Sorry about the day about the ability to sign up for classes. The course should now be listed and you can register. Below is the announcement that Primrose just sent out.
All the course offerings for Summer 2010 are now on the Graduate Studies Lecture Schedule.
Graduate Students,
Please be reminded that the registration deadline is May 3, 2010 (see below). After this deadline, the Faculty of Graduate Studies will be charging a late fee of $200.
All important deadlines for graduate students is available at the Faculty of Graduate Studies webpage: www.yorku.ca/grads
It is your responsibility to be aware of these deadlines
Sincerely,
Primrose Miranda
Graduate Program in Mathematics & Statistics
Friday, April 16, 2010
Saturday, March 13, 2010
Updated Announcements on Upcoming Classes
Upcoming summer courses offering by the math department for the M.A. in Mathematics for Teachers program:
May 10-June 16 (Monday and Wednesday) 6-9pm
MATH 5210 3.0 credits: Problem Solving I
by Yun Gao.
June 21-July 28 (Monday and Wednesday) 6-9pm
MATH 5500 3.0 credits: Topics in Mathematics for Teachers
on the topic of Naive Set Theory by Bob Burns.
For the Fall-Winter 2010-2011 terms the Mathematics department will be offering:
MATH 5450 6.0: Geometry for Teachers with Walter Whiteley, Thursdays 6-9pm
MATH 5400 6.0 History of Mathematics with Stan Kochman, Mondays 6-9pm
2010-2011 Important Dates for Graduate Students have now been published on FGS website, http://www.yorku.ca/grads/index.htm
The following announcement was sent towards the end of February and I am re-posting it here. If you did not receive this message, write me with an updated email and I will get you on any missing mailing lists I can.
February 22, 2010
Dear Graduate Students,
Summer Term 2010 registration begins Monday, March 1st, 2010! Be sure to register and enroll in courses early!
As a graduate student, you are required to maintain continuous registration in your program of study; this means you must register in each term (Summer, Fall and Winter) until you complete all the requirements of the degree as either full-time or part-time.
To register, enroll in courses and view the York course website, please go to: http://www.yorku.ca/yorkweb/cs.htm. The registration deadline for Summer Term 2010 is May 3, 2010, if you register beyond this date, a late fee of $200 will be applied to your student account. Therefore it is important that you register for the term early and enroll in courses early (if applicable).
Please note that unless your Graduate Program Director and the Dean of the Faculty of Graduate Studies have approved a change of status, you must remain in the category of registration to which you were admitted. To request a change of status (i.e. leave of absence, change to part-time, extension of time) for the Summer 2010 term, students must make their request through their Graduate Program and complete an Academic Petition Form or a Program Approval Form at least six weeks in advance of the start of the term.
For more information on; Registration; Important Deadlines and Dates; Faculty Regulations; Forms and the FGS directory, please visit the Faculty of Graduate Studies website at: http://www.yorku.ca/grads/.
The Faculty of Graduate Studies wishes you all the best in your academic progress and success in your current program of study, if you have questions, please contact your Graduate Program or the Faculty of Graduate Studies.
Sincerely,
Sharon Pereira
May 10-June 16 (Monday and Wednesday) 6-9pm
MATH 5210 3.0 credits: Problem Solving I
by Yun Gao.
June 21-July 28 (Monday and Wednesday) 6-9pm
MATH 5500 3.0 credits: Topics in Mathematics for Teachers
on the topic of Naive Set Theory by Bob Burns.
For the Fall-Winter 2010-2011 terms the Mathematics department will be offering:
MATH 5450 6.0: Geometry for Teachers with Walter Whiteley, Thursdays 6-9pm
MATH 5400 6.0 History of Mathematics with Stan Kochman, Mondays 6-9pm
2010-2011 Important Dates for Graduate Students have now been published on FGS website, http://www.yorku.ca/grads/index.htm
The following announcement was sent towards the end of February and I am re-posting it here. If you did not receive this message, write me with an updated email and I will get you on any missing mailing lists I can.
February 22, 2010
Dear Graduate Students,
Summer Term 2010 registration begins Monday, March 1st, 2010! Be sure to register and enroll in courses early!
As a graduate student, you are required to maintain continuous registration in your program of study; this means you must register in each term (Summer, Fall and Winter) until you complete all the requirements of the degree as either full-time or part-time.
To register, enroll in courses and view the York course website, please go to: http://www.yorku.ca/yorkweb/cs.htm. The registration deadline for Summer Term 2010 is May 3, 2010, if you register beyond this date, a late fee of $200 will be applied to your student account. Therefore it is important that you register for the term early and enroll in courses early (if applicable).
