## Friday, December 18, 2009

### Program Overview

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

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?

## Monday, November 16, 2009

## Friday, November 13, 2009

### Move over calculus, now there's something meatier

In Ontario there is a data management stream of the mathematics curriculum. I do not know if that is the case elsewhere in the U.S. so Ontario is ahead of the game on this one. I believe that what this means is that his idea is not as novel as he makes it seem. Moreover, in Ontario we are facing the same educational challenges and such a change does not create a dramatic difference. To see math scores dramatically improve I think that we will have to look to other solutions.

## Tuesday, November 10, 2009

### Math waterboarding

## Monday, October 26, 2009

### The Nobel prize in economics

Mathematics is closely linked to many branches of science. Although not necessarily highlighted, the

*use*of mathematics is recognized, especially with the physics and economics awards. I listen to NPR's Planet Money podcast which covers all things economics and they recently posted an interview with Elinor Ostrom, one of the winners of the Nobel prize in economics, where she explained some of her research.

Her research is about 'the tragedy of the commons' (highlighted in a 1968 paper by Garrett Hardin) which says that if there is a common resource (e.g. fishing grounds that the public makes their living from by harvesting fish), then if this resource is limited then it will ultimately be overused until it is destroyed even if it is not in the public's interest for this to happen. The only way that this can be prevented is for the government to step in and through enforcement, stop the overuse of this resource. In the example of the fishing grounds, the government must come in and put limits on the amounts of fish that people can catch otherwise the grounds will be over-fished.

The discussion in this podcast with Elinor Ostrom highlights that this is a mathematical model which can be used to predict the behavior of a group's action. As she says in this podcast, this model "can be used mathematically to predict outcomes when the problem is pure private goods and you have a highly competitive market." She then goes on to say that it is a challenge to prevent this tragedy from occurring, but since humans are highly complex, this model does not always apply and that solutions for solving the tragedy of the commons can come from within the group.

Some of the ideas about her research became crystal clear in this interview when she was asked about the communal fridge in the office. The 'tragedy of the common fridge' is that it must eventually degrade into a repository for moldy sandwiches. The research of Elinor Ostrom says that, no, in fact human beings are complex enough that they can develop a solution from within the group.

I'm a fan of the podcast because I would like to be more economics literate. Economics reasoning is complex and this podcast looks at a lot of different aspects of it. If you are interested in hearing another good story about how mathematical models can be used to predict behavior, the podcast a day earlier talked about monkey behavior seems to follow a mathematical model for supply and demand, even though monkeys probably don't understand the meaning of 'one half of.'

## Sunday, October 18, 2009

### Too many being left behind

It is worth reading the blog entry posted by the New York Times because I found it was way better than any of the news articles that just summarized the test score results.

Here is my summary of the opinions that they highlighted on this blog post

* the education policy professor says: "The culture of standardized testing ... has served to de-skill and demoralize our best teachers." But he offers no real suggestions about what to do.

The other 4 seem to fall into two main groups. The first says improve the skills of teachers:

* the education studies director says: "Teaching methods, curriculum, lack of adequate subject matter knowledge among math teachers and lack of real consequences in school accountability systems, rather than tests and standards, could be the real culprits for low scores." He highlights research that says that eighth graders taught by math majors did better than eighth graders taught by teachers without a math major.

* the mathematician says: "Many elementary teachers have strong backgrounds in reading and writing, but will readily admit their discomfort with math." This is yet another call for ensuring that math teachers have a strong math background. This pundit also falls into the second group and has comments about the math curriculum.

The second group says improve the curriculum:

* the parent says: "If we want to improve mathematics education, we should banish nonsensical curricula like Trail Blazers, Everyday Math and Investigations and make sure that our teachers are properly educated and proficient in math content." To paraphrase, this 'fuzzy math' is a joke and it is not preparing students to succeed in math (what is the color of infinity?).

* the policy analyst says: "What is needed is not another test, but sound mathematics instruction that stresses content over process." He says that content is very important and we are not doing students any favors by not teaching concepts and just teaching problem solving.

My opinion is that both of these are correct. Just saying 'no child left behind' three times, clicking your heels together, and waving a bunch of money around isn't going to solve any problems. A real solution is improving teacher training and getting a content rich curriculum in place.

## Saturday, October 10, 2009

### Teach the rule or teach how to derive the rule

"Why is a negative of a negative a positive?" and "why is a negative number times a negative number a positive number?" My first thought when she asked me these was 'because' and my second thought was 'chicken sandwich now or a donut to tide me over until I get home?' It was particularly ironic that she was asking me these questions because on the first day of the 'Fundamentals' class I ask at least one similar question to gauge how well students are able to respond as part of their math background (and I probably got a few answers like 'chicken sandwich').

These are probably types of questions that teachers of basic math get asked all the time. Mathematicians encounter statements like this all the time too and we need to justify them in terms of the axioms.

To answer the question "Why is a negative of a negative a positive?" we need to know what 'negative' means: for any element x, -x is the element for which x + (-x) = 0.

This also means that -(-x) + (-x) = 0, so x = 0 + x = (-(-x) + (-x)) + x = -(-x) + ((-x) + x) = -(-x) + 0 = -(-x).

The second question, "why is a negative number times a negative number a positive number?" can be justified using the fact that a negative of a negative is a positive: 0 = a*0 = a*(b + (-b)) = a*b + a*(-b)

so a*(-b) = -(a*b) because a*(-b) is the thing you add to a*b to get 0.

