But it is not a simple, smooth path always, and it is a bit of a mystery what makes the path smooth and what makes it bumpy. Visual models help, and can often turn an abstract, confusing idea into an engaging puzzle. What does it mean to figure something out? And how does one build understanding for oneself? And then, of course, I want to know how to externally facilitate that internal process. Through posing questions? Through instructing them on the use of tools? Through sharing my solutions and modeling my thinking processes? How much struggle is necessary and/or useful?
I wonder what is useful to have the students think about? For example: the skill for finding a percentage of a number. How useful or important is it for them to figure out that one way to find 17% of 95 is to use multiplication in some form? The really important outcome is for them to be able to recognize when the skill is needed, and how to apply it appropriately, say in a problem that involves compound percentages. Do they really have to have derived or "figured out" the skill themselves in order to go on to applying it? How does that help them? How does it hinder them if they don't?