PDF of the survey results.
I wanted to make available the results of the survey that I asked you all to fill out. I had roughly 14-15 responses and I think that the feedback that I got ranges over a full spectrum of why people enrolled in the program and what they are finding the program is about.
I fear that the people who are not happy with the program (responses 11&12 answered 'no, they would not recommend') got the impression from outside influences what this program is about and not from us (both make reference to such). I am making changes to the program description, so if you have suggestions about how to make it clearer please let me know.
I am listening to your suggestions but I know I can't meet all of them. Those that have asked for 'more education' courses, we can only try. This program is a mathematics degree so we have (a) second choice on education courses and (b) the math courses are supposed to go above and beyond what you learn in high school so that as teachers you have a better perspective of what is in the elementary/high school/college curriculum and why.
We are trying to offer more semester long courses, this will improve over the next year. We can offer summer courses during the day *in theory* but all students who would be taking the course have to agree given our mandate of being a part time program. This year we wanted to offer one of the summer courses during the day but ran into a conflict with a student who works during the summers and needed the last course to finish.
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To those who are complaining about this program not meeting their expectations, I must rant:
ReplyDeleteColleagues, I’m disappointed.
What would you say to one of your students who is choosing his/her university program because of what a friend said or because of something they read in a glossy promotional brochure? Come on now. The fact is YOU DIDN’T DO YOUR HOMEWORK. How many of you took the time to come and visit York, sit in on a class or two, talk to students CURRENTLY in the program, meet with professors or at least with the program director, reviewed the list of courses offered in the past few years, etc.??? If you had done some or all of these things, I think that you would have had much more realistic expectations of what this program offers.
Secondly, you enrolled in a math program and are now complaining that the program contains too much math. Good grief! If you wanted an education degree, you should have signed up for one. If you so badly want to take the education courses, then why not switch to that program instead. That way you would get first dibs on the courses you actually want to take and more course selections to choose from. Moreover, cross-listing works in both directions. You could still pick and choose from the math offerings, sticking to math topics that suit your interests and/or comfort level. But please stop trying to turn a math degree into an education one. Believe it or not, there are teachers who want and enjoy the math focus in this program.
Finally, concerns were expressed about the professors not understanding the nature of their audience. I’m not sure that the “audience” understands the nature of the professors. Let’s take a look at this from the other side of the coin. This is a MATH program. The majority of profs in this program work in York’s math department and have no connection whatsoever with the education people. They are mathematicians first and foremost, rather than educators. As in most university departments, they were likely hired predominantly for their research potential, and not for their teaching strength. Many were educated outside Ontario. They are being asked to teach relevant mathematical content from their discipline (in a way that is watered down enough to be accessible, but not so much to be demeaning) to a group whose background ranges widely: from those who have taken only a handful of university math courses to those who have themselves instructed graduate level math courses. They are being asked to accommodate those who have been away from university for several years, while still challenging those who just graduated recently. They must cope with quite diverse levels of commitment and motivation among the students. And all the while, the one unifying characteristic of their students is their unending capacity to criticize, whine and complain (about EVERYTHING … it’s too hard, it’s too easy, not enough math, too much math, I don’t like Mondays, I want my summer vacation, I want more summer courses, I hate proofs, that’s not how my undergrad profs said to do it, where am I going to use this in the real world …). It’s analogous to a high school classroom with everything from the weakest 1L’s to the strongest 4U’s, all with the entitlement attitude of a 2P. Would YOU want to teach that? It’s a wonder there aren’t more profs. beating down Mike’s door begging to give up their time with family and friends in evenings and summertime(& summertime evenings) in order to teach these courses!
A little more gratitude and appreciation is in order. These profs. are doing the best job they can to teach math. It is NOT their job to learn the Ontario curriculum and find connections for you. Do that yourself -- the connections are there if you approach these courses with an open mind and willingness to learn. This program is about stretching the boundaries of your mathematical comfort zone as you learn some new things (isn’t that really what all educational programs should do? ). If you signed up for the wrong program, I have little sympathy. As educators who constantly counsel students about the importance of making good course choices, you should have known better.
