Book Review: "How to Solve It" by G. Pólya
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Introduction
“How to Solve It: A New Aspect of Mathematical Method” is a classic work by Hungarian mathematician George Pólya, first published in 1945. The book is a guide on problem-solving, especially mathematical problems, and it is written with the intent to develop problem-solving skills in the reader. It has influenced many areas of thinking and problem-solving beyond mathematics, including computer science, programming, artificial intelligence, and more.
Pólyas book appears on many reading lists about software, maths and algorithms, and, although it can be a bit hard to read at times, it’s definitely worth the try. The writing style is sometimes odd an abstract and feels slightly outdated, which doesn’t come as a surprise given the year the book was published. “How to Solve It” is mostly written in the format of a dialogue between teacher and student. Pólya doesn’t just try to teach the reader how to solve problems, but also how to become a better teacher. He imparts how to interact with a student and what questions to ask, in order to make them solve problems and learn to solve them on their own.
Summary
The book is divided into four parts and revolves around the four steps of the problem-solving method. Those steps can be found at the very first part of the book.
The Pólya Method
1. Understanding the Problem
The goal is, obviously, to understand the problem, but sometimes that’s not as easy as it seems. Pólya gives us many approaches to understand the problem as good as possible, mainly by asking the right questions. The problem should be divided into its parts, condition, data, and the unknown. Those can be divided further by asking if the condition is sufficient to determine the unknown or even contradictory.
At this stage it’s often helpful to draw a figure, write down the data, or restate the problem with own words, to get a good overview and understanding of the problem. The problem statement can often be abstract, and we have to separate the meaningful and unnecessary parts of the data.
2. Coming Up With a Plan
Pólya advises us to, yet again, think deeply about the problem, once we got familiar with it. We should try to think of related problems, try to search for anything that sparks ideas on how to solve it. Sometimes that involves a related problem, or a way of working backwards from the solution.
At this point we are advised to use heuristics. Those are different practical approaches to problem-solving that aren’t ideal or optimal, but effective. There are many heuristics described in the book and some are more or less specific to a certain kind of problem, others can be generally applied to almost any problem. The approach could involve solving auxiliary problems, using logic, trying to find a pattern or plain and simple guessing.
3. Executing the Plan
This stage is all about strictly following the plan we came up with in the previous step. Each step must be clearly understood and executed, and we should be able to prove that it’s correct.
4. Looking Back
This is a crucial and sometimes overlooked step of the problem-solving process. We have to check if the solution is correct, if we solved the problem completely or if we missed a part. But Pólya urges us to do more than that. The solution should be reconsidered to see what could be done differently. We can also think about other problems that this solution can be used for.
Parts 1 and 2: The Classroom and the Method
George Pólya starts the book with explaining his philosophy of the classroom, how to interact with the students, how to talk and motivate them. He gives many simple examples from mathematics how to guide the student through the whole process, from finding the unknown to looking back at the solution and trying to find a better or more interesting way of getting there.
He cycles through different steps of the method, each time exploring different heuristics and examples. This is done by asking a lot of questions from the perspective of the teacher and then trying to act upon those questions from the perspective of the student.
Part 3: Short Dictionary of Heuristic
For the largest part of the book, Pólya writes about key points of heuristics. This is actually much like a real dictionary in the way that it’s written. There is a lot of repetition and asking the same questions we already learned from the method and the earlier parts.
Part 4: Problems, Hints, Solutions
The last small part of the book lists a set of problems and their hints and solutions as exercises for the reader.
Reflection
I got the impression that Pólya was a man of great wisdom and intellect. His language is clear and concise, the wording of the book is slightly dated, but easy enough to understand. However, reading the book was not as easy as I’d hoped. The writing style takes a while to get into, but once you understand that this is mostly an imaginary discussion between teacher and student, it’s manageable. The formatting and structure of the topics are not ideal and there is a lot of repetition.
The simple dialogue style actually helps a lot with making abstract processes easy to understand, and Pólya delivers a lot of knowledge while making sure it’s repeated often enough for everything to properly sink in. The problem-solving process described is not a mind-blowing revelation though. Most of the approaches are quite generic and well-known to anyone working in a technical job, but it doesn’t hurt to read about them from a different point of view nevertheless.
Conclusion
Problem-solving is an essential skill for any software developer, but it’s also something that comes naturally with experience. “How to Solve It” was an interesting new perspective for me, but I wouldn’t say it’s an essential read. It might be a good book to pick up for anyone interested in general mathematical problem-solving or the philosophy of teaching it, however, I think there are other, more modern and more easily digestible books on those topics available.