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.'
Monday, October 26, 2009
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.
Monday, September 7, 2009
Back to school complaining about students
Welcome back to the Fall term. I mostly took a break from writing to this blog over the summer since I didn't think I needed to be posting when everyone was off for the summer. I started writing a few postings when a news article caught my eye but didn't post anything and may come back to them if I have fresh ideas.
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.
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
Simply from the title of the course called "Fundamentals" I wouldn't know what material would be covered. The course description says it is about number theory and combinatorics. While these two topics are not necessarily interrelated, they do have some things in common. The most interesting is that both topics do not require very much background in order to encounter very difficult problems.
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.
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
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.
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
An article titled A Calculating Web Site Could Ignite a New Campus 'Math War' was posted to the math department mailing list and it got a bit of response. The author seems to suggest that WolframAlpha is going to change the way that we teach math because it can solve calculus problems and other types of 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.
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
The budget crunch has caused some rough times for cities and municipalities in the U.S. (I don't know what it is like in Canada because Toronto, my own municipality, hasn't been able to balance its budget for years now) and this has caused some of them to teach students on the cheap. Arnold Schwarzenegger has canceled the contract with textbook companies and switched to lower cost electronic versions of textbooks. Surprisingly I haven't seen teachers and school districts in the state of California screaming. Instead the news seems to imply that schools elsewhere may use California as a model and use the same technique to lower their school budgets. We may be seeing electronic textbooks in our classrooms in a few short years if this experiment is successful.
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.
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
I was at a day panel that was organized by Walter Whiteley in relation to the CMESG meeting that is being held at York University this weekend when I wrote this. The first talk was about the competing goals of mathematics education and mathematicians.
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.
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
A report on the state of U.S. education in the Proceedings of the National Academy of Sciences says that a performance gap between girls and boys has narrowed significantly in recent years. The researchers also found a correlation between cultural gender equality and performance by girls in mathematics. There are still some achievement gaps but this study indicates that past achievement disparity between gender is likely attributable to factors other than the popular belief that 'boys have a math gene and girls don't.'
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
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
Back in March I posted an article about some research on opportunities in mathematics for women. Yesterday the New York Times published a summary about a report by the National Research Council on the state of opportunities for women in science and engineering. In summary (quoting from the article):
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 have noticed posted in the York University math department near the main office (and more recently elsewhere) a number of very good single page poster/summaries of examples of where mathematics has managed to solve some very practical problems. I took a look online and these posters are freely available from the American Mathematical Society.
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.
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
Posted May 14
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/
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?
I missed an article in January's AMS Notices titled "Is the Sky Still Falling?" The author takes a close look at the number of incoming mathematics and engineering students to universities. A followup letter was published in the April 2009 issue of the AMS Notices titled "Why the Sky is Still Falling" that was written by our very own Walter Whiteley along with Peter Taylor (Queen's University) and Ed Barbeau (University of Toronto).
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).
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
Certain of the courses that are in the M.A. for Teachers program are offered by the Eduction department (those which are not exclusively mathematics courses). The three on the list include: MATH 5840 3.0: Mathematics Learning Environments, MATH 5900 3.0: Thinking about Teaching Mathematics, and MATH 5910 3.0: Quantitative Research Methods.
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.
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
There is a really good meta-math blog that I have been using as the source for some of the articles that I have posted here.
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/
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 had a discussion with someone this past weekend. She said that she has this dream about math. She is in a exam for which she has not studied and is not prepared.
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?
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?
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