## Fundamentals of Mathematics for Teachers

 Professor: Mike Zabrocki Office: Ross N518 Hours: Thursday 4-5:30pm phone: 416-736-2100 x66085 e-mail: Class meets Thursday 6-9pm in Ross N812 Text : Number Theory by George Andrews
 Description: Number theory and combinatorics are branches of mathematics in which challenging problems can be explored that require a background with which most students are familiar. This course deals 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 and generating functions. It is one of two required courses for the Mathematics for Teachers program. An emphasis in this course is placed on writing and explaining mathematics clearly. Prerequisite: Permission of the instructor is required for students who are not in the Graduate Programme in Mathematics and Statistics.

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

 Class presentation 10% Un-exams (4 x) 5%+10%+15% Forum exercises Fall 20%+Winter 20% Class participation (incl attendance) Fall 10%+Winter 10%

(September 10, 2015) Moodle site for this course
(October 1, 2015) videos: Vi Hart, Numberphile, Khan Academy, Patrick JMT, ASAP Science, hyperlapse
(November 5, 2015) Widget and doodle matching
(November 5, 2015) Widget and doodle non-matching
(November 9, 2015) Math video from Breakthrough Junior Challenge
(November 12, 2015) Coefficients in generating functions
(November 19, 2015) first unexam
(November 23, 2015) A YouTube playlist for Math 5020 consisting of your videos
(January 14, 2016) Examples of generating functions
(January 15, 2016) Number theory review problems
(January 15, 2016) oeis.org questions
(January 28, 2016) More number theory questions
(February 4, 2016) generating functions from recursions
(February 4, 2016) bigger library of generating functions
(February 26, 2016) combinatorics and generating functions
(March 8, 2016) Next unexam (we are skipping #2)
(March 17, 2016) A nice article explaining Fermat's Last Theorem
(March 24, 2016) Matching sets of partitions with their generating functions
(March 24, 2016) Generating functions for sets of partitions
(March 24, 2016) F. Franklin's proof of Euler's pentagonal number theorem
(April 2, 2016) My proof of the identity in the book

## Schedule

 Date Topic Notes Sept 10 Induction, Telescoping sums, Sections 1-1, 1-2 HW do some induction problems, telescoping sums too Sept 17 Sections 2-1, 2-2, reading mathematics Sept 24 Fake theorems, linear diophatine equations finish proof, pick topics for videos Oct 1 addition and multiplication principles Oct 8 generating functions, distributions, Fermat's Little theorem + generalizations Oct 15 Review Fermat's Little Theorem, equivalence, card shuffling Oct 22 Card shuffling, euler-phi, generating functions have drafts of script ready to record Oct 29 co-curricular days Nov 5 Euler phi, more widgets and doodles Nov 12 generating functions, moebius inversion Nov 19 watched the videos videos due Nov 26 mobius pairs, necklaces, more coeffs in g.f.s. Dec 3 more coeffs in g.f.s, derive a library of g.f.s. first unexam due Jan 7 a taste of cryptography, finish library of g.f.s. (except for #11-13 h.w.) Jan 14 number theory practice, oeis.org and example forum question assigned Jan 21 primitive roots, discrete log, number theory exercises, sage Jan 28 factoring, discrete log, raise to power, primality testing, sage more number theory questions Feb 4 more number theory problems, generating functions from recursions, library of gfs do the last number theory problem, work on the library of gfs problems Feb 11 finished the big library of g.f. Feb 18 Reading week Feb 25 identities, combinatorics and generating functions Mar 3 combinatorics and generating functions, adjacent quadratic residues next unexam given Mar 10 partitions and generating functions Mar 17 unexam bonus, partitions, partition matching worksheet, fire alarm unexam due Mar 24 Weather emergency: class postponed Mar 31 Euler's pentagonal number theorem, gaussian binomial coefficients, more partition generating functions Apr 4 infinite sums and and infinite products Makeup class, class evaluations, unexam assigned

## Announcements

(September 10, 2015) We have a Moodle site for this course which you will need to access soon, but I will place most announcements and handouts on this page because I have more control over the content and archive process.

(September 10, 2015) I use this page as a running calendar, but I often fill in the precise schedule after the fact (because what we actually accomplish in a given evening and what I plan to accomplish are not always equal). I have done the same in previous years and if you would like to get a rough idea of schedule and content, refer to previous course web pages (2003-2004, 2005-2006, 2007-2008 2009-2010, 2011-2012).

(September 17, 2015) In past years I've had students do in class presentations. I want to continue that practice because the main point of this course is about communication, but with an effort to modernize the tradition. Instead of presenting the work in class, I will ask you to make a video. This exercise involves scripting your presentation, adding pictures and visual effects (and we can talk about different ways vloggers are doing this), and recording a voiceover. The tools to do this are available for free on most computers and even some phones. I will provide you with additional details and a list of potential topics in the next couple of weeks but I wanted to start talking about it now. This is similar to a contest that you could get your own students to participate in called the Breakthrough Junior Challenge.

