Instructor

Generalities

For Fall 2008 Mathematical Inquiry (MTH 487) at the University of Massachusetts Dartmouth will be based around a number of general mathematical topics and unsolved problems.

The aim of the course is to give you experience in thinking and working like a mathematician.

The way the course will work is through the class working on specific problems. It will help, during the semester, to keep the following words of Jurgen Moser (1928 – 1999) in the fore-front of our minds:

“… it seems idle to argue whether to prefer solving of challenging problems, building abstract structures, or working on applications. Rather, we should keep an open mind when we approach new problems, and not forget the unity of mathematics. “

In this course you will increase substantially the amount of thinking on how you think about mathematical problems. So, while working on, or solving, problems is indeed the goal, there is another goal, which is to make as explicit as possible everyone’s thinking about how to tackle mathematical problems.

Problems will come either from me, as instructor, from you, or from guest instructors.

Assessment

Assessment will be based on:

• Active class participation (as many points for being wrong as for being right)
• A diary of work on problems, including thinking about solution strategies
• Write-up of solutions

So, to get a really good grade for this course, think of, and express, lots of ideas for attempting the problems, write up your and other people’s ideas and always keep trying to draw lessons from solution strategies, and bring interesting problems to class for the rest of the class to work on.

This is probably the first class you will have attended where you can get good – very good grades by being wrong! Not that being wrong is a desirable state of affairs. However, being wrong is better than being ignorant. Far better: if we are wrong in a guess then at least we had an idea.

So, in this class, you will be rewarded for having ideas, and for working on and polishing those ideas that lead somewhere.

Class expectations

1. Participation is expected. This means actively speaking up when you have an idea, when you see how to extend or modify someone else’s thinking, when you think someone else – instructor included – is in error. You are expected, at all times, to express your opinions, feelings and thoughts. This class is a cooperative venture in learning how to think mathematically in a community of like-minded people. Active participation is essential. Keeping quiet and thinking things through later, outside class, is failure to participate fully in the community, unless those later thoughts genuinely feed back into the discussions. This is a different sort of class, in which failure is not possible, unless it is failure to participate.

2. You must keep regular, dated, records of your thoughts, conjectures, guesses, and calculations. These MUST be in LaTeX format – a simple way to do this is to add your comments etc. to this set of pages. To do this you need to have a WordPress account. Please arrange to do this no later than the second class meeting. Email me with your WordPress username and email and I will register you as a contributor to this site.

3. Absences, unless due to illness or other unavoidable situations, such as death of a family member, are unacceptable. Failure to attend class regularly – 3 or more unexplained absences – may result in a grade of F, without further discussion. This sounds stringent, but this class is about people meeting regularly to actively participate in mathematical exploration and inquiry. Missing classes on a regular basis will not cut it: the community will be weaker without each person’s full and active participation. Please come to class each class session, and participate actively.

Relevant advice from Terry Tao

1. Solving mathematical problems

2. There’s more to mathematics than grades and exams and methods

3. Ask yourself dumb questions – and answer them!

4. Write down what you’ve done

5. Write a rapid prototype first

6. Make your work available

7. Work hard

8. On time management

9. Continually aim just beyond your current range

Resources

The problems and issues to be considered in this course are all open to a computational approach – particularly to the gathering of evidence to suggest what might be true and provable.

The class will be held in the Department of Mathematics computer lab, room 218 in the Liberal Arts Building. The Apple machines in this room are equipped with a variety of mathematical software – Maple, Mathematica and Matlab in particular.

Sometimes a programming language such as Perl or Python can be very useful.

Additionally, Matlab is available in an Open Source version called Octave, from GNU.

All write-ups must be in LaTeX (or a version of TeX). Useful TeX editors are TeXnicCenter and LyX (pronounced “Lick“). LyX is probably the easiest to use, particularly in relation to graphics. However, WordPress has a built-in Latex editor. All you have to do to get TeX in WordPress pages is use the command

(remember the “latex” which you do not normally insert when you use a TeX editor). Images are also very use to insert into WordPress pages.

A latex_manual in pdf format is available, and a short list of commonly used mathematical symbols is available here.