
Mathematics Through
Problem Solving
by Margaret Taplin
Institute of Sathya Sai Education, Hong Kong

What Is A 'ProblemSolving
Approach'? 
As the emphasis has shifted from teaching problem
solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos
Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a
problemsolving approach to teaching mathematics. The focus is on teaching mathematical
topics through problemsolving contexts and enquiryoriented environments which are
characterised by the teacher 'helping students construct a deep understanding of
mathematical ideas and processes by engaging them in doing mathematics: creating,
conjecturing, exploring, testing, and verifying' (Lester et al., 1994, p.154). Specific
characteristics of a problemsolving approach include: 
 interactions between students/students and teacher/students (Van Zoest et al., 1994)
 mathematical dialogue and consensus between students (Van Zoest et al., 1994)
 teachers providing just enough information to establish background/intent of the
problem, and students clarifing, interpreting, and attempting to construct one or more
solution processes (Cobb et al., 1991)
 teachers accepting right/wrong answers in a nonevaluative way (Cobb et al., 1991)
 teachers guiding, coaching, asking insightful questions and sharing in the process of
solving problems (Lester et al., 1994)
 teachers knowing when it is appropriate to intervene, and when to step back and let the
pupils make their own way (Lester et al., 1994)
 A further characteristic is that a problemsolving approach can be used to encourage
students to make generalisations about rules and concepts, a process which is central to
mathematics (Evan and Lappin, 1994).

Schoenfeld (in Olkin and Schoenfeld, 1994, p.43)
described the way in which the use of problem solving in his teaching has changed since
the 1970s:
My early problemsolving courses focused on
problems amenable to solutions by Polyatype heuristics: draw a diagram, examine special
cases or analogies, specialize, generalize, and so on. Over the years the courses evolved
to the point where they focused less on heuristics per se and more on introducing students
to fundamental ideas: the importance of mathematical reasoning and proof..., for example,
and of sustained mathematical investigations (where my problems served as starting points
for serious explorations, rather than tasks to be completed).

Schoenfeld also suggested that a good problem
should be one which can be extended to lead to mathematical explorations and
generalisations. He described three characteristics of mathematical thinking:
 valuing the processes of mathematization and abstraction and having the predilection to
apply them
 developing competence with the tools of the trade and using those tools in the service
of the goal of understanding structure  mathematical sensemaking (Schoenfeld, 1994,
p.60).
 As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just
to solve specific problems, but to 'encourage the interiorization and reorganization of
the involved schemes as a result of the activity' (p.187). Not only does this approach
develop students' confidence in their own ability to think mathematically (Schifter and
Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own
theories about mathematics and the theories of others (NCTM, 1989). Because it has become
so predominant a requirement of teaching, it is important to consider the processes
themselves in more detail.