Thanks for sending the results of the on-line survey. The intent of the programme is not clear to all who took the survey. I know we had a discussion in Fundamentals last year and there was discussion on how the programme would impact one's effectiveness as a teacher -- this continues to be a concern. I, for instance, (and mentioned on the survey) am not sure of what is meant by "Mathematics for Teachers". It is not evident how the mathematics being learned is relevant for teachers. Several people mentioned the desire to learn mathematics to have a deeper understanding to teach high school mathematics -- I agree! This makes sense. I found it interesting that several people mentioned that the courses should be rigorous so you must earn the degreee -- this I agree with since this programme is viewed as "everyone passes if you do your time" -- but "rigorous" to me doesn't mean learning isolated mathematics void of any relevance to teachers. Perhaps profs need to make direct reference to how the mathematics being learned is useful FOR teachers -- I think the assumption is made that teachers will know how the mathematics being learned is relevant -- but I know I don't :-| Lastly, I agree with the comment made that profs need to know their audience -- I think teachers want to know what credibilty the profs have for teaching them -- this is important as not anyone should be allowed to teach in this unique programme.
ReplyDeletePerhaps we need to go back to the grassroots of this programme and see what the original intent was. I remember being able to do mathematics in Martin Muldoon's Problem Solving course while also reflecting on how to use the math learned in the classroom -- through the bi-weekly assignments and the course project. Martin seemed to enjoy my reflections on how I could use this in my teaching. My hope is more profs will follow Martin's lead.
To add to my previous comments...
ReplyDeleteAll the MA courses for mathematics teachers are also cross-listed as EDUC with the faculty of education. So, MEd students can take our MA courses and would expect to learn about relevance to education.
Before applying, I did speak with the previous Coordinator of the programme who informed me that there would be opportunities to tie in the university level mathematics learned with relevance to high school. That was accomplished in Martin Muldoon's Problem Solving course by making relevance to teaching and pedagogy a criteria on the 30% project; also, Martin enjoyed reading how I tied in relevance to my work on the bi-weekly assignments.
What I need to know (and it sounds like others who took part in the survey), is what is the the intent/purpose/outcomes of this MA programme FOR mathematics teachers. It needs to be clearly stated since I must admit I do not know.
* If it is to learn university level mathematics for the sake of learning, void of relevance to one's professional practices (or the connections are up to the student to determine), then that needs to be stated upfront -- others may want to opt for the MA degree in mathematics (can this be done part-time?). There's nothing wrong with this, but what is FOR mathematics teachers to mean?
* If the intent is to make one more effective as a mathematics teacher by allowing opportunities to develop a deeper understanding of the high school mathematics taught (through the lens of learning university mathematics), then that needs to be stated. If so, opportunities need to be provided for reflection on connecting what is learned in lectures to what goes on in the high school classroom. It cannot be assumed. For credibility, the profs need to show interest in the high school curriculum and be aware of what's going on. If they don't know, they need to find out or ask.
I would love to see lots of 3.0 credit courses. I find the full-year courses to drag on and on. Why not offer all the courses as 3.0 credits so all students in the programme would be able to take all the courses offered?
Or make the programme requirement 5.5 courses with the remaining 0.5 courses an exit portfolio where you document how the programme has/will impact your professional needs -- i.e., accountability for the title, "MA programme FOR mathematics teachers".
Just some ideas as the survey results clearly indiciate the programme is not meeting everyone's needs.
I'll try to post this again.
ReplyDeleteHi Louis,
Thanks for taking the time to comment. Here are some scattered thoughts on what you wrote.
1. Courses were cross-listed a long time ago. The program has evolved since then. If what you say is true, then I should have expected to learn MATH in the cross-listed EDUC course I took??? That certainly wasn’t my experience. Perhaps cross-listing needs to be re-examined.
2. Yes, Martin did include a component on the final project about making connections to curriculum. I think it is important to note though that he was very flexible about the portion of the 40 page paper that the student needed to devote to this. I’m likely at the opposite end of the spectrum from you, but I was able to meet the criterion with 3 sentences in my conclusion. I’m glad that you found the Problem Solving course rewarding; I did too, but for me it was simply because I found the problems interesting and learned some new math. I most certainly never mentioned any classroom connections on my bi-weekly assignments … and Martin seemed fine with that too. Maybe flexibility on this is the key. We all SHOULD be reflecting on the math we’re learning and how that connects with other math we’ve done and with our classroom practice, whether the prof. requires it or not. However, this reflection is very personal. We are all coming in with different backgrounds and are at different stages in our careers. Why does the prof. need to be a part of this? I still maintain that their job is to teach the math. Period. Perhaps this is one area where we (students) can take the lead in improving this program. We SHOULD be communicating more and sharing ideas -- in a positive way, not just complaining. For me, this opportunity to network is one of the meanings behind the phrase ‘for teachers’ in the program title. Why don’t we create a blog, forum, google group … where teachers who want to discuss curriculum connections have the opportunity to do so, while those who don’t care to are able to opt out? Would the profs. need to participate or even read it? Only if they want to. It would be for us.
