In a recent article about math education that ballyhoos the “latest approach” in how to teach math, this statement was made:
“If the teachers are telling students how to solve a problem, and then that problem isn’t exactly what’s on the test, it creates this disequilibrium for a student,” said Beverly Velloff, the math and science curriculum coordinator for the University City School District.
There is nothing new about the so-called breakthrough ideas the article discusses. Moreover, this quote is representative of how traditionally taught math is mischaracterized. The notion is that students are taught by rote, given a set of problems that are all exactly alike, and thus leaves them flummoxed when presented with a problem that is even slightly different. Such a caricature may be true for traditionally taught math done poorly, but it makes no allowance for it being done well.
Looking at an example from algebra: there are many varieties of distance/rate problems. There are problems in which two objects are going in opposite directions, going in the same direction (i.e., playing catch-up), round-trip problems, objects being influenced by wind or current, and so on. At some point students are given basic instructions for solving these various types. In opposite direction problems, students should be taught that we are dealing with two distances that are equal. That is, if two people are driving towards each other, then the distance each travels before they meet up is equal to the initial distance of separation between them.
In well written textbooks, such problems are scaffolded so that initial problems are solved by following the worked example. But subsequent problems might have some small variation. Instead of two cars coming towards each other at respective speeds of 60 and 40 miles per hour, we might be told the speed of one car, the distance between them initially, and the time it takes for them to meet. For example, two cars are 200 miles apart, and one car goes 60 mph. It takes 2 hours for the two cars to meet. What is the speed of the other car? We know that in two hours the 60 mph car has travelled 120 miles. The distance of the second car added to 120 miles equals 200 miles. That’s the “distance = distance” relationship so that if x equals the speed of the other car, then 2x+120 =200. The speed of the second car is 40 mph.
Students will need some guidance in going through these problems, but after given practice with these types, they learn what to look for.
Math reformers may look at this as spoon feeding and rote. They would rather give students problems for which they have not been given specific instructions, and need to synthesize prior knowledge. Alternatively, they are expected to learn in a “just in time” manner what is needed to solve problems. Thus, problems are given in a top down form in the belief that over time, students will develop a problem solving “schema”.
An article by Sweller et al (2011) states that such notion is mistaken:
Recent “reform” curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but are teaching them some form of general problem solving then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general and that will make them good mathematicians able to discover novel solutions irrespective of the content.
We believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics.
Nevertheless, reform/progressive math ideas rule the roost in education. Articles such as Sweller’s are thought of as fluff, not proven, no evidence to back it up, or dismissed in light of arguments such as “It has worked in my classrooms”.
Students need instruction, worked examples, scaffolding, ramp-ups in difficulty of problems, guidance, and much practice. Reformers view such steps as “inauthentic math” that produce “math zombies” who do not have “deeper understanding”. Ignored in all this is the fact that the so-called math zombies are the ones in college who by and large are not in need of remedial math classes.