Sweller, Clark, & Kirschner explain that problem-solving in math should be taught through carefully sequenced worked examples, not general problem-solving skills or strategies that Polya advocated. The skills approach has not worked well in classrooms, but the content knowledge approach has.
“Many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge. The best-known exposition of this view was provided by Pólya (1957). He discussed a range of general problem-solving strategies, such as encouraging mathematics students to think of a related problem and then solve the current problem by analogy or to think of a simpler problem and then extrapolate to the current problem. Nevertheless, in over a half-century, no systematic body of evidence demonstrating the effectiveness of any general problem-solving strategies has emerged.”
Sweller, Clark, & Kirschner point out that general problem-solving skills independent of content are not supported by evidence. Common Core with its Standards for Mathematical Practice advocates that math should be taught via general problem-solving skills. The pedagogy is wrong.
“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 rather are teaching them some form of general problem solving, then the 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.” The argument is not true!
“Whereas a lack of empirical evidence supporting the teaching of general problem-solving strategies in mathematics is telling, there is ample empirical evidence of the validity of the worked-example effect.”
Practicing problem-solving strategies independent of worked examples doesn't work. Students learn little arithmetic and algebra.
“Domain-specific mathematical problem-solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies. There is now a large body of evidence showing that studying worked examples is a more effective and efficient way of learning to solve problems than simply practicing problem-solving without reference to worked examples (Paas & van Gog, 2006).”
Students are novices, not little mathematicians. They need to learn content to support problem-solving.
“Studying worked examples is a form of direct, explicit instruction that is vital in all curriculum areas, especially areas that many students find difficult and that are critical to modern societies. Mathematics is such a discipline. Minimal instructional guidance in mathematics leads to minimal learning (Kirschner, Sweller, & Clark, 2006).”
Note. When I read Polya's book, I noticed that every example of problem-solving required knowing content knowledge. In short, without specific content knowledge (concepts, procedures, and applications) in long-term memory, you cannot solve problems in mathematics. In other words, you cannot solve a trig problem without knowing some basic trig.
Note. When I read Polya's book, I noticed that every example of problem-solving required knowing content knowledge. In short, without specific content knowledge (concepts, procedures, and applications) in long-term memory, you cannot solve problems in mathematics. In other words, you cannot solve a trig problem without knowing some basic trig.
©2016 LT/ThinkAlgebra
