Historically, learning mathematics and teaching it to all students at the school stage has been motivated by the belief that a study of mathematics helps students to learn to reason and apply such reasoning to everyday problems. It is believed that learning mathematics leads to learners’ cognitive development. Thus, one of the important questions that all mathematics educators must constantly ask themselves is: Does the mathematics that we teach (and that our students learn) lead to an enhancement of students’ cognitive abilities? This leads us to clarify what we mean by the understanding of mathematics that we seek to develop in our students.
The deeper understanding that we are looking for must enable students to look at and understand a new situation, delve into the repertoire of mathematical knowledge that they have in terms of concepts, processes, and ideas and adapt or modify those ideas so as to apply them towards resolving a new problem situation. Such understanding calls for building deep connections between concepts, a variety of lenses and representations with which to view the concepts, and flexibility that allows one to sufficiently modify concepts so as to apply them to a new situation. It requires students to develop a rich network of ideas that one may draw from when faced with a novel situation. In this process, students develop habits of the mind that enable them to analyze other situations that they may encounter in life, mathematical or otherwise. This critical blend of processes is what mathematics educators refer to as problem-solving. It is this kind of cognitive development that most modern societies would like their citizens to develop. Problem solving plays an important role in mathematics and should have a prominent role in the mathematics education of students. However, knowing how to incorporate problem solving meaningfully into the mathematics curriculum is not necessarily obvious to mathematics teachers.
It is a commonly accepted goal among mathematics educators that students should understand mathematics (Hiebert & Carpenter, 1992). The most widely accepted theory, known as constructivism, suggests that children must be active participants in the development of their own understanding.
There are several reasons why it is important to do problem solving or to take a problem solving approach to mathematics, especially with more able students. We discuss these under three headings below: to benefit the student, to better represent the subject and to benefit the teacher.
To benefit the student
Problem solving provides an opportunity for children to explore ideas and so gives them the chance to extend their creativity. Children are continually coming up with ways of tackling problems that we hadn’t thought of before. The interesting thing is that the children who are producing these ideas are not always the ones who we generally think of as being good at mathematics. Even apparently relatively weak children may have ideas that turn out to be fruitful. Sometimes though, it can take a little work on your part before the consequences of some ideas are seen. Probably the first way that a student’s exceptional ability comes to the fore is because they solve sums accurately, efficiently, and speedily. Certainly, in primary school, students stand out because they learn new concepts quickly, manipulate numbers intuitively and easily remember number facts.
Learning and applying algorithms in standard situations can quickly become not very interesting for them. Introducing a problem solving approach to mathematics can present bright students with a challenge. It gives them an opportunity to think outside the square and develop their confidence in themselves.
Not all bright students’ ability thrusts them to the fore. Some gifted children may, for whatever reason, quietly sit in a class and be content with the regular programme. Problem solving activities can give these students more motivation and challenge and provide them with an interest in the subject and renewed confidence in themselves.
To better represent the subject
Some observers see mathematics as a subject where success depends on learning rules that are to be followed without thought. This aspect is the creative side of mathematics and is what research mathematicians do on a day to day basis. The highs and lows of their creative activity are felt by mathematicians in the same way that they are by artists, musicians and sports people.
Solving problems parallels very closely the creation of mathematics. By encouraging problem solving in mathematics classrooms we are helping show that there is another aspect to mathematics that has a much more human face and is more interesting than simply following rules.
To benefit the teacher
There are also some advantages to implementing a problem solving approach for teachers. Being the teacher of a really exceptional student can be very challenging. It may be in the best interests of all if a mentor can be found to extend the students and relieve the teacher of some pressure. However, the problem solving may be of some immense help to a teacher ‘in need’. Once an able student is familiar with using a problem solving approach, they should be able to work independently on a mathematical problem for an extended period of time, choosing and exploring generalizations and extensions to the problem that interest them.
Using a problem solving approach in teaching and learning mathematics is of value to all students and especially to those who are high achievers. The reasons for using problem solving are summarized below:
- Problem solving places focus on the student who makes sense of mathematical ideas. When solving problems students are exploring the mathematics within a problem context rather than as an abstract.
- Problem solving provides ongoing assessment information that can help teachers make instructional decisions. The discussions and recording involved in problem solving provide a rich source of information about students’ mathematical knowledge and understanding.
- Problem solving is enjoyable. It allows students to work at their own pace and make decisions about the way they explore the problem. Because the focus is not limited to a specific answer students at different ability levels can experience both challenges and successes on the same problem.
- Good problem solving activities provide an entry point that allows all students to be working on the same problem. The open-ended nature of problem solving allows high achieving students to extend the ideas involved to challenge their greater knowledge and understanding.
- Problem solving develops mathematical power. It gives students the tools to apply their mathematical knowledge to solve hypothetical and real world problems.
- Problem solving better represents the nature of mathematics. Research mathematicians apply this exact approach in their work on daily basis.
In teaching through problem solving, learning takes place during the process of attempting to solve problems in which relevant mathematics concepts and skills are embedded (Lester & Charles, 2003; Schoen & Charles, 2003). As students solve problems, they can use any approach they can think of, draw on any piece of knowledge they have learned, and justify their ideas in ways that they feel are convincing. The learning environment of teaching through problem solving provides a natural setting for students to present various solutions to their group or class and learn mathematics through social interactions, meaning negotiation, and reaching shared understanding. Such activities help students clarify their ideas and acquire different perspectives of the concept or idea they are learning. Empirically, teaching mathematics through problem solving helps students go beyond acquiring isolated ideas toward developing increasingly connected and complex system of knowledge (e.g., Cai, 2003; Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Cobb et al. 1991; Hiebert & Wearne, 1993; Lambdin, 2003). The power of problem solving is that obtaining a successful solution requires students to refine, combine, and modify knowledge they have already learned.
It is important to point out that we are not saying that every task that students encounter must be problematic. If the goal of a lesson is to develop and master certain skills, some exercises are necessary. In addition, as we indicated before, teachers may modify existing less problematic problems to be “true” problems.
As is the case in the past, most people today still believe that mathematics is all about computation. However, computation, for mathematicians, is merely a tool for comprehending structures, relationships, and patterns of mathematical concepts, and therefore producing solutions for complex real life problems. This perspective of mathematicians has gained more attention and importance with rapid advancements in information and communication technologies. It has become a necessity for people of all ages to reach, analyze, and apply the mathematical knowledge effectively and efficiently to be successful citizens in our information age. In particular, students need to be well-equipped with higher-order mathematical knowledge.
The quality of teaching and learning in mathematics is a major challenge for educators. General concern about mathematics achievement has been evident for the last 20 years. The current debate among scholars is what students should learn to be successful in mathematics. The discussion emphasizes new instructional design techniques to produce individuals who can understand and apply fundamental mathematic concepts. A central and persisting issue is how to provide instructional environments, conditions, methods, and solutions that achieve learning goals for students with different skill and ability levels. Innovative instructional approaches and techniques should be developed to ensure that students become successful learners.
Cai J., Lester, F., (2010). Why Is Teaching With Problem Solving Important to Student Learning? National Council of Mathematics Teachers, Houston
Reston, V.A. (2000). Principles and Standards for School Mathematics. NationalCouncil of Teachers of Mathematics.
Preety, N. T. Problem Solving In Mathematics: A Tool for Cognitive Development (p. 1). State University of New York, Oswego, USA.
Van de Walle, J.A. Teaching Developmentally. Elementary and Middle School Mathematics (4th Ed.). Virginia Commonwealth University, New York: Pearson Longman.