Abstract
Everyone gets stuck sometimes, and it can sometimes be frustrating, even debilitating rather than stimulating. However, being stuck is an honourable and useful state because that is when it is possible to learn about mathematics, about mathematical thinking, and about oneself. This applies especially to teachers, because the best way to sensitise yourself to learners’ struggles is to experience parallel struggles yourself. People are usually eager to get unstuck, to locate and enact some hopefully helpful action, without attending to how they got stuck in the first place, nor how the fresh action arose. I propose to dwell in the states of becoming and being stuck, and to use this as a springboard to examine and amplify the notion of ‘having some possibility come-to-mind’ as a means to get unstuck. This will include an expansion of the notions of system 1 and system 2 (automatic-habitual reaction and considered response) from dual systems theory, so as to take account of the full human psyche and development of the role of reflection. My method will be as phenomenological as possible, drawing on specific accounts from my own experience, but hoping to resonate with the experience of readers.
Highlights
“A solved problem is as useful to the mind as a broken sword on the battlefield” (Shah 1970, p. 142)
To enjoy the state of hopeful not-yet-knowing but being on the edge of knowing, is valuable, while feeling stymied and hopeless, even helpless, is not so useful because once negative emotions kick in, the current state of feeling hopeless may transfer to a sense of oneself as hopeless, whether at doing this problem, at this topic, or even at mathematics altogether
Working on the issue of .9 with some third-year mathematics undergraduates at a prestigious university who were taking a mathematics education course, I recall vividly that one of them said “it may be 1 in analysis, but not out on the street”, and this was agreed to by many of the students
Summary
As a teacher it is vital to be aware of how learners become stuck, and what sort of prompts will enable them not to get unstuck but to learn from the experience, instead of amplifying the desire to ‘get unstuck at all costs’. It is necessary as a teacher to uphold the state of frustration in order to sustain students’ work. I want to dwell in MASON and on the state of being stuck, to consider briefly the psychology and sociology of that state in the light of various theories and commonplace clichés, and to elaborate on the role of reflection in developing a repertoire of mathematical actions which can inform future practice
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