What is a mathematical problem?

In mathematics education, the term ‘problem’ is probably one of the most commonly used terms. Despite the large body of research in this area, many researchers dealing with mathematical problems have used the term quite differently. Indeed, it is remarkable that when people refer to a problem in mathematics, it is possible that they do not have the same thing in mind and what is a problem for someone, may not be a problem for another (Borasi, 1986; Blum & Niss, 1991; Wilson, Fernandez & Hadaway, 1993).

According to Blum and Niss (1991), there are two kinds of mathematical problems. The first kind is applied mathematical problems, which refer to real world situations, “the ‘rest of the world’ outside mathematics” (p. 37). The second kind is purely mathematical problems, which are entirely connected to the world of mathematics. I consider this distinction of problems to be both logical and functional, since the same problem can exist in both domains, due to the way it is posed (context).

Problem solving in mathematics education, reports Chapman (1997), means different things to different people. Polya (1981) asserts that “solving a problem means finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable. Solving problems is the specific achievement of intelligence, and intelligence is the specific gift of mankind: problem solving can be regarded as the most characteristically human activity” (p. ix). Polya (1945) outlines four phases in problem solving: understanding the problem, devising a plan, carrying out the plan, looking back.

Mason, Burton and Stacey (Mason et al, 1985) analyze three phases for the process of tackling a question; Entry, Attack and Review. Entry begins when the problem solver is faced with a question. The Entry phase work is largely in formulating the question precisely in deciding exactly what should be done. Thinking, according to Mason et al (1985), enters the Attack phase, when the problem solver feels that the question has moved inside her/him and become her/his own. The phase is completed when the problem is abandoned or resolved. During Attack, several different approaches may be taken and several plans may be formulated and tried out. The final phase is what Mason et al call the Review. When the problem solver has reached a reasonably satisfactory resolution or when he/she is about to give up, it is essential to review the work made. Mason et al (1985) assert that in this phase it is time to look back at what has happened in order for the problem solver to improve and extend her/his thinking skills and try to set her/his resolution in a more general

context.

The fact that there is limited research on problem solving from a teacher’s perspective was observed by Chapman (1997) a decade ago, while he noted that the general focus is on teachers’ instructional effectiveness rather than teachers’ problem solving competence. A similar comment was made by Thompson (1985), according to which, “research related to instruction in problem solving has centered on the effectiveness of instructional methods designed to develop global thinking and reasoning processes, specific skills, and general, task-specific heuristics” (p. 281).  Thompson argued that the disproportionately small amount of attention that researchers have given to the role of teachers is disturbing.

Thompson (1985) highlights the need for teachers (1) to experience mathematical problem solving from the perspective of the problem solver before they can adequately deal with its teaching, (2) to reflect upon the thought processes that they use in solving problems to gain insights into the nature of the activity and (3) to become acquainted with the literature on research on problem solving and instruction in problem solving. According to Cooney (1985), studies suggested that teachers may not possess rich enough constructs to envision anything other than limited curricular objectives or teaching styles and hence may be handicapped in realising a problem solving orientation. The use of a problem-solving approach demands not only extensive preparation but also the development of ways to maintain at least a modicum of classroom control and, perhaps most importantly, the ability to envision goals of mathematics teaching in light of such an orientation.