Comments on Hung-Hsi Wu’s “What is school mathematics?”

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.

Confusion happens when we confuse fractions as linear measure or distances on a number line, and something like parts of a whole, such as 1/2 of a pizza. In other words, a fraction OF something is different from a fraction that is simply a number or a linear distance on a number line. Wu mentions this idea.

Something else I noticed from the time I was a child was that fractions appeared in music. If you look at the time signature of Paul Desmond’s “Take Five” or Jethro Tull’s “Living in the Past,” you will find 5/4. Again, the numerator 5 tells you how many beats per measure, because the purpose of the 5 is for counting. The denominator 4 tells you what kind of note gets 1 count, in this case, a quarter note (or a 1/4-note). I think it would be tough to explain to a student who is just learning about fractions what is meant by 5/4 of a pizza.

Not only is measurement on a ruler or a time signature in music easy enough to interpret, it can show you equivalent fractions. For example, on that ruler that has 16ths, one can notice that every 4th little line is a little longer than the others, and every 8th little line is longer still. If you focus on every 4th little line, you see that the inch is divided into 4 intervals, or 1/4 of an inch. That shows that 4 of those 1/16ths is equivalent to 1/4 and that 8 of those 1/16ths is equivalent to 1/2. In music, a song can be played in 3/4 time or 6/8 time and the melody is preserved. To me, this is absolutely fascinating, because 3/4 and 6/8 are equivalent fractions! I was captivated by this at a very early age, but I probably wasn’t the only one who noticed, though it was never brought up other than at piano lessons.

I was the district K-12 math coordinator in Iowa City for about 7 years, and when I was looking at K-6 textbooks, I was horrified to see “mixed numbers” expressed as numbers like 1 3/2. Mixed numbers actually are sums of whole numbers and proper fractions (numerator less than denominator) and expressions like 1 3/2 did not look right at all. The thing about mixed numbers is that, while we intend them to mean sums, we don’t write the “+” sign. I recall students who took, for example, -1 2/3, and tried to place it on a number line where -1/3 is, because they added a positive 2/3 to a negative 1. Because our mathematical language is, at times, ambiguous-looking, we have to be very clear what we mean by the symbols we use.

On a different topic, I was tutoring for an algebra 2 honors student a few days ago, and she was working on finding arc lengths on circles. Her central angle (intercepting the arc) was something like 2pi/3 and the radius was 5 inches. She said, “Oh, you just multiply 2pi/3 by 5 and get 10pi/3 inches for the arc length.” I asked her how she knew to do that and she said, “Well, it’s a formula our teacher gave us and it’s in the book.” I asked her if she could do that if the central angle was 120 degrees, so that 120 times 5 is 600. But a 120-degree angle has the same measure as an angle of 2pi/3 (just different units), and with the same radius, shouldn’t the arc length be the same? I then launched into what constitutes an arc – it’s merely some fraction of a circumference. If one uses the central angle divided by the measure of a full rotation, that should mark off an arc intercepted by the central angle. If we divide 2pi/3 by 2pi and multiply that by the circumference – 2pi times 5, we get 10pi/3, which is the same as 2pi/3 times 5 (the formula from the book). With degrees, it’s 120/360 times 2pi times 5, and again we get 10pi/3. Now I ask you – in ten years, will she remember S=theta times r for the length of an arc, or will she remember that an arc is a fraction of a circumference and then have the tools to figure it out?

I once had a student come back from college to tell me that he had to factor x^3-8 and forgot the factoring formula (on a test). However, he recalled my saying that you don’t need to memorize that factoring pattern. If you identify a zero (x=2), then x minus that zero (x-2) is a factor of x^3-8. Polynomial division of x^3-8 by x-2 will yield the other factor as the quotient. This is why I do not believe in memorizing everything in a shaded box in a textbook. As teachers, we need to teach the tools to derive or figure out what we need without the cognitive overload that comes from trying to memorize things like all points on the unit circle with special angle reference angles.

Sorry this is too long, but I have thought a great deal about the things Wu mentioned. As a teacher, I seldom used explanations given in the textbook because often, they were not clear, at least not without a lot of prior development or derivation.

One more thing — current teaching materials are often purchased online, with worksheets and note sheets. Often they are of poor quality. One I saw had a calculus problem that asked for an approximation of the area between the graph of a cubic polynomial and the x-axis between x=0 and x=8 using NINE subintervals (rectangles) and the midpoint rule. Seriously??? EIGHT rectangles, yes, but NINE?? This was for AP calculus AB. I was horrified about that as well. Teachers just assume that those worksheets are nicely stated, and they use answer keys that come with them. To me, this is a HUGE problem with today’s teaching and textbook or other published materials — teachers simply MUST work the problems before they are assigned, but many don’t want to take the time.

OK, I’m off my soapbox — for now…