A Ten Year Retrospective Michael Richards Problem Solving Task Centre Network Michael presented this address to the 32nd Annual Conference of the Mathematical Association of Victoria in December 1995. At the time he was a mathematics teacher are Mordialloc-Chelsea Secondary College. We are indebted to the MAV for keeping this history on their site from 1996 until 2008. The aims for the A.C.T. (Mathematics Task) Centre are to give students a successful mathematical experience, to develop problem-solving strategies within the students and to encourage teachers to use concrete materials to teach mathematics with students above Year 3. In achieving these aims we hope also to develop skills of careful reading and interpretation of cards and of applying basic mathematical knowledge. In a co-operative non-competitive way, with a partner, students discover new thinking patterns from each other by talking and expressing their ideas in words to consolidate the learning experience. Neville de Mestre and Bea Duncan (1980) p. 8

Pre-Task Centre Teaching

I taught in this way because I believed that my job was to pass on mathematical skills and standard mathematical applications to the students. I tried hard to explain things clearly and reasoned that if I could explain the maths ideas as simply as possible, and in a way that made sense to me, most students would be able to understand my explanations.

Remembering the boredom of mathematics classes in my school days, when opportunities arose, I would try to 'make maths interesting'. These efforts took the form of collecting and using everyday examples of mathematics, organising 'maths games days' and offering an elective subject of geometric constructions.

Maths Task Centres and their attraction to me in 1984

The Task Centre was a classroom containing more than one hundred 'tasks'. Each task was housed in a plastic box and consisted of a written mathematical problem and some related manipulative materials. Classes of students attended the Task Centre on a regular basis. They worked in pairs at solving the problems, with the aid of the materials. All pairs in a given class worked on a different problem. The problems were graded according to degree of difficulty and the teacher chose problems appropriate to each pair of students.

After hearing about and visiting the Task Centre in Brunswick, I, as mathematics coordinator at my school (Footscray Technical School), decided to apply for Disadvantaged Schools funding to set up a similar one. The application was successful and included the services of a teacher's aide.

There were two main attractions of the Task Centre to me; mathematical problem solving and cooperative learning. I had never tried mathematical problem solving with my students. I had heard about it at conferences and had read about it in journals. I was interested in it because it was new and because it seemed to be another way of making maths interesting. However I had lacked the stimulus to introduce it into my teaching. Also, I was threatened by mathematical problem solving. It seemed too different and too daring, and I was worried that I 'would not always know the answer' (as good maths teachers must!).

Cooperative learning was the other attractive teaching strategy. It seemed (seems) a most natural way to learn - through cooperating and sharing ideas. But again the idea of actually trying the strategy with a class of students was frightening. I was scared of losing control. I had no background knowledge of methods of classroom use of cooperative learning.

It is little wonder that the Maths Task Centre, a vehicle that allowed for the easy introduction of both mathematical problem solving and cooperative learning, was attractive to me.

The operation of the 1984 Task Centre

It is apparent to me now that the basic ideas behind Maths Task Centre use, as articulated by de Mestre and Duncan (1980), worked. All students were working on problems (just like puzzles) of interest to them at their own level; all were encouraged to achieve success; they were encouraged to talk (in a maths lesson!) about the problems; to work as a team, share ideas and arrive at joint solutions. (Richards, 1985).

Through observation, it was evident to me that student learning was occurring on a number of levels. In solving the problems, as well as applying previously learned mathematical knowledge, they were encountering obstacles (the essence of problems). These blockages were often overcome, and learning achieved, either through the brainstorming of possible solutions with their partner or by discussion with me or the teacher's aide. New knowledge gained was usually to do with a mathematical concept or process, or a problem solving strategy.

In a pilot study (Richards, 1994) involving a small number of students working with problem solving tasks, those involved were found to enjoy mathematics presented in the Task Centre format, because of their challenging 'puzzle' nature. The students stated that they believed problem solving skills improved slightly, but contrary to my observations, the students believed that their mathematical skills had not developed when working with the tasks.

It was my impression that student learning was also related to the value of teamwork. Students were constantly encouraged to work together, and observed doing so. Have you talked to each other about that? or How about each of you try to solve the problem in different ways and then check with each other how you are going later?, are the sorts of questions I asked. It was clear to me that students came to appreciate the benefits of working together as a team.

There was also a change in communication between teachers. Mathematics teachers who previously talked little about the way they taught, were enthused by the centre and talked openly about how their students were progressing. They also talked about the problems and how the lessons in the Task Centre were going.

My use of Maths Task Centres in 1995

Organisational Factors

• A Task Centre is no longer a separate room into which students move for a problem solving session. It is now merely the collection of tasks. For me it is housed in a storeroom adjacent to my classroom. I have not sought a separate room for the Task Centre, and at my present school, where space is at a premium, it is unlikely that a separate room would be available. There are benefits involved with the current location of the tasks. It is more convenient to collect the tasks from that nearby location compared with having to move the class to another room each time I wish to use them. Furthermore, in the students' minds, maths problem solving is an integral part of the maths course (as it is undertaken in the normal maths room), not a separate and perhaps unrelated activity, as might have been construed in the past.
• I now use a set of fifteen tasks at a time with my classes each semester, rather than the students having one hundred or more to choose from. A smaller collection, as well as being easier to move into the classroom, is much easier for me to organise when distributing and retrieving the tasks. I can much more readily become familiar with the mathematical intricacies of fifteen tasks compared with a larger number. I am hence better able to assist students as they work with the tasks.
• I do not have a teacher's aide to assist in the construction and maintenance of tasks, and to assist during the problem solving sessions. The difficulty of constructing the tasks has been resolved by choosing and adapting problems that require relatively simple equipment, such as matchsticks and counters. The choice of simple equipment also relieves the problem of maintenance. The incorporation of an 'equipment card' with each task, that the students complete at the start of each session, after checking the manipulatives, also reduces the maintenance required. If equipment is missing, I replace it immediately.