(October 1, 2015) Here are some examples of videos communicating mathematics and Science:
Vi Hart's channel
Numberphile channel
Patrick JMT
ASAP Science
A research video on hyperlapse
hmmm, were there more? Watch these videos with an eye towards certain elements. Notice how the explanations were scripted, how the images were generated, how they were timed, transitions and the role that the narrators play in the videos. You will need to communicate content in a way that holds the interest of your audience. My advice is that you should remember that a video can be replayed if it explains something too fast and it can (will) be stopped if it is too slow. Your assignment is to explain a topic in the book in a video that is less than 10 minutes long (and preferably less than 5 minutes in length). The videos are due Nov 12.

(1) Go to Sage Math Cloud and sign up for an account
(2) Open a new project
(3) Click on the (+) New tab and then select |x| LaTeX Document.
(4) You can start writing beautiful mathematics documents in the text area and the corresponding pdf appears to the right. Equations should start with two dollar signs and end with two dollar signs if you want them in a line by themselves, $$\int \hbox{like this} dx$$ one dollar sign on either side and they will be embedded in the text $\theta$ like this.

(October 9, 2015) I got a new Moodle page for this course and I have updated the links elsewhere on this page. If you have trouble logging in, let me know. I think that I have the power to enroll students on this page. It will mainly be used for posting solutions to problems. Remember when you do post a solution, make a new topic with a brief description in the header and write a full description of the question you are explaining.

(October 9, 2015) You have two homework problems for next time.
(1) find a value of $X$ such that $6$ divides $n^6 - n - 3X - 2{ n \choose 2}$. Your $3X$ should be the number of words of length $6$ in the numbers $\{1,2,\ldots,n\}$ such that there are three distinct cyclic shifts of these words.
(2) post a problem on the forum. I will try to get into a rhythm of checking the forum and responding to your solutions.

(October 16, 2015) You have two things to complete for next week. (1) post your problem on the forum (2) a draft of your script! I will need a hard copy and because there is no class on Oct 29 I will try to give you feedback for it over email.

(October 16, 2015) I was looking through the videos on Numberphile and I noticed there was a sequence of 3 (Part 1, Part 2, Part 3) on Persi Diaconis and another one about shuffling the best and worst ways. I know this guy too: Federico Ardila.

(October 29, 2015) I wanted to have the feedback for those of you who handed me your scripts by tonight. I have read them all, but I will have to send it tomorrow (I'm feeling under the weather). Those who did not finish on time, please email me your scripts. They are late and it will be tough for me to respond and for you to finish recording them on time.

(November 5, 2015) Please remember that next week the videos are due. We will be watching them in class, but please don't try to email them, bring them on a usb drive, or your phone or upload them to youtube.

(November 9, 2015) Back in September I added a link to the Kahn Academy's Breakthrough Junior Challenge. The winners were announced yesterday and there was only one video that I thought was in mathematics. I thought the student did a really good job. It is something to aspire to.

(November 19, 2015) I have to say that the videos you all did were AMAZING! All of them were a really good job. I will provide links to them later this week.

(November 19, 2015) I had said that I was going to assign the first unexam. This assignment will be due December 3. I've provided instructions in the file.

(January 14, 2016) Last week towards the end of class I asked you to do number 11, 12 and 13 from this worksheet.

(January 15, 2016) I assigned you all a forum problem. You will find that there is a limit of what you can compute by hand for your question. Next week we can talk about how to use computers to push that further (and check your answer). If there is something unclear about your question please ask, but I will ask you to be explicit about what sequence you are computing.

(February 26, 2016) Please work on your combinatorics and generating functions problem for the forum. I asked you to do part (b) for one question. I'm asking you to post this because I want the chance to read how you are explaining the relationship of combinatorics and generating functions. The unexam will be issued soon and you will need to explain these types of combinatorial problems.

(March 8, 2016) At the March 3 class, we talked about adjacent quadratic residues. I pushed the calculation of the number of adjacent quadratic residues to a point, but then there was a calculation on p. 130-131 that I thought was better not left to classroom time. I asked to read that section and do Exercise #1 on page 132.

(March 8, 2016) I gave you the next unexam. We will only have three this year. When you write your solutions please follow the type of advice that I am giving you on the forum. Precision in your language you use to describe is super important. One other piece of advice: follow other's advice about writing mathematics.
1. A Guide to Writing Mathematics
2. Tips on Writing Mathematics
3. Rules and Tips for Writing Mathematics
4. A Few Tips on How to Write Mathematics

(March 24, 2016) Class is cancelled this evening because of a weather emergency. It will be rescheduled by the university at a later date.

(April 2, 2016) The class cancelled due to the weather will be made up Monday April 4, 2016 6-9pm and will be held in S525.