tl;dw summary:<p>you know you're doing math wrong if students display (1) a lack of initiative, (2) lack of perspective, (3) lack of retention, (4) aversion to word problems, and (5) the few who understand the math just want to jump to a formula.<p>Math textbooks encourage teachers to teach math wrong. The way they present problems is with a complex visual with mathematical structure already imposed, step-by-step handholding through the problem, and asking a question at the end (a question that can often be solved just by figuring out which number to plug into which part of the formula, without necessarily understanding why.)<p>Suggested method for teaching right: (1) use multimedia. (2) encourage student intuition. Students will argue with each other about what they see and buy in to the problem. (3) ask the shortest possible question. Don't begin with a page full of numbers, measurements, and individual steps. Let the detailed questions come out through discussion. (4) Let the students build the problem. Students will recognize the need for mathematical structure (labels, coordinates, measurements, etc.) as they decide what information they will need to answer the question. They'll go through the steps on their own. (5) Be less helpful. The textbook helps in all the wrong ways, taking you away from your obligation for developing patient problem solving and mathematical reasoning.<p>Example: he completely rewrites a question from a math book about filling a water tank. He produces a video of a water tank being filled from a garden hose, which takes excruciatingly long to complete. Students get uncomfortable, complain about how long it's taking, and then put in their guesses as to how long it will take. Then they decide what information they'd need to calculate the end result, ask for the measurements they think are important, do the calculations, and watch the rest of the video to see if their calculation was right and how close their initial guesses were.