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 at MordiallocChelsea 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 problemsolving 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 cooperative
noncompetitive 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
PreTask Centre Teaching
Prior to 1984 I had taught in the traditional mathematics manner. That
is, I encouraged silence at all times; I explained mathematical concepts
from the front of the class and with the aid of the blackboard to write
examples of relevant problems; the students copied the examples, then
tried to replicate my method on other problems listed in their text book.


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
Unbeknown to me, in 1982, something called a Maths Task Centre had been
established at Brunswick Primary School. It was modelled on the first Task
Centre in Australia, the ACT Maths Centre, and was the first Victorian one
(Byrne and Zeplin, 1990).
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
Once our Mathematical Task Centre was set up I started to take my
classes there once each week. The use of the centre followed that of the
Brunswick centre, as described above. My students loved the sessions. They
could not believe that 'this was maths!' It was fun.
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).
Some students initially found the idea of using blocks, counters and
other manipulative materials 'beneath them', but these concerns were
overtaken by enjoyment. They were industrious at solving the problems and
at working cooperatively, but it was busy work for me (the teacher) and
the teacher's aide keeping up with them.
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
Changes have occurred for me in my use of mathematics tasks since 1984.
The changes have been mainly organisational or to do with teaching in a
way that better caters for mixed ability classes at a secondary school
level.
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 UpsideDown Triangle
(de Mestre and Richards, 1995).
Conclusions
I wrote in 1985 (Richards):
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)


The changes I have made in teaching with Mathematics Task Centres since
1984 have in general reflected this statement. The changes have been in
response to changed teaching circumstances such as different schools and
staff, and in response to the nature of the secondary school students
including the mixed abilities of students in a typical secondary
mathematics class. A number of the changes have come about following
suggestions from members of the Problem Solving Task Centre Network. This
group of practising Task Centre users was established in 1985 (de Mestre
and Richards, 1990) and still operates today.
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.
The teaching strategies I now employ in these areas, although distinctly different from those I
use with the Task Centre, reflect remarkably the aims of the first Maths
Task Centre, described at the beginning of this retrospective. While I am
uncertain of the extent of the influence of the 'Task Centre way of
teaching' on my overall teaching of mathematics, I am sure that those aims
would be included among those of many progressive mathematics educators
even today.
References
Byrne, J. & Zeplin, D. (1990). A problemsolving approach:
Brunswick Maths Task Centre. Melbourne: Ministry of Education.
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 handson 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, 89.
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), 35.
Continue discovering the history of problem solving task centres and the contributions of so many teachers to the evolution of the concept of learning to work like a mathematician in fascinating, captivating and absorbing classrooms in A Sense of History, Mathematics Centre's own ten year retrospective (2002).
Follow this link to Task Centre Home page.