The Role of Problem Solving
in Teaching Mathematics as a Process 
Problem solving is an important component of
mathematics education because it is the single vehicle which seems to be able to achieve
at school level all three of the values of mathematics listed at the outset of this
article: functional, logical and aesthetic. Let us consider how problem solving is a
useful medium for each of these. 
It has already been pointed out that mathematics
is an essential discipline because of its practical role to the individual and society.
Through a problemsolving approach, this aspect of mathematics can be developed.
Presenting a problem and developing the skills needed to solve that problem is more
motivational than teaching the skills without a context. Such motivation gives problem
solving special value as a vehicle for learning new concepts and skills or the
reinforcement of skills already acquired (Stanic and Kilpatrick, 1989, NCTM, 1989).
Approaching mathematics through problem solving can create a context which simulates real
life and therefore justifies the mathematics rather than treating it as an end in itself.
The National Council of Teachers of Mathematics (NCTM, 1980) recommended that problem
solving be the focus of mathematics teaching because, they say, it encompasses skills and
functions which are an important part of everyday life. Furthermore it can help people to
adapt to changes and unexpected problems in their careers and other aspects of their
lives. More recently the Council endorsed this recommendation (NCTM, 1989) with the
statement that problem solving should underly all aspects of mathematics teaching in order
to give students experience of the power of mathematics in the world around them. They see
problem solving as a vehicle for students to construct, evaluate and refine their own
theories about mathematics and the theories of others. 
According to Resnick (1987) a problemsolving
approach contributes to the practical use of mathematics by helping people to develop the
facility to be adaptable when, for instance, technology breaks down. It can thus also help
people to transfer into new work environments at this time when most are likely to be
faced with several career changes during a working lifetime (NCTM, 1989). Resnick
expressed the belief that 'school should focus its efforts on preparing people to be good
adaptive learners, so that they can perform effectively when situations are unpredictable
and task demands change' (p.18). Cockcroft (1982) also advocated problem solving as a
means of developing mathematical thinking as a tool for daily living, saying that
problemsolving ability lies 'at the heart of mathematics' (p.73) because it is the means
by which mathematics can be applied to a variety of unfamiliar situations. 
Problem solving is, however, more than a vehicle
for teaching and reinforcing mathematical knowledge and helping to meet everyday
challenges. It is also a skill which can enhance logical reasoning. Individuals can no
longer function optimally in society by just knowing the rules to follow to obtain a
correct answer. They also need to be able to decide through a process of logical deduction
what algorithm, if any, a situation requires, and sometimes need to be able to develop
their own rules in a situation where an algorithm cannot be directly applied. For these
reasons problem solving can be developed as a valuable skill in itself, a way of thinking
(NCTM, 1989), rather than just as the means to an end of finding the correct answer. 
Many writers have emphasised the importance of
problem solving as a means of developing the logical thinking aspect of mathematics. 'If
education fails to contribute to the development of the intelligence, it is obviously
incomplete. Yet intelligence is essentially the ability to solve problems: everyday
problems, personal problems ... '(Polya, 1980, p.1). Modern definitions of intelligence
(Gardner, 1985) talk about practical intelligence which enables 'the individual to resolve
genuine problems or difficulties that he or she encounters' (p.60) and also encourages the
individual to find or create problems 'thereby laying the groundwork for the acquisition
of new knowledge' (p.85). As was pointed out earlier, standard mathematics, with the
emphasis on the acquisition of knowledge, does not necessarily cater for these needs.
Resnick (1987) described the discrepancies which exist between the algorithmic approaches
taught in schools and the 'invented' strategies which most people use in the workforce in
order to solve practical problems which do not always fit neatly into a taught algorithm.
As she says, most people have developed 'rules of thumb' for calculating, for example,
quantities, discounts or the amount of change they should give, and these rarely involve
standard algorithms. Training in problemsolving techniques equips people more readily
with the ability to adapt to such situations. 
A further reason why a problemsolving approach
is valuable is as an aesthetic form. Problem solving allows the student to experience a
range of emotions associated with various stages in the solution process. Mathematicians
who successfully solve problems say that the experience of having done so contributes to
an appreciation for the 'power and beauty of mathematics' (NCTM, 1989, p.77), the
"joy of banging your head against a mathematical wall, and then discovering that
there might be ways of either going around or over that wall" (Olkin and Schoenfeld,
1994, p.43). They also speak of the willingness or even desire to engage with a task for a
length of time which causes the task to cease being a 'puzzle' and allows it to become a
problem. However, although it is this engagement which initially motivates the solver to
pursue a problem, it is still necessary for certain techniques to be available for the
involvement to continue successfully. Hence more needs to be understood about what these
techniques are and how they can best be made available. 
In the past decade it has been suggested that
problemsolving techniques can be made available most effectively through making problem
solving the focus of the mathematics curriculum. Although mathematical problems have
traditionally been a part of the mathematics curriculum, it has been only comparatively
recently that problem solving has come to be regarded as an important medium for teaching
and learning mathematics (Stanic and Kilpatrick, 1989). In the past problem solving had a
place in the mathematics classroom, but it was usually used in a token way as a starting
point to obtain a single correct answer, usually by following a single 'correct'
procedure. More recently, however, professional organisations such as the National Council
of Teachers of Mathematics (NCTM, 1980 and 1989) have recommended that the mathematics
curriculum should be organized around problem solving, focusing on: 
(i) 
developing skills and the ability to apply these
skills to unfamiliar situations 
(ii) 
gathering, organising, interpreting and
communicating information 
(iii) 
formulating key questions, analyzing and
conceptualizing problems, defining problems and goals, discovering patterns and
similarities, seeking out appropriate data, experimenting, transferring skills and
strategies to new situations 
(iv) 
developing curiosity, confidence and
openmindedness (NCTM, 1980, pp.23).


One of the aims of teaching through problem
solving is to encourage students to refine and build onto their own processes over a
period of time as their experiences allow them to discard some ideas and become aware of
further possibilities (Carpenter, 1989). As well as developing knowledge, the students are
also developing an understanding of when it is appropriate to use particular strategies.
Through using this approach the emphasis is on making the students more responsible for
their own learning rather than letting them feel that the algorithms they use are the
inventions of some external and unknown 'expert'. There is considerable importance placed
on exploratory activities, observation and discovery, and trial and error. Students need
to develop their own theories, test them, test the theories of others, discard them if
they are not consistent, and try something else (NCTM, 1989). Students can become even
more involved in problem solving by formulating and solving their own problems, or by
rewriting problems in their own words in order to facilitate understanding. It is of
particular importance to note that they are encouraged to discuss the processes which they
are undertaking, in order to improve understanding, gain new insights into the problem and
communicate their ideas (Thompson, 1985, Stacey and Groves, 1985).

