kitchen table math, the sequel: How to Solve It

Friday, July 17, 2009

How to Solve It

In light of our frenzy on what's wrong with problem solving in k-12 math education, below is the summary of the greatest book ever written about how to solve math problems. It's from How To Solve It, written by George Polya, a mathematician. Polya came to the States in 1940 where he joined the Stanford faculty. He was a probabilist or a combinatorist. Later in life, he cared a great deal about how to teach problem solving to students, and so wrote several books about mathematical reasoning, the most famous of which is How to Solve It.

Summary taken from here:

First. 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?

Second. 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 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 involved in the problem?

Third. 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?

Looking Back
Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument? Can you derive the solution differently? Can you see it at a glance? Can you use the result, or the method, for some other problem?

As you can see, this is pretty much a script. A script for problem solving a in a few heuristics! The great Polya's notion of how to solve a new problem constantly asks what problems you already know how to solve. You must therefore,
a) know things to mastery
b) know that you know those things
c) know them so well that you can see what parts of your solution are contingent on what parts of the problem, so you can
d) know what you can alter

In Polya's method, it's impossible to overstate the value of knowing results rote. His question of whether you know a theorem that might be relevant requires that you simply know a large number of theorems, know them so well that you can state them clearly (and likely prove them as well). Spiral methods of discovery learning won't ever lead you to the sureness required to state a theorem in a mathematically precise way.

His books work for some college students. It is inappropriate to hand these books to k-12ers and walk away. It might be valuable to explore the book with your child, depending on their maturity. I think parents would get a great deal out of these books, if only as a contrast to how their children's teachers do spiral curricular discovery learning. Polya's work should be known by every math or science teacher, period. Teachers who need to learn how to guide their students would gain a lot from seeing how Polya's notions of "discovery" and "ownership" of a solution entail prior knowledge over and over again.


ChemProf said...

I'm going to steal this, Allison. I like to start my class with a talk on problem solving, and I like this script. That's exactly where I see a difference between what I do and discovery learning -- I want them to be able to solve new problems, but expect them to do that by recognizing how the new problem relates to problems they have done already, not by flailing around and getting lucky! As a colleague of mine likes to say, you can't figure out chemistry during the test.

Allison said...

Please please read the book, too. Give it to everyone you can! (I've lent the book and never had it returned more times than I can count!) It's so terrific. It has these dialogues between student and teacher that really help show how and where students get stuck and how the heuristics help in real problems.

It sounds simple, but of course, real students don't know how to adapt from the tiny space of their mastery to bigger and different problems. They don't know *which* part of a problem to collapse, *which* part to ignore for now, *which* part is analogous to a problem they already know. "Solve a simpler problem"--but which way is the way to simplify the problem? All of that is confusing and takes practice practice practice, but here at least, they can start to actually practice.

Personally, my recitation sections in stat mech, and to some degree, quantum, were built around this model. And especially in those subjects, as there just aren't all that many problems that can be solved in a small amount of time, mastery of this model meant mastery of the test material, too. With practice, doing this in recitation meant it started to gel how to recognize the basic problems to master and how to tell what related problems were (microcanonical ensemble! grand canonical! isothermic system! Harmonic Oscillator! phonons!)

This also relates back to the modeling instructions stuff of Hestenes. The reason why modeling instruction can help is because it so strongly helps define "this is the basic problem to solve; this is a related problem". Learning how to see similarities and trust your mastery of the simple problem is 80% of the way there.

Anonymous said...

This relates directly to the points Barry Garelick made when citing the Kirschner, Sweller, Clark study.

Polya seems to be giving explicit directions on how to decide what information in long term memory is relevant to the problem so it can be retrieved. He is giving criteria to help determine what to pull into working memory.

K-S-C point out that "the aim of all instruction is to alter long-term memory".

It takes problem practice in math and decoding and blending practice in reading to alter long term memory.

Discovery or inquiry learning by a novice rarely alters long term memory.

Spiral learning is not sufficient to alter long term memory for most kids.

Basically instructional techniques like inquiry learning in math and the spiral approach ignore empirical evidence of what works and the cognitive architecture of the human brain.

It seems that too many educators in this country are pushing reading and math instruction techniques that are nothing more than superstition.

CassyT said...

I always discuss with teachers the distinction between Polya's "Look Back" as opposed to what is more commonly used: "Check your work."

Here's how it looks at a 3rd grade level:
Does my answer satisfy the question?
Does it make sense?
Is there another way to solve the problem? A simpler way?
Have I shown my work?
(and in my classroom: Can Mrs. Turner read it?)

dan dempsey said...

From Bertrand Russell or was it Alfred North Whitehead...."the goal is to have no problems only exercises".

By expanding long term memory one can turn potential problems into exercises.

It seems that the goal of many pushing the discovery/inquiry approach is to have no exercises but only problems.

I am still in shock from Dr. Ruth Parker's powerpoint given at NCTM national that the Standard Algorithm always harms conceptual understanding, which seems to be a recipe for lots of problems.

The person most everyone would like to hire is the one that has so much extensive background knowledge that most everything is an exercise.

Anonymous said...

