In a seminal chapter from 1987 entitled What’s all the fuss about metacognition?, Alan Schoenfeld discussed his observations of how undergraduate students solved problems. More than half, when faced with a non-standard problem outside a familiar context, followed the path illustrated in diagram 1. In this case, a pair of students read the problem, made a correct conjecture, but then got bogged down in calculations and ran out of time, even though they had the required knowledge to solve the problem. At no time did they stop and ask themselves “Is this getting us anywhere? Should we try something else?” If they had, Schoenfeld contends, they would have had a chance of solving the problem. As it was, their effort “is an all too typical example of the disastrous consequences of an absence of self-regulation.”
In contrast to the students’ attempt, a mathematician tackled a geometry question in a far more rigorous way to arrive at a solution within the time limit (diagram 2 – adapted from the original). The question was outside the mathematician’s field of expertise so, Schoenfeld argues, we cannot look to her superior knowledge as the reason for success. Instead, she was an expert problem solver who was ruthless about testing and rejecting the many approaches she considered. Moreover, the mathematician spent the majority of her time thinking (and, in particular, analysing) rather than doing.
However, the key difference is the inverted triangles in the second diagram and their absence from the first one. Each triangle indicates an example of self-regulation when the mathematician asked herself “How am I doing?” and decided, on the basis of the answer, how to proceed. The triangle at the start of
the 'analyse' phase, for example, corresponds to a comment about the need to make sense of the problem; the final one of that phase marks where she acknowledged her readiness to attempt a solution. The triangles towards the end of the implement phase related to a recognition that she was coming to the end of a solution and would need to check it. Crucially, in the analyse phase, the mathematician was able to reject speculative approaches, saying at one point, “But I don’t like that. It doesn’t seem the way to go.”
In Inquiry Maths, students learn to regulate their activity by justifying the choice of a card to their peers and, in turn, engaging with their peers’ justifications. They also benefit from the teacher’s rejection of non-mathematical suggestions and arbitration between contending legitimate ideas.
Showing the two diagrams above to students has provided a powerful learning experience. A year 7 student responded, "I've been like the student today, I need to be more like the mathematician." The diagrams help students understand why the teacher is asking them to stop exploring and to consider how to proceed.
Schoenfeld asserts that the main point about self-regulation is as follows: “It’s not only what you know, but how you use it (if at all) that matters.”