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February 1, 2004
Vol. 61
No. 5

Marvelous Math!

Teachers can get students involved in the fascinating side of mathematics by building a repertoire filled with the intriguing and the unexpected.

Marvelous Math! -thumbnail
When we adults think back on our school days, we usually recall a few dynamic teachers who brought novel ideas into the classroom, material that came from outside the textbook. This strategy not only motivated students but also enabled teachers to enthusiastically express their interests. That sudden surge of enthusiasm was contagious.
Teachers are often leery about making such digressions for fear of taking time away from the prescribed curriculum. But perhaps we should look at the situation differently and consider the strategy as an investment. Taking the time to cultivate interest and enthusiasm in students about relevant topics helps mold them into receptive learners. Moreover, such a practice could result in a more effective and faster-paced instructional program.
Unfortunately, we seldom remember math class as the place where we experienced those learning highs. When I reveal in a social setting that I am in the field of mathematics, my disclosure invariably triggers a negative reaction in other adults. “Oh, I was always terrible in math,” they quickly point out. For no other subject in the school curriculum is an adult so proud of having done poorly; failure in mathematics is almost a badge of honor.
Why this distaste and disinterest? Perhaps we aren't making a point of showing students the beauty and wonder of mathematics. Addressing this problem might well be one of the keys to student achievement. Achievement isn't always about the choice of curriculum materials, the mandated instructional philosophy, or the strength of a given class. Achievement can also stem from resourceful teachers who bring relevant topics to their students with genuine enthusiasm. Successful instruction often depends on the delivery, and we tend to get enthusiastic when we show something clever. So why not use this strategy in your teaching? Compile books on recreational mathematics, on problem solving, and on the history of math; assemble a personal library that provides you with a plethora of fascinating ideas to bring into the classroom. Some suggestions from my collection follow.

Probing Into the Past

If you are working with students on the number system, mention the origin of the place value system. Students will most likely find it surprising that the digits 0–9 first appeared in 1202, when Leonardo of Pisa, more commonly known today as Fibonacci, introduced them in his book, Liber Abaci (Sigler, 2002):The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these figures and with the sign 0, which the Arabs call zephyr, any number whatsoever is written.
As for π<!--Note: numeric entities for "pi" are &#928; for capital case and &#960; for small case.-->—the ratio of the circumference of a circle to its diameter—the wise teacher might take a moment to discuss how, in the course of 4,000 years, people have struggled to determine π's value. One might show how they have misunderstood for centuries the biblical reference to π (Posamentier &amp; Gordon, 1984). Students always relish the notion of hidden codes revealing long-lost secrets.
A certain sentence in the Bible appears twice, identical in every respect except for the Hebrew word for “line measure,” which the two citations spell differently. Both 1 Kings 7:23 and 2 Chronicles 4:2 describe a pool in King Solomon's temple:And he made the molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line measure of thirty cubits did compass it round about.
The circular structure described has a circumference of 30 cubits and a diameter of 10 cubits. (A cubit is the distance from a person's fingertip to his elbow.) From this information, we calculate π as 30/10, or 3. In the late 18th century, however, a scholar named Elijah de Vilna studied the discrepancy between the two sentences. In 1 Kings 7:23, “line measure” appeared as ה ז ק <!--Note: Unicode entities for Hebrew letters aren't working with our SGML/Omnimark configuration. For reference, the Unicode entities for Hebrew letters: "he" is &#x05D4; ... "zayin" is ;&#x05D6; ... and "qof" is &#x05E7; . Numeric entities for them: "he" is &#1492; ... "zayin" is &#1494; ... "qof" is &#1511; . Also used below.-->, whereas in 2 Chronicles 4:2 it appeared as ז ק. Using a biblical analysis technique called gematria, which assigns a specific numerical value to Hebrew letters depending on their sequence in the Hebrew alphabet, Elijah noted that 1 Kings 7:23 spelled line measure as ה ז ק (ה ז ק = 5 + 6 + 100 = 111) and 2 Chronicles 4:2 spelled it as ז ק (ז ק = 6 + 100 = 106). Elijah took the ratio of these two values: 111/106 = 1.0472 (to four decimal places). He considered this the necessary correction factor, because multiplying 1.0472 by 3 (the value the Bible seems to attribute to π) results in 3.1416. This is π, correct to four decimal places! The usual reaction? Wow!

The Magic of Numbers

You might also entertain your classes by showing that our base 10 number system exhibits some unusual patterns. For starters, take a look at a pretty multiplication table (see fig. 1, p. 46).