3. I do have a serious problem, however, with your continued use of the phrase “void of relevance”. Making connections is easier in some courses than others. But, I’m not sure what math in this program could be categorized as ‘void of any relevance to teachers’. Although it was not my goal at the outset, I have found several things in EVERY course that I could take back (in some form) and which ultimately have improved my classroom practice. Being a program FOR math teachers, I think that the (entirely reasonable) assumption is that the students in these courses will find mathematics intrinsically interesting and rewarding, without always needing to have the answer to the question “where am I going to use this in my day job?” spelled out for them. Remember - what's relevant (or not) for you may be very different from what's relevant (or not) for me.
4. Exit portfolio? Yikes. Been there, done that. It was called teacher's college. I can unequivocally say that I would NOT have ever enrolled in this program if that was a component since for me that would have been a hoop-jumping exercise ... this is not supposed to be a MATH AQ course.
My suggestion: How about an exit exam instead? Purpose: To add credibility to the MATHEMATICS part of the program title. I’m only sort of kidding.
I do like your 3.0 credit idea for adding flexibility in the program. I see that this is already in the works.
Hi Alice;
ReplyDeleteIt's nice that we can have a critical dialogue :-) Cross-listing of all the MA courses happened about 5 (?) years ago for the math education diploma so I'm not sure how/why the programme has evolved. As others mentioned in the survey, the confusion is what is meant by FOR teachers. If it's as what you suggest (i.e., networking with other teachers), I must reply with "Is that all?" I don't have a problem with the courses being about learning mathematics -- in fact, I enjoy that :-) but it comes back to what is meant by FOR teachers -- it implies (well to me, anyways) that it'll have an impact on our effectiveness as mathematics teachers. There is not much (any?) opportunity to talk with others in the course since the classes are all lectures -- so getting to know your classmates is tough. I chose the programme to get a deeper understanding of the mathematics I teach (through seeing the broader picture) -- for me, that hasn't been accomplished. But that's just me, of course.
You are correct in saying that what is relevant to me may not be relevant to you. At the graduate level, there should be (in my opinion) flexibility to the assignments -- e.g., to pursue your interests -- like the 30% paper in the Problem Solving course. I also agree with you that it is a MATHEMATICS degree. I want to learn mathematics but also have lots of opportunity to connect it to my work. All I'm finding I'm doing is learning mathematics with no connections made.
I'll be done all the courses for this programme by late July. I must admit that what I have learned has not impacted my professional practice -- what has been learned has been long forgotten. I just feel (and this might offend some, but it's not meant to be offensive) that the university has taken my money and granted me a degree for "doing my time". To me, that is quite sad.
Hi Louis (is there anyone else out there?)
ReplyDeleteLet me clarify how I interpret the ‘for teachers’ part of the title since I don’t think you are fully understanding my point of view.
To me, ‘for teachers’ means:
a) the ‘bums’ in the seats will all have completed a teacher training program and would qualify to be currently employed as teachers
b) all students have some undergraduate background in math or a related discipline
c) the students find math intrinsically interesting and are motivated to learn more
d) courses will not be offered at times which conflict with the typical Ontario school day schedule; there is understanding around job-related issues such as missing class due to parent interview night
e) the students will have the opportunity to network with colleagues, who may share common interests and challenges
As one survey respondent succinctly put it, this program is for teachers; it is not about teaching per se.
I notice that I can join my local gym using a membership plan ‘for teachers’ or I can purchase my insurance/mortgage through a program ‘for teachers’ or take a vacation to Costa Rica with a travel group ‘for teachers’. I would not expect the businesses offering these ‘for teachers’ options to outline connections to the curriculum or explain how my use of their service would impact my classroom practice. I view this program the same way … it is offered by the math department; therefore, I expected to learn math. Anything beyond that was just an unexpected bonus (and as I mentioned in a previous message, I do find that my professional practice has been improved as a result of my participation in this program).
