NOT JUST ANOTHER THEOREM: A CULTURAL AND HISTORICAL EVENTAuthor(s): Michael McDanielSource: The Mathematics Teacher, Vol. 96, No. 4 (APRIL 2003), pp. 282-284Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871311 .

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3 H AR ING TEACHING IDEAS

NOT JUST ANOTHER THEOREM: A CULTURAL AND HISTORICAL EVENT

Mathematics

teachers can

impress these future

doctors,

lawyers,

politicians and

journalists with some

jewels of mathematics

The typical student rarely gets a chance to hold an ,

old, powerful piece of history. In mathematics, teachers have the opportunity to present just such museum pieces; and furthermore, the class can ver

ify their truth. Too often, however, students cringe in the face of proof and thus miss their chance to

appreciate the treasure. Mathematics teachers can

impress these future doctors, lawyers, politicians, and journalists with some jewels of mathematics; and teachers do mathematics a disservice if they skimp on the presentation. Since students and teachers have the time, the intelligence, and the materials to demonstrate the validity of a theorem, they should take advantage of this opportunity and

privilege. Significant results in the history of thought that students can understand should be occasions for great drama. The high school mathe matics sequence includes proofs of the quadratic formula, the Pythagorean theorem, the fundamen tal theorem of integral calculus, and other results. In this article, I suggest rolling out the red carpet for the proof of an important theorem. I focus on

the Pythagorean theorem; the interested reader can

easily adapt the treatment to any theorem that is

worthy of unusual notice.

Letting the statement of an important theorem and its proof diffuse into the flow of the course is like bringing the original Rosetta Stone into a history class, waving it around, and then moving on. Obvi

ously, such an artifact deserves more care and atten tion. The teacher would slowly place such an object down and say, "Be careful, it's very old." Everyone would speak softly and with respect. Any teacher can imagine enjoying such a moment in school; too few teachers recognize that mathematics offers the material for such an occasion. The challenge is to create a similar atmosphere around a showstopper like the Pythagorean theorem, which is at least 2500 years old. The teacher needs to deliver the

proof in a memorable way and to use the theorem in

ways that demonstrate its importance. In short, by controlling the context, the teacher can turn the the orem into an event. The occasion requires the best

presentation that a teacher can arrange. The follow

ing are some ideas to stimulate the reader's thinking. Changing the location of the class when the proof

is introduced adds to the excitement. Perhaps the school has an auditorium or a place that is reserved

for recitals. The library might have a room with a

scholarly atmosphere. Sometimes an impressive administrative room can be used. The selection of a room is important because it will probably influ ence the technology that can be used for the presen tation. To create an appropriate atmosphere in any room, the teacher can bring old mathematics books in languages other than English, a map of the

world at the time of the proof, a time line or a

poster, busts of mathematicians, such design tools as a T-square or a sextant, and other props. The historical context catches the attention of students whose specific interests are mentioned: wars,

music, literature, art, and food. The first challenge in arranging a special day for

the Pythagorean theorem is setting the date. The teacher often does not know the exact day even a

week in advance when discussion of the proof will be appropriate. One sticky question?or a particu larly nasty set of problems?and the pace of the course can slow. Fortunately, students can be

adaptable, especially when a treat is on the way. The teacher should try to forecast a day on which the proof can be discussed and save the proof for that day. The teacher can then reserve the special room for the presentation. After obtaining the room

and selecting a date, the teacher has a deadline. To

convey the theorem's importance, the teacher must balance content, context, the schedule, and propri ety with the urge to show off a gem of mathematics. Such detailed work takes time.

A teacher might want a seminar setting to draw the proof out during a directed discussion. I bor rowed an idea from trading-card games and wrote each rule, definition, and theorem necessary for the

proof on individual cards. Students had the state ment of the theorem to be proved, a sketch depict ing the approach to the proof, and these cards. We built the proof as a class, with the cards providing

"Sharing Teaching Ideas" offers practical tips on teaching topics related to the secondary school mathematics cur riculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the begin ning and the experienced teacher. Of particular interest are alternative forms of classroom assessment. See the masthead page for details on submitting manuscripts for review.

