# Mathematical Creativity: Multiple Solutions to the Pencil Sharpener Problem

I enjoy watching students exploring a problem that forces them to come up with their own structure for solving it. Today, a group got a chance to mess around with The Pencil Sharpener Problem, which is a problem I posted a month or so ago. (I’ll leave you to read it if you are curious what the problem is.)

From my perspective, what makes this problem interesting for the students is the ease with which it is communicated and the complexity with which is it solved. It seems quite easy. The answer is fairly predictable, but the students quickly found out that if they were going to solve this problem accurately, they were going to need two things:

1. A way to organize their thoughts and,

2. a way to verify their answer.

As long as the solution process included those two things, the students ended up fairly successful in the process of this problem.

I submit to you four examples of student solution structures. They all look different, but they all have one thing in common: The students could tell you with absolute certainty what was on the page and that they were right. (I withheld judgment from the correctness, but I will say that their final answers were all pretty much the same.)

This one includes a bit of a floor plan

In this first one, shown above, this pair of students decided to draw a floor plan with each of the pencil sharpeners (and the bucket they were tossing their leftover pencil nubs). The solution process progressed from there.

Guessing and Checking

These two students are classic minimalists. They worked to guess-and-check, which requires a bit more skill than it might seem. They needed to decided what number they were going to guess (sounds like an independent variable) and how to check their answer (sounds like a function producing a dependent variable). They weren’t using the vocabulary. It didn’t seem to be a problem for them.

Guessing and Checking

These two were a little more formal with their guess-and-check process, and they were using the terms independent and dependent variable. For the record, the time was independent and the number of pencils sharpened was the dependent variable.

A graph of multiple equations (where the solution does not include finding a single intersection point)

These two struggled a bit to convince themselves that they were on the right track as they completed this graph. They were discussing the need for the graph to be accurate (essential), how to interpret the three sets of points (one set to represent each pencil sharpener). Eventually, the were able to find the spot on the graph when the numbers added up properly. The number along the x-axis represented the final time. (Once again, they didn’t call time the independent variable, but it looks so nice along the x-axis, doesn’t it?)

Each of these students would have likely looked at the others with confusion, yet their answers largely agreed. What it required for them to solve it wasn’t me, as teacher, telling them how. On the contrary, it required me, as facilitator, designing a problem they would engage with and then giving them the resources to make sense of the problem with each other.

## One thought on “Mathematical Creativity: Multiple Solutions to the Pencil Sharpener Problem”

1. Reblogged this on nebusresearch and commented:
TheGeometryTeacher has here the four kinds of results gotten from a class given a word problem (about the time needed for a certain event to occur). I like not just the original problem but the different approaches taken to the answer. It seems to me often lost to students, or at least poorly communicated to them, that nearly any interesting problem can be solved several ways over. Probably that’s a reflection of wanting to teach the most efficient way to do any particular problem, so showing more than one approach is judged a waste of time unless the alternate approach is feeding some other class objective.

Given the problem myself, I’d be inclined toward what’s here labelled as the “guessing and checking” approach, as I find a little experimentation like that helps me get to understand the workings of the problem pretty well. If the problem is small enough this might be all that I need to get to the answer. If it’s not, then the experience I get from a couple guesses and seeing why they don’t work would guide me to a more rigorous answer and one that looks more like the graph depicted.
Guessing and checking gets little respect, probably because when you’re trying to train the ability to calculate like “what is eight times seven” it’s hard to distinguish informed guessing from a complete failure to try. (The correct answer is, of course, “nobody knows”; the sevens and eights times tables are beyond human understanding.)
But when you’re venturing into original work for which there may be no guidance what a correct answer is (or whether there is one), or when you’re trying to do something for fun like figure out “What are the odds my roller coaster car will get stuck at the top of a ride like Top Thrill Dragster?” guessing and correcting from that original guess are often effective starting points.