The Mathematical Discovery Process
Author: Jeff Denny, Summer 2004
Purpose: This extended case exposes students to the mathematical discovery process, the human side of mathematics research, and allows them to make a discovery in their group.
Materials:
- Students will need a simple calculator to perform the discovery exercise.
- Discovery exercise worksheet
- The Proof video
- The Proof discussion questions
- Introduction to modular arithmetic
- Article by Keith Devlin from the internet on the importance of equations (http://www.maa.org/devlin/devlin_12_02.html)
- Discussion questions for Devlin’s article
- Wikipedia article from the internet on Fermat’s Last Theorem (http://en.wikipedia.org/wiki/Fermat's_last_theorem)
Suggested Extended Case Schedule:
Day 0 – Assign the Wikipedia article to be read before watching the video.
Day 1 – Watch The Proof video.
Day 2 – Discuss the video and the discovery exercise preliminaries.
Day 3 – Do the discovery exercise, and assign the Devlin article to be read.
Day 4 – Spend more time on the discovery exercise if necessary, and discuss the Devlin article. (Optional – discuss academic lineage and assign students to use the mathematical genealogy website at http://www.genealogy.ams.org/to find the genealogies of some famous or local mathematicians.)
Details for each Class Day:
Day 1:
The video The Proof explores the road to Andrew Wiles’ proof of Fermat’s Last Theorem. This video was directed by Simon Singh who wrote Fermat’s Enigma, which one or two of your students may have read. Program notes, the transcript of the video, and other materials can be found at http://www.pbs.org/wgbh/nova/proof/
Hand out the discussion questions before the film and let the students know that you will be discussing them on the next class day.
Wiles’ effort is unique in ways that you should emphasize to your students.
- First, he worked entirely by himself for seven years on this problem. Although students may think this is typical of mathematics research, it is not. This opens an opportunity to point out that modern mathematicians do nearly all of their research in collaborations.
- Second, the result in Fermat’s Last Theorem does not provide any particular applications nor does it add to the world of mathematics theory; however, the effort to prove the Theorem has generated many, significant new mathematical ideas. Indeed, Wiles’ proof added a surprising number of new ideas to the mathematical literature.
- Third, pursuing the proof of Fermat’s Last Theorem was quite risky for Wiles. He was well known in his field, and his effort would have been quite closely watched and critiqued if he had worked publicly. Students will often wonder how Wiles was being paid while he worked for these seven years. Of course, he was supported on grants and did very little teaching, if any. Rumors existed that Wiles was close to losing his NSF support for his other research; indeed, he published other work relatively slowly while he was working on Fermat’s Last Theorem.
One interesting sidelight in the video that often fascinates
students is the seminar that Wiles offers on “Elliptic Curves.” The students that attend just wander off
after a few weeks. How can they do
that? This is a good time to point out
the nature of a research seminar and that
Day 2
Remind students to bring a calculator for Day 3.
Finish the video if necessary and discuss the questions on the handout. In the time remaining, you will want to begin introducing the concepts of modular arithmetic, which will be necessary for the discovery exercise. An explanation of modular arithmetic is attached. You will want to present these ideas carefully and have the students work a few examples. The ideas are simple (really just division and remainders), and students usually pick them up quite quickly. (My presentation of this material is not formal in any way for these students, and I emphasize the “clock arithmetic” idea heavily.)
Day 3
Remind students of the basics of modular arithmetic by asking them to compute a few examples. See the modular arithmetic explanation for a few suggested examples.
Hand out the “A mathematical discovery” worksheet to your students and have them work on making their own discovery. They hopefully will uncover a theorem from number theory known as “Fermat’s Little Theorem.” This theorem states that
,
where n is a prime number. The key that the students must find is that a neat pattern emerges when the exponent is prime. To find this pattern, the students must compute a large number of examples and record them on the worksheet. Encourage the students to compare examples in the chart in which n is prime and n is not prime. See the Introduction to Modular Arithmetic for a sample of a completed worksheet.
Ask the groups to select a leader to direct the research and
to assign examples to each group member to compute. The worksheet hints that the examples are
easier to work with if 
. It is helpful to
suggest that the students choose relatively small (less than 20, say) values of
n to start. For some reason, students often try to start
with huge values for n, which the
students cannot recognize as prime or not.
This activity should let the students experience the feeling of “walking blind in a room” that Wiles describes in the video. Be sure to tell them that the frustration, confusion, and “blind” feeling is an important part of the exercise. Also, point out that mathematicians spend a lot of time observing mathematical phenomena just as the students are doing as they generate examples.
Finally, ask students to make a hypothesis (or conjecture)
based on their examples. Some may see
the theorem quickly, but others will likely need some prodding. Also, be aware that some students may
discover number theory facts other than Fermat’s Little Theorem. For example, 
, and if a is odd, 
.
There are lots of patterns for the students to discover. Encourage and praise the students for whatever patterns they find, and then push them in the direction of Fermat’s Little Theorem.
Day 4
Allow the students to complete the discovery exercise if necessary. Discuss the questions for Devlin’s article.