Tag Archives: Problem Solving

How to Solve It

Steven Volk (Director, CTIE)

December 9, 2013

An article I was recently reading (“Teaching Learning Processes – to Students and Teachers,” by Pamela Barnett and Linda Hodges) reminded me of a 1957 book on mathematics by George Pólya, How to Solve It (2nd ed., Princeton: click on link for a partial pdf of the volume). The issue is a central one for all teachers: Rather than solving problems for our students, we provide them with strategies for problem solving. Or, as Pólya put it, we are always “trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others…” (vi).  Pólya is quite clear that while his book “pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of invention and discovery” (vi). “Invention and discovery” – what better to words to describe what we want to inspire and develop in our students?

G. Polya, How to Solve It, Princeton Science Library

George Pólya’s Approach

Pólya’s approach has four parts, which I’ll copy here from his text before suggesting some changes I have made when approaching problem solving in history, and which others can similarly adapt to their specific discipline.

I. UNDERSTANDING THE PROBLEM

You have to understand the problem: What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down?

II. DEVISING A PLAN

Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution. Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involve in the problem?

III. CARRYING OUT THE PLAY

Carry out your plan. Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

IV. LOOKING BACK

Examine the solution obtained. Can you check the result? Can you check the argument? Can you derive the result differently? Can you see it at a glance? Can you use the result, or the method, for some other problem? (From 2nd ed., p. xvi).

Revising Pólya’s Approach for History and other Social Sciences & Humanities

Pólya’s approach is well designed for math and other quantitative disciplines. But, with a few tweaks it can be equally useful for the (non-quantitative) social sciences and humanities where we are not looking for proofs, and where experiments cannot be repeated. Rather, we are after the strongest arguments (the best readings) that take account of the evidence at hand. Problems in history are quite different than math problems, but “solving” them (i.e., putting forward an empirically based, well reasoned argument) can be approached in a similar manner. The steps below are revised from an approach put forward in the Pamela Barnett and Linda Hodges article (which, in addition to the link can be found on CTIE’s Blackboard site), and are based on answering the following question: “Why did the Tupac Amaru II rebellion of 1781 fail and were there any circumstances in which it could have succeeded?” (For those so enthralled with the question, check out my “flipped class” lecture on the topic: “The Great Andean Revolts.” )

Tupac Amaru II (Flickr creative commons: seriykotik1970)

I. Understanding the problem: What information do you have to begin with (secondary sources, primary sources, lecture notes, other information gained in different courses or non-assigned readings, etc.)? What information do you still lack in order to be able to address the problem? Can you restate the problem in your own words, or in a way that helps you understand it better? Is Tupac Amaru II’s failure in 1781 similar to or different from the failure of the first Tupac Amaru’s rebellion? What characteristics in the information you have strike you as potentially important? Why do you think they are important?

I have found over the years that this first part of problem solving is critical. Nine times out of ten, a poorly argued paper comes back to the fact that the student hasn’t understood what is being asked. Advise students – many times!! – not to begin writing their papers unless and until they are clear that they understand what is being asked. This is a good time to consult the teacher, a peer instructor, or a colleague from the class.

II. Setting out your plan: Trace out your initial ideas: Could it be inter-ethnic rivalries? Lack of broader sets of allegiances? Lack of military strength? Problems of communication?

  • Initial ideas. To the extent that the question implies some comparative data (have other rebellions succeeded or failed?), look for similarities, differences, other kinds of approaches that have worked for you in the past.
  • Following up with these ideas: Begin to gather data on your initial points. Will they help you answer the question? Do they make sense (no, it had nothing to do with Spain’s ability to control death rays from the Planet Xynthar)? Are they going to lead to either a dead end or a tangential issue that has nothing to do with Tupac Amaru II’s failure? (Go back to the initial question: Do you understand it?)
  • Avoiding Rube Goldbergian approachs: Yes, it’s a plausible answer, but are there, um, more straightforward approaches? Think of breaking the problem into smaller pieces that can help in the solution (List all the elements that can account for Tupac Araru II’s failure; list all the elements of Tupac Amaru I’s failure; what contextual events were similar or different in 1781 compared with 1572? Any contingent events to think of?)

