Math 5020: Fundamentals of Mathmatics for Teachers

Math 5020

Fundamentals of Mathematics for Teachers

Description: Number Theory and Combinatorics are branches of mathematics in which theorems and problems are usually easy to state but often difficult to prove or resolve. This course will deal with topics in these two fundamental mathematical fields, including modular arithmetic, linear and quadratic diophantine equations, continued fractions, permutations and combinations, distributions and partitions, recurrence relations, generating functions, formal power series. The use of computers for mathematical exploration will be encouraged through the computer program Maple. The course will cover material from 'Number Theory' by George Andrews and supplementary material on generating functions and species.

Professor Mike Zabrocki
Office: TEL 3046
Hours: Monday 4-5:30pm
phone: 416-736-2100 x33980
e-mail: zabrocki at mathstat dot yorku dot ca

Class meets Monday 6-9pm in Ross S701

Text : Number Theory by George Andrews

A random picture by M. C. Escher that looks cool

We have a FORUM for math 5020. This is a web site that will allow you to post questions/comments/communications about the course. I will try to keep track of postings and answer them on a regular basis.

I am asking you to keep a notebook/journal consisting of homework problems. I will ask you to hand this in at the end of the semester so that I may look through it and give you a grade on it. Your 'notebook' may be just looseleaf paper but please obtain a folder of some kind before you hand it in.

I will continue to give you problems as the class progresses and assign more problems than I think that you will have time to do. I am doing this on purpose so don't feel discouraged that you never feel caught up. Just keep working! The best way of learning a language is to continue to speak it, and mathematics, like any other language, is no different.

Assignments:

Week 1: Read sections 1.1, 1.2, 2.1, 2.2. Do at least three proofs by induction (I want you to be comfortable with them, 3 is a minimum, do more or all from section 1.1 if you like). Do the first 5 problems from the photocopy of the counting problems that I gave you.
Week 2: Read sections 2.1, 2.2 with a lot of detail and sections 2.3, 2.4, 3.1, 3.2 (see the remarks I made on the Forum discussion). Do your problem and post it to the Math 5020 Forum. Do as you have time: section 1.1 #1-18, section 1.2 #2,4,5, section 2.1 #7, section 2.2 #2, 3, 6, 9, 12, handout on counting problems #1-15. Solutions to these problems should go into your notebook.

Grades for the course will be based on the following criteria:

Homework and other assignments:	50%
Journals:	20%
Final(s):	30%

The course will require the use of the computer program Maple. If you have a home computer you may want to obtain a copy of this program it will be useful to become familar with it. I am in the process of applying for the 'Maple Adoption Program' and we shall discuss the details in class. If you want to order Maple through this program then please e-mail me and I will send you the order code.

Announcements

(Nov 10, 2003) I have made up a web page with some of the maple input that we will be doing in class tonight. There is an HTML version and a Maple worksheet version.
(Nov 11, 2003) Here is the web page/lab that we did last night. HTML version and a Maple worksheet version.

There is an interesting radio show on the BBC by the writer Simon Singe. He wrote a book called "The Code Book" and I am currently reading it. This radio program is really well done. I haven't listened to them all, but I recommend for our class the story about the number 7.

(December 3, 2003) The final exam (part I) that I have assigned to do during the winter break is due Jan 5. The version that I handed out in class had a mistake on the statement of question #9. This version is corrected. I recommend that you sit down and try to do this exam in a three hours sitting, otherwise you could easily spend many more hours trying to complete it piecemeal.

(December 3, 2003) For those of you who are interested in having a list of questions that I asked you to do for your journals I have gone through my notes looking for questions that I have assigned. Here is the list in PDF format because some of the questions were not directly from the book.

(January 12, 2004) Here is the handout on the RSA system with three homework problems I would like you to put in your journals. You will have to do this in Maple or some other computer algebra system. You might want to copy the numbers directly from the pdf file rather than typing them in directly. If there is a problem I will post them here so you can copy them from your browser.