I’d like to preface this last comment by saying that I have a lot of respect for your passion for math education and agree that it is sad that you didn't find this program rewarding. However, like most things in life, I found that with this program you ‘get out of it what you put into it’. I approached the courses with an open mind and willingness to work hard and take what I could from each one. As a consequence I found the program rewarding. However, (and I readily admit that I am somewhat of an anomaly in this regard), I took this program as an ESCAPE from my day job … some people garden, some like to jog, some go to movies – I take advanced math courses for credit … I needed something to keep my math brain active, to challenge myself, to have fun (a place where I didn’t have to think about prep, marking, curriculum, my students, etc). My reasons for taking this program were entirely selfish, personal ones. Perhaps this program (in its current state) is a much better fit for weirdos like me, than it is for dedicated, conscientious professionals like you.
Now, does the program description need to be changed? Seems many interpreted it differently than I did, so probably yes. After reading your suggestions on the previous message, I would comment that we need to keep it positive. We can’t have the program described entirely by what it doesn’t do. It already says that it doesn’t prepare for a math PhD nor lead to teaching certification (2 negatives). Here’s a challenge for you: How can we describe what it does do in a positive manner?
I'm traveling so I can't comment in detail as I would like. There are parts of this discussion that I need to say something about.
ReplyDeleteThe survey may point out that we are not meeting the needs of all the students, but I can't offer a different program. I want to tell you what we offer, not what others want us to offer. I can change some things to try to make this program better, but I can not turn this into a degree that will make everyone happy. Please read the survey results with this in mind. You can make some of the people happy some of the time...
I think that there is a fundamental disagreement that is not being explicitly pointed out here and so I am going to state an extreme point of view (which is very close to what I actually believe) and I would like to hear your comments. "EVERYTHING YOU LEARNED IN ALL OF YOUR CLASSES IN THIS PROGRAM IS RELEVANT TO YOUR TEACHING." If you disagree with that statement please give me some examples. Louis, you seem to be saying that the things you learned in the classes you took are not relevant, but I cannot address this remark if you are not specific. If you want me to give some examples from the classes that I taught I will (FYI, I will never be able to prove my conjecture because a finite list of examples is generally not sufficient to justify a statement that must hold for all elements in a set...."point" for me because I said this about 1000 times in Fundamentals).
The mathematics classes were crosslisted recently (within the last 5 years) and the faculty who teach them and those that arranged that they be crosslisted consider the material very relevant to teachers, even (or especially) those that are interested in mathematics education. Again, if you disagree with this statement please explain what you mean because we are on completely different pages if you think that they are not relevant to someone who is taking a program in mathematics education.
Hi Alice;
ReplyDeleteYou calling me a "bum" ;-) Haha... You a "weirdo"? I think not!
Seriously, though, I like your interpretation of "For Teachers". I think the title of the programme needs to be changed -- can't we just say it's an MA in mathematics, with the audience restricted to teachers? Make it clear that the intent is to pursue mathematics beyond that of the undergraduate programme -- which may or may not have a direct impact on one's teaching. As you said, it depends on the topic. I, for one, did not see how diophantine equations or generating functions could be carried over to my professional practices (maybe it's just me and my fault for not asking the professor) -- but believe it or not, I enjoyed the Fundamentals course -- especially the 2nd half -- it reminded me of university mathematics -- it was challenging and very rewarding when I experienced success. It was nice to do mathematics, which I don't do as a teacher. If that is the true intent of the programme, then it needs to be clearly stated -- and you'll get mathematics teachers applying to the programme and others not applying -- this will help ensure people are not misplaced in the programme. And I would still apply to it, knowing what I'll get out of it. As is, I'm not sure what I'm supposed to get out of this programme.
I do think it's great to be able to have this discussion in order to make the programme better for those that come after us.
Hi Mike;
ReplyDeleteI would not sell the programme by saying, " "EVERYTHING YOU LEARNED IN ALL OF YOUR CLASSES IN THIS PROGRAM IS RELEVANT TO YOUR TEACHING," since you'll get lots of people enrolling in this degree programme wanting to be better teachers. Can we remove "FOR teachers" and say it's an MA degree in mathematics (audience restricted to teachers)? The intent of the programme is to learn mathematics beyond that of undergraduate level mathematics, which MAY OR MAY NOT have a direct influence on one's professional practices. The key word is DIRECT -- so it takes the pressure off the math professors from feeling pressured to seek connections to high school teaching or to know their audience. The outcomes of the programme are to:
1.) Learn mathematics beyond that of the undergraduate level;
2.) Engage in the process of doing mathematics (through communicating clearly and concisely; problem solving);
and giving absolutely no indication that the programme will make better mathematics teachers or impact one's teaching.
That works for me -- I would apply with the lens of learning mathematics and doing mathematics -- and not wanting to seek how this will impact my professional practices.