Now (-a)*(-b) = -((-a)*b) = -(b*(-a)) = -(-(a*b)) = a*b

When I was asked this question, the 'fundamentals' student stated it as "Why is 3-(-5) = 3+5?" And to me the good answer to tell a elementary/high schooler would be "well, because that is the rule" unless I really wanted to drive them away from math. To me students just have to follow the rules of basic arithmetic and they have to learn them forwards and backwards. Her explanation was something like "We know 3-(-5) = 3+(-1)*(-5) = 3+1*5 = 3+5." And this explanation was no more convincing to me than my "because that is the rule" answer.

The more I think about it, the more I realize I am not sure how I would respond when asked by an elementary or high school student.

So here is where I get all philosophical (or perhaps I am getting not philosophical, but practical). Should you teach your students how to derive the basic rules from the definitions so that they are easier to remember, or do you just tell them the answer is "because that is the rule" ? As a teacher, do you need to know enough basic algebra to derive these rules yourself or can you accept them "because that is the rule" ?

## Wednesday, September 16, 2009

### Math Task Force

The interviewer asked me about 5 questions which I responded to by email. I thought I would post my (mostly) complete answers to the quotes that he extracted for the article. He asked, "what sort of skills (if any) are students lacking or what seems to be their primary weakness?"

My response was:

I know you are going to be disappointed with this answer, but I won't tell you that I perceive patterns of weakness in students. I am not convinced that anybody knows what is causing a high dropout rate from math courses. I know that people are trying to determine what it is and I will refer you to them to let you know what they found.

For almost any mathematics course we will assume that students have mastered basic algebra, trigonometry and arithmetic. This is true for the calculus and statistics courses and it will be true Math 1200 course that I will be teaching. There are other important implicit skills and mathematical notation that are harder to describe such as pattern recognition and manipulation of symbolic expressions. Students who have not mastered these skills will face more struggles than those that have.

The bridge we hope to provide for students with the Math 1200 course is to improve their of writing, logical reasoning and explanation. In order to succeed in some of the courses for their major, students will be required to explain why a statement is true rather than simply calculate or apply formulas.

One thing that I have found is that high school mathematics tends to prepare students to anticipate solving certain types of problems and look for a pattern or a set method that they can apply. This is an important skill that students must master, when they arrive at university they will also need and develop other skills and other types of reasoning. It is not unusual that students face a bit of a shock when they are forced to encounter problems that have open ended solutions and are perhaps designed to break all previous expectations of questions they encounter. A good example of this type of question might be something like:

if at a meeting of 10 couples everyone in the room shakes the hand of everyone except for their spouse, how many handshakes occurred at this meeting?

There is a period of adjustment to this type of mathematics.

In response to the question:

"And in your opinion is this a result of the education system, the curriculum, the teaching style, or some say it's due to technology?"

I answered:

There are larger factors involved than the education system, the curriculum or changes in teaching style. Although I do believe that these do change over time, I perceive that their effect is minor compared to social and economic factors.

There is a certain level of math phobia that exists in the general public and that sentiment spreads. Students are told from an early age that 'I can't do math' is an acceptable attitude while such a comment about basic reading or writing skills is unheard of. When they arrive at university, math then becomes the expendable subject.

I will say that technology has introduced a level of informality that did not exist in the past and could be causing serious harm. Across all disciplines we hope that incoming students to be able to write clearly with proper grammar and spelling, understand their basic algebra, arithmetic and science concepts. Technology has made it harder for us to assume that all students will have mastered these skills. What I fear is that it has taught students to give up after they are unable to find the answer on Google.

## Thursday, September 10, 2009

### Fall Term Registration Reminder

For more information on registration, go to www.yorku.ca/grads/registration/index.htm.

If students have questions or require assistance, they may contact the Faculty of Graduate Studies, Student Affairs Office at 416 736-5521 or visit us in 283 York Lanes.

## Wednesday, September 9, 2009

### U.S. President's Speech to Students

I especially liked this part of his speech (the full text is available here):

*No one’s born being good at things, you become good at things through hard work. You’re not a varsity athlete the first time you play a new sport. You don’t hit every note the first time you sing a song. You’ve got to practice. It’s the same with your schoolwork. You might have to do a math problem a few times before you get it right, or read something a few times before you understand it, or do a few drafts of a paper before it’s good enough to hand in.*

The attitude that I hear about math from a good portion of the general public is that you are either "born with it or you ain't." I think that we need to motivate our students to anticipate and believe that although mathematics is hard work, it is something that they can succeed and benefit from.

## Monday, September 7, 2009

### Back to school complaining about students

I couldn't however pass up commenting on the latest headline in the Toronto Star "Internet teens failing math : Multi-tasking lifestyles, abolition of Grade 13, leaves 'i-generation' ill-prepared for university" from the September 6 paper.

The most interesting thing about this article were the comments online. Articles like these generate a ton of finger pointing: high school teachers, elementary school teachers, the university curriculum, the parents, abolition of grade 13, the curriculum, math itself, textbooks, politicians, standards for teachers, etc.

No short term fixes are going to change the situation. I am teaching a first year class that is supposed to be a bridge for students coming from high school and going into a university level math course (Math 1200: Problems, Conjectures and Proof). One thing I hope that I am able to convey to students in this course is that if they put in the work then they will learn and succeed and this is something that the article acknowledges is lacking with this generation. However I don't doubt that many of the problems with our education system identified in the article and the comments are real.

