THE IMPORTANCE OF PROBLEM SOLVING

“Problem solving is not only a goal of learning mathematics but also a major means of doing so.”

                                                        (NCTM, 2000, p. 52)

An information- and technology-based society requires individuals who are able to think critically about complex issues, people who can “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998, p. 2-1). To prepare students to function in such a society, teachers have a responsibility to promote in their classrooms the experience of problem-solving processes and the acquisition of problem-solving strategies, and to foster in students positive dispositions towards problem solving.

In promoting problem solving, teachers encourage students to reason their way to a solution or to new learning. During the course of this problem solving, teachers further encourage students to make conjectures and justify solutions. The communication that occurs during and after the process of problem solving helps all students to see the problem from different perspectives and opens the door to a multitude of strategies for getting at a solution. By seeing how others solve a problem, students can begin to think about their own thinking (metacognition) and the thinking of others and can consciously adjust their own strategies to make them as efficient and accurate as possible.

In their everyday experiences, students are intuitively and naturally solving problems. They seek solutions to sharing toys with friends or building elaborate structures with construction materials. Teachers who use problem solving as the focus of their mathematics class help their students to develop and extend these intuitive strategies. Through relevant and meaningful experiences, students develop a repertoire of strategies and processes (e.g., steps for solving problems) that they can apply when solving problems. Students develop this repertoire over time, as they become more mature in their problem-solving skills. The problem-solving processes that Kindergarten students use will look very different from those that Grade 6 students use. Initially, students will rely on intuition. With exposure, experience, and shared learning, they will formalize an effective approach to solving problems by developing a repertoire of problem-solving strategies that they can use flexibly when faced with new problem solving situations.

“We want children to take risks, to tackle unfamiliar tasks, and to stick with them – in short, to try and persevere. We want children to be flexible in their thinking and to know that many problems can be modeled, represented, and solved in more than one way.” (Payne, 1990, p. 41)

In fostering positive dispositions in their students towards problem solving, teachers deal with the affective factors that have an impact on student behaviour in both positive and negative ways (Schoenfeld, 1992). Students who believe that they are good problem solvers are not apt to give up after a few minutes when faced with a challenging problem. Because beliefs influence behaviour, effective mathematics programs always consider students’ beliefs and attitudes, and teachers work to nurture in students confident attitudes about their abilities as mathematical problem solvers and their beliefs that everyone can make sense of and do mathematics. As students engage in problem solving, they participate in a wide variety of cognitive experiences that help them to prepare for the many problem solving situations they will encounter throughout their lives. They:

• learn mathematical concepts with understanding and practise skills in context;

• reason mathematically by exploring mathematical ideas, making conjectures, and justifying results;

• reflect on the nature of inquiry in the world of mathematics;

• reflect on and monitor their own thought processes;

• select appropriate tools (e.g., manipulatives, calculators, computers, communication technology) and computational strategies;

• make connections between mathematical concepts;

• connect the mathematics they learn at school with its application in their everyday lives;

• develop strategies that can be applied to new situations;

• represent mathematical ideas and model situations, using concrete materials, pictures, diagrams, graphs, tables, numbers, words, and symbols;

• go from one representation to another, and recognize the connections between

representations;

• persevere in tackling new challenges;

• formulate and test their own explanations;

• communicate their explanations and listen to the explanations of others;

• participate in open-ended experiences that have a clear goal but a variety of solution paths;

• collaborate with others to develop new strategies.

“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.” (Burns, 2000, p. 29)

Problem solving is central to learning mathematics. Problem solving is an integral part of the mathematics curriculum and is the main process for helping students achieve the expectations for mathematics outlined in the curriculum documents because it:

• is the primary focus and goal of mathematics in the real world;

• helps students become more confident mathematicians;

• allows students to use the knowledge they bring to school and helps them connect mathematics with situations outside the classroom;

• helps students develop mathematical knowledge and gives meaning to skills and concepts in all strands;

• allows students to reason, communicate ideas, make connections, and apply knowledge and skills;

• offers excellent opportunities for assessing students’ understanding of concepts, ability to solve problems, ability to apply concepts and procedures, and ability to communicate ideas;

• promotes the collaborative sharing of ideas and strategies, and promotes talking about mathematics;

• helps students find enjoyment in mathematics;

• increases opportunities for the use of critical-thinking skills

(estimating, evaluating, classifying, assuming, noting relationships, hypothesizing, offering opinions with reasons, and making judgements).

Problem solving needs to permeate the mathematical program rather than be relegated to a once-a-week phenomenon – the “problem of the week”. In this guide it is not considered to be one approach among many; rather, it is seen as the main strategy for teaching mathematics. Problem solving should be the mainstay of mathematical teaching and should be used daily.

Not all mathematics instruction, however, can take place in a problem-solving context. Certain conventions of mathematics must be explicitly taught to students. Such conventions should be introduced to students as needed, to assist them in using the symbolic language of mathematics. Examples of mathematical conventions include operation signs, terms such as numerator and denominator, the decimal point, the numerals themselves, the counting sequence, the order of the digits, and the is less than (<) and is more than (>) signs.

“Developing mathematical power involves more than simply giving students harder problems. It means asking them to focus on

understanding and explaining what they are doing, digging deeper for reasons, and developing the ability to know whether they can do a better job on working on a task.” (Stenmark & Bush, 2001, p. 4)

Although students are natural problem solvers, they benefit from guidance in organizing their thinking and approaching new problem solving situations. Teaching about problem solving focuses on having students explore and develop problem-solving strategies and processes. Teaching about problem solving allows students and teachers to create strategies collaboratively and, at all stages of the problem-solving process, to discuss, informally and formally, the thinking and reasoning that they use in determining a solution. When teaching about problem solving, teachers provide students with opportunities to solve interesting and challenging problems.

In many cases, teaching about problem solving occurs simultaneously with teaching through problem solving. As students are engaged in a problem that focuses on a mathematical concept (as described earlier in the chapter), the skilled teacher integrates the discussion of problem-solving strategies and processes into the discussion of the mathematical concept. For some students, this approach may not be sufficient; such students may require additional and more focused opportunities to learn about problem solving. Whether the approach to learning about problem solving is an integrated or a more isolated one, teachers should ensure that, in their instruction about problem solving, they encourage students to develop their own ways of solving problems. To find their own ways, students must be aware of the variety of possible ways. They learn about new strategies by hearing and seeing the strategies developed by their peers, and by discussing the merits of those strategies.