Why are math word problems SO difficult for elementary school children?

Most children love stories, and even problems and puzzles. So why do they have such a hard time with math word problems? I feel the answer lies in the TYPES of word problems they solve in the very first years of school (grades 1-4).

These difficulties don't start in 1st grade with such easy story problems as: There are five ducks on the lake and three on the shore. How many ducks are there in total? Often the math book has a nice picture to accompany it. Instead, it is typically from grade 3 onward that that many students cannot apply math into even the most simplest situations presented in words.

I feel it all boils down to this "recipe" used in MULTITUDES of math lessons:

LESSON X

---

Explanations and examples.
Numerical exercises.
A few word problems.

Notice the following characteristics:

• The word problems are usually in the end of the lesson. Thus, if there is no time, they get omitted. Plus, since they are placed last in the lesson, it looks like they are the least important part... correct?
• Very important: have you ever noticed... If the lesson is about topic X, then the word problems are about the topic X too!
  
  For example, if the topic of the lesson is long division, then the word problems in the lesson are extremely likely to be solved by long division.
• Another common characteristic is that often the word problems only have TWO numbers in them. In other words, they are one-step problems. (One-step problems prevail in some curricula all the way through 7th grade!) So, even if you didn't understand a word in the word problem, you might be able to solve 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 sábanas en 9 colores diferentes. Hay la misma cantidad en cada color. ¿Cuántas sábanas de cada color tiene la tienda?

My thought is that over the years, when children are exposed to such lessons over and over again, they kind of figure out that it's mentally less demanding not to even read the problem 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 those simple routine problems will cause students to learn this unspoken "rule":

Yet 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 − 15. Then if the word problem doesn't follow a 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, none of these calculations gives you the answer: 17 − 30, 17 + 30, or 17 × 30.

What can be done?

• Occasionally, give students a bunch of short routine word problems, but instead of asking them to find the answers, ask them to find WHICH OPERATION(S) are needed to find the answers. The students' task is therefore to analyze the problems, as explained below.
• Most of the time, give students a mixed collection of word problems to solve, so that if the lesson is about operation X, only some of the word problems are solved by that operation.
• Let students solve non-routine word problems and generic brain teasers and puzzles, and devote some time to them. Most students will learn to LOVE puzzles and puzzling problems if given opportunity. Besides math textbooks, you can use this list of problem solving websites to find more good word problems.
  
  This will require an atmosphere where students are not afraid of making mistakes, though.

Analyzing elementary math word problems

Let students analyze word problems WITHOUT calculating the answers, so that they think and find which operation is needed to solve each problem. Here is a list of situations that are associated with certain operations:

• The total is divided into so many parts/containers, each part having the same amount.
  
  This is a multiplication/division situation:
  (number of parts) × (amount in each) = total
  
  
  • If you know how many parts and the amount 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.
  
  
• The total is divided into unequal groups.
  
  This is an 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. Here are some example word problems:

    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 in these situations. They are actually addition situations:

    color pictures + black-n-white pictures = all the pictures

    ripe pumpkins + non-ripe pumpkins = all the pumpkins

    Yet they are solved by subtraction.
  
  


• Something goes away or is taken away. This is the classic subtraction situation.
  Jenny had $14.56 and she bought a doll for $2.55. How much money is left?
   
• "How many more" situations (= difference) are solved by subtraction. Note that once again, nothing is "taken away" or going away.
  Joe has 24 stamps and Bill has 13. How many more does Joe have?.

The above situations cover the basics of how the four operations are used in word problems, which covers most types of word problems used in grades 1-4.

See also:

A list of websites focusing on word problems and problem solving

The Do's and Don'ts of teaching problem solving in math