## Monday, August 24, 2009

### About Math 5020: Fundamentals of Mathematics for Teachers

What we find early on in this course that there are still things that are not known about the integers. For instance, while we know for sure that there are an infinite number of primes, we do not know if there are an infinite number of pairs of prime numbers of the form p and p+2 (e.g. (3,5), (5,7), (11, 13), (17,19), (29, 31), ...). Primes themselves are very hard to pin down and describe. It is still very surprising to me that there doesn't seem to exist a formula for an infinite sequence of primes.

One of the main applications of number theory is that of cryptography (sending coded messages so that an interceptor is unable to read them). One of the commonly used codes in modern cryptography is based theorems about integers that were discovered hundreds of years before the computer age. It still amazes me that the pieces of number theory that were discovered by mathematicians that could not dream of a modern computer became so relevant to making sure that your credit card number is not stolen when you send it over the internet. Although we only briefly touch on this application of number theory (the textbook was written before its invention!), most of what we learn is relevant for further study in cryptography.

Combinatorics is similar to number theory in that most people are familiar with the basic ideas of counting. However, it is easy to describe a set of things which is hard to count. For instance, how many ways are there of seating 10 couples (or more generally, n couples) around a table if no person is allowed to sit next to his or her spouse? To learn some of the basic rules of counting we build up arguments out of simple rules which readers accept and agree upon.

What makes these two topics "fundamental?" The main thing is that they provide a perfect setting for practicing a skill which clearly is fundamental to all of mathematics, writing and expressing yourself clearly and basing your arguments on principles that your reader accepts. This course takes some fundamental ideas (that explanation, justification, logic, proof and reason are behind mathematics) and ties them into some classical mathematics. Number theory is at its base addition, multiplication and division of integers. Combinatorics is about enumerating sets of objects. Both subjects involve familiar concepts to students and are taught at all levels. We will go deeper into these subjects, but what will make them 'fundamental' is we will also practice explaining why one applies basic rules to build up clear arguments.

You can look back at past web pages for this course to see what happened in past years. I've taught this course 3 times already but I tend to adjust it slightly each time.

### About MATH 5410 Analysis for teachers

In mathematics the concept of a real number comprises a rigorous treatment of the intuitive notion of `quantity'. The intuitive idea of a `relation' between quantities is encapsulated in rigorous mathematical terms by the

concept of a function of a real variable. The study of real numbers and functions of a real variable, together with higher concepts built upon them---such as measure and integration, metric spaces, families of functions, and so forth---is collectively referred to as analysis. Complex numbers and functions of a complex variable are closely related to real numbers and functions of a real variable, and study of the former is also referred to as analysis, or, more precisely , complex analysis.

MATH 5410 aims to convey a flavor of analysis as seen from a more sophisticated mathematical perspective than that of the Ontario high school curriculum, in the hopes that teachers will gain a new, enriched understanding that allows them to view the high school curriculum as part of a bigger picture.

Another objective of MATH 5410, is to convey a sense of the place of mathematics within the wider world. After all, why should math be taught as a basic subject from primary through secondary school? It turns out that the information age which is now upon us is largely based on the fact that a mathematically rigorous treatment of `quantity' is applicable to just

about anything: sounds, images, language, music, motion. The particular branch of mathematics that treats the `quantification' of sounds, pictures and communication is known as information theory, and it is the basis for digital communication as manifest in the world wide web, with its myriad applications such as youtube and iTunes, as well as cellular telephones and other wireless (and wired) media. But in order to have anything beyond a superficial understanding of information theory, one has to have a mastery of basic analysis. Any technical discussion of mp4 or coding theory, for

example, makes use of concepts from analysis. It is thus hoped that teachers who take MATH 5410 will be better equipped to convey to their students the place of mathematics in the broader world.

The format of the course will be weekly lectures and worksheets, with evaluation based on regular assignments, two in-class

tests, and a presentation.

## Wednesday, June 17, 2009

### WolframAlpha solves math problems

From the article:

*“I think this is going to reignite a math war,” said Maria H. Andersen, a mathematics instructor at Muskegon Community College, referring to past debates over the role of graphing calculators in math education.*

...

*"I still think that anyone who is not a little scared by the changes that WolframAlpha brings hasn’t thought about it enough yet,"*

I would strongly disagree with this statement even though I think that WolframAlpha is an amazing tool. If you have not seen WolframAlpha you should check it out. It is not a search engine but acts similar to one. It can solve equations and analyze data and has data-mined massive amounts of information. Try "integral sin x dx" and you will see what this web site can do for calculus.

Does this significantly change the way we teach? My answer is "no more than other tools that are out there." I think that we have not correctly adapted to the use of the internet in learning since I perceive that students are leaving high school and do not know how to write and research properly (it could be my "old fogy-ism" kicking in).

I do not think that WolframAlpha is going to change the way that students arrive at a computational answer any more than a CAS (Computer Algebra System) does and it will have less impact than graphing calculators. Perhaps the authors worry that now we have a CAS (WolframAlpha is based on Mathematica) which is freely available. We have to be aware that students have access to this tool, but we had to adapt to the way Google changed the way students do mathematics and we will have to adapt to the use of smart phones as they become ubiquitous. This is nothing new and it is a good thing that this tool increases access.

## Tuesday, June 9, 2009

### E-textbooks- all the rage

Advantages that I can see:

(1) lower cost

(2) able to update textbooks quickly and inexpensively

(3) convenience and smaller size

Disadvantages

(1) lower income students may have less access (its not clear how they plan to get around that one)

(2) electronic formats are WAY less convenient than their printed counterparts (what about notes in the margin?)

