> The usual trite crap: > ** This is, in fact, the exact opposite of how most students experience ** math. Too often, math is an inscrutable recipe book, describing foods ** you?ve never sampled (and probably wouldn?t care to). You must follow ** each recipe to the letter, because you have no clue how the ingredients ** taste, or what will happen if you combine them in new ways. And when you ** finish cooking, you throw the meal in the garbage disposal, and begin again. > > Presenting the quadratic formula, for example, without first thoroughly > handling completing the square is just bad mathematics. > Algebra avoidance yet again.
I liked his (the blogger, Ben Orlin) approach with the silly drawings, but it did strike me that his portrayal of most students experiences in math sounds more like something coming from the students themselves than something that actually takes place. Often students just want to know "how to do the problem" and they are not interested in why some method works or in alternate methods, both of which are likely presented by the teacher but go in one ear and out the other without being processed.
One thing I like about his approach is that rather than immediately lead up to some all-purpose distance formula, we work each distance evaluation in the way the formula can be derived: dist^2 = (horiz)^2 + (vert)^2. Or at least, do it this way for the first day or two of covering it. Doing this will likely help counteract students' tendency to process "why explanations" by having them go in one ear and out the other.
Speaking of black boxes, I don't like the way some teachers (and books) have students solve nonlinear inequalities such as occur in applying the first derivative test for calculus max/min problems. Say you want to determine where (x^2 - 2)(x+1)^2 is positive and where it is negative. The method I don't like has you label the zeros on a number line, then you pick a point in each (maximal) interval these points form, then you evaluate the expression (typically using a calculator) at the chosen point to see whether at that point the expression is positive or negative, which tells you, by the intermediate value property of continuous functions, the sign of the expression throughout the interval.
A method that puts you more in charge of the mathematical reasoning is to label the zeros on a number line, then determine the signs separately of x^2 - 2 and (x+1)^2 in each of the intervals (each of which is easy; x^2 - 2 has signs +-+ between its roots from the fact that the graph of y = x^2 - 2 is a parabola opening upwards that crosses the x-axis twice), then multiply these signs in each of the intervals to get the signs of the product of x^2 - 2 and (x+1)^2 in each of the intervals.