(January 28, 2004) About half of the class was not there on Monday because of the snowstorm. I agreed to put the material that we covered on the web. I was planning on going to the computer lab that night but it didn't happen.

(January 28, 2004) Here are the solutions to the exam.

(January 31, 2004) I have a first draft of the explanation of Lotto 6/49. Please look at the forum to see how I would like to get your input on the document.

(February 2, 2004) I gave you a first question for the final exam. Note that I changed the due date to two weeks from today (Februrary 16, 2004).

(February 10, 2004) The asssignments that I handed out in class can be found here. I have corrected the mistakes that I found on them. Note that I assigned everyone in class to post one answer on the web. Please do it by next week. If you were not in class please e-mail me and I will give you a question to post.
Some connections between sequences and sets of objects: Part I
Some connections between sequences and sets of objects: Part II
Some connections between algebraic expressions and sequences : Part I

(February 17, 2004) The second part of the second part of the final exam was given out last night.

(February 17, 2004) I also gave you some new problems to do to add to your journal. I only didn't assign these for the FORUM because I thought you had enough to worry about with the exam. I will post the answers to the
Some Connections Between Algebraic Expressions and Sequences : Part II
Some Connections Between Algebraic Expressions and Sequences : Part III
Some Connections Between Algebraic Expressions and Sequences : Part IV
Some Connections Between Algebraic Expressions and Combinatorial Descriptions : Part I

(February 19, 2004) I took some of the comments that I got on the forum and incorporated them into a new draft of the Lotto 6/49. I didn't take Jennifer's comments word for word but instead added some of the phrases and edited what I wrote about another 20-30 times before I put it here. I found that the explanation has increased in complexity but is much more complete now. Can anyone help bring it back down to Earth? Anyway see my remarks on the forum.

(February 24, 2004) I gave you a set of problems to add to your journal and I asked you to post one problem from either "Some Connections Between Algebraic Expressions and Combinatorial Descriptions" that I gave you last time or a new set of problems:
Some Connections Between Algebraic Expressions and Combinatorial Identities
Someone accidentally (I assume it was accidental) walked away with the list of problems that I assigned. If it was you, please post this information in the thread on the Forum. If this information does not show up, we may need ordered chaos in order to get the answers to these questions posted (I don't know how many people did not get to see the list before it disappeared) and we can do a first come first serve solve the problem that you like.

(March 1, 2004) I revised the draft of the explanation of Lottery 6/49 to produce version 3. At this point I don't have much momentum on this project, but please offer your comments on the forum (I got none this last week except for one negative one). See my remarks on the forum. I will bring this up in class tonight.

(March 2, 2004) Byron Wall sent me the updated reading list for Math 5400: History of Mathematics for Teachers. This course will be offered Summer of '04 starting in May.

(March 9, 2004) I gave you two handouts last night. One of them we did the exercises in class and was an exercise of matching a set of partitions with the generating functions. The second handout is a set of exercises that I wanted to give you last week on proving combinatorial identities using generating functions. This second handout is for your journal because I want you to be able to answer these types of questions, consider them as 'bonus.' I will not require that you have done these problems when I evaluate the journal but I would like you to attempt some of them. I asked you to draw the Young diagrams for all partitions of 9 with less than or equal to 3 parts and the Young diagrams for all partitions of 9 with parts of size 1,2 or 3. I also asked you to explain the following formula:

(March 21, 2004) At least two people have now asked me for a list of homework/journal assignments that I will be looking for when you hand in your notebook. This list may be updated in the next week or so. The journals will be due on March 29, 2004.

(March 22, 2004) I want to assign you one more Forum question before the end of the term on generating functions for numbers of partitions. I have made up a list of 30-some odd random questions, some are slightly harder than others so don't be afraid to ask me for a hint.

(March 23, 2004) I will be looking for an explanation of three identities in your journals. I have provided a proof of the first of these so that you can see how to do this. The other two are very similar and I hint how they are done in this explanation.