(3) inconvenience and larger size (a laptop or desktop is significantly larger than a textbook, but an iphone or a kindle are both smaller...which is it going to be?)

(4) there is a learning curve involved in using this technology

The NY Times also highlighted a school district in Connecticut. Connecticut District Tosses Algebra Textbooks and Goes Online. They switched from textbooks to e-books for a reason other than budgetary.

## Friday, June 5, 2009

### Mathematics v. Mathematics Education

The speaker was Peter Liljedahl, a professor of Mathematics education at SFU who teaches mathematics to future teachers. He emphasized that you cannot truly separate math and math ed and that there are common goals but he also talked about the differences between the goals of math and math education.

Some interesting ideas that came up in this talk:

1. The first is that sometimes people who are interested in learning math education are interested in learning how to teach mathematics and not about learning the mathematics content. This can be a conflicting goal with mathematicians who tend to believe that teaching mathematics well requires that the teacher understand the subject with a real depth.

2. The other idea that I thought was interesting was he addressed how teaching courses for math teachers (for example in this program) is different than other mathematics classes. The undergraduate classes that he gives in a 'for Teachers' program do not go into depth. The courses I have taught in this MA program always keep in mind two ideas (1) the students taking the course do not have the same background as each other or even a core basis that I can assume everyone knows and (2) the content that we do teach is not required for other courses. In other graduate and undergraduate courses the purpose is to teach a subject with a depth that can only be reached by assuming the students have understood and remember the material that came before.

3. He also touched on some of the skills that we need to provide students with for them to succeed in mathematics. For him it was less about what content we teach students, but what skills we provide them with. Some examples of skills that he mentioned are: how to read a math textbook, how to recognize when a teacher is skipping steps (and why), how to approach a problem that cannot be solved in 5 minutes, etc.

From my iPhone so this may be edited later.

## Thursday, June 4, 2009

### Girls v. Boys now a tie in Math

Here's some of the buzz:

Math: It's not a gender thing

Women Bridging Gap in Science Opportunities

Culture, Not Biology, Underpins Math Gender Gap

Girls Vs. Boys At Math

Sharon Begley: The Math Gender Gap Explained

Subject: The (Math) Gap - a critique

### Women in Mathematics II

In recent years “men and women faculty in science, engineering and mathematics have enjoyed comparable opportunities,” the panel said in its report, released on Tuesday. It found that women who apply for university jobs and, once they have them, for promotion and tenure, are at least as likely to succeed as men.

## Saturday, May 30, 2009

### Mathematical Moments from the AMS

I am writing this blog entry about them to suggest that you take a look and consider hanging them as math eye candy to in your classrooms and departments. The summaries are written for a general audience and are very good explanations of how mathematics is used in solving practical and interesting problems. Sometimes the posters are accompanied by additional teaching material such as podcasts and papers.

One example given was about a mathematician who won a Grammy for recovering recordings of Woody Guthrey using signal processing techniques. Another was about how it is possible to detect if a digital photo has been altered. Other examples in the area of digital technology include data compression, speech recognition, GPS, and digital movie animation. There are other examples from science, sports, art and langugage.

If you are looking for something to spark students interest in mathematics or even idea for classroom material this is a great resource. I recommend that you find a color printer if you do since they are not nearly as flashy in black and white. If you want to explore these topics further the online resources also seem very good.

### Some recent announcements to the mailing list

Subject: Extensions of Program Time Limits for Degree Completion and the CUPE Back-to-Work Protocol

---------------------------------------------------------

Please find the following communication from the Faculty of Graduate Studies, regarding Extensions of Program Time Limits for Degree Completion and the CUPE Back-to-Work Protocol.

The document is posted on the FGS website at the following link: http://www.yorku.ca/grads/cupe3903_back_to_work_protocol2009.

Posted May 27

Reminder:

Graduate students are required to maintain continuous registration. This means they must register in each term (Summer, Fall and Winter) until they complete their degree as either a full-time or part-time student. Summer 2009 registration commences on April 27, 2009 and the deadline for registering is June 15, 2009. Students who register after the deadline will be charged a late fee of $200.

Students should visit the FGS web site: www.yorku.ca/grads/registration/index.htm to register and view the summer 2009 course timetable.

Please note the following: unless a change of status has been approved by the Graduate Program Director and the Dean, Faculty of Graduate Studies, students must remain in the category of registration to which they are admitted.

If you have any further queries, contact FGS Student Affairs Office at 416 736 5521.

Posted May 28

RE: FGS Summer 2009 Convocation

Important, please note (June 2009 Convocation)

The online RSVP to a ceremony for the June 2009 convocation in not yet active, the online rsvp system (which includes robe ordering) will be activated approximately 4-6 weeks prior to the ceremony. Students who wish to to attend the ceremony will be required to rsvp their attendance and order guest tickets.

Convocation Date:

Summer 2009 - June 24 - 30th, 2009

For more information on Convocation, please visit the convocation website: http://www.yorku.ca/mygraduation/

## Monday, May 18, 2009

### Calculus is not the solution to all our problems?

Let me give a list of pointers to the buzz on the blogs:

Math Forum: 1 2 3 4 5 6

Ars Mathematica

The Math Underground

Nabble

Matthew Leingang

Rational Mathematics Education

A followup article on MAA

My two cents: I will agree that calculus isn't for everyone and I would agree that there should be at least a couple of math paths for university bound students.

