How to Make Mathematical Thinking Concrete

This note is about making “mathematical thinking” usable.

When a teacher says “learn the thinking behind the problem,” it can sound vague. A more practical version is: a mathematical idea is a viewpoint you form after noticing a feature in the problem; that viewpoint leads to a concrete first move.

The Core Flow

Read the problem
  -> identify the feature
  -> form several viewpoints about that feature
  -> map each viewpoint to one possible move
  -> choose the move most likely to push the problem forward
  -> test it by working through the next steps

The three useful layers are:

• Feature: what visibly appears in the problem.
• Viewpoint: how you choose to understand it.
• Move: the first written action that follows from that viewpoint.