Dragons, Gryphons, and Math Mistakes

This post is adapted from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It.

Do you ever take your kids’ math tests? It helps me remember what it is like to be a student. I push myself to work quickly, trying to finish in about 1/3 the allotted time, to mimic the pressure students feel. And whenever I do this, I find myself prone to the same stupid mistakes that students make.

Even teachers are human.

In this case, it was a multi-step word problem, a barrage of information to stumble through. In the middle of it all sat this statement:

“…and there were 3/4 as many dragons as gryphons…”

My eyes saw the words, but my mind heard it this way:

“…and 3/4 of them were dragons…”

What do you think — did I get the answer right? Of course not! Every little word in a math problem is important, and misreading even the smallest word can lead a student astray. My mental glitch encompassed several words, and my final tally of mythological creatures was correspondingly screwy.

But here is the more important question: Can you explain the difference between these two statements?

If Johnny Can’t Read, Then He Can’t Do Math

Few people realize that a math test is also a test of reading comprehension. In some cases the reading level required far exceeds the level of math.

Before they can solve a word problem, children must be able to understand what is written. They must hold that information in their working memory as they think about how to find a solution. They need to comprehend what they read, paraphrase it—concentrating on the relevant facts—and translate that information into a mathematical expression. Frequently they have to read between the lines and understand something that is implied, not explicitly stated.

Paraphrasing is one of the most important skills we can teach junior high and high school students. Often they want to rush into interpreting and reacting to a text even before they know what it means.

We teachers sometimes suffer from the delusion that since a student can read the words on the page, he or she understands what’s been read. But that’s not always true.

—Nick Senger, Quotes about Reading for Students to Paraphrase

That quote is from an article at Teen Literacy Tips blog. Does a literature teacher have anything useful to say about solving math problems? Well, the fact that word problems are also called story problems should clue us in to a significant connection.

When students struggle with word problems, more often than not it is a language issue that confuses them.

Consider, for example, the maddening confusion that may be caused by these two simple statements:

• Eight divided in half is four.
• Eight divided by one-half is sixteen.

These sentences seem to contradict each other, yet both are true. Can you explain the difference? If your students keep a math journal, you might use this question as a writing prompt.

(I’ve included one possible answer at the bottom of this post.)

My Problem with Dragons and Gryphons

In my word problem, it turned out there were fifty-six fantasy creatures in all. I got that part of the answer, but then I needed to find the number of gryphons.

This is how I did it…

“… and 3/4 of them were dragons …”

Imagine all fifty-six creatures in a zoo, split evenly into four pens. Three of the pens hold various breeds of dragon, and the fourth pen has the gryphons.

4 pens = 56 creatures

1 pen = 56 ÷ 4 = 14 gryphons

But that was not at all what the problem said. If I had been paying better attention to what I read, this is how I should have solved the problem…

“… and there were 3/4 as many dragons as gryphons …”

This zoo is wiser than the first one, because it splits the troublesome creatures into smaller groups, so there’s less chance of a fight that might endanger visitors. The gryphons are split into four pens, so each pen holds one-fourth of the gryphons. The dragons have three pens all to themselves, which makes three-fourths as many as the gryphons. Thus, we have a total of seven fenced-in areas.

7 pens = 56 creatures

1 pen = 56 ÷ 7 = 8 creatures per pen

4 pens = 4 × 8 = 32 gryphons

Math Journal Prompt

To make the reading issue more difficult, consider this: All the statements in the following list are equivalent. Compare each statement to the second diagram above, the correct one. Can you see each relationship?

• There are 25% fewer dragons than gryphons.
• For every four gryphons, there are three dragons.
• The ratio of dragons to gryphons is 3:4.
• Four out of every seven creatures is a gryphon.
• There are one-third more gryphons than dragons.
• There are one-fourth fewer dragons than gryphons.
• If you tag a creature at random from the group, the probability of choosing a dragon is three-sevenths.

Can you think of any other ways to say it? This would be another good math journal writing prompt.

“As important as mathematics is, it is a distant second to the need for good reading comprehension. We teachers so often hear students summarize a course by saying, ‘I could do everything except the word problems.’

Sadly, in the textbook of life, there are only word problems.”

—Herb Gross,
quoted by Jerome Dancis in Reading Instruction for Arithmetic Word Problems

The more we can work with our children on reading, paraphrasing, and translating word problems into mathematical expressions, the better prepared they will be to face the challenges they will meet in future math classes and in the textbook of life.

One Possible Answer to Dividing in/by Half

When you divide a number in half, you split it into two equal parts.

But when you divide a number by 1/2, you are finding how many halves it takes to make that number — that is, you are cutting it into half-size pieces and counting how many there are.

And in that case, because each whole thing is two halves, there will be twice as many pieces as the number you started with.