However, if one advances far enough along in mathematics, calculus becomes a fundamental tool that cannot be ignored. As a university math teacher I assume calculus as a tool which is as basic as logarithms and exponentiation (e.g. basic concepts of calculus come up regularly in just about all the courses I have been teaching: cryptography, gambling, computer graphics, combinatorics and number theory...subjects where one would not expect to need calculus at first glance).

I would not rush to force high school students through calculus in order to encourage them to do well at university. Too many learned little or nothing from an inadequate calculus education and need to retake it when they get to university anyway.

When I was in high school I found calculus to be absolutely fascinating and loved learning it. I don't think that it can be completely dismissed as a university preparation course because for some people it works well. I remember in 11th grade my "Functions" teacher answered a question that I had asked by writing down the formula for the area of an ellipse by integrating the equation for the curve a year before I would learn what differentiation and integration were about. For some reason that memory became a vivid example of the exceptional education I was exposed to (thanks Mrs. Keim!).

Moving away from calculus being a capstone course for college preparation is going to require much better preparation of our teachers. They will need exposure to a broad range of mathematics if they are going to be able to convey to students the applications and importance of mathematics (e.g. see the topic list for the advanced mathematics course discussed in one of the blogs).

## Friday, May 8, 2009

### Access to courses mounted by education

We are finding that students in the M.A. in Mathematics for Teachers program do not have priority or sometimes access to enroll in these courses. This happened this past Fall/Winter where we had limited spaces in Margaret Sinclair's classes and now it is happening this summer.

I just checked the enrollment for the course EDUC 5210 3.00/MATH 5910 3.0 and there are currently 19 enrolled. None of those students seem to be from our program. When I discussed upcoming courses with the graduate program director in education earlier this year she failed to mention the upcoming MATH 5910 3.0 course and I wonder if she intentionally left it off the list.

At least one of our students said he tried to enroll and somehow he was blocked from enrolling (although Primrose checked and could not find a reason for this block).

It is likely that for the Fall courses that the graduate program director for mathematics and I will secure from the education department some limited space in the courses that the education department is planning to offer. To determine priority in those limited spaces we will ask students to write a rationale for wanting to take the course and give priority to the students who have taken the most number of credits.

I have listed in another blog entry the expected course offerings for the Fall term.

Update: I sent an email on April 28 to the grad program director for education asking if there would be enrollment for Math students and she responded: "Yes, there should be space. For the CAS filters I have allowed some space. Not sure of RO has saved the spaces. There is still space for 3 atudents." I don't know what "CAS filters" and "RO" are but it seems like we should be able to get some students in this class.

## Wednesday, May 6, 2009

### All the math news that fit to digitize

I want to acknowledge it because it has links to all types of math articles, from those that appear in major news sources (NYT, Scientific American, Google News) to blogs on a variety of math topics such as puzzles, math education, general interest and related branches of science.

http://math.alltop.com/

## Tuesday, May 5, 2009

### The origin of math phobias

I know that she is not alone in associating math with exams and anxiety and not doing well. Discussions like this are rather uncomfortable for me because I often feel as though I am math's only advocate in a room full of animosity towards a maligned subject and the direction of a conversation often takes a tack where I say something equivalent to "its good to know more math" and then we have awkward silence and we change the topic.

I decided to pursue the conversation anyway and asked an obvious question "why is math different than other subjects in causing anxiety? Why do students fear this subject more than others?" She said something that I think may get to the heart of the matter. With math, students are split into different streams and more or less subtly labeling them as "smart" and "not smart." In grade school, there isn't "English for the smart kids" and "regular English"....there is essentially just "English" and everybody takes it.

I will pose this question: is it a good idea to "stream" math students when it causes a lifetime of anxiety? Can we teach the students who are faster and study more without subtly labeling other people as dumb? Thoughts?

## Tuesday, April 28, 2009

### Advising appointments with Augustine Wong

Evening appointments for advising:

Wednesday, May 6, 7 - 9 p.m.

Wednesday, May 13, 7 - 9 p.m.

Advising should take approximately 10 mins.

Please e-mail Prof. Wong (august@mathstat.yorku.ca) and let him know what time-slot is suitable for you.

ADVISING APPOINTMENTS - SUMMER 2009

Wednesday, May 6, 2009

7:00 7:10 7:20 7:30 7:40 7:50 8:00 8:10 8:20 8:30 8:40 8:50

Wednesday, May 13, 2009

7:00 7:10 7:20 7:30 7:40 7:50 8:00 8:10 8:20 8:30 8:40 8:50

Thanks

Primrose Miranda

Graduate Program in Mathematics & Statistics

Mike's note: I recommend anyone should set up an appointment to speak with Augustine or myself. I will be less formal and not set times, email me and give me a chance to get your file (remember I am not in the same building as Primrose and Augustine so give me advance warning). Since most of you are coming in on Monday and Thursday for classes rather than Wednesday I would recommend arranging to do the advising appointment by phone rather than making a trip here on another evening.

## Wednesday, April 22, 2009

### Summer 2009 registration begins Monday, April 27, 2009!

Dear Graduate Students,

I hope this note finds you well as you enter into summer 2009 term on June 8, 2009.

As a graduate student, you are required to maintain continuous registration in your program. This means you must register in each term (Summer, Fall and Winter) until you complete your degree as either a full-time or part-time student.

The deadline to register is June 15, 2009. If you register after this date, you will be charged a late fee of $200.