Crap, Louis. I can't let this go since you mentioned it 3 times in the previous two postings ...
ReplyDeleteI do not believe that the math in this program is "beyond the undergraduate level".
While some of the topics may have been new to me, and some of the exercises challenging, I found that with hard work all were accessible. I don't think we should be giving the false impression that one needs a specialized honours undergrad degree in pure math in order to handle the program content. Having spent some time recently looking into 6000-level math courses, I frankly don't see anything in our program that would qualify as graduate level mathematics. Borrowing a page from Mike, if you disagree with this (as I presume you do based on your postings), could you please provide a concrete example for me?
Louis, I disagree with your statement "giving absolutely no indication that the programme will make better mathematics teachers or impact one's teaching." I still maintain my "EVERYTHING" statement about relevance and by relevant I mean that I think it will impact your teaching and make you a better math teacher.
ReplyDeleteAs for the fundamentals class "An emphasis in this course is placed on writing and explaining mathematics clearly." This is a skill that every math teacher should learn to do better (yes, me too). You should have at least gotten some practice at doing this in 5020 and hopefully you learned some ways in which you can continue to practice this by turning the simple questions that are no more difficult than those you teach in high school into writing exercises where you need to explain every detail. Certainly thinking about how you explain every detail must make you question how you explain math to your own students. Or am I way off track?
I can explain to you the context in which generating functions and diophantine equations are relevant to teachers, but I am afraid you are missing something if I need to explain "you are going to teach x in your classes so you need to learn y." I can do this, but this is not the point of the M.A. for teachers program. I hope that when you finish this program that you have the skills to do this yourself.
What I can hope is that after an entire year of studying diophantine equations (well a FW term) that you have many examples of where solving diophantine systems arises in places that your students will be asking you about because they heard about it in the newspaper or in a movie or even in their textbook: Fermat's last theorem, cryptography, the Chinese remainder theorem, pythagorian triples, euclidean algorithm are all things that I am pretty sure do come up in the average high school classroom. If not, they should and you should be bringing them there!
I think I'm more confused than ever about the purpose of this degree programme. Mike, please write a paragraph on what is the intent/purpose of the programme is. You're saying the programme impacts teaching practices and makes one a better teacher [this implies a focus on teaching and education], yet Jen is saying it's about learning mathematics: "it is offered by the math department; therefore, I expected to learn math. Anything beyond that was just an unexpected bonus."
ReplyDeleteI
Hi Mike;
ReplyDeleteNone of the examples you've mentioned for diophantine equations is in the high school mathematics curriculum -- Fermat's Last Theorem (once could discuss this as an extension of the Pythagorean Theorem, but what else??); crytography (matrices are out of the curriculum, maybe factoring?); Chinese Remainder Theorem; Pythagorean Triples; Euclidean algorithm. So I still don't know why diopantine equations are learned and how they would impact my teaching and make me better in the classroom.
Also, I still have no idea what relevance generating functions has to high school mathematics. The same goes with widgets and doddles. What I did take with me from your course was the rewards of doing some challenging mathematics and your enthusiasm as a math prof. Yes, communication was emphasized as well as problem solving. But these are the mathematical processes and not the actual content learned.
Wow...interesting discussion. I came into the program thinking that it was a mathematics education degree, and was actually quite surprised to find out that it wasn't. That being said, it is my decision to stay in the program for what it is, not what I would ideally want it to be. Nobody's forcing me, or anybody else, to stay; I'm sure it would be possible to transfer into an M.Ed program if what one really wants is more of an education focus.
ReplyDeleteGranted, I've only completed one course and am in the process of completing two more, but so far I do see the relevance to what I do professionally. The thing to keep in mind is that we are not teaching our students the same material we are learning in this program, which, so far, is mainly at an undergrad level. I know people who did abstract Algebra in undergrad at York, and they learnt much more that we are going to cover. We all did an undergrad math degree knowing that we would most likely never be teaching game theory to our students, since we didn't encounter it until university. It's the same thing here. I don't expect to go back and teach my students about groups, but I can, for example, expose them to proofs and techniques for proving things that they are learning, even though it may not be precisely in the curriculum. I have all my classes doing cryptograms for bonus marks, because it's fun and it teaches logical thinking skills. You can do fun things with exponents and the towers of Hanoi problem.
The point is that we're never going to be able to go to class on Monday night, and Tuesday teach the same thing to our students. Maybe it's just me, but most of what I learnt as part of the ED degree didn't directly help me either, but it did give me a broader perspective of what I do and I've taken what I've learnt, tweaked it to my situation, and applied it that way.
