Across the grain#

Based on the idea that mathematics has inherent structure. That learning mathematics is best achieved by engaging with and learning to use that structure. Something not often achieved. Watson (2001) suggests that good mathematicians are aware of this structure. They see global similarities in local examples.

The problem is that some/much of the examples in mathematics hide this structure. They go with the grain of the "wood". Watson (2001) promotes the metaphor of "going across the grain". i.e. cutting wood across the grain to reveal the internal structure, highlight it to students, and help them engage with it.

Going with the grain is iterative/recursive and focuses attention on generating the next example based on previous examples (for example). Whereas going across the grain required generalising from the structure.

Watson (2001) suggests three principles (for working with low attaining students, but perhaps can be generalised)

Draw attention to the structure through observing patterns which go across the grain of the work.
Ask students to give examples of the structure to get a sense of the structure, generality, and extent of possibilities.
Prompt students to articulate similarities in their work and prompted to represent similarities in symbols

Echoing: contemplate-then-calculate, productive-failure

References#

Watson, A. (2001). Instances of mathematical thinking among low attaining students in an ordinary secondary classroom. The Journal of Mathematical Behavior, 20(4), 461--475. https://doi.org/10.1016/S0732-3123(02)00088-3