MATH 2590 - Thinking Mathematically I
Professor: Mike Zabrocki
Office: TEL 2028
Office hours: Monday 4-6pm and Tuesday 3-5pm
Textbook: Thinking Mathematically by Mason, et. al.

Course description (from the calendar copy): The main objectives of this course include providing opportunities for students to achieve success in thinking mathematically and to reflect on the learning and practice of mathematics. Intended primarily, but not exclusively, for Education students in the PJ and JI streams. Prerequisite: Successful completion of at least 24 credits or permission of the course director. Note: This course is not open to any student who has taken or is taking another university mathematics course unless permission of the course coordinator is obtained. NCR Note: This course may not be taken for credit by any student who has taken SC/MATH 1580 3.00 prior to Fall 2009: NCR Note: This course may not be taken for credit by any student who has taken AS/SC/MATH 1580 3.00.

We will explore the meaning of "Thinking Mathematically" with several components of this course. First, we will explore some problem solving following the book "Thinking Mathematically" by Mason, et. al. The techniques developed in this book to attack 'easy-to-state hard-to-solve' problems are a good demonstration of the 'art' of mathematics. Solutions develop over a process of understanding the problem, followed by writing and explaining to express to others why your answer is correct. The other main component of this course will be to show places in everyday life where mathematics is used in technology, art and science to use as a starting point to discuss mathematics in the classroom. This will be sort of 'A Mathematician Reads the Newspaper' (John Allen Paulos) but with other types of topics.

Grades: The grade for this course will be broken down into three main components. Attendance is mandatory and your class participation is required. I will partially measure this by taking attendance and your overall grade will be marked down if you are not in class. Each evening course covers 3 hours and should you miss two courses this would consist of a significant amount of material.

Projects: 40%
Journals: 30%
Assignments: 30%
Descriptions components are listed below and more details will be provided in class.

Journals (ongoing, final version December 7): This will be your record of the problems that you work on in class and your progress on them. You should obtain a bound notebook that you plan to hand in at the end of the term (because I will be unable to evaluate ongoing assignments). You should divide this notebook into sections where you write about the problems that you are assigned and work on for this class. Your journal will be evaluated at the end of the course and I will ask you to hand it in on December 7. The journal should: (1) show that you made progress and solved all or most of the problems you were assigned (2) demonstrate how you arrived at your solutions by the examples you did and ideas that you pursued, (3) list resources (including fellow students) you consulted (4) at the end of the problem, provide a summary which explains the answer to the question to someone that does not know what the problem is about (5) your entries should consist of writing which is clear and grammatically correct (6) explains the most general statement you can make about the question (e.g. if the question is to explain something about an 8x8 grid, don't just tell me what the answer is for an 8x8 case, I want to know what happens for an nxn checkerboard for any positive integer n) (7) a complete solution should go beyond simply copying what we did in class since I will expect you to do work on these problems at home and extend beyond what we did together.

Assignments (ongoing, roughly every other week): These will consist of a small set of problems that I will give you in relation to the topic under discussion. Usually these few questions will develop after our discussions and will require you to learn some mathematics and hopefully the assignments will make it easier to see the relationship between the applications of mathematics and our topic.

Projects (topic title and description due Oct 5 (15%), written versions due Nov 2 (revised due date Nov 9) (40% script, 20% math questions/exercises), final audio version due Nov 23 (revised due date Nov 30 (25%)): This component will consist of an oral and written report that will be submitted either alone or in pairs. I would prefer if you work with someone on this project, you will find that it will be easier to learn new technology of making a podcast with two heads rather than one. The written component of this assignment is a script of an explanation of a topic (of your choice) and its connection with mathematics. This should be accompanied by mathematics exercises that ideally ties math concepts to the topic of your report (one or two exercises should be fine). I am looking for you to pick something from everyday life, something that you wouldn't normally associate with mathematics. I would like you to try to understand your topic well enough to explain how mathematics is related. To make this project out of the ordinary, I want it to be submitted as a podcast and the written component will be the script. Discussions in class will be similar to the type of projects that I would like you to explore.

