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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.

a surprising source of mathematical problem-solving and teaching

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.

When we tell stories about creativity, we tend to leave out this phase. We neglect to mention those days when we wanted to quit, when we believed that our problem was impossible. Instead, we skip straight to the breakthrough. We tell the happy ending first.

A study now suggests that simply taking a break does not bring on inspiration — rather, creativity is fostered by tasks that allow the mind to wander.

From an evolutionary perspective, mind-wandering seems totally counterproductive and has been viewed as dysfunctional because it compromises people’s performance in physical activities. However, Baird’s work shows that allowing the brain to enter this state when it is considering complex problems can have real benefits. Zoning out may have aided humans when survival depended on creative solutions.

scientists from the University of Chicago reported that young children who understand how shapes fit together are better able to use a number line and to solve computation problems.

The theme here is fun math books. Thus, if a book says something like “this is a division sign,” you will not find the book on this list. These books get kids thinking about numbers and problem solving in neat ways without realizing that they are learning math.

“One stereotype we have not been able to break in the United States is that ‘faster is smarter,’ ”

“Sometimes it’s the person who is more reflective and introspective about thinking through the problem and might take a little longer to get to the answer who illustrates more understanding of the mathematics involved.”

students learn math best when the focus is on exploration and understanding, not just regurgitation and computation

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 quickly figured out how to play that game successfully – success in that case being measured by my being able to solve under exam conditions, problems like the ones the teacher had shown us and we had practiced in class and done for homework.

In fact, you can’t separate real teaching from learning. They are simply two perspectives of the same human interactive process.

For whereas technology can provide instruction and can provide teachers and students with resources to assist them, what is cannot do on its own is teach them.