<blockquote>For example, a teacher trying to explain multiplication might ask a student to first draw three rows of dots with two dots in each row and then imagine what the picture would look like with three or four or five dots in each row. Guiding the student through the exercise, the teacher could help her see that each march up the times table 3x2, 3x3, 3x4 just means adding another dot per row. But if a teacher doesn’t use the dots to illustrate bigger ideas, they become just another meaningless exercise. Instead of memorizing familiar steps, students now practice even stranger rituals, like drawing dots only to count them or breaking simple addition problems into complicated forms 62+26, for example, must become 60+2+20+6 without understanding why. <strong>This can make for even poorer math students.</strong> “In the hands of unprepared teachers,” Lampert says, “alternative algorithms are worse than just teaching them standard algorithms.”</blockquote>
via <a href="http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html">Why Do Americans Stink at Math? - NYTimes.com</a>.