How to Solve It is the key work in the two initiatives that, so for as I can tell, occupied the second half of Pólya's remarkable career. The first was to create a practical guide for teaching reasoning and problem-solving, focusing on mathematics. The second was to reintroduce the field of heuristics. From the glossary:
Pólya's intention was to build on the work of Pappus, Descartes, Leibnitz and Bolzano while, in some cases, scaling back their ambition (Descartes and Leibnitz both tended to think big). He was attempting to lay out a framework for a discussion that could be productive but was unlikely to be resolved.
The primary focus was something Pólya called plausible reasoning (a term he used in the title of his first two follow-up volumes). The idea was that while the final product in mathematics is based on rigorously proven statements, the process of getting there is usually a messy combination of induction, analogy and intuition, propped up with informal and incomplete proofs until something rigorous can be erected. In order to be good at their profession, mathematicians need to be (or become) skilled at coming up plausible conjectures.
Pólya's prose is plain-spoken and direct, which has sometimes caused trouble for less careful readers because, though the style may be simple, the ideas are not. Pólya often makes fine distinctions and his assertions often are only valid in their carefully laid out context.
With its heavy reliance on Socratic dialogues and its extensive discussions of philosophy and the history of mathematics (all of which are directly relevant to the main points), How to Solve It does not lend itself to bullet points and executive summaries. Unfortunately those have become very much the language of education today. I have seen a lot of education proposals -- particularly those promising to teach critical thinking and problem solving -- that appear to come from people who saw the bullet points but never read the book. That leads to bad things.
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