I’ve been helping a student prepare for the math portion of a standardized test. We had identified a few problem concepts to review from working through word problems in the sample test. The list for review was: ratios, percentages, adding fractions, geometry. As we struggled through the second of these topics (after deciding to take a break from fractions because it was proving frustrating) I realized that there was an underlying concept making everything difficult. Division.
I’ve long suspected that this is a big turning point for many students as they go through their math education. Some students build up a strong conceptual understanding of division and then are able to leverage it on concepts such as percents, fractions and ratios. Others don’t quite make the connection between multiplication and division and become dependent on algorithms and calculators. They are then haunted by a lingering confusion that comes from not quite grasping the concept of division in full. It carries through algebra, in solving linear equations, in factoring, in exponents and roots. There is so much to be baffled by without a strong grasp on division that concepts like logarithms are completely out of reach.
Once I realized where my student’s real problem spot was, it prompted us to isolate and focus on division for a while and it was many times (get it?) more productive than our former strategy.
I’ll have to look and see how strongly the reasearch supports this hypothesis of mine. If anyone knows of relevant studies, please send them my way!