Some advice about your scripts:
(1) When you write the script, start with an outline and break the topic you are trying to explain into short compartmentalized ideas.
(2) Use simple language and simple sentences. Each sentence should contain one idea.
(3) When I did mine I recorded one sentence at a time. If I didn't like it, I erased the sentence and redid it.
(4) One page formatted as `sentence, space, sentence, space, ...' seemed to be about 2 minutes of recording.
(5) Start with a script and read your script as it appears on the page (don't try talking and then write your script afterwards). It will probably be the easiest way to complete the podcast.
(6) You are reporting, tell a linear story with your script. Start by introducing the subject and develop it so that you make a connection with mathematics.
(7) Make sure that your facts are accurate and precise. You should provide references with your script if you reference facts which cannot be verified easily.
(8) Try not to do any calculations in your story. In my second podcast it was very hard to follow because I tried to convince the listener by explaining a calculation. This sort of material does not lend itself well to a podcast. For this reason you should just try to avoid calculations altogether.
(9) Try not to use complicated vocabulary. If you do need to use words that people are unfamiliar with, then explain it. Generally, explaining the vocabulary will probably help advance your story too.
(10) Don't just assume that because you use buzzwords like "mathematics," "sine wave," "symmetry," etc. that you are telling me how math is used. You will almost definitely have to do some real research to learn something new to understand the connection between math and the subject you chose. It is that element that you want to communicate to your listener in the podcast.
(11) Have your friends listen to/read your story and have them ask you questions to help you fill in any blank spots.

I will evaluate your scripts for the following criteria: introduction of the topic, explanation of the topic, development of the story, making the connection with mathematics, correctness (factual), grammar and spelling. Overall I expect there to be continuity in the story, that there are no really vague or imprecise statements, and that new vocabulary is explained.

Some advice about the corresponding exercises:
(1) Try to build on mathematics that you are familiar with, but don't be afraid to introduce something new.
(2) You can either use your exercises to introduce your topic or as a follow up after the listener has a chance to hear your podcast. Try to make it clear which you intend. (e.g. in my first podcast I included exercises about bit manipulations and colors which are almost independent and would best follow my podcast, in my second podcast I introduced the exercises first and even used the results as part of how I explained the topic).
(3) Don't worry if your exercises are tangentially related, but they should try to give an idea of the type of mathematics that is involved in your subject. If your subject happens to be something like 'public transportation planning' then actually finding the optimal route between Downsview and York University is directly related to your topic, but it does not make a good exercise. However, a few questions on graph theory or finding the line that minimizes distances between points is something you can research in textbooks and is related to your topic.

I will evaluate your exercises on the following criteria: questions related or tangentially related to your topic, everything clearly defined (perhaps supplementary material included), solutions should be provided separately from the questions.

Important Links: York University Important dates, Mathematical Moments from the AMS (ideas for your projects), Mike's web page.
Other material:

(Sept 14, 2010) Pictures from class :pic1, pic2, pic3, pic4, pic5, pic6, pic7, pic8.

(Sept 14, 2010) Videos: video 1 - queen starting point for discussion, video 2 - computer generated 3d graphics.

(Sept 21, 2010) The Binary System, Wikipedia: entry on binary or base 2, Wikipedia: XOR, AND, OR, NOT, IMPL.

(Sept 21, 2010) Assignment #1 - due Oct 5

(Sept 24, 2010) Problem #2 for your journals that I told you about Sept 20. We will discuss this Sept 27 class.

(Sept 28, 2010) A copy of the syllabus that I handed out last week.

(Sept 28, 2010) My podcast on digital images when you watch this look for the images which are embedded in it (without a display screen you will only hear the audio). Here is the script. Here is my proposal, however this one was written after the fact.

(Sept 28, 2010) Problem #3 - think about this one for Oct 5 class

(Oct 6, 2010) The audio clip from BBC radio that I played last night. This is a single episode of a radio program by the author Simon Singh about the number 7. He also did 14 other episodes that are all very interesting.

(Oct 6, 2010) The second assignment that I gave you about modular arithmetic and card shuffling. Next time I will try and show you what they have to do with each other.

(Oct 6, 2010) Examples of Modular Arithmetic: Reference 1 (wikibooks: High School Mathematics Extensions/Primes/Modular Arithmetic), Reference 2 (BetterExplained), Reference 3 (University of Regina), Reference 4 (Rutgers), Refence 5 (Tutor Vista).

(Oct 25, 2010) The podcast on shuffling and the script (see 'announcements' for my comments about this)

(Oct 25, 2010) Problem #4 for discussion in class on Oct 26.

(Oct 26, 2010) Problem #5 for discussion in class on Nov 2.