Please note that unless your Graduate Program Director and the Dean of the Faculty of Graduate Studies have approved a change of status, you must remain in the category of registration to which you were admitted. To request a change of status (i.e. leave of absence, change to part-time, extension of time) for the Summer 2010 term, students must make their request through their Graduate Program and complete an Academic Petition Form or a Program Approval Form at least six weeks in advance of the start of the term.
For more information on; Registration; Important Deadlines and Dates; Faculty Regulations; Forms and the FGS directory, please visit the Faculty of Graduate Studies website at: http://www.yorku.ca/grads/.
The Faculty of Graduate Studies wishes you all the best in your academic progress and success in your current program of study, if you have questions, please contact your Graduate Program or the Faculty of Graduate Studies.
Sincerely,
Sharon Pereira
Friday, March 12, 2010
Someone else's presentation on the Multiplication Principle
There is a blog Math Notations that covers various topics about math. This weeks topic is titled "Counting, Multiplication Principle, Pigeonhole Principle and Reasoning for Middle Schoolers and Beyond."
I know I have a different perspective on this subject that many of you have been forced to see in Fundamentals, but one of the reasons I like it is that this idea comes up in almost every one of my classes at some point.
No matter what I teach, at some point I find myself arguing that the number of something is the product of two values and the reason is the multiplication principle. In number theory, this simple idea allows us to derive very deep results from drawing a few pictures.
I know I have a different perspective on this subject that many of you have been forced to see in Fundamentals, but one of the reasons I like it is that this idea comes up in almost every one of my classes at some point.
No matter what I teach, at some point I find myself arguing that the number of something is the product of two values and the reason is the multiplication principle. In number theory, this simple idea allows us to derive very deep results from drawing a few pictures.
Women over 40 good at math
An article in the Toronto Star from March 10 sites research from Graham Orpwood, a York University emeritus professor, that women over 40 do significantly better than men in college math courses. The article suggests that the difference might be due to better time management skills but it seems that that particular conclusion is more of a guess.
Toronto Star article
The study also says that half as many women study math as men. There was little attempt in the article to explain why this might be the case.
What I don't like about the Toronto Star article is that they then get a bunch of (apparently random) people to comment on the results apparently picking the narrative that the reader might take away from this study.
What is important are the three summary points of the conclusion of the study:
It is hard to make such sweeping statements about gender differences over time, but data should be collected to see what the possible causes of large scale failings and shifts of attitudes towards math. I can say that when I was going through high school the number of girls in the advanced math courses were cut in half each year from years 9-12 until there were almost none in the calculus course in my high school. However, in the years after I was in school this changed dramatically. What this could mean is that in the next few years more women over 40 will start studying math. In my first year courses I see a roughly 50-50 split between the sexes.
Toronto Star article
The study also says that half as many women study math as men. There was little attempt in the article to explain why this might be the case.
What I don't like about the Toronto Star article is that they then get a bunch of (apparently random) people to comment on the results apparently picking the narrative that the reader might take away from this study.
What is important are the three summary points of the conclusion of the study:
- Many students identified as being at risk of failing math have a poor grasp of basic functions taught in elementary school, such as fractions, ratios, proportions and percentages, so students should be provided more practice in these.
- College and school staff should hold a round-table discussion on how to streamline which high school math courses are required for admission to college courses and not have such a disconnect from school to school.
- Schools should convince parents and students to focus on time management and self-discipline.
It is hard to make such sweeping statements about gender differences over time, but data should be collected to see what the possible causes of large scale failings and shifts of attitudes towards math. I can say that when I was going through high school the number of girls in the advanced math courses were cut in half each year from years 9-12 until there were almost none in the calculus course in my high school. However, in the years after I was in school this changed dramatically. What this could mean is that in the next few years more women over 40 will start studying math. In my first year courses I see a roughly 50-50 split between the sexes.
Saturday, February 20, 2010
Some states will allow graduation after 10th grade
Eight states in the U.S. are planning to participate in a program that will allow students to graduate 2 years early assuming that they pass the appropriate exams. The tests would cover math, history and science. The program is being introduce by the National Center on Education and the Economy.
The purpose of the program is to allow students who are interested in a vocational or college degree to not have to take the full four years of high school. Even if students pass the tests they may opt to during their junior and senior years of high school to take college prep courses Massachusetts decided not to implement the program because they felt it was aimed at students interested in vocational training and they already have a vocational program developed in their schools.
New York Times coverage
The purpose of the program is to allow students who are interested in a vocational or college degree to not have to take the full four years of high school. Even if students pass the tests they may opt to during their junior and senior years of high school to take college prep courses Massachusetts decided not to implement the program because they felt it was aimed at students interested in vocational training and they already have a vocational program developed in their schools.
