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Dr. Alfred Posamentier: April 2012 Archives

# April 2012 Archives

## Enriching Mathematics Instruction, Rather than Teaching to the Test

With the impasse about teacher evaluation dominating our thinking about education, and the controversy over using test results to determine a good teacher, it might be time to take a step back and consider another way that one of America’s most tested subjects — mathematics — can be more effectively taught.

Let’s take just one example that demonstrates how the subject can be enlivened.

Imagine, as you begin to collect your tax information for 2011, if we still used only Roman numerals. For starters it wouldn’t be 2011. It would be MMXI.

Looks strange, right? Our Hindu-Arabic number system must have looked just as strange 810 years ago when it was first introduced to the Western world by an Italian named Leonardo of Pisa, who was later known as Fibonacci. In 1202, he wrote Liber Abaci, a book of calculations, which introduced the numbers 9, 8, 7, 6, 5, 4, 3, 2, and 1. “With these nine figures,” Fibonacci wrote, “and with the sign 0, which the Arabs call zephyr, any number whatsoever is written.”

Europeans didn’t embrace this strange new idea right away. It took nearly 50 years before other parts of Italy began to accept the base-ten number system. Merchants were suspicious of those who used these clever numbers instead of the more familiar Roman numerals. It wasn’t until more than two centuries later that the use of numerals caught on across Western Europe, about the time the Leaning Tower of Pisa was completed.

This was not Fibonacci’s only — or even most famous — contribution to the world’s body of mathematical knowledge. In chapter 12 of that very same book, he considered the procreation of rabbits. Fibonacci sought to find the number of pairs of rabbits that would result over the course of a year, if a pair required one month to mature and to produce an offspring pair. His calculations of rabbit pairs produced month-by-month led to the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144.

This sequence of numbers became known as the Fibonacci numbers — the most ubiquitous numbers in all of mathematics and beyond. We find these numbers in nature — for example the numbers of the three types of spirals on a pineapple are 5, 8 and 13 (Fibonacci numbers). The Fibonacci numbers are also found in the proportions of famous architecture – including the Parthenon, the United Nations building, and the doors of the Cathedral of Chartres. They provide a direct connection to the golden ratio that in itself has a plethora of applications. They even have some practical features: They allow an instant (approximate) conversion of miles into kilometers and the reverse.

Fibonacci explored mathematics at a time when it was not at all a subject to be considered on its own but merely as a tool. We need to infuse his creative and playful spirit into our teaching of mathematics. With children as young as 7, there are puzzles, games, and astonishing relationships that can be brought into the classroom and made an integral part of our teaching of mathematics — taking this important subject out of the realm of “a strange language that must be memorized but not necessarily embraced.” The Fibonacci numbers about which many books have been written (including one by this writer) have boundless applications to enrich mathematics (and beyond) at all academic levels.

All too often, we hear the lament that our children are growing up disliking mathematics — and then, as adults, boasting about how bad they were in the subject all through school. Perhaps our teachers are so often obsessed with preparing students for tests they lose sight of the pleasures and applications that help students to learn about mathematics in different ways. If teachers were to expand their knowledge to the more creative aspects of mathematics, we may instill a more inquisitive spirit into the classroom and enrich our students’ experience with mathematics.

At age 810, the Fibonacci numbers still provide many topics for the creative teaching of mathematics. Who knows what strange, new, and groundbreaking insights might result.
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