On being stuck on a mathematical problem: What does it mean to have something come-to-mind?

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.