Elementary Education Online, 6(1), 50-61, 2007. lköretim Online, 6(1), 50-61, 2007. [Online]: http://ilkogretim-online.org.tr Learning To Solve Non-routine Mathematical Problems * Çidem ARSLAN ** Murat ALTUN *** ABSTRACT. Recent studies have pointed out that many schoolchildren do not master the skill of solving non-routine mathematical problems. In this article, a trial study designed to encourage seventh and eighth grade students to learn and use problem solving strategies is discussed. The strategies consist of six heuristic strategies known as Simplify the Problem, Guess and Check, Look for a Pattern, Make a Drawing, Make a Systematic List and Work Backward. Classroom activities consisted of a short whole-class introduction, group studies and a final whole-class discussion on the given problem. The teacher’s role was to encourage and guide the pupils towards engaging in the problem. It is observed that in these classes some of these strategies are effective in learning, others are not. Key Words: Problem solving, problem solving strategies, non-routine problems, mathematics teaching. INTRODUCTION Many research studies and projects have pointed out the importance of learning problem solving in school mathematics courses (Ford, 1994; Higgins, 1997; National Council of Teachers of Mathematics, 1989; Verschaffel, De Corte, Lasure, Van Vaerenbergh, Bogaert, Ratinckx, 1999). One of the major goals of mathematics education is the acquisition of the skill of learning how to solve problems. There are, however, conflicting views about the attainability of these goals (Verschaffel et al., 1999). Despite long years of instruction many research studies show that children are insufficient and not confident in having the aptitudes required for approaching mathematical problems, especially non-routine ones, in a successful way (Asman and Markowitz, 2001; Higgins, 1997). The reasons for these deficiencies in primary and secondary school children can be attributed to two factors. The first of them is the lack of specific domain knowledge and skills (e.g. concepts, formulas, algorithms, problem solving). The second factor is shortcomings in the heuristic, metacognitive and affective aspects of mathematical competence. When confronted with unfamiliar complex problem situations, children mostly do not spontaneously apply heuristic strategies such as drawing a suitable schema or making a table, etc. The students usually only glance at the problem and try to decide what calculations to perform with the numbers. Besides this, many pupils have inadequate beliefs and attitudes towards mathematics itself, learning mathematics, and problem solving. These beliefs exert a strong negative influence on pupils’ willingness to engage in a mathematical problem. Some examples of such beliefs and attitudes are that there is only one correct way to solve a problem, that a mathematical problem has only one right answer, and that ordinary students can not solve non-routine problems. These insufficiencies in pupils’ beliefs are related to the nature of the problems given in the lessons and the classroom culture. Pupils are mostly confronted with routine problems which require only basic operations and calculations. Non-routine problems which reflect the relations between mathematics and reality are rarely presented. Classroom activities can also contribute to unwanted attitudes towards learning outcomes such as the use of strategies for coping with word problems and to beliefs about what mathematics and problem solving is (Verschaffel et al., 1999). Activities such as these do not give opportunities to students for investigation, reasoning or deciding on the solution process and do not improve problem solving skills. Taking into consideration the problem solving process for this study, a brief summary of this topic is presented below. There are several approaches to explain the problem solving process. Polya (1957/1997) proposes four stages, which have sub stages, to explain the problem * “Learning to Solve Non-routine Mathematical Problems” presented orally at the 10 th International Congress on Mathematical Education, Denmark, July 2004 ** Uludag University, arslanc@uludag.edu.tr *** Assoc.Prof. Dr., Uludag University, maltun@uludag.edu.tr