(Oct 26, 2010) Assignment #3 - due Nov 16 - (assuming that we have time to finish Nov 2)

(Nov 7, 2010) The podcast on cryptography and the script.

(Nov 9, 2010) The podcast on International Banking Account Numbers and the script.

(Nov 9, 2010) Assignment #4 - due Nov 23

(Nov 9, 2010) Problem #6 - for discussion in class Nov 16.

(Nov 22, 2010) Problem #7 - for discussion in class Nov 23. 'Square Take Away' from Mason. Too many things to do so we didn't get to this Nov 23. Will do Nov 30.

(Dec 5, 2010) Assignment 5 - do problem #1&2 for Dec 7, we will work in groups on the rest.

(Dec 7, 2010) The recorded experiment from last week - I sped up the recording a bit so that it doesn't sound like it is in slow motion. It didn't turn out as good as I would like since the claps seem to drown out the other sounds in the recording. another experiment with pasted sounds.

(Carmen Plank and Nicole Lee) a podcast on the mathematics of origami and an application to medicine Finding Math in the Folds in MP3 script exercises solutions.

(Mary Michael, Sharla Niroopan & Jocelyn Santos) podcast on a facial detection algorithm podcast in MPEG 4 script and exercises solutions to exericses.

(Stephanie Colangelo and Sara Russo) podcast on digital and analog hearing aids podcast MPEG 4 script exercises solutions

(Maria Demude) podcast on mathematics and art podcast MP3 script exercises solutions solutions pt 2

(Sarah Tang and Caroline Wadid) podcast on kidney transplants video on YouTube script exericses

(Mark Fernandez, Kimberly, Vhenus) podcast on digital sound podcast in MP3 script exercises solutions

(Nikole Daws) podcast on Shazam podcast on YouTube script exercises

(Julia Warner, Fatemah Dhirani, Katrina Booth) podcast on digital sound podcast wav script and exercises solutions

(Arthur Frustaci) podcast "Taking the Luck out of Chance, and replacing it with - Math!!" podcast mp3 script and exercises solutions

(Dimitrios Kalfopoulos) podcast on computers use of boolean logic podcast mov script and exercises

(Jamila Husbands) podcast on music podcast mp3 script exercises solutions


(Sept 14, 2010) The textbook for this course "Thinking Mathematically" currently has a new edition. If you have a older edition of this textbook it should suffice, however if you find large differences between editions you may need to let me know so that I can provide you with missing information.

(Sept 14, 2010) I have some inside sources on Mathematics education events. Here are two:
Fields Mathematics Education Forum, next event is Saturday, September 25, 2010, 10AM - 2PM Fields Institute, 222 College Street, Toronto.
2010 Mathematics Conference for Elementary Teachers at York University Wednesday, October 13, 2010

(Sept 24, 2010) Posting from the department: The following information about detailed online resources covering the basics of algebra and trigonometry might prove of some use to those students who are struggling as a result of deficiencies in their high school backgrounds.
(1) There is a quite extensive set of algebra tutorials, covering a wide range of topics, maintained by West Texas A&M University. The URL for the main page of this resource is though a quicker way to get to this page is via the link on the Bethune College Math Help page,
(2) There is also a detailed online course in Trig basics, which starts right from the beginning, the URL for the main page of the course being (as you can guess from this URL, the Bethune Math Help page has a link which takes you to the Trig course main page). Other potentially useful information is also listed on the Bethune Math Help web page, and further online resources will be added there as they become available.

(Sept 24, 2010) Katrina Booth is a student in our class and is having a family emergency and was unable to make it to Tuesday's class. She would like someone to send her the lecture notes from Tuesday's class if possible. Her email is

(Sept 27, 2010) Don't forget that on October 5 I want your title and a brief outline about your project that will be due later this term. The title and outline is worth 15% of the total grade for the project. I have placed a few more details in the description above about what I expect on this project and now you have a completed example that you can (roughly) follow as a guide. You will want to find a partner and discuss with him/her about the topic you want to cover. Although it does not need to contain specific details, the connection with mathematics should be clearer than: "I will look at the algebra/trigometry/calculus related to my subject." Your outline should make clear how you plan on connecting mathematics with your topic. I am placing on the web page my first podcast as an example of what I am looking for. FYI, I included images in my podcast (a) because it was very easy to do with Garage Band and (b) because the subject of my podcast was very visual. This is not a necessary component and might even be very difficult to do if you don't have access to "Garage Band" on a Mac.

