The Problem with Mathematics Education

There are numerous essays and newspaper blurbs lamenting the poor state of mathematical education in the US. Here is a typical example: Presidential panel bemoans state of math education.

What I see as the problem is that advanced mathematics is introduced in language that is unfit to inspire any but the few that were genetically destined to be mathematicians (or physicists).

Ask a recent college grad what an Eigen value or Eigen vector is. I give you 100:1 odds you'll get a blank stare. Okay now ask them to read this explanation from a popular Math web site. I bet their face will be even blanker. Now ask them to read this wonderful little explanation. Chances are the lights came on.

This is not to say that the later explanation will allow a person to do the math. But this is certainly where Math education, even at the highest levels, should begin. Illustrate why the problem is important, give a sensory picture to go along with the abstractions. Some might believe that this is how most Mathematicians teach but that is simply not the case. Mathematics is a very macho profession and many mathematicians believe its beneath them to offer intuition prior to rigor. The sad truth is many of them could not come up with compelling intuitive explanations even if they wanted to. It was not the way they were taught either.

Mathematica on LinkedIn and on a Wiki

This is my third post today (penance for not posting for so long!)

I recently started a LinkedIn Group called Mathematica Users Group. If you are a member of LinkedIn you can join the group by clicking here. After creating the group I thought it would be cool to have a Mathematica Wiki and soon discovered that Luc Barthelet thought this was a good idea too but thought of it a few years earlier than I did!

Semantic Wiki's

I attended a Semantic Web Meetup this past Thursday (Mar 13) where the topic was Semantic Wiki's. Although the presentations were not as focused as I would have liked, the topic is an interesting one. The two talks focused on the Semantic Media Wiki and the presentations can be found here and here..

Semantic Media Wiki is an extension to Media Wiki, the wiki engine that powers Wikipedia. The basic idea is that the Wiki supports an underling Triplestore (product example). Triples model subject, predicate, and object relationships (For more Semantic Web background see this, this and this).

The problem with a regular Wiki is that the information is largely unstructured. Some may argue this is a feature and there is something to the argument that the popularity of the Wiki stems from not forcing authors to use cumbersome syntax to structure the data for the benefit of computers. However, this lack of structure makes the information in a Wiki hard to re-purpose and also makes Wiki's harder to maintain (consider the fact that there is no automation in Wikipedia to keep lists like this one in sync with new pages).

Semantic Wiki's solve this problem by tagging data with known relationships that the computer can automatically leverage to cross-reference, collate and re-purpose data.

I think this idea is a natural progression of the Wiki concept but it remains to be seen if Semantic Wikis ever reach a critical mass comparable to Wikipedia. My personal view is that the work of organizing mounds of textual information needs advances in computer processing (AI) and that only a select few fanatics will engage in "tripling up the web" manually. Although, when it comes to web trends my crystal ball has been rather clouded.

Readers of my older posts know that I have proposed similar ideas under the moniker WISDI. I am still interested in the WISDI idea but circumstances have forced me to turn my attention elsewhere for the near term (I'll update readers in future posts) .

Ultimately, triples are just a syntax for the logic of relations (which is not even first order logic) so, to me and many others, the Semantic Web initiative is using really low fidelity tools to attack a high fidelity problem. However, in the agile spirit of "the simplest thing that can possibly work" they may achieve a more usable and reusable web in the near term.

Nested Dreams

I have always found dreaming a fascinating subject. I think dream research is key to unlocking data about consciousness. However, it is one of those areas of research where there is a very low signal to noise ratio.

In the past year I have had a very vivid class of dream that I don't recall ever experiencing earlier in my life. I don't know if there is a technical term for it but I call it a "nested dream". This is a dream that I apparently wake up from, retrospect about the dream but am, in fact, waking up into another dream. I am not talking about simply transitioning from one dream to another but actual dreaming, dreaming about waking up from that dream, things occurring in the new "dream stack frame", and then ultimately really waking up and remembering details from both frames.

