Problem solving is central to mathematics. Yet problem-solving skill is not what it seems. Indeed, the field of problem solving has recently undergone a surge in research interest and insight, but many of the results of this research are both counterintuitive and contrary to many widely held views. For example, 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)....[I]n over a half century, no systematic body of evidence demonstrating the effectiveness of any general problem-solving strategies has emerged. It is possible to teach learners to use general strategies such as those suggested by Pólya (Schoenfeld, 1985), but that is insufficient.
The alternative route to acquiring problem-solving skill in mathematics derives from the work of a Dutch psychologist, De Groot (1946–1965), investigating the source of skill in chess. Researching
why chess masters always defeated weekend players, De Groot managed to find only one difference. He showed masters and weekend players a board configuration from a real game, removed it after five seconds, and asked them to reproduce the board. Masters could do so with an accuracy rate of about 70% compared with 30% for weekend players. Chase and Simon (1973) replicated these results and additionally demonstrated that when the experiment was repeated with random configurations rather than real-game configurations, masters and weekend players had equal accuracy (±30%). Masters were superior only for configurations taken from real games.
The superiority of chess masters comes not from having acquired clever, sophisticated, general problem-solving strategies but rather from having stored innumerable configurations and the best moves associated with each in long-term memory.
[L]ong-term memory, a critical component of human cognitive architecture, is not used to store random, isolated facts but rather to store huge complexes of closely integrated information that results in problem-solving skill. That skill is knowledge domain-specific, not domain-general. An experienced problem solver in any domain has constructed and stored huge numbers of schemas in long-term memory that allow problems in that domain to be categorized according to their solution moves.
[D]omain-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).
For novice mathematics learners, the evidence is overwhelming that studying worked examples rather than solving the equivalent problems facilitates learning. 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).
Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics (pdf file)
by John Sweller, Richard Clark, and Paul Kirschner
Wednesday, November 3, 2010
Knowledge is good
Just heard from Barry - new article from Sweller, Clark, & Kirschner is out --