[Photo by Betsssssy.]
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
To solve word problems, students must be able to read and understand what is written, and they must be able to follow directions. They need to comprehend what they read — to paraphrase it, concentrating on the relevant facts — and then to translate that information into a mathematical expression. Many times, they must be able to “read between the lines” and understand something that is implied, not explicitly stated.
When students struggle with word problems, more often than not it is a language issue that confuses them.
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.
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.
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.
[The entire article by Dancis is worth reading, and you may want to explore the rest of his webpage as well. I will be using Supposedly Difficult Arithmetic Word Problems as ratio practice with my MathCounts students later this semester.]
For a simple (yet often confusing) example, consider these two statements. Can you explain the difference?
- Eight divided in half is four.
- Eight divided by one-half is sixteen.
If your students keep a Math Journal, this would be a great writing prompt. An answer is given at bottom of this post.
Now, Let’s Analyze My Mistake
In my word problem, it turned out there were 56 creatures in all. I got that part of the answer just fine, but then I needed to know how many of those creatures were gryphons.
This is how I did it:
But that was not at all what the problem said. There should have been several more gryphons than dragons. If I had been paying better attention to what I read, this is how I should have solved the problem:
Just to make the language issue more difficult, consider this: All of the following statements are equivalent. Compare each statement to the second drawing above (the correct one). Can you see each relationship?
- There are 3/4 as many dragons as gryphons.
- For every 4 gryphons, there are 3 dragons.
- The ratio of dragons to gryphons is 3:4.
- 4 out of every 7 creatures is a gryphon.
- There are 1/3 more gryphons than dragons.
- There are 25% fewer dragons than gryphons.
- If you tag a creature at random from the group, the probability of choosing a dragon is 3/7.
Can you think of any other ways to say it? This would be another good math journal writing prompt.
Ratio problems like this are some of the most confusing word problems our pre-algebra students will face. The more we can work with them on reading, paraphrasing, and translating these problems into mathematical expressions, the better prepared our students will be to face the word problems they meet in “the textbook of life.”
[Edited to add: This problem follows students beyond middle school. Jackie is struggling to get her high school math students to read carefully. See her post Mis-Reading in Mathematics (and the comments section).]
One Possible Answer to the Question About Dividing by 1/2
When you divide a number in half, you split it into two equal parts. But if 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 double as many pieces as the number you started with.