On Being Stuck: A Preliminary Investigation into the Essence of Mathematical Problem Solving

Abstract

Expert mathematicians often refer to the state of being stuck as a valuable experience. Some even say that it is their natural state – that being stuck is the essence of doing mathematics. On the contrary, many students think of being stuck as a sign of failure, and as an indication that they do not have what it takes to do mathematics. This mismatch is a problem, because if being stuck is the natural state of doing mathematics, but students regard it as a sign of failure, then students are likely to avoid rather than embrace being stuck. This thesis is a preliminary investigation into the condition of being stuck in mathematical problem solving – a first look at how a group of novice mathematicians behaved within the state of being stuck. The emphasis is not so much on getting unstuck, as it is on effectively managing and coping with the condition of being stuck. A group of third-year undergraduate mathematics students were observed while working (and being stuck) on a sequence of tasks developed using the principles of modeleliciting activities (MEAs). The data gathered were primarily analysed using the frameworks of attention and goal-oriented behaviour. The results obtained from the study revolve around: (i) the sorts of things that were not helpful to the group when they were stuck (e.g., an overemphasis on dominant, but often inadequate ideas); (ii) general unfavourable characteristics of the group's thinking that were perhaps induced by the condition of being stuck (e.g., increased immersion in instant gratification); (iii) the sorts of things that were helpful to the group when they were stuck (e.g., re-presenting the problem in a different manner); and (iv) some things that might aid any attempt to develop and enhance the novice's ability to cope effectively with being stuck (e.g., developing the skill of knowing-to by sensitising oneself to the condition of being stuck).