I use the notion of a stack loosely since there is no remembrance of pushing down from dream 1 to dream 2 but rather there is the remembrance of popping out of 2 and into 1.

Has anyone experienced a similar kind of dream?

Prof. Ray C. Dougherty's Research

A few weeks ago I attended a Wolfram Research event called Mathematica Publishers Day. The goal of the event was to highlight the capabilities of Mathematica 6 as a platform for technical publishing. I really enjoyed this event but was also pleasantly surprised by a talk that did not quite fit into the overall theme of the event but nevertheless was quite fascinating to me.

The presenter was Prof. Ray C. Dougherty, NYU Linguistics researcher. He used Mathematica to model all possible sine wave based communications systems. The presentation is available via Wolfram. Unfortunately, as with many interesting presentations, you needed to hear the talk to get the most out of it. Here are some interesting excerpts that I remember:

• The Cochlea is computing the second derivative of the auditory input.
• The most mathematically complex communication system is one where the transmitter and receiver have the same anatomy (e.g., wings of insects).
• Bats can hear phase changes because they can rotate their ears. A human can not hear a change in the rotation of a tuning fork but a bat can.
• Prof Dougherty believes he has a Chomsky generative grammar that enumerates all possible animal communication systems.
• He also believes he can map each possible system onto the integers in a natural way.
• From this he concludes that evolution must proceed in jumps.
• He relates this idea to the evolution of all possible Tic Tac Toe Games to illustrate the notion that all such games are not unique and similarly the space of all possible communication systems contains many redundant systems as well.
• He goes on to visualising distributions of the primes to illustrate that there are systems that are not random but whose patterns are too complex for us to model in a simple fashion. Explains how this is related to the ideas in Stephen Wolfram's NKS.

Stephen Wolfram on Software Design and Naming

In this post Stephen Wolfram talks about design reviews and the importance of naming things correctly. I sought of envy him for working in a company where one can afford to spend a total of 10,000 hours in design review and where you sweat over the details of naming every last function. As much as I can see the merits of Agile Development, I have never worked on an Agile project that creating anything nearly as elegant as Mathematica. Sure, Agile focuses on working software and "good enough" software and the economics of most software projects makes this mode of development necessary. But it would be nice to one day work on a project where time to market was less important than the end result.

Ubiquitous Eigenvectors and Quantum Computing

Recently I have been studying Quantum Computation (QC). If you want to get anywhere with QC you need to master Linear Algebra and Vector Spaces since QC is baically an exercise in applied vector space theory.

Being a bit rusty in the topic myself, I decided to pick up the book Finite-Dimensional Vector Spaces by Paul Richard Halmos. Although Halmos is one of my favorite math authors, I picked this book primarily because it has a Kindle edition. It turns out that another one of Halmos's books, Linear Algebra Problem Book, is a far better choice for the non-mathematcian. The later book walks you step by step through bite sized problems and provides hints (and also all the answers if you get stuck). A free resource with answers can also be found here.

One of the central mathematical techniques at the heart of Linear Algebra is the concept of Eigenvectors and Eigenvalues. The term Eigen is derived from German and means "characteristic". An Eigen Decomposition is a method of reducing a square matrix to into a constant (eigenvalue) and a vector (eigenvector). This decompostion is central to many problems in physics.

It turns out that the study of Eigen decompostion can yield deep insight into problems that are in the realm of computer science. Consider, for instance, this paper about Google's page rank algorithm and the Eigenface technique for facial recognition.

Coming to grips with the mathematics behind Vector Spaces is one of the single most rewarding experiences for anyone interested in advanced problems in computer science. It is a must if you ever want to graduate from the comprehension of clasical algorithms to the comprehension of quantum algorithms. However, if are curious about QC but the thought of learning advanced linear algebra sounds like too big of a comitment, then you might want to check out Quantum Computation explained to my Mother. This is the most approachable paper I have ever read on the topic that is also mathematically accurate.