The following is a part of the Flow Channels brainstorm in the Natural Math wiki.

Paul Lockhart, in his “Lament” (the book version) talks about the importance of math as its own context, without trying to motivate it through any applications. I think it’s not about applications – it’s about metaphors! You can use “math as its own metaphor” so to speak, or approach it non-metaphorically, as autists supposedly do, which works great – for those who are willing to, and can, follow that path. In my observation, a tiny minority of people self-initiate or choose this self-contained approach to math, given the choice among multiple contexts as metaphor sources.

Here is how metaphors work. At first, there is a single entity, which only retrospectively can be named a metaphor. For example, a kid can think that division IS fair sharing. People frequently feel uneasy and even offended if you call their metaphors “metaphors” in this stage. It’s a defense mechanism, allowing the metaphor to support enough “roleplay” for the person to develop rich images that can later sustain formal structures. Forcing early formalization breaks the play, disrupts the natural learning rhythm, and leads to despondent feelings average math classes currently invoke.

If we manage to sustain rich image making in the all-important early stage, and gently help students move on to noticing some patterns in what they do, the patterns become math, and the metaphor turns into a simile. Patterns at this stage can just as easily become science, philosophy, or any other pattern-based discipline, but let’s focus on math patterns for now. In our example, once the kid starts noticing, say, that you can’t share certain quantities fairly without splitting, she is doing math, namely division, and sharing becomes LIKE division. When the context of sharing becomes unimportant (though the vocabulary may remain), and the focus fully shifts on quantities and their properties, the metaphor “dies” (Lakoff) and the new math structure, now self-sustained, is born.

To recap the life of metaphor in the context of the model of mathematical learning created by Pirie and Kieren:

• Metaphor promotes Image Making and supports Image Having
• Metaphor turns into a simile during Property Noticing, when its source and its target visibly separate
• The source of the metaphor dies, and the newly born math structure stands alone, in Formalizing

If you happen to love a context other than math – dragons, marine biology, car racing, fashion design – using it as a source of your math metaphors can be as powerful as using math as its own context. However, using math as its own context allows for mathematical elegance and depth not available otherwise. It has to happen, as well.

Bonus: Madison’s poem

know what i figured out?