> If math knowledge is limited to “when I see this sort of problem, I do that” it’s inflexible; if problems are phrased just a little differently, the student i

> If math knowledge is limited to “when I see this sort of problem, I do that” it’s inflexible; if problems are phrased just a little differently, the student is often stumped. And once students hit algebra, math increasingly demands thinking through a sequence of steps and picking the right mathematical tool for each. That’s hard if you don’t understand what the tools actually do. That's easy to say, but hard to do. I find in my math experiences that I often don't truly master a concept until I am a level or two beyond it. I learned fractions when I was in grade 5 or 6 and could demonstrate adding, multiplying, reducing, etc. I think if you asked me 'why can you multiply across when multiplying fractions' I may have said I knew, but I don't think I actually mastered that until high school--when I had the skills to derive the fraction operation rules. Ditto with things like FOIL, completing the square, etc which I don't think I mastered until I was out of high school. And of course, everything seems easy and obvious in hindsight. I suppose what I mean is: I don't think there is anything new about what the author lauds as a goal. It's what everyone has been trying to do for 50 years. It just isn't easy, and so far no one has found a magic bullet aside for lots of hardwork and practice.