Polya, George - Mathematical Discovery

The Structure of Concern Project compares many theoretical models from many disciplines to the Adizes PAEI model, arguing that they must all be reflecting the same underlying phenomenon. One concern structure model is described below.

George Polya was Hungarian-born mathematician and educator interested in problem-solving techniques. His first book on mathematical reasoning How to Solve It (Polya, 1945) is credited as the document which popularized the term ‘heuristic’ (Baron, 1994). In a later work on mathematical reasoning (Polya, 1965) Polya identified four tactics of problem-solving consistent with the structure of concern. They are listed below in PAEI order:

P – Mobilization: By struggling through a problem, an investigator activates more knowledge and gathers more material towards a solution.

A – Isolation: Reducing complexity by focusing on one small detail at a time, shifting a very narrow attentional spotlight through the problem structure and staying with each item until it is a fully evaluated as possible.

E – Combination: The assembly of parts into wholes, into more harmonious Gestalts.

I – Organization: Connecting together mobilized knowledge, organizing separate parts into a purposeful whole.

Polya represents these operations as points on a diamond, connected by edges describing further mental operations that connect the four points. Isolation and Mobilization are connected through Recognition, such that we focus in on something recognizable and work at it, or work at something and recognize a pattern to examine more closely. Mobilization and Combination are joined by Remembering – a pattern-completion function that is often sparked by recognized elements to support further mobilization.


On the Organization side, Combination and Organization often Supplement each other, with part-to-part relationships and part-to-whole relationships each clarifying the other. Isolation is useful for Organization by clarifying distinctions and allowing elements to be Regrouped. Regrouping can also single out targets for Isolation.

This is like a cue-triggered version of PAEI responses, rather than impulse-driven or perspective-driven.

1. Polya, G. (1945). How to Solve It: A new aspect of mathematical method. Princeton: Princeton University Press.
2. Polya, G. (1965). Mathematical Discovery Volume II: On understanding, learning, and teaching problem solving. New York: John Wiley and Sons Inc.
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