The word on arithmetic

When we think of genre, it is often in the sense of literature or film. However, rhetoricians will tell us that genre is a concept that includes any sort of writing that has well-defined conventions, such as business memos, grant proposals, obituaries, syllabi, and much more. Even such workaday genres present complexities, pitfalls, and opportunities for refinement.

So consider the humble math word problem, a genre you may not have thought much about since high school. Word problems are the stuff of educators’ attempts to fit arithmetic and mathematics to real world situations. The concept goes back to ancient times and Babylonian math exercises connected to engineering, agriculture, and business.

Some of today’s problems are pretty elementary: If Sue has five dollars and spends three of them, how many dollars does she have left? But even for very simple arithmetic, there is a vocabulary of addition and subtraction; you translate the question into the rudimentary five minus three by knowing that spend means subtract.

But there are other ways to express subtraction.

If Sue has five dollars and gives three of them to Rick, how many will she still have?

The problem requires someone to know that giving something away indicates subtraction. But give can also indicate addition:

If Sue has five dollars and Rick gives her three more, how much money will she have?

An elementary school problem-solver needs to understand the difference between being the subject of giving or the object. More generally, they need to gain proficiency with genre clues, like “in total” or “in all” for addition, like “have left,” “more than,” “less than” for subtraction, and “each,” “times,” “doubled” for multiplication.

A number of studies have shown that providing good verbal clues improves children’s performance and allows them to get right to the arithmetic. Even a little specificity can help. Which of these do you suppose is easier for a first-grader?

There are five marbles. Two of them belong to Mary. How many belong to John?

There are five marbles. Two of them belong to Mary. The rest belongs to John. How many belong to John?

A 1991 study by Denise Dellarosa Cummins found that the second was easier because it contained more specific information about how many marbles John had. And the mean proportional accuracy increased from 30% for the first problem to 85% for the second. 

Sometimes too, inconsistency can trip up a word problem. Take this one, adapted from a study by Anton Boonen and colleagues: “At the grocery store, a bottle of olive oil costs $7. That is $2 more than at the supermarket. How much will a bottle of olive oil cost in the supermarket?” Here the use of “more” may prompt an addition strategy rather than a subtraction strategy, giving the wrong answer of $9 rather than the correct $5.

Like any genre, word problems get more sophisticated with age. Some require translation into mathematical concepts plus a knowledge of mathematical problem-solving tricks. How about this one:

Two cars are 300 miles apart on Interstate 80, which has a speed limit of 65 miles per hour. One is driving 10 miles per hour faster than the other and they pass each other after three hours. How fast is each car going?

Here you need to know the formula distance = speed time. And you need to know the trick that the combined speed can be expressed as the speed of the slower car plus 10. In other “words”, 300 miles = 3 hours (slower speed + slower speed + 10). And that means that the speeds are 45 and 55 mph. The information about the highway and the speed limit are irrelevant, but if you follow the genre convention that all the numbers in a problem are important, it’s possible to get distracted.

If you can see through the distractors, remember the formula, and realize that the combined rate is 100 mph, the problem is not too hard.

Sometimes, the expectations of the genre can entice children to look for answers that are not there. Yet another study gave 97 first and second-graders this problem:

There are 26 sheep and 10 goats on a ship. How old is the captain?

Seventy-six students used the numbers to “solve” the problem. The expectation that there should be a solution and that you should do something with the numbers drove young learners to find one.

When students struggle with word problems, they are often struggling with wording. Adults too, perhaps.

_{Featured image by Roman Mager on Unsplash}