I read a document by Hung-Hsi Wu entitled “What is school mathematics?” I recall reading about some of Wu’s ideas back in the 1990s when I worked for the Department of Mathematics at The University of Iowa. Since that time, I did spend 23 years teaching high school mathematics, 22 of which were teaching honors precalculus to Iowa City’s (Big 10 University town and home of a teaching medical school) best and brightest. A few topics caught my eye enough to want to comment.
Regarding fractions, I became aware after teaching high school and before that, working in the Mathematics Tutorial Laboratory at U of Iowa for 7 years, that students really didn’t understand what fractions were. Clearly, it also shows up if one works, for example, in industry, (I turned cast iron and steel parts on automatic lathes in a machine shop off and on) only to find that new workers struggle with measurement — in particular, how to read a tape measure. I once taught a group of 6th graders at the Belin-Blank Center for gifted education (U of Iowa) and we talked about the meanings of “numerator” and “denominator.” In my mind, the numerator is the counter — it says how many. The denominator indicates what is being counted. For example, 3/4 means three quantities of 1/4. In other words, it means 1/4+1/4+1/4, or 3(1/4). On a tape measure, if the distance, say, between 1 and 2 inches is divided into 16 intervals (count those little lines between the 1 and 2 on a ruler), that means that each little space between consecutive little lines represents 1/16 of an inch. If the distance being measured shows 5 of those little lines after the 1 inch mark on the tape measure, then it measures 1 5/16 of an inch. We didn’t use pizzas or areas to do this. We actually used linear measure, much like a number line.