282 MATHEMATICS TEACHER

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some direction. If the teacher has planned a

straightforward presentation, slides on a computer work well; I like projecting the slides onto a white

board because I can emphasize important parts of

the illustration and erase these marks without

affecting the original figure. Presentation matters. Students appreciate a set

of slides that look attractive. A detailed script con

taining the sequence of slides can be invaluable

when a student requests another look at a previous slide; a good list includes a sketch of each slide and notes on the mathematical content. The teacher can

easily create such a script and slides with a software

package such as PowerPoint. Transparencies have a nice advantage over digital slides: the teacher can

put them on a table for easy referencing. Each class has its own personality, and I try to

tailor my presentation to the class. For example, a

music major in one class was a brilliant flutist, so I

played some Vivaldi in the background. Another stu

dent was a quiet history major, so I checked a histor

ical time line to see what else was going on in the

world and looked for interesting events and condi

tions to include in the presentation. I consider the

diversity of my students and then simply investi

gate a few ideas to incorporate into the presentation. Two experiences did not work as well as I had

expected. Old books and handheld models did not

create a strong sense of history for one class; stu

dents considered the old texts unclear and there

fore inferior to our books. Unfortunately, my stu

dents regarded "modern," "digital," and "best" as

synonyms. Another experience in a different class

revealed that classical music during the proof decreased student attention.

Students themselves can contribute to the pre sentation through their appearance. The teacher can ask the class to dress up a bit out of respect for

the material and to add to the fun. When students

prove the Pythagorean theorem, they participate in

the verification of a great truth. It is certainly sig nificant that the students themselves can under stand why the theorem is true. Such access to

ancient, powerful knowledge is worth clean socks and perhaps even a tucked-in shirt. Establishing the absolute validity of an idea does not happen often in disciplines outside mathematics.

Figure 1 shows my approach to the proof on one

occasion. The proof that goes with the figure origi nated in Persia and was accompanied with the sin

gle word "Behold." The actual proof is the one in which the four copies of the given right triangle are placed as pictured, with a quick look at why they can be placed this way. Then the area of the square is computed using two methods: as a side squared and as a sum of the two expressions for the same

area. Algebraic simplification leads to what is

known as the Pythagorean theorem.

This proof uses a few good geometric properties and some algebra. I like using it because of the "Behold" hook. It serves as a good example of a proof written with an economy of words, and it requires serious unpacking. I do not prescribe a certain proof or method of proof, and I have done the proof in dif

ferent ways. I have used cue cards that gave a reason

for each step of the proof for one class. That method

works well when I have a limited amount of time.

Students get cues from the reasons. Having the area

formula for a triangle in sight reminds the students to use that formula. The presence of the formula for

squaring a binomial hints that something up there

consists of two terms and needs to be squared. I

have followed the students' development of the

proof with an example drawn to scale that can be

measured. Then, depending on the group, I may

present a more difficult proof, like Euclid's.

Examining at least one other proof allows stu

dents to compare methods. For the Pythagorean theorem, I have some flexibility, since so many civi

lizations left a record of their knowledge of this the orem. Different cultures have proved it in many ways. If the teacher treats the proof as a museum

piece, then he or she can present a false proof and ask the students to evaluate the possible "forgery."

Tb avoid monopolizing the day's presentation, the teacher can plan questions in advance. Planting

questions with students before class guarantees student participation. In one such attempt, I want

ed the students to use the Pythagorean theorem to

show that a right triangle cannot be equilateral. Even after we found that the equilateral right tri

angle would have sides of length 0 and thus be a

point, the student who I had requested to ask the

planted question remained skeptical. This advocate for the existence of the special tri

angle slipped out of the crushing grip of logic, rebut

ting, "It could still happen," and offered an example. He asked us to think of ZABC as a vertex of ABC, where the hypotenuse AC is the shortcut. The dis tance from A to C is then less than the distance from

A to B plus the distance from to C. We all agreed.

The teacher can present a

false proof and ask the students to

evaluate the

possible "forgery"

Vol. 96, No. 4 ? April 2003 283

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The class

participates in

intellectual

life, discerning

why a theorem is

actually true

The student then pointed out that if all three sides were the same length, the distance from A to C would be half the sum of the distances from A to and from to C, so the length of the hypotenuse would be shorter than the sum of the lengths of the sides. Somehow, the algebraic step from there to

showing that the length of each side is 0 did not convince this student. This mathematical discourse was a welcome contribution to the presentation.