I use an organizer that approaches this technique. I think it's adapted from an ELA template called four square. Here's what I have the kids do.

Take a piece of paper and fold into quarters. Open it up and draw a rectangle in the center. Now you've got four quadrants and a rectangle which is not exactly 'four square' but in the interests of marketing I guess four square sounds sexier.

Read the problem twice. Then in the rectangle restate the question in the form 'find blah blah blah in units of xxxxxx'.

Read the problem again and in the upper left quadrant identify and define all the symbols (variables and constants) you'll use in your solution.

Read the problem again and use the lower left quadrant for a diagram/picture.

Read the problem again and use the upper right quadrant to define your strategy. This can be words or preferably a set of equations to solve.

The lower right quadrant is where you show your arithmetic.

Finally, the back of the paper is used to justify your answer.

It works well for entry level problem solving of the kind you would encounter through maybe grade six. If kids master this it should instill some good habits for more complicated things.

Anonymous said...

It occurs to me that this is exactly the process one has to use to solve any problem, not just math.

It's also precisely what does not happen when folks seek to solve the problems of public education. I wrote about this a year ago in The Maze and it still seems relevant today. The 'repairs' that I've experienced are always in the form of isolated fads, like throwing M&Ms at your hunger when what you really need is a Snickers.

OrangeMath said...

Perhaps the biggest problem educational theorists DON'T KNOW THEY HAVE is that Polya and people that try to use this work (including me), have never been able to find what content goes with what strategies - both are needed, but... In other words, there is no general theory of heuristics.

The obvious fact that in all of the hubbub surrounding strategies and content, "heuristics" is never (good luck finding it) mentioned merely indicates how pathetic ed researchers are. It's not the billions ($) lost, but the unwillingness to confront the limitations of teaching heuristics.

Humility goes with using heuristics in instruction. It's the best we can do, but the multiple-choice world is a dead end

Catherine Johnson said...

The 'repairs' that I've experienced are always in the form of isolated fads, like throwing M&Ms at your hunger when what you really need is a Snickers.


I'm (currently) with Richard DuFour on this: 'it's the culture, stupid.'

You can bring in every good reform there is, but if you don't have a 'student-focused' culture none of them is going to work.

Catherine Johnson said...

Could you imagine a more accessible related problem?

At some point, way back when, I figured this one out on my own. It's MAJOR for anyone who does **not** have a serious education in math.

Interestingly, it's quite hard to teach this approach to your own child (or so I found).

At least two of my friends said the same thing.

Some of the issue, of course, is that kids don't want their moms re-teaching them math.

But there was something else, too...

I'm guessing we all tried to get our kids to use this approach when they were trying to do a problem that was far too difficult for them.

Catherine Johnson said...

wow - Paul!

I love that!

Barry Garelick said...

Could you imagine a more accessible related problem?

It takes many years to get to that point on your own. But you can get there in baby steps. When teaching division by fractions, a problem might be "How many 3/4 inch pieces of ribbon can be cut from a piece of ribbon that's 9 inches long?"

When confronted with the blank stares such a problem engenders, then I would switch and say, "How many 3 inch pieces can you cut from a 9 inch piece?" They would quickly figure it out. "How did you do it?" I would ask. "Divide 9 by 3".

"OK, now let's go back to the first problem. How would you do it?"

Hesitant, unsure answer: "Divide by 3/4?"

"Yes, absolutely".

The student has to build up a repertoire of problem solving techniques which then can be drawn upon, as well as being shown as I did above, how to look for similar problems. It isn't by any means easy for students to do this.

Catherine Johnson said...

great example

I'm sure all of us were just desperately trying to teach our kids this "truc" as a last-minute survival strategy.

Anonymous said...

Another way to 'conquer' fraction division is to demonstrate a problem using lowest common denominators. It's an unconventional use of LCD but effective in getting the light bulbs turned on at times.

Here's an example, say 2/3 divided by 1/4. Well this is 8/12 divided by 3/12. At this point I usually just say that the denominators are just like units for the numerators, so this problem is really just 8 whatevers divided by 3 whatevers or 2 2/3.

It works pretty well because by the time they tackle fraction division they are fluent with the whole LCD concept and are used to it being treated like 'units' for the numerators.

I don't teach it this way of course but it's a useful alternative for kids who are convinced that the standard method is just magic and try to call BS on you.

OrangeMath said...

Liping Ma is famous for asking: list problems where division by a fraction make sense. Few American elmentary teachers can name any. Chinese ones can name many.

Allison said...

Catherine, the notion of a related accessible problem is really hard in practice. That's why Polya's book is worth reading, not just the summary. He walks through trying to show a student just *what* related problems the student *does* know.

But as Barry pointed, knowing what you know takes years. The whole idea of knowing which way to modify a problem to make it look like a problem one already knows how to solve is subtle--not all ways are equally good. At the beginning, watching your teacher do it is mystifying. Their intuition leads them in certain ways based on the sheer volume of problems they've already managed to see as interrelated. Building up that level of flexible knowledge requires massive practice. No wonder that you couldn't get your kids to see what a related problem was--you probably could not describe why you knew which related problem was the right one, too.