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Learning to practice, this book vividly illustrates, takes time and effort, trial and error. It won’t happen tomorrow. But even a small movement in the direction of more practice will reap benefits, in teaching and many other things we do.

The goal is always to emulate best practice.

The content does not determine the flow, the goals of meaning and transfer do. But content matters; it can’t just be projects. It takes artful design.

What's this have to do with lectures and my students' complaints? Well, far too often, lectures and recitation sessions are just like the conversations Steve and I had with Paul. When somebody else is presenting a detailed explanation of how they solve some problem, it's very easy to nod along and say "Yes, yes, of course, that's the thing to do." You leave the room perfectly convinced that you've understood everything, but when you try to apply what you think you know by explaining it to someone else, you find that you didn't really understand a thing.

That's the problem with good explanations: they're incredibly seductive, convincing you that you understand things that you don't understand at all.

For many students, no matter their age, math begins with an answer. You then form a question, jeopardy style, to help disguise the number.

Most students learn to expect math questions and problems to be short, quick, to the point, solvable and structured around "clean" answers (often related in some way to integer components). They anticipate the answers before they anticipate the questions. I am not sure if they even consider the math.

They completely miss the point and the empowering strength of math process and pattern.

I petitioned my counselor to let me skip Algebra II and Pre-Calc to go straight into AP Calculus.

A common problem throughout math education is the "When am I ever going to use this?" attitude. I already knew where I was going to use the math I was learning.

it's misleading to depict math as the chief villain in America's high dropout rate.