Here's where the cliche part comes in: if you choose to stay in the program, take the most you can from it and do the best you can to apply it in school. If nothing else, you'll have a renewed appreciation for what our students go through at exam time ;p
Louis,
ReplyDeleteThese are not contradictory statements:
(1) YOU CAN EXPECT TO LEARN MATH IN THIS PROGRAM. (Me)
(2) EVERYTHING YOU LEARNED IN ALL OF YOUR CLASSES IN THIS PROGRAM IS RELEVANT TO YOUR TEACHING, i.e. will impact your teaching and make you a better math teacher (Mike)
In fact, assuming both (1) and (2) are true, then we can conclude that the math you learn in this program (being a subset of EVERYTHING), will impact your teaching and make you a better math teacher.
So far, I haven’t seen a valid counterexample to show that either (1) or (2) is false.
Hi Sarah
ReplyDeleteI'm trying to understand how you might have come to the conclusion that this was a math ed degree.
I pretended to be an interested applicant with no prior knowledge about the program, and meandered around on York's website.
Here are 3 descriptions of the program that I came across.
(1) We offer an MA in Math for Teachers – this part-time program is unique in Ontario – designed to help high-school teachers to be more effective in their classrooms and assist with curriculum development.
http://www.yorku.ca/web/futurestudents/graduate/programs/Mathematics_and_Statistics/#degrees
(2) In existence since 1975, this program focuses on giving the student an overview of various mathematical fields and discussing mathematical issues of relevance to teachers. It will thus provide the teacher with a broader perspective on school mathematics. It does not prepare students for study at the Ph.D. level.
http://www.yorku.ca/web/futurestudents/graduate/programs/Mathematics_For_Teachers/
(3) The program focuses on giving students an exposure to a variety of mathematical subjects providing to those that are teachers a broader experience that they can bring to their own classrooms. The range of this program gives students an historic perspective as well as chances to practice techniques of problem solving, writing and presenting mathematics. These elements are relevant for teachers of mathematics at any level.
http://www.math.yorku.ca/new/grad/maTeacher.htm
My opinion: (2) and (3) seem accurate. I could see someone getting a false impression from (1). It should be changed.
Louis
ReplyDeleteSigh. guess what...
All of the things that I mentioned are in the Ontario high school curriculum or in almost every high school classroom and it is not just relevant to your teaching, I claim it is fundamental that you know where math is in your own surroundings and be able to discuss it with your students.
But the fact is you are getting a masters degree, you should know MORE than the Ontario curriculum.
1. How do you teach your students fractions? Do they have to find the greatest common divisor? In order to understand how to calculate the greatest common divisor for large examples the Euclidean algorithm is really the only way we have to do this. You may not teach your students this directly, but you need to know it exists and how it works and not just what a high school student knows.
p 30 9th/10th grade:
simplify numerical expressions involving
integers and rational numbers,with and
without the use of technology;
2. Have you seen this problem before? It is probably in a problem set that you will come across:
Odd Oranges
Greengrocer C. Carrot wants to expose his oranges neatly for sale. Doing this he discovers that one orange is left over when he places them in groups of three. The same happens if he tries to place them in groups of 5, 7, or 9 oranges. Only when he makes groups of 11 oranges, it fits exactly.
The Question: How many oranges does the greengrocer have at least?
This is the Chinese remainder theorem.
3. Do you use the internet in your school? When you contact your bank your computer is calculating 100's of times a minute the Euler-Fermat theorem, gcds, and lots of modular arithmetic. I bet you have bright students who know about this (it is in books like "The Code Book"). Even if it isn't in curriculum, it is in the classroom and you should have some competence to discuss it.
Do you discuss the show Numb3rs in your class? Apparently season 5, episode 13 scene 9 covered some of what we covered in fundamentals.
4. Show of hands, how many of you showed the NOVA program about Andrew Weils and Fermat's Last theorem in their high school at some point? It is a diophantine equation. Does it have to be explicitly in the curriculum book for it to appear in your classroom?
One would hope that you would come away from this program and have some ability to discuss the importance and the influence this work and how it is related to other mathematics.
5. Diophantine equations are any system of equations solved over the integers. Any word problem where the answer is a whole number is probably a diophantine equation.
6. When math comes up in the news do you discuss it in your classes? If you don't know enough math then you will ignore these current events when they are relevant to showing your students that math is everywhere. e.g. When GIMPS finds another large prime? A hashing algorithm is found to have a security fault in it? A computation team finds a way of breaking RSA?