To register and view the summer 2009 course timetable, visit the Faculty of Graduate Studies (FGS) web site: www.yorku.ca/grads/

Please note that unless your Graduate Program Director and Dean, Faculty of Graduate Studies have approved a change of status, leave of absence academic petition, you must remain in the category of registration to which you were admitted. If you are having difficulty registering please do not hesitate to contact FGS.

The Faculty of Graduate Studies wishes you all the best in your academic progress. If you have any question, please contact your graduate program or call or visit the Faculty of Graduate Studies at 283 York Lanes or 416 736-5521.

Sincerely,

Sharon Pereira

Student Affairs Officer

Faculty of Graduate Studies

### Summer 2009 Registration

Graduate students are required to maintain continuous registration. This means they must register in each term (Summer, Fall and Winter) until they complete their degree as either a full-time or part-time student. Summer 2009 registration commences on April 27, 2009 and the deadline for registering is

**June 15, 2009**. Students who register after the deadline will be charged a late fee of $200.

Students should visit the FGS web site:

__www.yorku.ca/grads/__registration/index.htm to register and view the summer 2009 course timetable.

Please note the following: unless a change of status has been approved by the Graduate Program Director and the Dean, Faculty of Graduate Studies, students must remain in the category of registration to which they are admitted.

If you have any further queries, contact FGS Student Affairs Office at 416 736 5521.

Thank you,

Karen Reid

Student Affairs, Assistant

York University

Faculty of Graduate Studies, 283 York Lanes

416.736.2100 Ext. 55521

kreid@yorku.ca

www.yorku.ca/grads

## Monday, April 20, 2009

### We pretend to teach 'em, they pretend to learn

The article clearly says that students are not motivated, are not stretched academically, get As and Bs just for showing up, are immature and disorganized, lack math and writing skills, can't think critically, and believe that research is synonymous with Wikipedia.

Count me a member of the "Old Farts Club" that is mentioned in the article. Now, what can we do about it?

This article is confirmed by the Toronto Star on April 6, 2009. What is this, take pot shots at students month? Its like shooting fish in a barrel.

## Sunday, April 19, 2009

### A novel approach to grading

This professor at University of Ottawa is being punished for his assigning all A+ grades in a class. Oh how I wish he were in the norm. I am not as eloquent as he is and not able to defend such a position but I would support a move to a non-grading or a pass-fail system.

Towards the end of the interview the interviewer says in summary:

"We are too focused on grades and not enough on learning."

Ah-ha! He gets it!

I should add a link to the petition in support of Professor Denis Rancourt:

http://rancourt.

## Friday, April 17, 2009

### Deadline for applying to graduate at the end of winter term

Please also note the deadline for submission of final copies of theses/dissertations to the Faculty of Graduate Studies.

Thanks

Primrose Miranda

Graduate Program in Mathematics & Statistics

## Sunday, April 12, 2009

### Digging For The 'Prime' Jewels Of Numbers

## Wednesday, April 8, 2009

### Upcoming summer courses

Math 5350 (June 8 - July 16) An introduction to Mathematical Modeling - Discrete Time & Probability

J. Heffernan

MR 6:00-9:00 Ross N836

Math 5360 3.0 (July 20 - Aug 27) An introduction to Mathematical Modeling - Continuous Time & Probability

J. Heffernen

MR 6:00-9:00 Ross N836

Math 5910 3.0* (EDUC 5210) Quantitative Research Methods in Education

P.D. Millett

MTRF 9:30-2:30

W 9:30-3:30

Math 5350 3.0 - An introduction to Mathematical Modeling – Discrete Time and Probability

This course provides an introduction to discrete time and probabilistic mathematical models. The course focuses on the mathematical methods underlying scientific inquiry and discovery. Through hands-on exploration and reflection, students will examine topics such as historical connections between mathematics and science, empirical modeling, model validation, proportionality, and simulation. The course starts with an overview of the modeling process and a review of relevant technology - Texas Instrument TI-92, the Internet and the World Wide Web, Java applets and computer algebra systems. Strategies to initiate modeling in the secondary classroom and classroom assessment of modeling activities are introduced and discussed. Topics include difference equations, Markov chains, and leslie matrices, applied to problems in biology, the environment, and finance i.e. modeling infectious disease spread, species extinction, power delivery. Particular attention is given to topics in the intermediate and senior Ontario curriculum.

Math 5360 3.0 - An introduction to Mathematical Modeling – Continuous Time and Probability

This course provides an introduction to continuous-time and probabilistic mathematical models. The course focuses on the mathematical methods underlying scientific inquiry and discovery. Through hands-on exploration and reflection, students will examine topics such as historical connections between mathematics and science, empirical modeling, model validation, proportionality, and simulation. The course starts with an overview of the modeling process and a review of relevant technology - Texas Instrument TI-92, the Internet and the World Wide Web, Java applets and computer algebra systems. Strategies to initiate modeling in the secondary classroom and classroom assessment of modeling activities are introduced and discussed. Topics include differential equations, Markov processes, and leslie matrices, applied to problems in biology, the environment, and finance i.e. modeling infectious disease spread, predator prey, heat flow. Particular attention is given to topics in the intermediate and senior Ontario curriculum.

Instructor: Jane Heffernan (jmheffer at yorku dot ca)

Math 5350 and Math 5360 are new and I am really excited that we are offering these courses because they promise some interesting content that you will not find in too many other programs (undergraduate or graduate). The other great feature that these courses introduce into the program is the flexibility of the 3.0 credit option. Most of our previous classes have all been 6.0 credits and I plan to try to offer a few more like this in the future.

