Friday, November 11, 2016

Problem Solving in Mathematics

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 PĆ³lya (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.” 

Note. The quotes are from “Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics” Sweller, Clark, & Kirschner (Doceamus, November 2010).

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. 

©2016 LT/ThinkAlgebra

Tuesday, November 8, 2016

Achievement Inequalities

Children are not the same. "Ability varies widely," says Charles Murray (Real Education). So, let's stop pretending that all kids are the same and can learn the same math. Children need different math curricula based on their cognitive abilities--not the same curriculum as in Common Core and state standards. In contrast, Nobel-prize winning Physicist Richard Feynman writes, "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities." Education increases inequality! 

"Fairness as the equal treatment does not produce fairness as equal outcomes," writes Thomas Sowell (Dismantling America). Some kids have more cognitive horsepower than others. Some kids do better at math than other kids. Some kids run faster than others. Some kids play the piano better than others. Some kids are more persistent than others, ad infinitum. Inequalities abound everywhere. It is unfortunate, says Sowell, that "virtually any disparity in outcomes is almost automatically blamed on discrimination." It's not discrimination! Sowell suggests jokingly that "tests discriminate against students who don’t study." 

Even if it were possible to equalize school resources, then there would still be an achievement gap in schools based on standardized tests. It seems that inputs from the family and community, not just school inputs (e.g. expenditures, facilities, teacher quality, etc.), account for much of the achievement gap. We do not live in Lake Wobegon where all the children are above average. In fact, half are below the median in intelligence. I think we can make progress in math performance, but there will always be inequalities no matter what is done. Also, the fact that math is taught poorly in many schools has been a major factor. Consequently, the result has been very slow, incremental progress or flat learning, which I consider, unacceptable. 

The top math students have skilled instruction and excellent practice with feedback, but they also have higher levels of cognitive horsepower than average kids. However, cognitive ability, by itself, is not enough to become a good math student. Kids need to practice to automate fundamentals, which requires effort and purposeful practice. Indeed, normal kids can learn arithmetic and do algebra. Also, school outputs, such as test scores, graduation rates, college readiness, etc. will always vary.

©2016 LT/