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Bring me your high school textbook (almost any one will do, but 'functions' might be stretching it) and I will show you how to solve problems in it using generating functions. I think if you had learned generating functions better you would be able to do this yourself.
Do you remember the day that Mario brought a problem from a problem set that he used about paths on a grid. He solved it as fast as he could and I used the addition and multiplication principle. I remember that I won by having an expression for the answer first. Go widgets and doodles.
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You think that what makes you a better teacher is that you reflect on your teaching. I think what makes you a better teacher in mathematics is that you know more math, show enormous amounts of passion for it, and are able to communicate both the excitement and the material to your students. Show your students that you can and should be excited about science and mathematics and this will begin to show in the next generation of students.
That way when I go to a party I won't have to answer the question "So what good is math anyway."
Thanks, Mike! You've shaped my view that one needs to keep a broad perspective (i.e., an outside view looking in) when taking these mathematics courses to find the relevance (if one seeks it). Often times, one may immerse themselves in the concept (i.e., without looking out). The very fact that you had to explain to me how the mathematics I learned fits into the high school curriculum, tells you something -- I hope it's not just me :-|
ReplyDeleteWe can talk about how the math learned leads to applications, for instance, factoring is needed for cryptography, but the applications is lost when one is struggling with the procedures needed for factoring. Maybe it's just the way I teach but I do talk about cryptography (3 minutes) but what is valued is that students learn the techniques to factor (several days). Students who cannot factor will be frustrated and wouldn't care about its application to cryptography -- does knowing it's important for cryptography motivate students to learn these procedures? That's how I've felt in taking the courses -- I had no idea how the particular math concepts I was learning was meaningful (although there is a possible connection at the end) -- that's similar to how frustated high school students can become -- e.g., how to make factoring trinomials relevant WHILE they are immersed in the procedure; how to make simplifying polynomials in grade 9 relevant WHILE they are immersed in the procedure. Can't students just learn mathematics for the sake of its beauty and elegance?
I still am uncomfortable with the programme being sold that it will help with mathematics teacher effectiveness [this was how it was presented to me and I was disappointed when I didn't get it]. As Alice said, then take an MEd degree (I have this and it truly impacted my teaching directly). Instead, I can now see through this wonderful on-going discussion the MA degree impacting my teaching INDIRECTLY through the opportunity to do mathematics -- I don't see the direct impact of each class but it's there through the experience of taking MA courses -- i.e., having experienced the mathematical processes. That's what has made the degree programme "bearable" for me -- I have given up long time ago to expect my profs to make direct connections to the high school curriculum. When I think of this programme is about learning more mathematics, it helps me view the programme positively. Expecting to impact my teaching (as I did when I took it) lead to frustrations and my belief the programme has failed in that regard -- I still see nothing with learning mathematics for its own sake and beauty/elegance. It's a tough sell as a high school teacher to explain DIRECT relevance to every mathematical concept I teach -- and students have given up asking since they know what is valued is what is taught.
It's good discussion we've had :-) Thank you.
Hi Sarah;
ReplyDeleteYou're not alone in feeling this MA programme is a math education degree -- with a focus on the mathematics. I, too, thought that. It was in fact "sold" to me (or how I interpreted it) when I spoke with the previous coordinator of the programme. I, too, have gone through the transformation in thinking that you have. I think if one has high expectations for this programme on direct impact to one's teaching, you are setting yourself up for disappointment. Just so you know...I took this degree programme after withdrawing from the PhD programme in education -- if the programme was sold as learning more mathematics, then it has done its job :-) I am sure there are teachers out there who are shying away from this degree programme since they don't think it's too "mathy" (i.e., they want it to be "mathy"). There is no need to talk about relevance to teaching [as more folks such as yourself and I will apply to get in and then find it not what we expected] -- reminds me of the current situation at my school with students misplaced in academic and applied courses! We're not alone as the survey results indicate. BTW -- I am #2 in the survey!
Alice,
ReplyDeleteI heard about the program when I was in my third or fourth year at York, I can't remember how. I initially did think it was just math, but when I went to see Walter Whitely in my fourth year about applying, he told me that most of the people in the program had been teaching for many years, and that I would need teaching experience first. The impression I was left with was that it was focused on math education, for current/former educators only. That's why I was surprised to find out that it was mainly math, and that teaching experience didn't make a difference. The first description you listed, along with some of the (admittedly few) descriptions helped cement that veiw.
btw, 'pretended to be an interested applicant'???