III. Carrying out the plan: Beginning to draft the paper. With your arguments and data in place, begin to draft the answer, always making sure that the points are leading to an answer to the question that was posed and not answering some tangential issue, are supported by evidence, and are presented in a logical (and, in this case, chronological) order. Make sure to support your evidence with footnotes/endnotes in the proper format.

IV. Revising the draft. There are a lot of questions you can ask yourself after you’ve completed a draft: Does it make sense? Is it plausible? Does it conform to the evidence? Have you left anything of importance out? Is there a piece of historical evidence that doesn’t fit – kind of like the bolt that’s still lying on the floor after you’ve put your desk together? Have you documented your evidence and used the proper formatting? Is something nagging at you about your work that you haven’t come to grips with? Can you share your work at the Writing Center, with a peer instructor, or a classmate (if allowed in that class)? Have you checked spelling, format, grammar, etc.?

V. Reflection. While reflection is not necessarily a part of problem solving, it is an essential part of learning and should always be a part of an assignment: What did you learn in this project: not about the topic per se, but about how you approached it? What steps did you take to solve the problem, to answer the question? What did you learn in this process that you will use again? What approaches led you to a dead end and were ultimately unproductive? Do you feel pleased with your paper? Why? Why not?

Heuristics

Pólya also offers a set of heuristics that can help students (and faculty) solve problems with reference to different approaches.

Heuristic Informal Description
Analogy Can you find a problem analogous to your problem and solve that?
Generalization Can you find a problem more general than your problem?
Induction Can you solve your problem by deriving a generalization from some examples?
Variation of the Problem Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem?
Auxiliary Problem Can you find a sub-problem or side problem whose solution will help you solve your problem?
Here is a problem related to yours and solved before Can you find a problem related to yours that has already been solved and use that to solve your problem?
Specialization Can you find a problem more specialized?
Decomposing and Recombining Can you decompose the problem and “recombine its elements in some new manner”?
Working backward Can you start with the goal and work backwards to something you already know?
Draw a Figure Can you draw a picture of the problem?
Auxiliary Elements Can you add some new element to your problem to get closer to a solution?

Final Considerations

I found two things interesting when returning to Pólya after so many years:

(1) how similar problem solving techniques can be across the disciplines, and

(2) how important it is to keep disciplinary differences that do exist in mind when instructing our students.

These diverging points often come back to the “experts vs. novices” problem. As experts in our fields and disciplines, problem solving, particularly at a relatively basic level, is so ingrained in our thinking that we don’t think about the fact that it is not second-nature to our students. When we hit a road block, it will happen at a much higher level than will be confronted by students.  So, early on in our classes, particularly in introductory (100-level) classes, it is always good to formally trace out problem solving strategies in our disciplines. But it is also important to be explicit about the fact that many of these strategies can be used when solving problems in other disciplines (e.g., Pólya’s math problem solving strategies are quite applicable in physics or economics), and when certain approaches are specific to one’s discipline and cannot be used in precisely the same way in other disciplines. History is not an experimental science: we don’t  look for proofs in the manner of mathematicians or biologists.

Finally, this leads to a greater understanding of the rare opportunity we have at a liberal arts college. By teaching in a place where we know our students will be receiving instruction in a variety of approaches and disciplines, we can strengthen their learning (and their problem solving abilities), as well as our own approaches to the problems that we set out to solve, by consciously engaging in activities that bring disciplines together, asking: how would a physicist solve this problem? A biologist? How would a literary critic pose the question? A sociologist? What would happen if an artist were a part of a biology lab? If a physicist taught in the museum?

George Pólya and Alexander Ostrowski (Photograph: Paul Halmos, 1958)