Conclusion 
It has been suggested in this chapter that there
are many reasons why a problemsolving approach can contribute significantly to the
outcomes of a mathematics education. Not only is it a vehicle for developing logical
thinking, it can provide students with a context for learning mathematical knowledge, it
can enhance transfer of skills to unfamiliar situations and it is an aesthetic form in
itself. A problemsolving approach can provide a vehicle for students to construct their
own ideas about mathematics and to take responsibility for their own learning. There is
little doubt that the mathematics program can be enhanced by the establishment of an
environment in which students are exposed to teaching via problem solving, as opposed to
more traditional models of teaching about problem solving. The challenge for teachers, at
all levels, is to develop the process of mathematical thinking alongside the knowledge and
to seek opportunities to present even routine mathematics tasks in problemsolving
contexts.

References 
Carpenter, T. P. (1989). 'Teaching as problem solving'. In R.I.Charles
and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving,
(pp.187202). USA: National Council of Teachers of Mathematics. 
Clarke, D. and McDonough, A. (1989). 'The problems of the problem
solving classroom', The Australian Mathematics Teacher, 45, 3, 2024. 
Cobb, P., Wood, T. and Yackel, E. (1991). 'A constructivist approach to
second grade mathematics'. In von Glaserfield, E. (Ed.), Radical Constructivism in
Mathematics Education, pp. 157176. Dordrecht, The Netherlands: Kluwer Academic
Publishers. 
Cockcroft, W.H. (Ed.) (1982). Mathematics Counts. Report of the
Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's
Stationery Office. 
Evan, R. and Lappin, G. (1994). 'Constructing meaningful understanding
of mathematics content', in Aichele, D. and Coxford, A. (Eds.) Professional Development
for Teachers of Mathematics , pp. 128143. Reston, Virginia: NCTM. 
Gardner, Howard (1985). Frames of Mind. N.Y: Basic Books. 
Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon,
V.M. and Raymond, A.M. (1994). 'Learning how to teach via problem solving'. in Aichele, D.
and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp.
152166. Reston, Virginia: NCTM. 
National Council of Teachers of Mathematics (NCTM) (1980). An Agenda
for Action: Recommendations for School Mathematics of the 1980s, Reston, Virginia:
NCTM. 
National Council of Teachers of Mathematics (NCTM) (1989). Curriculum
and Evaluation Standards for School Mathematics, Reston, Virginia: NCTM. 
Olkin, I. & Schoenfeld, A. (1994). A discussion of Bruce Reznick's
chapter. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp.
3951). Hillsdale, NJ: Lawrence Erlbaum Associates. 
Polya, G. (1980). 'On solving mathematical problems in high school'. In
S. Krulik (Ed). Problem Solving in School Mathematics, (pp.12). Reston, Virginia:
NCTM. 
Resnick, L. B. (1987). 'Learning in school and out', Educational
Researcher, 16, 1320.. 
Romberg, T. (1994). Classroom instruction that fosters mathematical
thinking and problem solving: connections between theory and practice. In A. Schoenfeld
(Ed.). Mathematical Thinking and Problem Solving. (pp. 287304). Hillsdale, NJ:
Lawrence Erlbaum Associates. 
Schifter, D. and Fosnot, C. (1993). Reconstructing Mathematics
Education. NY: Teachers College Press. 
Schoenfeld, A. (1994). Reflections on doing and teaching mathematics.
In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 5369).
Hillsdale, NJ: Lawrence Erlbaum Associates. 
Stacey, K. and Groves, S. (1985). Strategies for Problem Solving,
Melbourne, Victoria: VICTRACC. 
Stanic, G. and Kilpatrick, J. (1989). 'Historical perspectives on
problem solving in the mathematics curriculum'. In R.I. Charles and E.A. Silver (Eds), The
Teaching and Assessing of Mathematical Problem Solving, (pp.122). USA: National
Council of Teachers of Mathematics. 
Swafford, J.O. (1995). 'Teacher preparation'. in Carl, I.M. (Ed.) Prospects
for School Mathematics , pp. 157174. Reston, Virginia: NCTM. 
Swafford, J.O. (1995). 'Teacher preparation'. in Carl, I.M. (Ed.) Prospects
for School Mathematics , pp. 157174. Reston, Virginia: NCTM. 
Thompson, P. W. (1985). 'Experience, problem solving, and learning
mathematics: considerations in developing mathematics curricula'. In E.A. Silver (Ed.), Teaching
and Learning Mathematical Problem Solving: Multiple Research Perspectives,
(pp.189236). Hillsdale, N.J: Lawrence Erlbaum. 
Van Zoest, L., Jones, G. and Thornton, C. (1994). 'Beliefs about
mathematics teaching held by preservice teachers involved in a first grade mentorship
program'. Mathematics Education Research Journal. 6(1): 3755. 


