Solving Math Problems

The following article presents a number of key concepts for successful math problem solving.
From the introduction:
“This paper describes a method for solving math problems.The basic idea is to combine two things:

• First, a simple method for making handwritten notes while thinking about a problem.This method is aimed at supporting
  – a step-by-step approach to problem solving and
  – reflective thinking: Better understand and control what you do while solving a problem.
• Second, a densely packed cheat sheet with broad advice on math problem solving. At present, this sheet focuses on general methods for problem solving. Later versions may contain material on specific domains like calculus or algebra.

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How to solve problems? Building a technology of problem solving

It seems difficult to overemphasize the importance of problem solving skills. I find it all the more puzzling that many powerful problem solving techniques are much less common than one should expect.
Moreover, little attention is directed towards a comprehensive technology of problem solving.

Such a technology may comprise

• “hardware” (from simple pens and writing pads to sophisticated interactive whiteboards and beyond),
• software,
• routines of behaviour (usage of hard- and software) and
• a process of improving these elements.

(This list is not exhaustive. We could add the brain and ways of manipulating it – by training, drugs or, another brainstorming item, by genetical engineering.)

This post presents some basic ideas on a technology of problem solving. Please comment and add your ideas!

We describe a basic model, introduce some elementary concepts and present a number of modifications to the basic model.

The seminal ideas

Our most basic model looks like this:
We make parallel use of two paper mind maps:

• a “problem map”, where we examine the given problem, and
• a “tool map”, which is a repository of problem solving tools.

To make things more transparent, here are some ideas how a tool map may look like.
We use 5 stages, for example the IDEAL model:

• I = Identify the problem
• D = Define alternative goals
• E = Explore possible strategies
• A = Assess and act
• L = Look back and learn

We may attach a couple of problem solving tools to each of these stages, e.g. creativity methods like brainstorming for exploring possible strategies.

If we get stuck while working on our problem, we may consult the tool map for inspiration; or we may add a useful tool to the tool map.

Here are some general elements of this basic model:

• We have some way of representing our thoughts on the problem – in this case mind mapping. We will call this the “substrate” of problem solving.
• We may have a second substrate for representing tools. In the above case however, we use again mind mapping.
• We have the problem solving tools itself. I like the idea of having only one “master tool” – in the above case the master tool consists of the stage structure and the single tools attached to the stages.

Modifications of problem solving substrate and tool representation substrate

Here is an incomplete list of alternative substrates.
Instead of mind mapping we could use…

• ordinary written notes, either on paper or computers,
• spreadsheets,
• computer mind maps,
• alternative arrangements, e.g. in a system of linked nodes,
• whiteboard pages on the internet for online collaboration,
• spoken words, e.g. talk to yourself,
• collaborative versions, e.g. discuss with others and let them suggest tools for tackling the problem.

Modifications of the master tool

Here are some ideas – instead of the elementary IDEAL model described above we could use

• a more specialized model, e.g. for solving math problems,
• a master tool that comprises several basic architectures, e.g. one collection of tools arranged by stages (as in the IDEAL model), another collection arranged by elements from the problem domain (in the math example, we could collect tools for dealing with sequences, series, vectors, matrices) and a third collection for dealing with difficult problem solving situations (frustration, getting stuck, running out of time).

The process of gradual improvement

To recap the paragraphs above – in the process of gradual improvement we can try to improve

• the hardware – e.g., large screens are a blessing when using mind mapping software,
• the software – a variety of layouts (maps, tables) in a single package is nice,
• the substrates of problem solving and tool representation – again, a mix of representations is best,
• the routines of behaviour – e.g. how often to change between direct work on the problem and some “metacoginitive” reflection,
• the “master tool” in its structure and its single items – e.g. things like asking “why” in several iterations often prove to be astonishingly powerful.

What makes solving math problems difficult? Diagnosis

“One of the most important factors [in deficient mathematical problem solving] is poor mental management:

• Students did not pay attention to the winding path of their activities in solving a problem.
• They often did not think to use heuristics they knew and could have applied.
• They often perseverated in an approach that was not yielding progress rather than trying a new tack.
• They often gave up without rummaging in their repertoire for another point of entry.
• Amidst the trees, they lost sight of the forest.”

(I found this diagnosis in David Perkins’ book “Outsmarting IQ: The emerging science of learnable intelligence”. Perkins reports some of the findings of mathematician-psychologist Allan Schoenfeld (p. 87).)

One promising way of mastering these difficulties lies in combining two major approaches to problem solving:

• heuristics in the tradition of George Polya’s “How to Solve It”, and
• mapping techniques, like mind mapping (or concept mapping).

Here are more details.