New York Times coverage
Saturday, February 13, 2010
New York Times blog about a second chance at math
The New York Times is publishing in the opinions section a blog by Mathematician Steven Strogatz. The purpose of this series that he is writing is to take a look at math from elementary school to grad school for an artist friend of his who knows very little about mathematics. He hopes to introduce to this friend the meaning of what we say when we call a proof elegant and to convey the importance of mathematics.
In his first two entries he is starting with some very elementary ideas, numbers and arithmetic, but he intends to build up to bigger things. Already he has given a proof (using rocks) that
1+3+5+...+(2n-1) = n^2 .
Here is a TED talk by him about how synchrony arises from very simple mathematical rules and in nature.
In his first two entries he is starting with some very elementary ideas, numbers and arithmetic, but he intends to build up to bigger things. Already he has given a proof (using rocks) that
1+3+5+...+(2n-1) = n^2 .
Here is a TED talk by him about how synchrony arises from very simple mathematical rules and in nature.
Friday, January 29, 2010
Fear of Math passed from Teachers to (female) students
I have to admit that I am not one to acknowledge fear of math since that is far away from my experience. But there are plenty of websites that discuss math phobia and how to get over it.
example 1, example 2, example 3.
Now in an article in the Proceedings of the National Academy of Science researchers tested the hypothesis that fear of math is learned from elementary school teachers that have weak math backgrounds and that themselves have a fear of math. What they found is that fear of math seems to be passed onto the female students and not the male students. How this happens is not known.
The experiments were conducted by first measuring math anxiety within both the teachers and the students at the beginning of a school year. By the end of the school year, the more anxious a teacher was about mathematics, the more likely that girls (and not boys!) were to agree with the stereotype that boys are good at math and girls aren't.
One thing I did not know that 90 % of elementary school teachers are women (this is a statistic of U.S. schools, but I believe that it is probably similar for Canada), although I have spent some time in elementary schools here in Toronto and I should have been aware of that.
There are obvious followup questions such as "does fear of math get passed from male teacher to male students?" (which doesn't get addressed because the teachers this study were all female), and "does this only happen at the elementary school level?"
Here is some news coverage of this study:
A very short summary of the results from Chicago Public radio
NPR
Science News
Science Daily
Here is a 2007 story on fear of mathematics:
CBC - Fear over math marks reaches academic proportions
example 1, example 2, example 3.
Now in an article in the Proceedings of the National Academy of Science researchers tested the hypothesis that fear of math is learned from elementary school teachers that have weak math backgrounds and that themselves have a fear of math. What they found is that fear of math seems to be passed onto the female students and not the male students. How this happens is not known.
The experiments were conducted by first measuring math anxiety within both the teachers and the students at the beginning of a school year. By the end of the school year, the more anxious a teacher was about mathematics, the more likely that girls (and not boys!) were to agree with the stereotype that boys are good at math and girls aren't.
One thing I did not know that 90 % of elementary school teachers are women (this is a statistic of U.S. schools, but I believe that it is probably similar for Canada), although I have spent some time in elementary schools here in Toronto and I should have been aware of that.
There are obvious followup questions such as "does fear of math get passed from male teacher to male students?" (which doesn't get addressed because the teachers this study were all female), and "does this only happen at the elementary school level?"
Here is some news coverage of this study:
A very short summary of the results from Chicago Public radio
NPR
Science News
Science Daily
Here is a 2007 story on fear of mathematics:
CBC - Fear over math marks reaches academic proportions
Thursday, January 21, 2010
A couple of announcements
Important dates
I got the following message to forward on to the program. If you didn't receive it and would like to be added to the mailing list please let me know:
Dear Graduate Program Directors & Graduate Program Assistants,
Please be advised that the 2010-2011 Important Dates for Graduate Students have now been published on FGS website, http://www.yorku.ca/grads/index.htm.
About upcoming courses for summer session:
We are still waiting on final approval for the summer courses. We will be offering in the Summer 1 session (roughly May - June) "Problem Solving I," a 3.0 credit course with Yun Gao. Yun has in past years been responsible for the Putnam practice sessions for our undergraduate students.
In the Summer 2 session (roughly June - July) Bob Burns will be teaching a topics course on set theory using the book "Naive Set Theory."
Note that both "Problem solving I" and the topics course have yet to be approved so we have been slow to announce them.
About upcoming courses for FW 2010-11:
The mathematics department will be offering Math 5400 6.0: History of Mathematics for Teachers with Stan Kochman on Monday evenings 6-9pm and Math 5450 6.0: Geometry for Teachers with Walter Whiteley on Thursday evenings 6-9pm.
