we will not easily feel free to ignore the syllabus and satisfy ourselves (and the inspectors!) that it does not matter that the end-point of the activity is not recognizable as mathematics.

What sort of activity can we set that will allow us to meet these requirements as to our role? I believe that there are at least three different levels at which we can choose to work, and for each of them some guidelines are available as help.

1. Investigations. Lingard (1980) presents an account of the use of mathematical investigations in the classroom. An investigation is typified by the presentation of a situation whose question is posed as an open-ended invitation to investigate. This leaves the pupil with the power to select an aspect of the problem that she finds interesting, identify and define her own parameters and rules, and to decide when the task has been completed, e.g. Draw 4 straight lines on a piece of plain paper so that you get the maximum number of crossing points. How many inside regions are there? Outside regions? Investigate for other numbers of lines.

The advantage of this type of activity is that the problems are interesting and give the teacher an opportunity to practice using a different, listening role. The disadvantage is obviously that the topics covered tend not to be in the syllabus, and anyway, if the teacher has withdrawn from a position of authority, she cannot guarantee what route will be taken nor can she know the destination.

Nevertheless my experience of using the ATM books listed below as resources for an introduction to pupil-centered lessons both for myself and for my students has been extremely positive, and I strongly recommend their use to anyone seriously contemplating this approach.

2. Do, Talk and Record activities. The Open University has prepared an excellent course entitled “Developing Mathematical Thinking.” In a reader (Floyd 1981) and a series of five topic books (using accompanying sound and video accessories) they develop the idea of designing activities that pupils can stay with for a long time, forcing them to become involved in doing, then talking to one another about what they did, and finally attempting to record their work. A particularly useful booklet is “Topic 5,” on fractions. The booklet takes the student through the process of designing and refining a set of suitable activities using available resource material as a starting point. Promising ideas are identified, discussed and refined until the presenter feels that the activity is reasonable.

For example, to play the “Shade-In Game,” you will each need four pieces of scrap paper. You’ll have to fold each piece of paper in half and then in half again. Then fold the piece twice more. When you unfold the paper you should find that the fold lines mark out sixteen equal sections. Each group will then need one die marked with one-sixteenth, one eighth, one fourth, two eighths, three sixteenths, and one half. Each of you should throw the die in turn. The score on the die tells you how much of one piece of paper to shade in. Gradually, the first piece of paper will become completely shaded-in and you will need to move on to the next sheet of paper. The first to shade all four bits of paper is the winner of round one. By turning the four bits of paper over, you can play a second round.

BUT nobody is allowed to shade in any part of their paper without first telling everybody else in the group what areas they are going to shade and why. That’s why the game is called “What-and-Why-Shade-in Game.”

Student teachers have found that this approach to designing suitable resource materials for mathematical activities has been extremely useful and informative.

3. Deeper Structures. While both the previous sections make a start in the search towards genuine pupil-centered mathematical activities, I will not feel content until an attempt has been made to investigate the activity into the key concepts that are to be found in the syllabus. The task now becomes extremely difficult.

For me, the critical understandings that I have to show in trying to penetrate to this deep structure of the topic are:

(a) What are the key concepts in the topic,

(b) What awarenesses are required to gain access to the topic,

(c) What entry points will help the student?

Gattegno (1982) discusses the dilemma of designing activities. “How can I present this challenge so that

[a] I make sure everyone will find an entry into it,

[b] Everyone will find it engaging and rewarding, and

[c] It will be easily extended into other challenges?”

Wheeler (1975) expresses it slightly differently: “We must accept the responsibility of presenting them with meaningful challenges:

–Not too far beyond their reach

–Not so easy as to appear trivial

–Not so mechanical as to be soul killing

–But assuredly capable of exciting them.”

“This sounds very daunting and abstract. What does it mean when we turn our attention to the syllabus, for example, the teaching of geometry? Perhaps the sort of questions we should be asking are:

“‘What do children already know, before we try to teach them geometry, that we could use? What appropriate functionings or powers do children bring with them?

“‘Given that children already have relevant experiences and the capabilities to work with them, what special structurings of their experience will lead to geometry?'” (Wheeler 1975)

The progress from these questions to carefully worded instructions for a mathematical activity that forces each pupil to become involved with the key concept will undoubtedly be slow and painful. In order to show any progress at all we will need to become learners and acknowledge our ignorance.

I believe that this is the final stage in the search toward genuine pupil-centered mathematical activities. Perhaps we will never be able to tackle the challenge, but at least in making the attempt, we move away from the pseudo activities that really constitute nothing more than a sugar coating.

References

A. T. M. (1980) *Points of Departure 1*. Derby: ATM

A. T. M. (1982) *Points of Departure 2*. Derby: ATM

Floyd, A (ed) (1981) *Developing Mathematical Thinking. London*: Addison-Wesley

Gattegno, C. (1970) *What We Owe Children (**Sli**deShare)*. London: Routledge and Kegan Paul

__________ (1981) “Children and Mathematics: A New Appraisal.” *Mathematical Thinking*, 94, 5-7.

__________ (1982) “Thirty Years Later.” *Mathematical Thinking*, 100, 42-45.

Kent, D. (1978) “Linda’s Story.” *Mathematical Thinking*, 83, 13-15

Kierkegaard, S. (1854) *The Journals.* Oxford: Oxford University Press

Lingard, D. (1980) *Mathematical Investigations in the Classroom*. Derby: ATM

Trivett, D (1981) “The Rise and Fall of the Coloured Rods.” *Mathematical Teaching*, 96, 37-41

Wheeler, D (1970) “The Role of the Teacher.” *Mathematical Thinking*, 50, 23-29

Wheeler, D. (1975) “Humanistic Mathematical Education.” *Mathematical Thinking*, 71, 4-9

© C. J. Breen

Education Department, University of Cape Town

Cape Town, South Africa

[A slightly different version of this paper appeared previously in the *Proceedings of the Mathematical Association of South Africa*, 8th National Congress]

*The Science of Education in Questions* – N° 12 – Une Education Pour Demain, France. February 1999.

## Biography

Chris Breen was introduced to the work of Caleb Gattegno while a student with Dennis Crawforth at Exeter University in 1974.

He taught mathematics education in the School of Education at the University of Cape Town from 1982 – 2008 and was President of the International Group for the Psychology of Mathematics Education from July 2004 – July 2007. He currently works as a freelance teacher/consultant in the field of executive education – details of which can be found on his website www.chrisbreen.net.

Email: Chris.Breen@uct.ac.za

“Humanising Mathematical Education” by Chris Breen is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.