*[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.

— 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.

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 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.**

and

**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:

…and 3/4 of them were dragons…

4 units = 56

1 unit = 56 ÷ 4 = 14 gryphons

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:

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

7 units = 56

1 unit = 56 ÷ 7 = 8

4 units = 8 x 4 = 32 gryphons

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.

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How timely! My students are doing word problems right now and converting from prose to mathematical expressions is challenging. I’ll watch out for tricky wording like you pointed out.

These things are hard to read. Even for good readers. I slow kids down, it probably helps.

I was just working on this (again) with my pre-algebra students. They admitted they just skip the word problems – especially on standardized tests. Our new goal is not to be “tricked” by them anymore.

I love your quotes – especially the one by Herb Gross.

sorry – somehow your comment was marked as spam – not anymore!I started out tutoring algebra, then switched to remedial reading using phonics. Once you get into higher math where you need to be able to read the explanations and the word problems, reading is important to math. Without the ability to read well, you’ll never excel in your other subjects.

Eight divided in half is four.

Eight divided by (one-)half is sixteen.

Okay.

Eight divided in two is four.

Eight divided by two is four.

(brain explodes)

LOL!

I did it again this week. My son gets a laugh out of my mistakes on his homework, especially when

hegot the problem right. Last time, I ignored the word “more” in a MathCounts problem, and this week I missed the word “additional.” You’d think I would have learned by now…As for comments, Jackie, I have given up on rescuing them from the spam folder. I have been getting way over 100 spam a day, and I just don’t have that much time to sort them. But I did fix your blog link for you!

I teach a lot of literacy in my class, even when it seems it’s weird to do so in a math class. It helps when kids actually read with me the problems they have to do. That’ll be especially important for a year in which I have a class full of ELLs. In any case, good post.

Very good post. This is where we had many fun discussion in college about the validity of tests and knowing what you are really testing.

But the problem is, if we want to prepare our children for life, they need this kind of reasoning as well.

You have some good examples here about the critical importance of knowing how to translate from words to math. I also liked the quote about the paraphrasing.

Of course, I like those because my book, “Solving Word Problems,” explains exactly these things (and more)!

The dragons-gryphons sentence is actually a very difficult one. Your mistake was a simple one but the sentence structure “3/4 as many dragons as gryphons” requires either an ability to manipulate the two objects, dragons and gryphons, in one’s head and understand that dragons = 3/4 grypohns, or to know how to rephrase it to an easy to understand sentence. The first is EXTREMELY hard to do and most students will write 3/4dragons = gryphons. The second can be easily taught! [See my book of course LOL].

To the one with the exploding brain: you can’t divide eight in two. You can only use this grammatical structure for fractions. It took me a minute to understand that this is the issue, but maybe I noticed it because English is not my first language so I’m more sensitive to the translation issue

Hi.

This is Herb Gross and I am now putting toglether all of my arithmetic and algebra materials (including textbooks, videos and slide shows) on my website for anyone ot use free of charge. The website is just temporary and in a short time it will be made more user-friendly in terms of being able to access items quickly.

Please feel free tot use the material on my site (www.adjectivenounmath.com) in any ways that you wish. You may email me at hgross3@comcast.net

PS

Please excuse any typos. At age 81 the small prnt is my nemesis.