Figure 1. The Patterns in Mathematics

Number relationships manifest themselves in arithmetic but need algebra to determine why they work. Show algebra students the following “trick.” Have each student select a three-digit number for which all three digits are different. Tell the students to reverse the digits of their three-digit number and subtract the smaller number from the larger. The students must then reverse the order of the digits in their answer and add this new number to the difference they got in the previous step. Ask one student to read out the answer. It should be 1,089. Other students will exclaim: “I got the same answer!” How can this be, when everyone presumably began with an individually selected number? The ensuing discussion can address the algebraic justification of this question.
When you tackle the issue of probability, dazzle your students with the mind-boggling “birthday problem.” This problem shows the probability of any two people in a group having the same birthday (in terms of day and month). For a group of 30 people, the probability of a match is approximately 7 out of 10, or 70 percent. That is quite remarkable, considering that there are 365 dates from which to choose. It gets even more impressive with a group of 40: The chance of a match is approximately 90 percent. And with a group of 55, a match is almost certain. Establishing these probabilities requires only the most elementary knowledge of probability.

Zeros, Lines, and a Six-Inch Mouse

Resourceful teachers will not just tell students that they shouldn't divide by zero but will show students what happens when they do. Students will learn that dividing by zero leads to the contradiction of the accepted fact that 2 ≠ 1 (see fig. 2). They will love the absurdity of it.

Figure 2. Dividing by Zero

Or have the students consider an example showing how lines play tricks with our vision (see fig. 3). This demonstration proves that mathematics requires sound strategies. Although people often rely on intuition, students cannot make assumptions about a problem on the basis of how lines or angles look on a page.

Figure 3. The Tricks Lines Play on Us

Seemingly insurmountable problems can fascinate a class. The cleverness of the solution to the following problem—and its inherent elegance—will impress students.A rope is tied along the earth's equator, circumscribing the entire sphere. Now lengthen this enormously long rope by 1 meter. It is no longer tightly tied around the earth. If we lift this loose rope equally around the equator, uniformly spacing it above the equator, can a mouse fit beneath the rope?
We are looking for the distance between the circumferences of these two circles. Because the sizes of the circles are not given, suppose the small (inner) circle is extremely small, so small that it has a radius of 0 and is thus reduced to a point. Therefore, the distance between the circles is merely the radius (R) of the larger circle. The circumference of this larger circle is 2πR = C + 1, or 2πR = 0 + 1 = 1, where C is the circumference of the earth (reduced to 0 for the sake of this problem) and C + 1 is the length of the rope. The distance between the circles is R = 1/2 π = 0.159 meters, which would enable a mouse to comfortably fit beneath the rope.

New Perspectives on Old Problems

When your students consider the following problem, how do they generally go about solving it?In a single elimination basketball tournament (where one loss eliminates the team), 25 teams are competing. How many games must they play until there is a single tournament champion?
Typically, students will begin to simulate the tournament: 2 groups of 12 teams play the first round, eliminating 12 teams (12 games played). The remaining 13 teams play, 6 against another 6, leaving 7 teams in the tournament (18 games played). In the next round, 3 of the 7 remaining teams are eliminated (21 games played). The 4 remaining teams play, leaving 2 teams for the championship game (23 games played). This championship game is the 24th game.
But there is a much simpler way to solve this problem, one that most people do not naturally see. Ask the students the key question, How many teams must lose in a tournament with 25 teams to produce one winner? The answer is, naturally, 24. So there you have it, simply done.
If you are asking yourself, “Why didn't I think of that?”, it's because the strategy is contrary to the type of training that many of us received. Familiarize students with the possibility of looking at problems from different points of view. This strategy can sometimes suggest the more direct solution.
There are endless examples of motivating ideas that teachers can bring into the classroom to demonstrate the “gee-whiz” aspect of mathematics. Examples must, of course, focus on the topic and the lesson's objectives.
A final word about routinely trying to apply mathematics to everyday life: Teachers shouldn't use this strategy to the exclusion of others. Students often quickly detect the artificial nature of the selections and end up seeing the “useful” application as merely another textbook exercise. Strive, rather, to demonstrate the beauty of mathematics. We may yet succeed in creating a population of adults who will boast about their accomplishments in math.

Posamentier, A. S., &amp; Gordon, N. (1984). An astounding revelation on the history of π. Mathematics Teacher, 77(1), 52.

Stigler, L. E. (2002). Fibonacci's Liber Abaci: A translation into modern English of Leonardo Pisano's Book of Calculation. New York: Springer-Verlag.

Alfred S. Posamentier is Professor of Mathematics Education and Dean of the School of Education of the City College of the City University of New York. He is the author and coauthor of more than 40 mathematics books for teachers, secondary and elementary school students, and the general readership. Dr. Posamentier is also a frequent commentator in newspapers on topics relating to education, and he works with mathematics teachers and supervisors, nationally and internationally, to help them maximize their effectiveness.

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