The do's and dont's of teaching problem solving in math
Many math students in the U.S. are scared, if not horrified, of math word problems. In general, they are thought of as difficult.
Why would that be? It doesn't totally make sense. I can't imagine children not liking word problems just because they need find an answer to something (a problem), or because the problem is explained in words. Even most of us adults are fascinated by puzzles, for example.
Also, this fear of word problems surely cannot start in the 1st grade. Story problems in the first grade are very simple, such as "There are five ducks on the lake and three on the shore. How many ducks are there total?" Often the math book even has a picture there to accompany it. I can't imagine children feeling it is difficult.
I feel the causes for this difficulty are many-fold:
- One-step word problems prevail in the end of lessons practicing a specific operation in elementary grades. These encourage children to simply find the numbers and use the operation studied in a linear fashion, as if all word problems were solved by using a "recipe".
- Many school books don't have enough GOOD word problems. They typically include lots of one-step problems, and then some isolated problem solving lessons which usually highlight a specific problem-solving strategy (so that once again, you have a "rule" that solves the problems in that lesson).
- Teachers are afraid of word problems so they skip them.
Let's look at 1 and 2 in more detail.
1. One-step word problems prevail in the end of lessons practicing a specific operation
You see this a lot in elementary grades. Children are practicing perhaps multi-digit multiplication, perhaps borrowing in subtraction, perhaps dividing decimals. After the calculation problems come some word problems, which oddly enough are solved by using the exact operation just practiced!
It extends beyond the lessons on the four operations, too. Haven't you ever noticed: if the lesson is about topic X, then the word problems are about topic X too!
When children are exposed to such lessons over and over again, they figure out that it's mentally less demanding to not even read the problem too carefully. Why bother? Just take the two numbers and divide (or multiply, or add, or subtract) them and that's it.
This is of course further encouraged by the fact that the word problems in the end of such lessons typically only have two numbers in them. So, even if you didn't understand A WORD in the problem, you might be able to do it! Just try: the following made-up problem is in FINNISH... and let's say it is found within a long division lesson. I assume now that you do NOT know Finnish — but can you solve it?
Kaupan hyllyillä on 873 lakanaa, 9:ää eri väriä. Joka väriä on saman verran. Kuinka monta lakanaa on kussakin värissä?
Drag your mouse over the white space below to see the translation (highlight it).
The store has 873 sheets in 9 different colors. There is the same amount of sheets for each color. How many sheets of each color are there?
Using lots of those kind of problems soon brings a problem: children "learn" (intelligently) this unspoken rule:
"Word problems found in math books are solved by some routine or rule that you find in the beginning of that particular lesson."
How can you avoid this terrible situation? Mix up the word problems so that not all of them are solved by the operation just studied. Another idea is to give students a bunch of short word problems to analyze so that instead of inding the answers, they find which operation(s) are needed to get the answer.
2. Many school books don't have enough GOOD word problems.
By good problems, I mean multi-step problems that advance in difficulty over the grades, and foster children's logical thinking.
One-step problems are good for 1st and 2nd grades, and then here and there mixed in with others. But children need to start solving multi-step problems as soon as they can, including in 1st and 2nd grades.
Look at this example problem from a Russian fourth grade book:
An ancient artist drew scenes of hunting on the walls of a cave, including 43 figures of animals and people. There were 17 more figures of animals than people. How many figures of people did the artist draw?
A similar problem is included in the 5th grade Singapore textbook:
Raju and Samy shared $410 between them. Raju received $100 more than Samy. How much money did Samy receive?
Now, these are not anything spectacular. You can solve them for example by taking away the difference of 17 or $100 from the total, and then dividing the remaining amount evenly:
$410 − $100 = $310, and then divide $310 evenly to Raju and Samy, which gives $155 to each. Give Raju the $100. So Samy had $155 and Raju had $255.
A far as the figures, 43 − 17 = 26, and then divide that evenly: 13 and 13. So 13 people and 30 animal figures.
BUT in the U.S., these kind of problems are generally introduced in Algebra 1 - ninth grade, AND they are only solved using algebraic means.
Here is another example, of which I remember feeling aghast, found in a modern U.S. algebra textbook:
Find two consecutive numbers whose product is 42.
Third-grade children should know multiplication well enough to quickly find that 6 and 7 fit the problem! Why use a "backhoe" (algebra) for a problem you can solve using a "small spade" (simple multiplication)!
I know some will argue and say, "Its purpose is to learn to set up an equation." But for that purpose I would use a bigger number and not 42. Don't such simple problems in algebra books just encourage students to forget common sense and simple arithmetic?
Another example, a 3rd grade problem from Russia:
A boy and a girl collected 24 nuts. The boy collected two times as many nuts as the girl. How many did each collect?
You can draw a boy and a girl, draw two pockets for the boy, and one pocket for the girl. This visual representation easily solves the problem.
