Math Problem Solving Promotes Confidence in Learners

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Solving problems in math promotes confidence in students as learners because of the interactions between students/students and teacher/students (Van Zoest et al., 1994), because of the mathematical dialogue and consensus between students (Van Zoest et al., 1994), because of the 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), because of the teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991), because of the teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994), and because of the
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). Solving problems in math promotes confidence in students as learners because of a further characteristic, which is that a problem-solving 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).

Solving problems in math promotes confidence in students as learners because, as it has already been pointed out, mathematics is an essential discipline because of its practical role to the individual and society. Through a problem-solving 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.

Solving problems in math promotes confidence in students as learners because according to Resnick (1987) a problem-solving 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 problem-solving 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.

Solving problems in math promotes confidence in students as learners because 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.

Solving problems in math promotes confidence in students as learners because it can develop skills and the ability to apply these skills to unfamiliar situations, it can develop the ability to gather, organize, interpret, and communicat information, it can develop the ability to formulate key questions, analyze and conceptualize problems, define problems and goals, discover patterns and similarities, seek out appropriate data, experiment, transferr skills and strategies to new situations, and develop curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).

Solving problems in math promotes confidence in students as learners because a problem-solving 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 problem-solving approach can provide a vehicle for ...