For this month’s column, I am going to share one of my
favorite student activities, one based on the MAA video Let Us Teach Guessing
by George Pólya [1]. It is an activity that I use
in our Discrete Math course, a first-year course required for Math and for
Computer Science majors. The intention of the course is to give students an
introduction to interesting mathematics, to get them solving problems, and
to give them practice in analyzing and creating proofs. The course attracts
a significant number of non-majors and is one of our most successful recruiting
tools. As I teach this course, it revolves around four major projects. These
change from year to year, but I always include the Pólya project because
it is so successful.
In Let Us Teach Guessing, Pólya uses the problem
of determining the number of regions in a space cut by n planes.
He waits until he is pushed by the students, and then clarifies the problem
by stating that the planes are to be taken as random (thus, no two are parallel,
no three meet on a single line, no four at a single point). I have found it
easier to describe the problem as finding the greatest number of regions that
can be formed by n planes.
Pólya begins by getting the students to guess how many
regions are formed by five planes. Students get into this. Every time I use
this activity and pause the video, students eagerly make guesses that usually
range from 10 to 32. Pólya’s point is that this is wild guessing.
It may be fun, and you might even get the right number, but you do not learn
anything.
He now leads the students through more careful guessing. One
plane creates two regions, two planes give four regions, three planes give
eight regions. I now pause for my own students who almost always agree that
four planes should give 16 regions. This is still a guess, but it is now an
educated guess, based on an observed pattern. The bulk of the video is now
spent testing this guess, counting the number of regions formed by four planes.
Pólya builds up geometric intuition by looking at a line cut by points,
then a plane cut by lines, and finally space cut by planes. And, of course,
he leads the students to discover that four planes create 15 regions, not
16.
The point that Pólya is making, one that I make very
explicit for my students, is that if you make an educated guess and it is
correct, that is fine and it may indicate that you understand what is happening.
But if you make an educated guess and it is wrong, then something interesting
is happening. If you can analyze the situation to understand why your guess
was wrong, then you are going to learn something. Educated guesses are fine,
but they need to be tested so that you do not miss an opportunity to discover
something new.
At this point, the process of building intuition has created
a table of the number of pieces of a line, plane, or 3-dimensional space cut
by n objects. The table is very suggestive.
n |
line |
plane |
space |
0 |
1 |
1 |
1 |
1 |
2 |
2 |
2 |
2 |
3 |
4 |
4 |
3 |
4 |
7 |
8 |
4 |
5 |
11 |
15 |
5 |
6 |
16 |
|
I always have several students who quickly pick up that the
number of regions formed by 5 planes equals the number of regions formed by
four planes plus the number of areas on a plane cut by four lines, and that
this pattern should continue. The number of regions formed by five planes
should be 15 + 11 = 26.
This is where the Pólya video ends, but for my class
it is just the beginning. One can check that the answer is correct for five
planes, but the real question now is whether the perceived pattern is valid.
Can these students justify the claim that the number of regions formed by
n planes equals the number of regions formed by n–1
planes plus the number of areas on a plane cut by n–1 lines?
Before setting them this task, I get them into small groups
[2] and I give them an additional challenge, to find
a closed formula for the number of regions formed by n planes. I
give them a hint. I ask them to compare their numbers to those in Pascal’s
triangle. It does not take them long to discover the formula ( n
choose 0 + n choose 1 + n choose 2 + n choose 3
) and to guess what happens as you move into higher dimensions. But now they
have to prove that this formula is correct. I require an inductive proof.
The key to the proof is justifying the inductive step. This
is where they have the most trouble. Some groups see it. Some groups make
appointments to get help from me. For some groups, it is not until after they
have turned in an unsatisfactory first draft and I have insisted that they
come to talk with me about it that they put it together.
But what I love most about this problem is the many different
directions in which we as a class or individual students can take it: What
does it mean to cut 4-dimensional space by 3-dimensional hyperplanes, or k-dimensional
space by k–1-dimensional hyperplanes? Why would anyone care?
(What a great lead-in to the problem of linear programming.) What happens
if you count just the bounded regions, or just the unbounded regions? What
is being counted when you first choose no planes, then one plane, then two
planes, then three planes, and why does this give the total number of regions?
What happens if you restrict the planes (see [3])?
You can find a worksheet that I have used for my class at www.macalester.edu/~bressoud/talks/mathfest2005/polya.pdf.
I like this project because it introduces my students to a real and very great
mathematician. I like it because it demonstrates the importance of guessing
as well as how much can be learned from guesses that are wrong. But most of
all, I like it because these students get meaningful hands-on experience with
the challenge of constructing an inductive proof (for k-dimensional
spaces, the proof requires double induction) of a result that they consider
interesting and that they have had to work to discover.
[1] Let Us Teach Guessing, George Pólya.
1966. Washington, DC: MAA Video.
[2] I assign the students into groups of four each. They
begin work on the project together. I also assign pairs of students within
each group who write up the project as a team. Most of the time, this works
well. I do allow students to re-distribute the grade they receive for the
project depending on the amount of work each has done. Thus, with a grade
of B+ = 88%, they can distribute the 176 points as they see fit. I have never
had a team not be able to come to agreement over how to split the points,
but I do let them know that if they cannot agree, then I will hear each side
and will be the final arbiter.
[3] Jean Pedersen, Platonic Divisions of Space, pages
117–133 in Mathematical Adventures for Students and Amateurs,
David F. Hayes and Tatiana Shubin, eds., 2004. Washington, DC: MAA.
David Bressoud
is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul,
Minnesota, and president-elect of the MAA. You can reach him at bressoud@macalester.edu.
This column does not reflect an official position of the MAA.