When is a problem not a puzzle?;Curriculum
For many years I have encouraged the participation of secondary school children in extra-curricular problem-solving competitions. At the same time, as part of primarysecondary liaison in science, we developed a problem-solving programme, aimed at primary 7 pupils.
The enthusiasm of the participants, and their original and ingenious solutions, fuelled my desire to investigate formally the problem-solving ability of all children in the 5-14 age group. My research centred on three problems, initially specifically designed to meet two criteria. All three had to appeal to the wide range of ability in this age range and the three problems had to facilitate the observation of the stages, strategies and skills associated with problem solving.
I felt that, although problem-solving has its roots in mathematics, the concept of problem-solving permeates almost every strand of the 5-14 curriculum, also featuring as a general aim in environmental studies and expressive arts. In addition, the closely associated skills of thinking, creating and designing, while perhaps more readily recognised as elements of science and technology, are also desirable in subjects as diverse as arts, music, drama, physical education and English.
The first task was to identify those strategies and skills pertinent to these elements, before considering which ones might be representative of the 5-14 curriculum. The next step was to create a few problems which would encourage pupils to demonstrate their strategies and skills. After initial trials, three problems were systematically tested on 60 pupils, six from each age level within the 5-14 span. Selected to reflect the mixture of the ability at each level and from schools diversified in academic and social background, the participants were representative of this age group.
Possibly the most compelling result was that all the children enjoyed taking part in the research. Moreover, with the exception of only two participants, everyone fully participated and committed themselves to the three tasks provided, despite the five- and 14-year-olds having to tackle the same problem. However, although the enthusiasm was universal, success differed greatly across the three problems.
Many of the strategies and the skills operated within a framework of discrete stages. Although I suggested four stages of problem-solving at the outset, I amended this at the end of the research to a critical five-stage model. Two of these stages appeared to present the biggest obstacles hindering a solver from reaching a successful solution.
Understanding the problem emerged as the greatest single difficulty. While the age of fully comprehending a problem varied with the problem set and the age of the solver, I noted that more than 50 per cent of those involved required more than one reading of the problem.
More telling was the fact that many of the solvers needed more explanation, despite telling me that they understood the problem. It was not only the younger children who had this trait. The older ones were equally guilty of quickly reading the problem, eagerly making a start, and then discovering they had misunderstood a key aspect of it.
The second stumbling block appeared after a child generated possible ideas or strategies to solve the problem. Ironically, while this stage proved to be the most successful, with the ideas of seven- and eight-year-olds comparable both qualitatively and quantitatively with the older age groups, the ability to choose the best idea proved elusive to most until the age of 10.
Indeed, only in the upper age groups of the 10- to 14-year-olds was the ability to select, justify and expand upon an idea demonstrated with 100 per cent success. Subsequently, this third stage of selecting one idea or strategy emerged as a critical stage in its own right and not merely as a subset of the previous one.
Although the fourth stage, that of testing an idea or implementing a strategy, revealed some positive aspects, several limitations to children's abilities in this area were also highlighted. Strategies such as simplifying the problem, guess-check and improving on a solution, and reasoning logically, were demonstrated with 100 per cent mastery, across more than one problem, by the age of seven or eight.
Conversely, some strategies, including the powers of elimination and forming structured questions, appeared to be beyond the younger children. These strategies were not displayed until the ages of 12 to 14. Even then, 100 per cent mastery was only recorded in the oldest age group. While subsequent settings of the same problem improved performance slightly, the inability of many children successfully to transfer learning from one event to another (even within the context of the same problem) was highlighted. This, allied to a difficulty in properly evaluating their idea or strategy, limited improvement, especially in the younger age groups. Among the five- to 11-year-olds only 38 per cent of the children could successfully transfer strategies or skills previously demonstrated to them, even on a second or third setting of the same problem.
This inability was reiterated by a sampling of the problem-solving exercises previously encountered by the children involved. A staff questionnaire revealed that despite the class teacher having taught and emphasised strategies, some formally through the package On the Track to Problem Solving (Nelson Blackie), several, including making a list, making a table and making a model, were seldom used by the children in solving the three problems.
Since problem-solving influences so many corners of the 5-14 curriculum (and beyond), it is not enough to provide problems for children to solve. While the problem, if carefully chosen, might serve to motivate children and allow them to demonstrate, an impressive array of strategies and skills, a passing interest in problem solving will not develop these skills sufficiently. Problem solving should be accorded a higher profile.
A child's ability to tackle problems might be enhanced in the following ways.
* Problems should not been seen as solely the domain of mathematics and more opportunities for problem-solving should be provided. Many of the strategies and skills needed are inter-disciplinary and transferable; they emanate from almost every part of the 5-14 curriculum.
* Recognition should be given to the fact that a problem is solved in a series of stages. The five stages identified by this research could be systematically applied to most problems, irrespective of origin, within or outwith the curriculum.
* A teacher should endeavour to find out why a child cannot progress through any stage in particular.
* It is equally important to ensure that a child understands how he or she reached the appropriate solution. As this research shows, understanding the problem poses the biggest difficulty, therefore it is important to present the problem in an unambiguous and concise manner.
* As in any learning situation, the child should be aware of the strategies and skills likely to be used and developed within any particular problem. These skills will be enhanced by repeated exercises.
In keeping with the rest of the curriculum, the input in teaching problem-solving is substantial. However, the return, as I found in my research, is highly rewarding.
Philip McGhee is principal teacher of physics at Our Lady's High School, Motherwell. This research formed part of his MSc from Strathclyde University.