## Sunday, March 29, 2009

## Monday, March 23, 2009

### TVO's "Big Ideas": Statistics: What's the point?

The beginning of this lecture is for anyone who would like a good introduction to what statistics is and why it is important. He gives a number of good examples about why numeracy is important. Towards the end of the lecture he explains vocabulary which, though important for his class, is not truly interesting for the general public.

## Sunday, March 22, 2009

### The difference between $.002 and .002cents

http://media.putfile.com/Verizon-Bad-Math

A Verizon customer tries to convince several supervisors at Verizon that .002 cents/per kilobyte is not the same as .002 dollars/per kilobyte. I think that he explains it very well. Apparently everyone at the Verizon office gets these two things confused and refused to even admit that there is a mistake.

At some point they started quoting "$0.002 per KB or $2.05 per MB" which is correct. Their employees read "$0.002 per KB" as ".002 cents per kilobyte" which is not correct. Where does this confusion come from?

I found this after reading http://xkcd.com/558/

## Wednesday, March 18, 2009

### The relevance of math to everyday life

There is no escaping the influence, and the virtues, of math

VANCOUVER SUN MARCH 18, 2009

In an effort to illustrate what he described as the "unreasonable effectiveness" of mathematics, the late Nobel laureate physicist Eugene Wigner used to tell a story.

The tale concerned two former high school friends, one of whom had become a statistician working on population trends. The statistician was explaining the meanings of various symbols he used when his friend asked about the meaning of pi.

When the statistician explained that pi was the ratio of the circumference of a circle to its diameter, his friend responded, incredulously, that "surely the population has nothing to do with the circumference of the circle."

The friend's incredulity is understandable, since it's reasonable to wonder what a relatively abstract matter like the circumference of a circle has to do with a population. But that is the unreasonable effectiveness of mathematics -- math seems to insinuate itself into everything, even things where it appears not to belong.

For example, everyone knows that math is absolutely essential to physics and, to a lesser extent, to the other natural sciences. But what many people don't know, and others choose to forget, is that math is crucial to the study of just about everything, including ecosystems, financial markets and sports scores.

Truly, everything is number, as the Pythagoreans said. And that is why everyone needs a decent understanding of mathematics.

The problem, of course, is that most people, particularly when they're in school, don't see how math is relevant to their lives. In an ironic way, this may be a direct result of its unreasonable effectiveness.

Heres what another blog had to say:

http://physicsandphysicists.blogspot.com/

## Saturday, March 14, 2009

### Results of the survey

I wanted to make available the results of the survey that I asked you all to fill out. I had roughly 14-15 responses and I think that the feedback that I got ranges over a full spectrum of why people enrolled in the program and what they are finding the program is about.

I fear that the people who are not happy with the program (responses 11&12 answered 'no, they would not recommend') got the impression from outside influences what this program is about and not from us (both make reference to such). I am making changes to the program description, so if you have suggestions about how to make it clearer please let me know.

I am listening to your suggestions but I know I can't meet all of them. Those that have asked for 'more education' courses, we can only try. This program is a mathematics degree so we have (a) second choice on education courses and (b) the math courses are supposed to go above and beyond what you learn in high school so that as teachers you have a better perspective of what is in the elementary/high school/college curriculum and why.

We are trying to offer more semester long courses, this will improve over the next year. We can offer summer courses during the day *in theory* but all students who would be taking the course have to agree given our mandate of being a part time program. This year we wanted to offer one of the summer courses during the day but ran into a conflict with a student who works during the summers and needed the last course to finish.

## Saturday, March 7, 2009

### Grades for a graduate program at York university

The following is an excerpt from the graduate calendar:

Grades will be awarded for every course in which a student is enrolled in accordance with the following system:

A+ (Exceptional)

A (Excellent)

A- (High)

B+ (Highly Satisfactory)

B (Satisfactory)

C (Conditional)

F (Failure)

I (Incomplete)

A student who received in total any of the following combinations of grades for graduate courses may not continue to be registered in the Faculty of Graduate Studies and in a graduate program unless this continuation is recommended by the graduate program director concerned and approved by the Dean:

(a) two C grades for full courses;

(b) one C grade for a full course and one C grade for a half course;

(c) a total of three C grades for half courses.

OR

(a) one F grade for a full course or two F grades for half courses; or

(b) one F grade for a half course and one C grade for a full or half

I encourage you to read all of the regulations at: http://www.yorku.ca/grads/calendar/fgs-calendar2007-09.pdf

## Friday, March 6, 2009

### Why one would want to take more statistics

Correlation doesn't imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing 'look over there'.

## Thursday, March 5, 2009

### More probable Fall-Winter 2009-10 offereings

R 6-9pm MATH 5020 6.0: Fundamentals in Mathematics for Teachers

M 7-10pm MATH 5410 6.0: Analysis for Teachers

We don't have Summer 2010 planned yet but are thinking about 'Math 5200: Problem Solving' or possibly I would like to offer it as another pair of 3.0 credit courses.

I have sent a message to the Education department and they just responded that they would be offering the following courses:

MATH 5900/EDUC 5841 Cr=3.00 Thinking about Teaching Mathematics - Fall 2009

MATH 5840/EDUC 5840 Cr=3.00 Mathematics Learning Environments - Winter 2010

EDUC 5215 Cr=3.00 Research in Mathematics Education - Summer 2010

The last class is not cross listed with our program but can be applied for through the graduate program director. Also I think that the Summer 2010 (since it is more than a year away) is still somewhat tentative. This past year our program was allowed limited enrollment in the education courses because they are not Math courses too. The limited enrollment was granted to those that had seniority in the program.