(Sept 27, 2010) Finish the first assignment for Oct 19 (remember that we don't have class the week of Oct 12) instead of Oct 5 as originally assigned. I will explain the third problem next time. I will also give you another assignment next week.

(Oct 6, 2010) In case you might be interested in some upcoming conferences for math educators. The Toronto Educators Association for Mathematics (TEAMS) is holding a mini-conference Thursday, October 28 4-8pm. (poster). Also, the Grand Valley Mathematics Association is holding a Fall conference Thursday, November 4, 2010 (Please register by Thursday, October 14) 3:45 - 8:45 p.m in Waterloo.

(Oct 25, 2010) Here is the podcast that I presented on Tuesday night. Remember, this one is a little more technical than I really liked and really required that you do the second assignment in advanced to really understand. Even then, I think it was difficult to follow the calculations. Live and learn. The podcast on shuffling and the corresponding script.

(Oct 25, 2010) The topics that you submitted were mostly good (with a few exceptions where we clarified those with individuals about those), but there were a few main issues that I had with the submissions. The first was that you should keep in mind that you are reporting on the use of mathematics in art, science or elsewhere. This is a report and I want you to make connections it with a few mathematical exercises. Start to give thought to what sort of exercises you are going to give. Some of the ones that you described sound really hard. I want to move the due date for the script back a week. Nov 2 is just a little too soon because I want to give you some indication about what I will be marking for on your scripts in class on Oct 26 and I didn't give them back until Oct 19. The script and exercises are due Nov 9 and the podcast is due Nov 30. I have increased the list of "advice" tips in the instructions for the podcast above.

(Oct 25, 2010) The web page did not get changed online until today! Sorry about that. I had all of these new announcements and files to add and I just had them changed on my local copy of the webpage. I added a lot of discussion on the podcast. Make sure you read it and we can discuss it in class tomorrow.

(Oct 25, 2010) Here is a really good example of what I WISH my podcasts sounded like. This is a 3 minute story on NPR that is an interview with a scientist that found a formula for how fast an animal must shake in order to dry off.

(Oct 26, 2010) I will try to tie together all the ideas that we discussed associated with the thrid assignment next week. I kind of showed you how to do the problems, but I am fully aware the "why" was a bit opaque in some of these problems. What you should realize is that every transformation that I did on letters (e.g. add 14 (mod 26), multiply by 5 (mod 26), multiply by 11 and add 20 (mod 26)) there was a transformation that reversed the process (e.g. subtract 14 (mod 26), multiply by 21 (mod 26), multiply by 19 and add 10 (mod 26)).

(Nov 7, 2010) For those of you who weren't in class, I moved the due date for the 3rd assignment back a week because we didn't have time to cover the last problem on either part of the assignment. FYI, the homework is not self-explanatory.

(Nov 9, 2010) I made some changes to my scripts and added some references (not much, but it is where I got my information). I will recommend the book where I got some of the information on cryptography. It is by Steven Levy called Crypto: How the Code Rebels Beat the Government Saving Privacy in the Digital Age and he tells a really good story.

(Nov 22, 2010) I made the announcement in class last week. It is not necessary to do ALL of Homework #4 for class this week. Just question number (1). We will do the other questions in class.

(Nov 29-30, 2010) To hand in your podcast you will need to send me your .mp3 (or other file format, but I think that mp3 is the preferred way ... so far I've gotten .m4a, .mp3 and .wav files). One way to do this is by email. Another way is by giving me a USB drive that I can plug into my computer and copy the files in class on Tuesday. Make sure that you also send me an electronic version of your script, exercises and solutions to those exercises (I would like the solutions in a separate file). I've gotten some podcast submission already and it turned out great! I will be posting it shortly so have a listen.

(Dec 1, 2010) Your journals are due next week. Follow the instructions above and you will do fine. If you have any questions on those instructions please email me.

(Dec 1, 2010) Please do question 1 (the survey for the course) and question 2 (find SID (mod 5) + 2 and SID (mod 3) + 7) for next week. We will get in groups to work on these problems. I have only one other subject for the agenda next week so I am hoping that it is a light evening.

(Dec 1, 2010) I got bombarded with emails to hand in your podcasts and I am having a hard time knowing for sure if I haven't missed a file or two. If one of the files that you sent me does not show up on the webpage please email me and let me know.