Summer School

In job applications and interviews I stress my technical abilities and desire to serve as a role model for lifelong, independent learning. To back this up, this summer I have plans for a number of independent learning activities. The first of these is learning Moodle, a free, open source Learning Management System used in many schools. I've installed it on my laptop (along with the associated software that makes up the platform on which it runs) and set up my first course: Python for Informatics. This is a just a copy of PythonLearn - Self-Paced Learning Python that I made for myself as an exercise in learning Moodle concepts and functions, but I will actually use the course to learn the Python computer language. At several recent interviews the schools have expressed interest in my programming background and if I'd be qualified to teach programming or other computer-related skills. From what little I've seen of it so far, Python seems well-suited for the high school classroom.

Dissing Math

Disrespect for mathematics was the theme of two small items that crossed my (virtual) desk this week. On Wednesday, Randall Munroe's webcomic xkcd.com featured an encounter between an algebra teacher and her former pupil:

"Hey, Miss Lenhart! I forgot everything about algebra the moment I graduated, and in twenty years no one has needed me to solve anything for X! I told you I'd never use it! In your face!"

On Thursday, Linda Gojak, the new president of the National Council of Teachers of Mathematics (NCTM), posted a similar message on the NCTM President's Corner:

"For too long we have heard the same thing from parents, adults, students, and the general population: 'I was never very good at math!' Too often at parent conferences I heard parents lament about their own inadequacies in mathematics, as though their experiences excused any difficulties that their children were having."

In both items there is an underlying tone that it's OK to dislike, not know, or not use math. John Allen Paulos, in the introduction to his 1989 book Innumeracy, goes further, claiming that many people don't just think it's OK, but take a perverse pride in their negative attitude toward math.

"Unlike other failings which are often hidden, mathematical illiteracy is often flaunted: 'I can't even balance my checkbook.' 'I'm a people person, not a numbers person.' "

Both Paulos and Munroe single out math as being somehow different from other areas of knowledge in this respect. In American culture there is not the sense of shame attached to innumeracy the way there is to illiteracy, for example. Bad grammar is frowned upon; bad probabilistic reasoning is not.

My Educational Philosophy

The hiring season has begun: schools in New Jersey have started posting open positions for high school math teachers for the 2012-2013 school year. In preparation, I dusted off a draft of "My Educational Philosophy" that I wrote for Larry at Traders-to-Teachers and cleaned it up for use on applications. I still consider it a working document, but it does express what I think are key points:

• Learning is neurologically based: education must be based on a sound theory of cognition. This is the old cognitive scientist coming out in me.
• There is an inherent tension between the needs of society and the needs of an individual, and a good teacher in the classroom does not avoid this tension.
• I teach for understanding, and my definition of "understanding" owes a lot to Howard Gardner. 

OK, here it is:

Number Theory for Teachers

This summer I'm taking a course in Number Theory designed for secondary school math teachers, specifically those of us who went through the Traders-to-Teachers program at Montclair State University. As such, it will cover a subset of the topics usually found in introductory number theory courses, with, I suspect, less rigor. From the syllabus: "Our class discussions will relate number theory topics to middle and high school level curriculum."

At the beginning of the first class the instructor asked what we hoped to get out the course. I kept silent, as my situation - unemployed - and interests differ from the rest of the class. However, I continued to ponder the question and hope to answer it here. Aside from the obvious and trivial response - to satisfy a requirement for full teacher certification - here's what I want to get out of the course:

My Math Bookshelf

Recently I cleaned up my public catalog of math books in my personal library as part of a larger effort to scrub and cross-link various public profiles: Facebook, LinkedIn, Twitter, etc. I use LibraryThing for my public catalog, but I've maintained my own private book catalog in a Lotus Notes database for almost twenty years. I've ported about two-thirds of my catalog into LibraryThing, including 164 entries tagged as "mathematics". Some of these are also tagged as "borrowed", indicating they are library books, not owned by me.

About a third of the math books are textbooks, some from my wife, some from my son, some recent acquisitions, some from my college days in the 70s. The rest are a mixture of history, biography, philosophy, math in popular culture, and recreational math. The author most represented is Martin Gardner. The book I've owned the longest is Courant and Robbins' What Is Mathematics? which was a present from my father on my 16th birthday. Other favorite authors include George Pólya, Philip Davis, Morris Kline, Douglas Hofstadter, and Keith Devlin.

Most of the books are unread: the collection represents aspirations, not accomplishments.

Errol Morris on Pythagoras, Incommensurability, and Whiggish History

In March 2011 Errol Morris, an award-winning documentary filmmaker and director of TV commercials, published a series of five opinion pieces in the New York Times with the collective title "The Ashtray". Those who know about Morris through his films and ads may not be aware of his philosophical writings about the nature of truth, knowledge, and history. If those topics put you off from reading Morris, perhaps it would help to know that the series title comes from the object thrown at the author by Thomas Kuhn, author of "The Structure of Scientific Revolutions", when Morris was a graduate student under Kuhn at Princeton. (If that's still not enough, then see his description of a Dan-Brownish library encounter, below.) The series as a whole is a critique of Kuhn's notion of "incommensurability" of scientific paradigms, but the part that appealed to me particularly is the third in the series, on the origin of the term "incommensurable" in mathematics and the legends surrounding the Pythagorean proof that the square root of two is irrational.

The Role of Technology in Education

A question on a recent job application asked about technology in education. Such questions often force me to reflect on the topic and focus my ideas; this one was no different. I take the standard view of "technology as a tool" and consider four kinds of tool use: 

1. Technology as a tool for communication and collaboration. This use is well-known and includes e-mail, using the web for access to information, collaborative software and social media for group projects, etc.
2. Technology as a tool for the expression of ideas. My best example is from mathematics: software that allows you not just to visualize mathematical ideas but to increase the number of dimensions through 3-D and animation; to emphasize or discriminate features using color and pattern;  to show the development of complex ideas using selective hide/show, etc. The best software makes this expressiveness available to students as well as teachers.
3. Technology as an extension of self. Through digitization and storage, technology enables a teacher to provide differentiated instruction both in the classroom, by increasing the number of modalities by which material is presented; and outside the classroom, by providing extra help or extended topics for which class time is not available.
4. Technology for ubiquitous data-gathering and number-crunching. This use of technology opens up possibilities for extended, fine-tuned assessments of learning and teaching; it also raises ethical issues.

In addition to all of these uses, exposure to the technology that will be encountered in higher education and the workplace is a necessary part of secondary education.

My original answer to the question follows.