Fractal Video from Teamfresh

I found this on Steven Strogatz's NY Times Math Blog

Classic newton fractal from teamfresh on Vimeo.

From a Spreadsheet Problem to the Umbral Calculus: A Mathematical Odyssey

I'm planning to write a paper where I describe how a colleague's challenge to come up with an Excel formula to compute a weighted average of grades led me to make a couple of mathematical conjectures, and how I was able to prove the conjectures and solve the problem. Along the way, I got a lot of help from many people and I discovered a lot of combinatorial mathematics that I had not known, including the Binomial Inversion Formula and the Umbral calculus. In describing this odyssey I will explore the social nature of mathematics and the different ways that people from different disciplines approach mathematical problems. Also, I hope to show that experiences of this sort can be replicated in the classroom through a problem-based method of learning.

My Problem Published

My problem about finding the kth largest element of a set has been published (with a very slight misprint) in the American Mathematical Monthly:
Monthly Problem 11520  

E18. A locus related to a rectangle

This problem is related to my earlier E16.

Let ABCD be a rectangle. Find the locus of all points P such at PA + PC = PB + PD.

The End of Ignorance: Multiplying our Human Potential

I've just finished reading The End of Ignorance: Multiplying our Human Potential, by John Mighton who has developed a mathematics education program called JUMP (Junior Unrecognized Mathematics Prodigies). His system has met with amazing success with a very wide range of elementary school students and considerable hostility from the Mathematics Education establishment in his native Ontario.

He challenges the NCTM orthodoxy, and the tenets of constructivist math education. I feared that this might be another "Mathematically Correct" screed, but it is far from that. Mighton has an enviable record of success in reaching the most "hopeless" students, and an admirable humility in recognizing that his system is not the only way to improve math education.

Mighton has a Ph.D. in mathematics, a career as a playwright, and a firm grasp of philosophy. He and a large cadre of volunteers have developed the program over a number of years, and refined it by trial-and-error. The major ideas are:

1. Learning takes place with a balance of concrete and symbolic, guided and independent, and procedural and conceptual.
2. Compared with constructivist methods, the teacher is expected to be a very active guide. Concepts are broken into small units, gaps in student understanding are detected and filled, lessons are carefully designed, sequential, and scaffolded. Weaker students are motivated by carefully graduated challenges, and stronger students are given extra challenges.
3. Whole-class lessons allow students to experience the thrill of discovery collectively.
4. Teachers give frequent and specific encouragement to all students.
5. Formative assessments are given continuously, and used to modify instruction. Students who don't know the material necessary to begin the lesson are given additional instruction before learning the new topic.
6. There is a strong emphasis of the development of  procedural knowledge through use of workbooks and individual work.

I think any educator who reads Mighton's calmly recollected stories of the hostility and closed-mindedness that his ideas have generated among certain math curriculum consultants (some of whom are subject to conflicts of interests due to their relationships with textbook publishers) is bound to feel a sense of embarrassment for our profession.

Mighton's book has caused me to rethink some of my pro-constructivist positions. I also will be following up on reading some of the work on cognitive psychology that he cites as having been seriously misinterpreted by the mathematics education establishment as supporting constructivist and situated learning approaches.

NES/MAA Meeting

The NES/MAA meeting I mentioned in my last post was held at Salve Regina University, which is located on the grounds of an opulent mansion on the ocean at Newport Rhode Island. There were a number of interesting invited presentations. I particularly enjoyed the talk by David Abrahamson and Rebecca Sparks on Baseball Statistics and Keith Conrad's talk on Check Digits (in credit card numbers, etc.). Both dealt with fairly elementary mathematics, but related the results to the everyday world in a compelling way.


Ed Burger's Battles Lecture on p-adic norms was a bit more technical, but Ed's high-energy and humorous style of presentation made the medicine go down very well.

My presentation on The Pythagorean Theorem (Revisited) was well received. I was a bit surprised and gratified that none of the mathematics professors or students previously knew the main theorem I was presenting, the Pythagorean Theorem for right tetrahedrons.

Presentation at NES/MAA Meeting, June 12

I will be attending the New England Section of the Mathematical Association of America meeting in Newport Rhode Island, June 11-12. I have had a paper accepted. It is an expository paper presenting some simple and interesting facts relating to the Pythagorean Theorem that many professional mathematicians do not know. To view my notes for the presentation, go to http://www.scribd.com/doc/32536190/Pythagorean-Theorem-Notes.

Comments on the paper are welcome.