International research on problem solving

In an oft-cited and influential article Schoenfeld (1992) states that problem solving has been used with multiple meanings which range from ‘working rote exercises’ to ‘doing mathematics as a professional’. In illustrating the former he cites a problem set by Milne dating back to 1897: ‘How much will it cost to plough 32 acres of land

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at $3.75 per acre?’ Although there are various ways to solve the problem, they all

involve fairly basic computational skills. Accordingly, solving the problem hardly necessitates the use of higher order thinking skills. Schoenfeld quotes from the

Everybody Counts report (1989) which recommended a renewed emphasis on

 seeking solutions, not just memorizing procedures;

 exploring patterns, not just memorizing formulas;

 formulating conjectures, not just doing exercises.

From this perspective, learning mathematics is empowering (Schoenfeld, 1992). Those who study mathematics are required to be flexible thinkers with a broad base of techniques and ideas for dealing with novel problems and situations. Here the reader will see that defining problem solving involves defining one’s view of the

epistemology or nature of mathematics knowledge. In quoting an earlier report by the National Council of Teachers of Mathematics (NCTM) entitled Agenda for

Action (1980), which recommended that problem solving be the focus of

mathematics, Schoenfeld (1992) declares that whereas ‘back to basics’ was the

theme of the 1970s, problem solving was declared to be the theme of the 1980s. Yet, he also cautions that problem solving persisted as one of the most overworked and least understood terms of that decade.

Indeed, during the 1980s it came to be viewed, according to authors such as Stanic and Kilpatrick (1988), as consisting of a hierarchy of skills. Top of the hierarchy was the solving of non-routine problems. Next in the hierarchy was the solving of routine text book problems and subordinate to that was the acquisition of basic computational skills. It seems that in the Irish context it is the lower two levels of the hierarchy that gain most attention. In America in 2012 the National Centre for

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Education Evaluation and Regional assistance issued a report entitled Improving Mathematical Problem Solving in Grades 4 through 8. These grades correspond to pupils aged 9 to 13 years in Ireland. The report made five recommendations for teachers:

1. Prepare problems and use them in whole-class instruction.

Teachers were requested to include both routine and non-routine problems. Routine problems were ones that could be solved by replicating previously learned methods in a step-by-step fashion. Such problems dominate Irish classroom teaching. Non- routine problems were ones for which there was no predictable, well-rehearsed approach or pathway suggested by the task, task instructions, or a worked-out example. Such problems are less prevalent in Irish classroom teaching and should form part of any reform agenda.

2. Assist students in monitoring and reflecting on the problem solving process. This involves modelling for pupils how to monitor and reflect on the problem solving process by using their thinking about a particular problem.

3. Teach students how to use visual representations. As an enthusiast of Jerome Bruner I have long been an advocate of using this strategy as it is reminiscent of his iconic mode of representation.

4. Expose students to multiple problem solving strategies.

This recommendation involves asking students to generate and share multiple strategies for solving a problem. In my opinion, this strategy should form part of any reform agenda as it is in line with a constructivist philosophy seeking to empower students to take control of their learning and enable them to construct their own meaning. It can start at a simple level. Consider the routine problem of finding 35% of 200. With encouragement from the teacher pupils might consider finding 35/100

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or 7/20 or 0.35 of 200. In line with recommendation number two above pupils can then reflect on how they completed the problem.

5. Help students recognise and articulate mathematical concepts and notation. The advice here is to ask students to describe relevant mathematical concepts and notation. This recommendation reminds me of the advice given by Schoenfeld (1992) earlier that children should be encouraged to work as mathematicians do. Being familiar with mathematical language and notation is part of the students’ enculturation process into the mathematician’s world.

Therefore, it behoves me to discuss the type of problem solving Schoenfeld himself envisaged. To do this, I need to go to the other end of the continuum Schoenfeld (1992) mentioned earlier i.e. to doing mathematics as a professional. Schoenfeld (1992) comments that in becoming a mathematician he, and other colleagues, had undergone a process of acculturation, in which they had become members of, and had accepted the values of, a particular community of practice. In other words, they had become mathematicians in a deep sense as a result of a protracted apprenticeship into mathematics. This helps to situate Schoenfeld’s (1990) view that “mathematics

instruction should provide students the opportunity to explore a broad range of problems and problem situations, ranging from exercises to open-ended problems and exploratory situations” (p. 345). Working as a mathematician does not involve

dealing with trivia; the problems encountered are often difficult and can take considerable time to solve. Schoenfeld (1990) conducted problem solving courses for students at college level. Roughly, one third of the time in these classes was spent with the students working on problems in small groups. In these classes Schoenfeld (1990) deferred teacher authority to the student community; both in withholding his own solutions to the problems and in developing in the class the critical sense of

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mathematical argumentation to lead it, as a community, “to accept or reject on

appropriate mathematical grounds the proposals made by class members” (p. 362). The justification for courses such as Schoenfeld’s comes from the research literature on practice change which reaches a consensus that “if teachers and prospective

teachers are to provide new and different sorts of learning experiences to their students it is important that they have such experiences for themselves as learners of mathematics” (Crespo and Sinclair, 2008, p. 396). Since the new millennium the

research literature has recommended that such experiences include problem posing as well as problem solving. Such a move is to be welcomed as it empowers students to take more control of their learning experiences. However, expecting pupils to pose problems spontaneously is not enough. English (1998) demonstrated that informal contexts, which are non-goal-oriented, are more conducive to the generation of diverse types of problems. She cites an example from her research of children being asked to look at a photograph of other children playing. This was followed by a request to make up a story problem which asked about something to be seen in the photograph. English (1998) found that children generated a more diverse collection of problems in this context than if they had been asked to generate a story problem that could be solved from a given number sentence (e.g. 12-8=4). This is a useful insight as converting a number sentence to a story problem is given prominence in the Irish curriculum (page 71). The use of the photograph as context led to an exploratory process which engendered the feeling of something being problematic enough to incite reflection and action. Crespo and Sinclair (2008) remark that this type of situation is essential for cognitive growth. However, they argue that there is a tension between the pedagogical and the mathematical fruitfulness or potential of problems. For instance, exploring Pythagorean triangles may be mathematically fruitful but teachers may decide that it is not a motivational topic for a particular

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cohort of pupils. In research terms, Crespo and Sinclair (2008) call for further studies to understand this tension better. Such studies would help us learn more about how teachers balance pedagogically and mathematically interesting questions in their classroom, and ultimately learn how to help prospective teachers pose problems that have both characteristics.

3.3 Problem posing and its connection with problem solving