Wednesday, January 13, 2010
Winter 2010 Term Graduate Registration - Jan 15 deadline!
The Winter 2010 Term has begun, the Winter 2010 Term Registration Deadline is January 15th, 2010.
As a reminder, graduate students are required to maintain continuous registration and register in each term (Summer, Fall, & Winter) until the completion of their degree as either a full-time or part-time student. Students are expected to remain in the category of registration to which they are admitted, unless a change of status is approved by the Graduate Program Director and the Dean of the Faculty of Graduate Studies. Graduate students (continuing and new) are required to register for the Winter 2010 term by January 15th, 2010, students who register after the deadline will be charged a $200 late fee.
For more information on Registration go to: http://www.yorku.ca/grads/registration/index.htm.
Students who have not yet completed all the degree requirements, who are making academic progress and continuing with degree requirements must be registered for the Winter 2010 Term. Failure to do so by January 30th may result in the student being withdrawn from their program of study for failure to maintain continuous registration.
For information on Important Dates for Graduate Students go to:http://www.yorku.ca/grads/calendar.htm.
As a reminder, graduate students are required to maintain continuous registration and register in each term (Summer, Fall, & Winter) until the completion of their degree as either a full-time or part-time student. Students are expected to remain in the category of registration to which they are admitted, unless a change of status is approved by the Graduate Program Director and the Dean of the Faculty of Graduate Studies. Graduate students (continuing and new) are required to register for the Winter 2010 term by January 15th, 2010, students who register after the deadline will be charged a $200 late fee.
For more information on Registration go to: http://www.yorku.ca/grads/registration/index.htm.
Students who have not yet completed all the degree requirements, who are making academic progress and continuing with degree requirements must be registered for the Winter 2010 Term. Failure to do so by January 30th may result in the student being withdrawn from their program of study for failure to maintain continuous registration.
For information on Important Dates for Graduate Students go to:http://www.yorku.ca/grads/calendar.htm.
Friday, December 18, 2009
Program Overview
I will be announcing in the near future here on the blog some changes to the M.A. in Mathematics for Teachers program that have passed through some approval steps. I am currently waiting for some final approval by the Faculty of Graduate studies. The changes are quite minor on the surface, but include new calendar text, admissions requirements, degree requirements and significant changes to the course offerings.
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.
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
One of the authors of the blog "Computational Complexity," William Gasarch, posted a computational challenge on November 30 and he is offering $17*17 = $289 if someone comes up with a solution. I like this problem because it is easy to state and hard to solve.
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?
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
I saw this video recently and thought that it was worth sharing. The idea is interesting, namely that we should change the focus of the mathematics curriculum from calculus to statistics and probability because most people use statistical reasoning on a daily basis and should know more.
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.
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
I think that even the most math cynical out there will like this video. I watched it 3 times last night and several of the videos made by other high speed cameras. The video is of a waterdroplet and what we observe is then explained by a mathematician. I would have perferred to see a little more discussion about how the mathematics is related, but you can't have everything.
Monday, October 26, 2009
The Nobel prize in economics
There is no Nobel prize in mathematics and the rumor that Nobel didn't create a mathematics prize because his wife/mistress/fiance had an affair with mathematician Gosta Mittag-Leffler is in all probability false. There doesn't seem to be a clear explanation of why the field of mathematics is overlooked as prize category.
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.'
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
Perhaps comparisons to the rapture are inappropriate, but recent news about math scores in the U.S. do not look good for future students. Once again a report about the progress of students in math gets every pundit wagging their finger at someone.
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.
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
So after class on Thursday a 'fundamentals' student asked me a couple of questions that I now think are worth exploring a bit on the blog because they can lead to some interesting philosophical questions about math, although at the time I was too hungry to answer clearly.
"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" ?
"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
Both Peter Gibson and I were interviewed for an article that appeared in the Excalibur today as a followup to the Toronto Star article related to the high dropout rate that we have been commenting about.
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.
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
The Fall 2009 term, registration deadline is September 15, 2009, students who register after the deadline will be charged a late fee of $200.
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.
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
U.S. President Barak Obama made a speech to students yesterday and, other than partisan political objections (which I hope are mostly ignored because it is irrelevant to his message), the speech seemed to be well received. Part of his appeal to students was to work hard as a patriotic effort for their country, but mainly he appealed to students sense of responsibility. As educators we can do everything in our power to provide students with a good education, but unless they are also motivated to learn, everything we can do will remain relatively useless.
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.
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.
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