Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment

Knowing-to is active knowledge which is present in the moment when it is required. To try to produce knowing-to, formal education focuses on forms of knowing which are easier to teach and to test: knowing-that (factual), knowing-how (technique and skills), and knowing-why (having a story in order to structure actions and from which to reconstruct actions). Together these constitute knowing-about the subject. Expertise is demonstrated by being able to respond to assessments: to write essays and to solve routine problems. The central problem of education is that knowing-about does not in itself guarantee knowing-to, as teachers have attested throughout the ages. For example, Edward Fitzgerald (Harrison, 1937) captures it beautifully in one stanza of his purported translation of the Rubaiyat of Omar Khayyam:Myself when young did eagerly frequent,Doctor and Saint and heard great argument,About it and about: but ever more Came out by the same door as in I went (p. 341).Instead of trying to reach definitions, we illustrate distinctions amongst kinds of knowing as used by various authors in the past. Then we turn to our own experience, for it is in one's own experience that one can locate and enliven sources of metaphoric resonances and metonymic triggers which constitute understanding. Drawing on our experience we distinguish knowing-to from other forms of knowing, and explore implications of that distinction for teaching and learning mathematics. We propose that knowing-to act in the moment depends on the structure of attention in the moment, depends on what one is aware of. Educating this awareness is most effectively done by labelling experiences in which powers have been exhibited, and developing a rich network of connections and triggers so that actions come to mind. No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform.This revised version was published online in September 2005 with corrections to the Cover Date.