## Tuesday, March 3, 2009

### Women in Mathematics

After wading through more than 400 academic articles on sex differences in math, they've concluded that it's not a lack of ability that keeps women from pursuing careers in engineering or computer science. Rather, it's a variety of factors, including a female preference for more people-oriented professions and significantly different lifestyle needs, such as having and raising children.

I'm not sure I like the article after that because the remainder tends to introduce more stereotypes than it tries to dismiss.

http://www.thestar.com/News/World/article/595540

More technical versions of this summary can be found at:

http://www.theregister.co.uk/2009/03/03/girls_in_stem_roundup_study/

http://www.sciencedaily.com/releases/2009/03/090303082807.htm

### Charlemagne’s Puzzles

http://tierneylab.blogs.nytimes.com/2009/03/03/a-prize-for-solving-charlemagnes-puzzle/?scp=1&sq=puzzle&st=cse

## Monday, March 2, 2009

### What Do We Need Algebra For?

http://www.npr.org/templates/story/story.php?storyId=101298505

Weekend Edition Saturday, February 28, 2009 · The mobile phone industry stuck a nerve among teachers this month. An industry trade group argued that cell phones should be allowed in the classroom, saying they can be used as a teaching tool to help children with their math skills — in particular, algebra.

Well, we're going to avoid that debate this week. But it did make us reflect: what exactly is algebra, and why do so many people find it hard to learn?

NPR's Scott Simon turns to Weekend Edition Math Guy Keith Devlin of Stanford University for help.

Here is the NY Times article about how the mobile phone industry is suggesting that cell phones are a good classroom tool to do algebra.

http://www.nytimes.com/2009/02/16/technology/16phone.html

### Online survey for the M.A. in Math for Teachers

You can take the survey by clicking on the link below or by copying and pasting into a web browser:

http://www.zoomerang.com/Survey/?p=WEB228V4QXHCPJ

You can take this survey once per computer. I will share the responses on this blog when the survey is done so that they can be discussed.

### Winter 2009 Registration Reminder

Students should visit the FGS web site:

http://www.yorku.ca/grads/registration/index.htm

to register and view the Winter 2009 course timetable.

An earlier announcement was made:

Note: Winter Tuition Fees are due March 10th, see below.

URL: http://www.yorku.ca/osfs/sos.shtml

Faculty of Graduate Studies revised dates and deadlines

http://www.yorku.ca/grads/calendar/calendarofevents.pdf

These are just reminders and everyone is responsible for knowing the dates if they get these messages or not so pass the word along to your fellow students!

## Friday, February 27, 2009

### Upcoming information session for prospective students

If you know anyone who would like to attend you may pass along a link to the following letter http://garsia.math.yorku.ca/~zabrocki/ma4teacherslet.pdf.

It has been suggested that we have an information session for the M.A. for Teachers program with current students as well. I teach on Mondays from 6-9pm and I can come after the algebra class one evening if you want to discuss any issues. Comments?

### Past courses in the program

FW2008-9

* EDUC 5848 3.0 Technology and Mathematics Education

* MATH 5400 6.0 History of Mathematics

* MATH 5420 6.0 Algebra for Teachers

Summer 2008

* MATH 5300 6.0 Computation in Mathematics for Teachers*

* MATH 5910 3.0 Quantitative Research Methods

FW2007-8

* MATH 5020 6.0 Fundamentals of Mathematics for Teachers

* MATH 5430 6.0 Statistics and Probability for Teachers

Summer 2007

* MATH 5410 6.0 Analysis for Teachers

* Math 5840 3.0 Mathematics Learning Environments

FW2006-7

* MATH 5910 3.0 Quantitative Research Methods

* MATH 5200 6.0 Problem Solving

* MATH 5400 6.0 History of Mathematics

Summer 2006

* MATH 5450 6.0 Geometry for Teachers

FW2005-6

* MATH 5020 6.0 Fundamentals of Mathematics for Teachers

* MATH 5500 6.0 Topics in Mathematics for Teachers

* MATH 5900 3.0 Thinking about Teaching Mathematics

* MATH 5910 3.0 Quantitative Research Methods

Summer 2005

* MATH 5420 6.0 Algebra for Teachers

* MATH 5910 3.0 Quantitative Research Methods

FW 2004-5

* MATH 5430 6.0 Statistics and Probability for Teachers

* MATH 5300 6.0 Computation in Mathematics for Teachers

* Math 5840 3.0 Mathematics Learning Environments

Summer 2004

* Math 5840 3.0 Mathematics Learning Environments

* MATH 5400 6.0 History of Mathematics

## Thursday, February 26, 2009

### What is this program about

The program focuses on giving students an exposure to a variety of mathematical subjects providing to those that are teachers a broader experience that they can bring to their own classrooms. The range of this program gives students an historic perspective as well as chances to practice techniques of problem solving, writing and presenting mathematics. These elements are relevant for teachers of mathematics at any level. This program does not prepare students for study at the Ph.D. level in mathematics nor does it lead to teacher certification in Ontario. Courses in the program will be scheduled in the evenings, usually with a three-hour session once a week for a course in the Fall-Winter term, and two three-hour sessions per course in the Summer term. Ordinarily, two courses are offered in the Fall-Winter session and one course in the Summer session.