And while generally I agree with Danielson that many students find the most interesting problems to be ones drawn from their own real experience--not these contrived, puzzle-y types of problems--I think he's wrong in saying that problems like Vincent's have no place in the curriculum.The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2409. What are the two numbers?This is a classic word problem of the sort that I hope to eradicate from mathematics instruction (at least from my own).
Why might they be important?
Consider the following situation we considered in geometry yesterday:
Students quickly identified that all three triangles are similar to each other, but to justify the statement, they needed to "chase angles" around the figure. So we labeled the measure of angle A as x. What's the measure of angle ABF?
The algebra here is not hard, but it requires translating an idea--two angle measures add to 90 degrees--into an equation or expression. There are lots of such situations in geometry:
To find the exterior angle sum of the pentagon, one of the easiest ways starts by writing:
a' + a = b' + b = c' + c = d ' + d = e' + e = 180 degrees.
Or how about this early optimization problem: what's the largest area for a rectangle of a given perimeter, say 60?
So while I think it's true that many students find problems like Vincent's uninteresting, the underlying skill--and here, there's really no issue of transfer--is crucial to setting up and doing interesting math on other problems. I learned this the hard way this week: we spent twenty minutes chasing angles I had thought we would find in just a few.
So what can we do? Two ideas:
First, we can separate "problems involving translating words into symbols" from "application" and "real-life" problems. And when we pose problems in the first category, we can be honest about the fact that we're trying to develop a skill students will use in attacking other math problems, not preparing them to do something most people do everyday.
But second, I think we can do a better job of honoring the category of puzzles. My experience is that kids like puzzles when they know they're puzzles. For example:
Here's a puzzle: a 25-foot ladder is propped against a wall 7 feet away. The bottom of the latter slips back 8 feet. How far down does the top slip?
The Jain mathematician Mahāvīra poses the following problem: There were 63 equal piles of plantain fruit put together and 7 single fruits. They were divided evenly among 23 travelers. How many fruit were there altogether?
In England, there's a tradition of trying to ring all possible permutations (or "changes") of a church's bells, without duplicating any permutations. Unfortunately, because of each bell's mass and angular momentum, the only way to change the order is to switch two successive bells: so ABCDE can go to ABDCE or BACDE but not ACEBD. Is it possible to ring all the changes on five bells without duplicating any?Indeed, puzzles are an important mathematical tradition; as teachers of mathematics, it's up to us to help students appreciate this vital mathematical aesthetic. (Great ideas about our role as teachers of mathematical aesthetics are in Natalie Sinclair's paper here.) We teach that aesthetic by including puzzles, riddles, and other mathematical games in our curriculum--not just what students "need", in some sense, "to use". [Not Danielson's quote or exact idea, to be fair!]
I'll go even further: I think that it would be great if we could develop our students' mathematical eyes so that they can notice and appreciate cool number patterns and symmetries just as much as we'd like them to notice and appreciate symmetries in geometric forms and patterns. That is: our goal is not just that students can do math or that they know math, but that, at some level, they appreciate it.
A last note on puzzles and aesthetics. Vincent's problem is truly a terrible problem. But what makes it bad is not that it's contrived or abstract. What makes it bad is that it presents a "puzzle" with nothing interesting in it to notice. Who would possibly care about a sum of 2479? There's no symmetry or pattern. It's like a painting that is composed of two discrete forms, but with no balance, interesting details. Even a tiny tweak makes it better--like using the year, or an interesting number, like this:
The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2011. What are the two numbers?or this
Don and Leo are going to visit their friend Vincent, who lives in an apartment building where units are numbered consecutively on each floor. As a puzzle, Don writes down the sum of Vincent's apartment number and his neighbor's, using a scrap of paper he finds in his pocket. When Leo gets the paper, he asks Don "Which way should I read this?" but then says "Wait, it doesn't matter!" If Don's number has four digits, what's the smallest possible apartment number Vincent could have?And then you can ask interesting, mathematical questions:
Assuming you don't run out of apartments, is every four-digit number a possible sum of adjacent apartment numbers? [Is this puzzle possible every year?]
What is the sum of the numbers of the apartments on Vincent's and his neighbor's other sides?
A visitor notices that each apartment is directly across the hall from another apartment, and that each pair of across-the-hall apartments has the same sum. How would you number the apartments so that this happens?Challenge to readers: in the comments, add your own "apartment problems"!