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The problem with story problems in elementary mathematics

WHY do so many kids feel that word problems or story problems are difficult? You know, most kids love stories and words, and even problems and puzzles. So what is the problem with word problems?

It surely can't start on 1st grade when you might have a story problem such as: There are five ducks on the lake and three on the shore. How many ducks are there total? Often the math book has a nice picture to accompany it. Surely kids don't think that as being difficult.

Consider this "recipe" of a typical math book lesson:

LESSON X
---------------------
Explanation and examples.
Numerical exercises.
A few word problems.

So the word problems are usually in the end of the lesson. AND, have you noticed... If the lesson is about topic X, then the word problems are about the topic X too!

For example, if the topic in the lesson is long division, then the word problems found in the lesson are extremely likely to be solved by long division.

And, typically the word problems 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 870 sabanas en 9 colores diferentes. Hay la misma cantidad en cada color. Cuantos sabanas de cada color tiene la tienda?

My thought is that over the years, when kids are exposed to such lessons over and over again, they kind of figure it out that it's mentally less demanding just 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.

I'm not saying that such word problems are not needed in the end of division lessons. I'm sure they have their place. But too much of those simple 'routine' problems can be a problem... Students may "learn" a rule: "Word problems found in math books are solved by some routine or rule that you find in the beginning of the corresponding lesson."

Another difficulty is that students tend to think linearly, step-by-step, and try make the numbers and the text match in the same order. For example, Jane had 25 pens and she gave away 15. How many does she have now? Answer: 25 - 5.

Then, when the word problems they encounter don't anymore follow any step-by-step recipe, they are lost. For example: After giving away some cards, Jane now has 17 cards left of her original 30. How many cards did she give away? This time, 17-30 or 17 + 30 or 17 × 30 does not give you the answer.

What you can do

• Occasionally, give students a bunch of short routine word problems, but this time DON'T ask them to find the answer. Instead, ask them to tell what operation(s) are needed to find the answer. Analyze the situations, as explained below.
• Have separate lessons with mixed word problems, including some non-routine, and devote some time to them.

Analyzing elementary math word problems

After studying a bunch of word problems WITHOUT calculating the answers but only thinking and finding which operation is needed to solve each problem, the student should start associating the types of situations with the appropriate operations:

• Total is divided into so many parts/containers, each part having same amount.
  
  This is the multiplication/division situation:
  (number of parts) × (amount in each) = total
  
  
  • If you know how many parts and how much in each, MULTIPLY.
  • If you know the total and the number of parts, DIVIDE.
  • If you know the total and the amount in each, DIVIDE.
  
  
• Total is divided into unequal groups.
  
  This is the addition/subtraction situation:
  (amount in group 1) + (amount in group 2) + (amount in group 3) + etc. = total.
  
  
  • If you know the amounts in groups but not the total, ADD.
  • If you know the total and the amounts in all but one group, SUBTRACT. This is the opposite of addition.

    Of 187 pictures, 45 were black-and-white. How many were color pictures?
    There were 57 pumpkins and 15 of them were ripe. How many were not ripe?

    Notice that NOTHING is 'going away' or being 'taken away'. They are typical "addition situations":
    
    color pictures + black-white pictures = total pictures
    ripe pumpkins + non-ripe pumpkins = all pumpkins
    They are solved by subtracting because the total is already known and in essence we're trying to find the missing addend.
  
  

Then there are some other subtraction situations:


• You know what the total used to be, but part of it went away or got used - the easy, classic 'take away situation'.
  Jenny had $14.56 and she bought a doll for $2.55. How much money is left?
   
• How many more (= difference)
  Joe has 24 stamps and Bill has 13. How many more does Joe have?
  Note nothing is 'taken away'.

I would also devote extra attention to problems involving time.

I'm not claiming this is a complete coverage of all the possible situations that are solved by single operation, but it can help if these ideas are explicitly taught to the students.

Resources

Word problems for kids
A great selection of word problems for grades 5-12. A hint and a complete solution available for each problem.

Aunty Math
Math challenges in a form of short stories for K-5 learners posted bi-weekly.

Problem of the Week home page
Links to 'problem of the week' websites organized by grade levels. These are excellent for finding more challenging problems and to motivate.

Primary Mathematics Challenging Word Problems
For grades 1-6 from Singapore Math. The books include answer key, worked examples, practice problems, and challenging problems. About $8 per book.

I have some more word problem resources listed here.