This last teaching procedure of checking each task's equipment may seem minor or of little consequence. It is organisational factors such as this, however, that can allow a method of teaching to proceed or not. Imagine the bedlam possible in a class where a number of students could not work at their task because of a lack of equipment.

Teaching & Learning Factors
Another change in my use of Task Centres was originally conceived for organisational reasons. I quickly came to realise, though, that the learning benefits for students were more valuable than the original reason for change.

Teaching with tasks can become hectic as pairs are not working at the same task, rather the class is working on up to fifteen different problems at one time. The lack of a teacher's aide means one less person to assist students with their tasks. In many primary schools, parents help in this way but it is more difficult to recruit parents at the secondary school level. As the only adult in the classroom, I found it extremely difficult to be able to assist students as required. I decided then, in order to reduce the demands for my attention, to require written reports by students of their progress with the problems. Previously, as with the Task Centre at Brunswick, the students were required to report their solution to the problems orally and with a physical demonstration.

The act of writing the reports has resulted in greater student learning. By writing, and talking to their partner about what they are writing, my students are being forced into more considered reflection of the problem and its solution. I require the reports to be written with set headings using Polya's (1945) four stages of problem solving.

The first heading, The Problem (in my own words), requires the students to understand the problem. Without understanding a problem, its solution is impossible. Prior to being required to use this heading, many students rushed in and worked on the problem without full understanding and consequently became confused and made errors. The second heading, My Plan (how I hope to solve the problem), requires the students to consider possible strategies of working out the problems. The third heading, My Solution (how I solved the problem), requires the students to describe clearly their solution, which often obliges the students to check the steps used. Under the fourth and final heading, Looking back or forward, some deeper reflection and analysis is needed. Students analyse the problem in a number of ways, for instance by checking the results or seeing if there are other methods of solution.

The pilot study referred to earlier (Richards,1994), involving students working with problem solving tasks, generally supported this notion of improved learning through the use of headings. When giving their reasons for using the headings, the student responses included:

• So you can make sure you understand it (the problem).
• Like you are writing it out and thinking about it.
• (We are) not rushing through it.
• It helps in organising.
• If we look back we can see how we did it.
• If you write it down it helps you remember.
• Headings are clues for recording.

A 'good' mathematical problem, by my definition, will intrinsically cater intellectually for a wide range of ability levels. An example of a good problem is 'Inverting The Triangle' (de Mestre and Duncan, 1980). Ideally some appropriate leading questions from the teacher will be required. If a pair of students was having difficulty, hints like these might be given: Try it with a smaller number of counters - say 3 or 5 instead of 10. Look for a pattern in the number of moves needed. To extend more able students questions such as Can you find the least number of moves for triangles consisting of more than ten counters? can be asked. Without some intensive teacher questioning, the problem in Figure 1 would not be demanding for many secondary students. However, it is often the case, as previously mentioned, that there is not enough time for sufficient questioning of and discussion with students. Consequently, I have adapted problems such as this so that, for a given problem, a number of questions are posed in increasing order of difficulty. There is also often an additional extension question attached.

The alterations are provided to allow most secondary students some success with the problem - without teacher assistance - and provides extension for the more able students. A problem similar to Inverting The Triangle and demonstrating this method is shown as Upside-Down Triangle (de Mestre and Richards, 1995).

Conclusions

It is vital that the staff to be involved in the (Maths Task) centre discuss issues (related to the running of the centre) ... In this way the centre will reflect the needs of the school, community, and the task centre's clients, the kids! (p.5)

What I did not refer to in 1985 were the changes in teacher understanding of the way students learn. I altered my teaching in Task Centres for this reason also. I believe that there is a need for further investigation into the type and extent of learning that takes place when students work with problem solving tasks.

Changes in my teaching that reflected an altered understanding of student learning, also occurred in my teaching of mathematics outside of Task Centres. Included have been an increase in the use in my teaching of:

• inquiry based learning
• language, both written and spoken
• cooperative learning and activity based learning.

References

de Mestre, N. and Duncan, B. (1980). Sixty Tasks from the ACT Mathematics Centre. Canberra: Curriculum Development Centre.

de Mestre, N., & Richards, M. (1995). 70 hands-on tasks: Mathematical problem solving for secondary schools. Gold Coast: Vengram.

de Mestre, N. & Richards, M. (1990). Problem solving task centres. The Australian Mathematics Teacher, 46, 8-9.

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.

Richards, M. (1994). Unpublished pilot study of students working with Problem Solving Tasks.

Richards, M. (1985). Problem Solving Task Centres: What? Why? How? Vinculum, 22(2), 3-5.

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