On Collisions: Math and Colloquial Speech

So I was remembering this time in, say, fourth grade when the teacher posed the following question:

How many numbers are between ten and twenty?

Of course, this problem is well within the skills of a fourth-grader. Simply subtract ten from twenty and you get ten. In fact, the only appreciable difficulty is in the interpretation of the question. Students, particularly children, struggle with these so-called “word-problems”: problems where the math is relatively simple, but the problem is phrased as a question. You know, using words and stuff.

Being who I am, I completely failed at understanding the question; I answered, “Eleven.”

So I got laughed at… But to this day I maintain that 11 is just as accurate as—indeed more accurate than—10. Why? It all lies in the interpretation of the word “between.” If “between 10 and 20″ means

Then yes, there are 10 such numbers: . But who the hell uses “between” to mean “including the upper limit but excluding the lower limit”? In my opinion, there are only two reasonable ways to interpret “between ten and twenty”:

1. 
2. 

But then the correct answer is either or . I chose option (1) in fourth grade, but option (2) is perfectly reasonable.

The problem is that this becomes no longer strictly a subtraction problem. What the teacher wants is for the students to compute . But this is wrong, so how do we reinterpret the question?

I propose the following: first consider the question, “what are the numbers between ten and twenty?” This question still has the ambiguity of the word “between,” but no person in his right mind would answer:

or

Then we can ask, after the set of numbers has been identified, “how many of them are there?” I think if the problem were posed this way, no one would ever arrive at an answer of eleven (unless they miscount).

Further Rambling

I’ve decided that word problems like the ones we’re given in grade school don’t really count as “math.” I think they’re physics. Consider the topics covered in college- and higher-level mathematics. They’re abstract and almost completely disconnected from the real world. A mathematician doesn’t care if we live in Euclidean space or a Minkowski space; if they can prove something about both, then that is interesting.

A physicist (or statistician maybe) concerns himself with questions like these word problems—problems dealing with possible real-world scenarios. When we answer such questions, we don’t really learn a lot about math; rather, we learn about the connection between math and the world we live in.

That’s not to say word problems aren’t good or helpful, but I think school teachers should be very careful both in posing questions and in receiving answers. What if my teacher had let me explain my logic? It would have been nice to shut up the kids who laughed at me for a “stupid” answer, and they might learn something at the same time.