Thinking about Problem Solving

rafael acorsi I just came across an interesting paper (via a Dan Meyer post I can’t seem to find again) titled “Teaching General Problem-Solving Skills is Not a Substitute for, or a Viable Addition to, Teaching Mathematics”.  Here’s the part that got me thinking:

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 conclusion they draw from this research is that:

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.

And their link to Problem Solving in Mathematics looks like this:

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.

So if the domain is the Math classroom students can improve their Problem Solving skills by studying worked examples of different kinds of problems.

Studying worked examples interleaved with practice solving the type of problem described in the example reduces unnecessary working memory load that prevents the transfer of knowledge to long-term memory.

This study resonates with me because I have been thinking a lot lately about how to really help my students become better problem solvers.  I have come to a few conclusions on my own and I think that this paper on chess might fill in a few gaps.  So far this is what I believe about teaching Problem Solving in the Math Classroom:

  1. It can’t be for marks.  Problem solving involves playing with ideas and following different paths that might lead to the solution.  If there is a mark attached to the process or the final answer then students can get so focussed on doing it right (for marks) that they aren’t willing to chose a path to if they aren’t certain that it is the right one.
  2. The usual problem solving steps aren’t enough.  For years I have tried to help my students solve Math Problems using some variation on this series of strategies combined with lessons on how to read a question correctly and pull out the necessary information.  I have never been all that happy with these strategies on their own and to be honest I have never had that much success with them.  To me these are great strategies for getting started with a problem, but they don’t help a student follow through to an correct answer that they can justify.  For that they need to be thinking about their thinking.
  3. Metacognition is really important.  Recently I have put a lot of effort into following my student’s thinking when solving problems.  When difficulties emerge it seems to be because they a) don’t know how to start or b) don’t know how to check their thinking for logical consistency.  Part b) intrigues me.  Often the reason one of my students gets an answer wrong is because they have done a calculation without really thinking about why they chose that particular calculation, or what the answer means.  They just saw two numbers and realised they could do a particular calculation with them so they did, thus leading themselves down a wrong path based on a logical inconsistency on their thinking.  I have a feeling that if I can help them be more aware of the way they are thinking as they tackle a problem I might be able to help them be more successful.
  4. Modeling worked examples is really important and technology could play an important role in making this happen.  This part just started to gel after reading about the research on Chess masters mentioned above.  I can see technology helping in two different ways. 
    • First, it could be used to capture examples of people working different kinds of math problems and thinking out loud.  Initially I imagine that would be me (the teacher) but the natural extension of this is to get students modeling their thinking for other students.  Some great examples of this process being used in classes already exist; Eric Marcos’s is the one I am most familiar with. An online repository of different kinds of Math Problems being worked out could be a really useful resource for helping students start to build problem solving schemas.
    • Secondly, it could be used to crowd source the act of finding and organising really good math problems.  I am imagining something like a Diigo group with a tag library that includes tags for the different kinds of problem solving strategies we usually ask students to use and maybe tags for the different kinds of thinking involved (admittedly this part needs to be flushed out).  These links could then be filtered by tag and embedded in a wiki somewhere.  My thought is that this would provide any teacher working on problem solving skills would have lots of math problems sorted according to thinking skills that they could use to first model the thinking involved and then give their students practice with the same kind of thinking;

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