The do's and dont's of teaching problem solving
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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 that kids don't like word problems just because they need find an answer to something (a problem), or because the problem is explained in words. Most of us even adults get fascinated by puzzles, for example.
Also, this fear of word problems surely can't 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 kids feeling it is difficult.
I feel the causes are probably many-fold:
- One-step word problems prevail in the end of lessons practicing a specific operation. These encourage kids 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 include the abovementioned one-step problems, and then some isolated problem solving lessons here and there, which usually highlight some problem-solving strategy.
- 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 in all elementary grades. Kids 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 it: if the lesson is about topic X, then the word problems are about the topic X too!
When kids are exposed to such lessons over and over again, they figure it 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 word problem, you might be able to do it. Just try: let's say that the following made-up problem is found within a long division lesson. Can you solve it?
La tienda tiene 873 sabanas en 9 colores diferentes. Hay la misma cantidad en cada color. Cuántas sabanas de cada color tiene la tienda?
Too much of those kind of simple problems soon brings a problem. Kids "learn" (intelligently) this unspoken rule, and use it to solve the problems.
How to avoid it? In the end of such calculation lessons, if you want word problems, mix them up so that not all are solved by the operation you just studied. Or, give students a bunch of short word problems for the purpose of NOT finding the answers but to find what 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.
Those one-step problems are good for 1st and 2nd grade, and then here and there mixed in with others. But kids need to start solving multi-step problems as soon as they can, in 1st and 2nd grade.
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's 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 kids should know multiplication well enough to quickly find that 6 and 7 fit the problem! Who in their right mind would ever 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 some some more difficult number and not 42. Doesn't using such simple problems in algebra books just encourage students to forget common sense and simple arithmetic?
(BTW, no matter what number you'd use ("Find two consecutive numbers whose product is 13,806"), I'd just take the square root and find the neighboring integers, and check.)
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 could draw a boy and a girl, and draw two pockets for the boy, and one pocket for the girl. This visual representation easily solves the problem.
Here's 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 too.)
Please see the resources section in the end of the article to locate sources 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" 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. We may go one route, notice it won't work, go backwards a bit, take another route.
In other words, devising plans and carrying them out can occur somewhat simultaneously, and we can go back and forth.
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 real example of real problem solving!
I've tried that a few times; see my thought processes at Proving is a process: proving a property of logarithms and Example of failed problem solving (on that occasion I had to give up because of time constraints).
What about problem solving strategies?
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.
The strategies or heuristics are of course very useful. However, I tend to dislike the problem solving lessons found in many school books that concentrate on one strategy at a time. You see, in such a lesson you'd have problems that would be solved with the given strategy, so it further accentuates the idea that solving word problems always follows some pre-established recipe.
I wonder if a better approach would be to instead solve good challenging problems weekly. 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 of course be 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'll 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!
Lastly a joke by Lynn Nordstrom:
Student's Misguide to Problem Solving
Rule 1: If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion.
Rule 2: Extract the numbers from the problem in the order they appear. Be on the watch for numbers written in words.
Rule 3: If rule 2 yields three or more numbers, the best bet is adding them together.
Rule 4: If there are only 2 numbers which are approximately the same size, then subtraction should give the best results.
Rule 5: If there are only two numbers and one is much smaller than the other, then divide if it goes evenly -- otherwise multiply.
Rule 6: If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.
Rule 7: If the rules 1-6 don't seem to work, make one last desperate attempt. Take the set of numbers found by rule 2 and perform about two pages of random operations using these numbers. You should circle about five or six answers on each page just in case one of them happens to be the answer. You might get some partial credit for trying hard.
I hope your students do not fit the above joke.
In my Math Mammoth books, I've tried to avoid problems that would lead children to the above scenario. I do not claim to be perfect in this; I feel I have lots to learn. But I will keep striving to make problems that do require many steps and that do not "dumb down" our children, but that progressively get more difficult as school years go by.
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.
George Polya, How to Solve It: A New Aspect of Mathematical Method. 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.
How to Solve Math Problems − advice by Denise at Let's Play Math blog.
Problem Solving Decks from North Carolina public schools
Includes a deck of problem cards for grades 1-8, student sheets, and solutions. Many of these problems are best solved with calculators. All of these problems lend themselves to students telling and writing about their thinking.
Math Stars Problem Solving Newsletter (grades 1-8) These newsletters are a fantastic, printable resource for problems solve and their solutions.
Open-Ended Math Problems
Collection of problems that lend themselves to more than one way of solving.
Math Kangaroo Problem Database
Easily make worksheets of challenging math problems based on actual past Math Kangaroo competition problems.
Learn to model or make visual representations of word problems with this interactive program.
Mathematics enrichment - nrich.maths.org
Open-ended, investigative math challenges for all levels.
Figure This! Math Challenges for Families
Word problems related to real life. They don't always have all the information but you have to estimate and think. For each problem, there is a hint, other related problems, and interesting trivia. Website supported by National Council of Teachers of Mathematics.
Learn to solve word problems
A collection of school algebra word problem solvers that solve your problems and help you understand the solutions. All problems are customizable, meaning that you can change all parameters.
word problems for Children - MathStories.com
Over 12,000 interactive and non-interactive NCTM compliant math word problems, available in both English and Spanish. Helps elementary and middle school children boost their math problem solving and critical-thinking skills. Need to pay for membership.
|Challenge Math For the Elementary and Middle School Student
There is a great lesson at the beginning followed by practice and then three levels of questions. The author has taken concepts that are generally saved for older kids (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.
"Problem of the Week" (POWs)
Problem of the week contests are excellent for finding challenging problems and for motivation. There exist several:
Problem of the Week at University of Waterloo
Each week, problems from various areas of mathematics will be posted here and e-mailed to teachers for use with their students in Grades 7/8, 9/10, and 11/12. In addition, there are hundreds of archived contest problems, math circle presentations and exercises.
Math challenges in a form of short stories for K-5 learners posted bi-weekly. Parent/Teacher Tips for the current challenge explains what kind of reasoning the problem requires and how to possibly help children solve it.
Grace Church School's ABACUS International Math Challenge
This is open to any child in three different age groups.
Primary Mathematics Challenging Word Problems
Books for grades 1-6 from Singapore Math. The books include answer key, worked examples, practice problems, and challenging problems. $8 per book.
Search for number, geometry, probability etc. word problems and challenges. Includes solutions.
Smart Skies - Distance Rate Time problems
Six Air Traffic Control (ATC) Problems, with downloadable curriculum materials and teacher guide. For grades 5-9.
Math League's Homeschool Contests
Challenge your children with the same interesting math contests used in schools. Contests for grades 4, 5, 6, 7, 8, Algebra Course 1, and High School are available in a non-competitive format for the homeschoolers. The goal is to encourage student interest and confidence in mathematics through solving worthwhile problems and build important critical thinking skills. By subscription only.