Math expressions are words, too

Writers often will say they are word people and not math people. And this defeatist pronouncement too often is reflected in their use of simple math. But mathematical expressions are words. They have specific meanings, commonly understood but often ignored in practice.

Most of us have the basic knowledge that allows to check to make sure the mathematical expressions we use hold true in context. If something is “twice as much” as something else, then half of that something also is something else. Right? If I have three apples and you have twice as many, you have six apples. If you give me half your apples and end up with four, we’ve screwed something up.

Don’t ignore the common-sense advice drilled into our heads in high school: Check your work.

This morning’s paper brought a story with more complex math than an exchange of apples, but nothing any of us can’t do in our heads. No calculator was used in the preparation of this blog.

The state of Ohio is capping the amount that insurance providers can charge people who are considered high-risk, such as those with diabetes or other medical conditions. The cost of high-risk plans are limited to twice the cost of the lowest-cost plan for people of the same sex and similar age. Some companies, the story tells us, charged four times the lowest rate, and some people paid $1,000 a month.

Fine so far. But the story also says “The cap is expected to reduce premiums by 50 percent to 70 percent.”

Now, let’s check the math. Starting with that poor fellow paying $1,000 a month, his fee is up to four times the base rate if we use the extremes. So the base rate is no more than $250, and reducing his fee to twice the base rate would have him paying $500 a month.

That assumes no increase in the base rate – a silly assumption perhaps, but we are solving for the extreme here. So, best case scenario, the $1,000-a-month fee is reduced to $500. We don’t need a calculator to know that $500 is 50 percent of $1,000.

Stated more simply, If we reduce something from four times as much to twice as much, we have reduced it 50 percent. No more. Change any of the assumptions, and the savings are less. Nothing we do will get us to 70 percent.

There is one other variable, but it requires that we assume a mistake in the story. The cap gradually decreases and reaches 1½ times the lowest-price premium in 2013. Assuming no change in the lowest-price premium, our $1,000 would be reduced to $375, a reduction of 62.5 percent. That’s short of 70 percent and still represents the best-case scenario. And the story never said those savings would come in 2013 instead of 2010.

Perhaps there is something more in the legislation that will increase savings, making all the assumptions and the quoted 50 percent to 70 percent savings correct. But based on what we’re given, the best anyone can hope for is 50 percent, and most consumers should expect less.

When we see expressions such as “twice as much” and “50 percent of,” we should pull the numbers out of our story and make sure they work together. Surprisingly often, they end up disagreeing.


2 thoughts on “Math expressions are words, too

  1. Great post, Mark! I wish mathematicians themselves would spread the word (beyond the “mathosphere”). One of my favorites along these lines is Nancy Rodgers’s book, Learning to Reason: An Introduction to Logic, Sets, and Relations. She’s wonderful when she translates mathematical symbols into words, and even into parts of speech, for instance verbs or verb phrases. Of the = sign, for instance, she states, “The most frequently used verb in mathematics is ‘equals’ .” (p. 7) (She translates many other symbols into parts of speech also, but the symbols are too complex to type here.)

    Of course, the grammar behind the symbols becomes clear as soon as we think or, especially speak the expressions – if the symbols allow spoken expression – but Rodgers brings mathematical grammar (I’m sure we can call it that), to the forefront.

    Along the same lines, a wonderful little book that explains how much mathematics actually infuses our everyday life is this one (which was written for aspiring mathematicians): Ian Stewart’s Letters to a Young Mathematician.

    Thanks for such an interesting post.


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