05 July 2017

A math problem solving strategy that's proven to work

As I said before, there's a tough set of requirements that any problem solving strategy that's worth teaching to students as a singular go-to overarching framework must satisfy. I have yet to see a simple problem solving strategy satisfy all of those requirements, but here below is something that's been proven to work [1].

The Formulaic Action Oriented Problem Solving Strategy

It's "formulaic" because the back-end of the strategy is centered around using formulas to derive numeric solutions. But also, this strategy is "formulaic" because it's step-by-step, bringing students from start to finish of solving a problem.

It's "action oriented" because it critically answers for students the question of "what should I do?"


It's a "strategy" and not a bag of techniques, and so perhaps it's better called a "framework". But only because so many other people call as "strategies" the techniques that one might use during problem solving.

Certainly problem solving require techniques, but just as important is learning terminology.

Terminology and terms are definitions. Terms have meaning that is denotational.

Technique is what to do, how and when.

Steps of the action oriented, formula centered problem solving strategy:

(1) diagram the problem or situation,
(2) label the diagram,
(3) choose or write a formula,
(4) fill the formula in with numbers,
(5) solve for the unknown.

Notice the first two steps, the front-end of this strategy, depend on knowing terminology. The last three steps, the back-end, depend on knowing how to work with formulas.

Working with formulas is the most advanced form of arithmetic techniques. It is also the most foundational form of algebraic techniques. Therefore knowing what to do with formulas is an essential skill, and set of techniques, to learn: especially to bridge students from arithmetic to algebra.

Thus this strategy's back-end could be equivalently used with arithmetic, formula, or algebra based math, depending on the student's mathematical maturity. Also equivalently, though it ought to be rarely (especially in grade school), the back-end here could be swapped out to use algorithmic based math as well --- and I really mean algorithmic as defined technically in computing science, not just any step-by-step non-mathematical short-cut that someone without higher mathematical maturity concocts.

The formula centre in that problem solving strategy's back-end

The formula centre consists of the last three steps. Recall they are:

(3) choose or write an appropriate formula
(4) Then filling into the formula the appropriate numbers to replace the variables in that formula.
(5) And finally, appropriately solving for the unknown variable.

The simpler the formula, the more it involves just arithmetic skills: including the arithmetic technique of number substitution which should've been previously learned in junior high school, and similarly also the arithmetic technique of balancing equations which is often confused or misunderstood as algebra.

The more complex the formula, the more it needs algebraic techniques: especially as you need better knowledge of the algebraic technique of cancellation for canceling terms, factors, and fractions until getting the formula into the desired form to solve for the unknown.

This problem solving technique works broadly, e.g. from topics like measurement (unit conversion, areas and volume), to trigonometry.

It even works for topics like roots and powers --- but then that involves some more advanced skills like the algebraic technique of formula substitution: replacing a whole, or part, of an equation with another expression, kind of like instead of filling an equation in with numbers, you fill it in with expressions.

It also works for Factors and products --- but students would need to learn either the abstract area model or algebra for things like distribution, factoring out GCD, binomial decomposition, etc. But all these are just applications of the basic algebraic techniques used in the Formulaic Action Oriented Problem Solving Strategy!

It also works for problem solving with Relations and functions. Of course, functions are just special formulas, and Graph Sketching and Interpretation techniques would be needed.

Similarly, Linear functions. And transforming between the 3 forms of linear equations with algebra mainly requires the algebraic technique of cancellation: cancelling terms, factors, and fractions until getting desired form.

Also Linear systems. Linear Systems can be solved using Graph Sketching and Interpretation techniques, and also with algebra (algebraic techniques of: formula substitution, and formula elimination).

Let's see: that's just 3 foundational algebraic techniques and two visual (diagrammatic, graphical) techniques! All working within one problem solving strategy.

And the many variations of those.
And a lot of terminology!

Turns out Math isn't that hard: the above covers a large number of basic fields of math that students have to learn, and we counted up just 5 foundational techniques and one problem solving strategy.

[1] I would know because I've seen it used by some very strong and some very weak math students, all to great success! In fact, the above was written in around 2014 August after such observations.

No comments: