Despite being written in 2008, this is an interesting and relevant paper, whether or not you're an "Apple person": Apple Classrooms of Tomorrow - Today, Learning in the 21st Century.

Also, it is worth the time to read about the Punahou School in Hawaii. Be sure to watch the video.

## Monday, November 28, 2011

### Pythagoras...

In addition to attending the conference at the Cedars School for Excellence in Greenock, Scotland which I blogged about earlier, I gave talks at two mathematics conferences. At both the Northwest Mathematics Conference in Portland, OR and at the California Mathematics Council - Southern Division Conference in Palm Springs, CA I gave a talk entitled

**Back to Basics - The Pythagorean Theorem**. The premise was that the Pythagorean theorem is not only one of the oldest theorems in mathematics but is also one of the most basic and widely used theorems in secondary school mathematics. If one peruses the mathematics materials used at Phillips Exeter Academy (once on the Mathematics department page, click on Teaching Materials), it is astonishing at how many times the Pythagorean theorem makes an appearance...from Alex in the desert to determining the depth of a lake by yanking on a lily pad to calculating just how far it is to the horizon when standing atop Mt. Washington. Some applications are obvious, some subtle. At any rate, whether you teach mathematics or not, I thought you might enjoy these Pythagorean Fun Facts:**The Pythagorean Theorem - In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs.**- The book
__The Pythagorean Proposition__contains 370 proofs. - An algebraic proof was published by U.S. President James A. Garfield
- There is debate as to whether the theorem was discovered once, or many times in many places.
- The earliest reference to the theorem was in the Egyptian papyrus Berlin 6619 written between 2000 and 1786 B.C.E.
- Pythagoras lived from 569 to 475 B.C.E.
- Pythagoras used algebraic methods to construct, um, Pythagorean triples.
- Around 400 B.C.E. Plato gave a method for finding Pythagorean triples using algebra and geometry.
- Around 300 B.C.E., in Euclid's Elements, the oldest known extant proof of the theorem is presented.
- A verse of the Major-General's Song in the comic opera The Pirates of Penzance makes an oblique reference to the Pythagorean theorem.
- The Scarecrow in the Wizard of Oz makes a more specific reference to the theorem, misstating it.
- Greece, Japan, San Marino, Sierra Leone, and Suriname have issued postage stamps depicting Pythagoras and the theorem.
- In 2000, Uganda released a coin with the shape of an isosceles right triangle. The coins tail has an image of Pythagoras.
- In Neal Stephenson's book
__Anathem__, the Pythagorean theorem is referred to as 'the Adrakhonic theorem'. A geometric proof is displayed on the side of an alien ship to display the aliens' understanding of mathematics.

### Focus on Calculus Newsletter article from November 1999

**Issues Facing Secondary School Mathematics Teachers**

It is a considerable understatement to say that mathematics education, particularly secondary school mathematics, has gone through significant change in the past fifteen years. Yet I believe that we have just glimpsed the tip of the proverbial iceberg and the biggest changes are about to happen. What follows are some of the reasons that have been at the core of the rapid and exciting change in secondary school mathematics and will continue to present secondary school teachers with even more challenging issues.

**Technology**

The advent of the personal computer in the early seventies and graphing calculator in 1985 forced changes in secondary mathematics. Suddenly the need for tables of logarithms and trigonometric ratios diminished rapidly and exciting topics and applications in mathematics became far more accessible to secondary school students. Computer programs like Green Globs, The Geometric Symposer, and the Kamischke Grapher led the way, giving us another strategy to use in helping kids understand mathematics. No longer did we have to use derivatives or Descartes Rule of Signs, among other methods, to get an accurate sketch of the graph of a function. Now students could use the graphing calculator to get an accurate graph with which they can better understand the behavior of functions. Also, problems that were dependent on calculus could now be done in precalculus. The continuing challenge facing secondary school teachers is to discover the correct balance of use of the technology. Just as total paper and pencil manipulations are obsolete, it is just as wrong to blindly push buttons with little thought about process and results.

**Curriculum**

There has been a move away from calculus as the pinnacle of the mathematics curriculum pyramid and this is good. While this may seem odd to state in a newsletter devoted to the focus on calculus, it is better for students and secondary mathematics programs in the long run. Discrete mathematics, statistics, and mathematical modeling now provide students with alternatives to the calculus. Students either tired of the focus or rigor of precalculus and calculus can continue to take mathematics that will help them become informed citizens. Moreover, calculus courses need not be watered down and those that are truly ready for college-level calculus will benefit. While many students can perform the manipulations required in calculus, few really understand the underlying theory and concepts required for moving further along in mathematics. The challenge facing secondary teachers is convincing students, parents and college counselors that all students need not take calculus in high school. Further, development of meaningful, rigorous alternative courses requires time and energy from already swamped teachers.

**The information highway**

There has been an incredible increase in the sharing of ideas through web symposia and the availability of free software. Considering that there are some 800 million websites and that the best search engines can reach just around 16% of them, the usefulness of the World Wide Web in secondary mathematics is just being recognized. Teachers are now challenged by how to make the best use of the web in their classes. Certainly the web is a source of data sets that can enhance the teaching of statistics but secondary teachers must continue to explore this vast resource.

**Text books**

Teachers have many more textbooks to choose from today. In addition to standard textbook series, there are many innovative series available: the ARISE project from COMAP, Interactive Math Program (IMP), Systemic Initiative in Montana Mathematics and Science (SIMMS) to name a few. In fact there are 13 NSF funded curriculum projects, five of them devoted to secondary school mathematics. The calculus reform projects at the college level have also given secondary teachers many more options for Advanced Placement Calculus. The challenge for secondary teachers is to continue to press for better and better textbooks. While textbook adoption guidelines in school districts make it difficult, secondary teachers must convince school administrators that it is in the best interest of students to select the most up-to-date texts possible.

**The TI-89**

The biggest challenge facing secondary school teachers is to determine the appropriate use of Computer Algebra System capable calculators. The TI-89 will force us to think about CAS curriculum like nothing before. While Hewlett-Packard has produced calculators capable of symbolic manipulation for nearly twelve years and computer programs like Derive, Maple, and Mathematica have all been available for years, now it’s cheap, relatively, and in your hand. Secondary teachers will be the ones to decide how to best use this technology and it would be irresponsible to not do so. Secondary teachers led the way with appropriate and creative applications of graphing calculator technology and will again do so with CAS. This time, however, the mathematics curriculum as we know it will change forever. With the TI-89 it will be more important for a student to recognize form than be able to manipulate from one form to another. There are applications of CAS at all levels of the secondary curriculum and the next few years should prove to be very exciting as we decide how best to implement it.

Tom Seidenberg, Phillips Exeter Academy

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