Here is an example of a Russian problem for grades 6-8:
An ancient problem. A flying goose met a flock of geese in the air and said: "Hello, hundred geese!" The leader of the flock answered to him: "There is not a hundred of us. If there were as many of us as there are and as many more and half many more and quarter as many more and you, goose, also flied with us, then there would be hundred of us." How many geese were there in the flock?
(I personally would tend to set up an equation for this one but it can be done without algebra, as well.)
Please see these resources for good word problems.
The purpose of word problems
One purpose of word problems is to prepare children for real life. This is true for example of shopping problems.
Another, very important purpose of story problems is to simply develop children's logical and abstract thinking and mental discipline. Note: one-step word problems surely will not do that!
Third one; some teachers use fairly complex real-life scenarios or models of such to motivate students. I've seen this for example in an algebra program.
The problem is, such problems take a lot of time and a lot of guidance from the teacher. The only true way of developing good problem solving skills is .... TO SOLVE LOTS OF GOOD PROBLEMS. They don't have to be real-life or involve awkward numbers (such as occur in real life). Realistic, complex problems might be good for a "spice", but not for the "main course". "Fantastic" (unreal) problems are fine.
A problem solving plan
Most math textbooks present some kind of problem solving plan, modeled after George Polya's summary of problem solving process from his book How to Solve It. These steps for problem solving are:
1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look back.
Those steps follow common sense and are quite general.
HOWEVER, I dislike presenting this plan to students. I think we could and should emphasize the first and the last steps, but also I feel that often we cannot "squeeze" problem solving into the two simple steps of devising a plan and carrying it out.
With challenging problems, the actual problem solving becomes a process whereby the solver keeps a mental "check" of the progress, and corrects himself if progress is not made. You may go one route, notice it won't work, go backwards a bit, and take another route.
In other words, devising plans and carrying them out can occur somewhat simultaneously, and the solver goes back and forth between them.
The steps outlined above are fine, as long as students understand that these steps are not always simple or straightforward, nor do they always follow sequentially. You might make a plan, start carrying it out, and suddenly notice something and realize that you hadn't even understood the problem right!
Consider the master/apprentice idea. Let your students be the apprentices who observe what you, the teacher, do while solving problems in front of a class. Choose a problem that you don't know the solution to beforehand. You might try a wrong approach first, but that's OK. Explain your thoughts. This will show the students a true example of real problem solving!
See for example my problem solving thought process here: Proving is a process: proving a property of logarithms.
What about problem solving strategies?
Problem solving strategies we often see mentioned in school books are draw a picture, find a pattern, solve a simper problem, work backwards, or act out the problem. Again, these are often taken from Polya's How to Solve It. He spends a lot of pages explaining and giving examples of various problem solving heuristics or general strategies.
These strategies or heuristics are of course very useful. However, I tend to dislike the problem solving lessons found in school books that concentrate on one strategy at a time. You see, in such a lesson you have problems that are solved with the given strategy, so it further accentuates the idea that solving word problems always follows some pre-established recipe.
A better approach would be to solve good challenging problems weekly or biweekly. Vary the problems and how they are solved. Use the various problem solving strategies naturally in the example solutions that you provide, but don't limit students' thinking by naming the lesson after some specific strategy.
So what should we do?
Teaching problem solving probably isn't as difficult as it might sound. The first step would be of course that you, the teacher, should not be afraid of problems. Read Polya's book.
Then, find some good problems to solve (see resources below), and have students solve problems as a part of their routine math education. Discuss the solutions. Explain to them various strategies in the context of problem solving. Don't be lulled into thinking that textbook word problems are good enough, because they might not be.
Model a problem solving process yourself sometimes, as explained above.
It will come together just fine. Like I said, the main thing that helps students become expert problem solvers is if they will get lots of practice in solving problems!
I hope your students do not fit the above joke.
Sources and further resources
Word Problems in Russia and America – an article by Andrei Toom. It is an extended version of a talk at the Meeting of the Swedish Mathematical Society in June, 2005.
A collection of favorite math puzzles for children, gathered from my puzzle contest. Most only require the four basic operations so work well for elementary school children and on up.
A list of websites focusing on word problems and problem solving
Use these sites to find good word problems to solve. Most are free!
How to Solve It: A New Aspect of Mathematical Method by George Polya.
A classic, and excellent book on problem solving. Polya's ideas are behind most problem solving "plans" and strategies presented in math books today. How to Solve It popularized heuristics, the art and science of discovery and invention. It has been in print continuously since 1945 and has been translated into twenty-three different languages.
Challenge Math For the Elementary and Middle School Student
Includes lessons followed by practice and then three levels of questions. The author has taken concepts that are generally saved for older children (and can be dry and tedious) and made them accessible to a younger age group. Some of the concepts are fairly simple but as you work through how to apply them with increasing difficulty to some real-world problems then it does get you thinking.