Friday, December 18, 2009

Program Overview

I will be announcing in the near future here on the blog some changes to the M.A. in Mathematics for Teachers program that have passed through some approval steps. I am currently waiting for some final approval by the Faculty of Graduate studies. The changes are quite minor on the surface, but include new calendar text, admissions requirements, degree requirements and significant changes to the course offerings.

In the past months I have found myself writing what this program is about over and over. Many of the times that I have written this program overview, the description goes into some document which is very unlikely to be read by anyone (and especially unlikely to be read by the students of this program). For this reason I thought I should record at least one version of this in electronic form. I also welcome comments on the following text. If you feel that it does not accurately reflect the reality of this program feel free to leave your feedback.

The M.A. in Mathematics for Teachers has been offered by the department of Mathematics and Statistics as a stand alone degree since the mid 1970s. The purpose of this program has been to offer an opportunity for elementary, high school and college teachers to increase the breath of their mathematical knowledge to give a broader context to the mathematics that they teach in their own classrooms.

Students are enrolled in the program on a part time basis. It is designed to welcome students that may have completed their university studies a significant time in the past. Courses for this degree are clearly distinguished through their numbering with the first digit 5 while the courses in our regular M.A. degree are numbered with first digit 6. In order to achieve the goals of the program, the requirements are more course intensive than our regular M.A. program. The range of courses offered for this degree gives students an historic perspective as well as chances to practice techniques of problem solving, writing, and presenting mathematics since these aspects of the discipline are relevant for teachers of mathematics at any level.

York University also offers a graduate diploma in Mathematics Education that may be taken concurrently with the M.A. in Mathematics for Teachers degree or an M.A. program with the Education department or as a stand alone diploma. The diploma focuses on mathematics education as an area of study grounded in critical examination of teaching practice, learning theories, and curriculum related to the teaching and learning of mathematics. It is designed to provide opportunities for graduate level study of theories and research in Mathematics Education and may be taken in conjunction with the M.A. in Mathematics for Teachers to prepare a student for study in Education at the Ph.D. level. Several recent graduates of the diploma in Mathematics Education have gone on to do graduate work in Education at York University and OISE/U of T.

This degree is part time program and classes are scheduled in the evenings to accommodate students who have employment teaching during the day. Prerequisites for the courses generally do not require that students have a recent training in mathematics, but the expectation is that the coursework will build on knowledge familiar to students that have had past experience in mathematics and are now required to teach mathematics. This program does not prepare students for study at the Ph.D. level in mathematics nor does it lead to teacher certification in Ontario.

Tuesday, December 15, 2009

Easy to state, hard to solve problem

One of the authors of the blog "Computational Complexity," William Gasarch, posted a computational challenge on November 30 and he is offering $17*17 = $289 if someone comes up with a solution. I like this problem because it is easy to state and hard to solve.

The n x m grid is c-colorable if there is a way to c-color the vertices of the n x m grid so that there is no rectangle with all four corners the same color.

For some good pictures of the definition see the blog: this blog posting.

Conjecture: the 17x17 grid is 4-colorable. (worth $289 if you prove this conjecture, worth nothing if you disprove it)

Theorem: the nxn grids for 1<=n<=16 are 4 colorable

Theorem: the nxn grids for n>=19 are not 4 colorable

William Gasarch is offering the $289 because "I REALLY REALLY THINK THAT IT IS 4-COLORABLE. I could still be wrong."

Here is a possible way to get students started on this problem:
1. Show that the 4x5 grid is 2 colorable. Here are some examples of the 2x2, 3x3, 4x4 grids.

Notice that once you have a c-coloring of an nxm grid then any subgrid of this gives a coloring of a smaller grid. Show that the 5x5 grid is not 2 colorable (hmmm, there are only 2^24 ways of coloring a 5x5 grid with white in the upper left corner).

2. Some extensions: How many ways are there of coloring the 2x2 grid with 2 colors? with 3 colors? with 4 colors? up to rotation and other symmetry? up to switching colors? Extend this to the 3x3. Can you extend it to the 4x4?

3. Find the mistake in the 17x17 grid below:

Hint: This 17x17 coloring was taken from here so this will give away where the mistake is.

4. Write a function (pick your favorite computer language) that tests if a c-coloring of an nxm grid is valid.

Any other ideas?