To explore the nature of logic, evidence, and proofs in the context of mathematics.
In this lesson, students will explore the nature of mathematical proofs and mathematical inquiry. This lesson would be appropriate after students are familiar with the Pythagorean Theorem. The nature of logic and evidence are topics that should come up frequently in science, history, social studies, and mathematics.
Research on students' development of the ability to construct proofs is somewhat conflicting. Piagetian research suggests that students can reason deductively from any assumptions once they reach the formal operational stage (around age 12 and beyond). Other research suggests that the ability to construct proofs depends on the amount and organization of particular knowledge they have. Still other research suggests that students may need to understand the nature of proof and how it differs from everyday argumentation before they are able to construct proofs.
Play the Pythagorean Puzzle on The Proof page, part of the NOVA Online website. A Shockwave and non-Shockwave version are available.
Have students do the part of this activity that demonstrates the Pythagorean Puzzle. They will do the part of the activity called "Using the Pythagorean Theorem" later in the lesson.
If necessary, refresh student understanding of The Pythagorean Theorem using this page from the Math Forum website.
Then ask students to read the interview of Solving Fermat: Andrew Wiles on The Proof website. If available, view the NOVA program on which the site is based.
Also, have students read Math's Hidden Woman, the true story of Sophie Germain, an 18th-century woman who assumed a man's identity in order to attempt to prove Fermat's Last Theorem.
When students have finished reading the selections, discuss the following questions:
- What is the difficulty in proving Fermat's Theorem? (Because there are an infinite number of equations, and an infinite number of possible values for x, y, and z, the proof has to prove that no solutions exist within this infinity of infinities.)
- Why did Wiles and Germain devote so much of their time to trying to solve this mathematical puzzle? Was there a practical application to solving the theorem? (The discussion should emphasize the non-practical aspects of this particular endeavor. Both of these stories emphasize that both Wiles and Germain were motivated by the intellectual challenge of a purely abstract problem.)
- Does Wiles believe that his proof is the same as Fermat's would have been? Does it matter? Does Wiles believe that there is only one way to prove the theorem? (Wiles does not believe that he found the same solution that Fermat had purported to find because he used techniques that were not available in Fermat's time. In fact, he believes that Fermat's probable proof was likely flawed.)
- How do non-mathematical proofs, such as in law or mystery novels, compare to mathematical proofs?
Follow the discussion by having students use the webpage Proofs in Mathematics from the Interactive Mathematics Miscellany and Puzzles for guided, firsthand experience in solving mathematical proofs. Working in small groups, students should select and explore one of the problems on this page and share the "proof" with the class.
Use the activities on the Using the Pythagorean Puzzle page to assess student understanding of the applications of Pythagorean Theorem.
Explore the Things Impossible page on the Interactive Mathematics Miscellany and Puzzles website.
Famous Problems in the History of Mathematics is a Math Forum resource, which includes problems that are suitable for middle-school and high-school math students, with links to solutions, as well as links to mathematicians' biographies and other math history sites.