I want the students to have the opportunity to feel academic, so I try to produce the atmosphere of a colloquium. I offer a little refreshment after the

big proof and invite the intelligent discussion of a mathematics topic. I have not yet tried period or

culturally influenced food. The usual fare is fruit, cookies, donuts, juices, or soda. Sometimes the

planted questions like, "Did you know that there is a three-dimensional analog to the Pythagorean the orem involving the corner of a rectangular solid?" break the ice, and more questions follow. Some times the cookies and beverages call more loudly than the students' curiosity. The break gives us a chance to look over the models and the books, and it provides the teacher with a chance for individual clarification and a buffer if the proof runs long or short. In a day that is supposed to be unusual, such a break adds to the effect.

I find that allowing students the opportunity to comment and offer suggestions soon after the proof is valuable. Their comments have included the fol

lowing: "Because you made such a big deal about it, I wanted to learn it even more," "... to go over to Holmdene [an administrative building]?that makes someone feel special and... we were invited to attend, which makes the occasion a big deal." This last remark refers to my issuing invitations as a way of reminding the students we would be in a different room that day.

Some readers will point out that the entire treat ment is a conceit, and I heartily agree. The com ments from students, including remarks from course evaluations taken months later, convince me that the conceit works.

Some teachers may complain that this presenta tion is yet another time-consuming activity that does not prepare the students for standardized testing. I admit that the process requires substantial time

initially. After the first presentation, however, the teacher has a prefabricated activity that improves each year. After completing the slides and reserving the room, the teacher can look forward to a winning proof with no time crunch. Considering a proof as a

significant moment in the lives of students makes this preparation more than an idealistic duty to

mathematics: the class participates in intellectual

life, discerning why a theorem is actually true. The idea is a simple one that a teacher can easi

ly adapt to his or her personal style. The effective

ness of the idea certainly depends on a teacher's inclination and expertise; yet simply presenting a

proof with solemnity or with a flourish makes it stand out. That is the point of this article. Students recall just a few minutes worth of material from each course. By arranging a session that salutes an

important theorem and by treating it as one of the seminal ideas of all time, the instructor guarantees that a few extra minutes of mathematics will lodge in each student's memory. Many students might remember details of the class session that have little to do with the theorem. Most will have, at

least, a positive memory associated with a mathe matical idea.

ON THE WEB Eppstein, David. "Geometry in Action." Available at

www.ics.uci.edu/~eppstein/geom.html. World Wide Web. This resource contains an interesting list of mathematical patents that might be useful when teaching geometry.

"Pythagorean Puzzle." Available at www.pbs.org/wgbh /nova/prooi/puzzle. World Wide Web. This site reflects the role that the Pythagorean theorem played in Nova's report on Wiles's proof.

"The Pythagorean Theorem Home Page." Available at www.geom.umn.edu /~demo5337/Group3. World Wide Web. This site gives a The Geometer's Sketch pad exercise using the Pythagorean theorem.

BIBLIOGRAPHY Baragar, Arthur. A Survey of Classical and Modern Geometries. Upper Saddle River, N.J.: Prentice-Hall, 2001.

Edwards, Barbara S. "The Challenges of Implement ing Innovation." Matfamatics Teacher 93 (December 2000): 777-81.

Katz, Victor J. A History of Mathematics: An Introduc tion. 2nd ed. Reading, Mass.: Addison-Wesley, 1998.

Kelly, Loretta. "A Mathematical History Tour." Mathe matics Teacher 93 (January 2000): 14-17.

Krantz, Steven G. How to Teach Mathematics. 2nd ed. Providence, R.I.: American Mathematical Society, 1999.

Quadrat, Jean-R, Jean-B. Lasserre, and Jean-B. Hiriart Urruty." Pythagoras' Theorem for Areas." The Amer ican Mathematical Monthly 108 (June-July 2001): 549-51.

Shirley, Lawrence H. "Sharing Teaching Ideas: A Visit from Pythagoras: Using Costumes in the Classroom." Mathematics Teacher 93 (November 2000): 652-55.

Michael McDaniel

mcdanmic@aquinas.edu Aquinas College Grand Rapids, MI 49506

Mr

284 MATHEMATICS TEACHER

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