28 June 2017

Many math problem solving strategies are problematic: here's why

(To see what a better problem solving strategy would look like, skip to the bottom... )


Many problem solving strategies in math are problematic in themselves.  Here are some typical problems with them:


(1) Some are too high level or abstract, and thus provide little to no traction for students to get into practically solving problems. Students either don't know what exactly they should do, or they misunderstand the intent of a step in the strategy.

e.g. one strategy calls for the student to describe any "barriers or constraints that may be preventing them from achieving their goal".  Would "the teacher hasn't taught us" be a valid barrier?  (The answer could be "yes", by the way.)


(2) It assumes a pretty smart student, rather than assume a barely passing one.  So the strategy works great for students who would've done fine without your help in the first place.


e.g. look at the above example again, and specifically that step calling on the student to describe any barriers: if a student was knowledgeable and cognizant enough to correctly and accurately identify the mathematical barrier to their solving a math problem, one could imagine that that particular student probably doesn't need much in being taught a simplified problem solving strategy anyway.

As another example, I like how many strategies, like this one, call on students to make a "plan to solve the problem".  If a student could make a usable and useful plan to solve a problem, then that student probably doesn't need your strategy!

On the other hand, some of them only mean for the student to plan on a technique to use, which is okay.  But "planning" is much too committal for techniques that may fail to work: that step should've been called, "try these and other techniques until you succeed".

 
(3) Some are too low level or narrowly focused, and thus provide little in framework to help students get from starting to solve a problem to closing out with a solution.  It turns out they're more like techniques rather than strategies.

e.g. another site calls using "Logical Reasoning" a strategy.  Certainly, students should use logical reasoning at some point, but what, specifically, should the student do in terms of logical reasoning?  What specific steps should a student follow if they use this strategy?

By the way, in most real problem solving situations, logical reasoning should not be used all the time anyway: often, a dose of creative, non-linear, lateral, thinking outside the cuboid, is necessary.  Thus, "logical reasoning" as a strategy is both too narrow in focus and too vague at the same time!

The same can be said of another strategy on that web site: "Guess and Check".  Can I just not even read the problem, and like just guess out of the blue then check?  Certainly all these strategies need to be taught well and used right, but even then, there's actually only a very narrow use-case for "Guess and Check", and so it can hardly be called a strategy.  Most of what's listed on that web site are more like techniques.


(4) Many are targeted towards a younger audience, and doesn't much help high school students learning algebraic solutions to solving problems.



Those examples of strategies above aren't bad or wrong; they are valuable to teach and learn in specific circumstances to particular students who needs it, and especially if none of them are singularly overly relied upon.  So long as students know those are rules of thumb that may or may not help, then it's okay, because they're problematic if taken too seriously as THE problem solving strategy to always use.


A better problem solving strategy

So any problem solving strategy that's worth teaching to students as a singular go-to overarching framework should:

- be useful to any student: both the best and worse student in math class should benefit

- grows with the student: the student could use it in grade 12 calculus as in grade 7 math

-  be broadly applicable and usable, whether with geometry, factoring, or in graphing 4th degree polynomials

- have specific steps to follow: it can't just be general aphorism to plan, decide, and act

- have general enough steps to follow so it's rightly a strategy or framework to solving a problem, not just a bag of techniques to try: it has to bring a student from the beginning to the end of solving a problem

This is a tough set of constraints, and I have yet to see a simple problem solving strategy satisfy all of these requirements.  But maybe there's one that works... (to be continued soon).

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