Louis, If you really do spend any time with cryptography in your high school class you would do well to take my 4th year undergrad course, Mathematics of Cryptography. I think in the first 4.5 hours we have enough material to keep high school students busy for weeks. Alice, you said you were teaching some of it to your students, what do you think about turning it into a 3.0 credit MA for teachers class? Sarah, you've taken it plus a 3.0 follow up reading class. Is it good "for Teachers?"
ReplyDeleteI knew that when I posted the list of diophantine equation related topics that I needed to do more justification. I was traveling and didn't have access to all the resources I needed. I can go into more detail on any of those points and add some if you would like. I maintain my original position: that I don't know what your classes are like and I have every confidence in your ability to teach high school classes and I claim no expertise in teaching high school mathematics.
I am coming from a school that believes that all mathematics teachers will bring more to their classrooms if they know more math. I think that this is the attitude that most mathematicians have. We study math day in and day out for decades and have a desire that the public has an appreciation for what we do. This is what I believe the mathematics side of this program is about (there is an additional education component that is part of the program too, but I am not representing that side).
When you say that in Martin's class that part of the assignments asked you to reflect on how the material relates to your teaching, what does this mean to you? Don't you always reflect on how these courses relate to your teaching?
I have this vague recollection of going through the high school textbook and I had an argument with you in Math 5020 and I felt at the time that I was clearly able to say that many problems in your book are exactly what we were doing in class. This made me upset at some point because you had successfully pointed out that fundamentals was no more difficult than a high school level course.
Sarah, since you are a recent York grad, I think that you have a different perspective than many of the applicants to this program, but I like what you said about applying what you learn in this program. I hope that you will be able to have discussions with your students about the "math of image manipulation" or "solving integer equations" or "where algebra is important" but we are not going to tell you how to teach this in your classes.
Alice found the bad program description because I also don't like how (1) is comes across in describing the motivation of this program.
Mike,
ReplyDeleteI produced this list of suggestions last September (at Augustine's request). Refer to #3.
Augustine:
You asked me to propose some ideas for next summer's course. I still think that it might be easier to find an instructor and then build a course around that person’s interests, rather than this way around, but here are my initial thoughts. Of course, I’m assuming that any of these would be taught at a sufficiently challenging level to make them reasonable courses for the program.
1) Financial Math – We teach it, but mostly in college and workplace level courses, which is probably a good thing since most of us have never formally studied the math of finance. People would likely find this interesting, if not for professional reasons, then at least for personal ones.
2) Problem Solving – This course is adaptable to suit a prof.’s interests and was offered long enough ago that there aren’t likely many people still around that took it the last time. Ideally I think that this course should incorporate teaching the use of SAGE or Maple and then using the power of the software to help in solving problems. Not that problem solving isn’t interesting in itself, but teachers would find the technology part something useful that they can take back to the classroom and use in their teaching.
3) Cryptography – We spent a couple of weeks on this in Fundamentals, but there are enough interesting aspects to this topic that it could (should?) be pulled out of that course and offered on its own. Might be best as a 3.0 credit course though.
4) Graph Theory – I like it. Some graph theory is introduced in the gr. 12 Data Management course. But you can work in enough theorems and proofs to make it sufficiently challenging. Also might be best as a 3.0 credit course.
5) Math of “Entertainment” – eg. math behind sports/sports statistics (baseball?), gambling (gee whiz, maybe Mike should just teach every course in this program), etc. Don’t know if this would be ‘difficult’ enough for the program, but it would definitely generate interest. Field trips would be fun.
6) Geometry – People seem to want this.
Alice thanks for reminding me about that email. I think that you forwarded a copy to me too. I'm thinking for Summer 2010 we might introduce two new 3.0 credit courses.
ReplyDeleteOne thing that I should comment on what Sarah said: "when I went to see Walter Whitely ... focused on math education, for current/former educators only."
It is true that we generally do not admit recent graduates to this program. These classes are specifically designed for people who have been teaching for a while and are not just continuing their education but instead are coming back to mathematics. Occasionally we allow an undergrad to take one of the 5000 level courses, but that is also a rare event (and when it has happened I found that the current teachers were mentors to the future teachers).
But the reason that we try to focus this program on returning students is not because this is a math education degree, but because the math that we teach someone who is finishing a B.A. math degree would be different than the material we would would teach to someone who hasn't been in a university classroom in 20 years but has been teaching high school math in the meantime.
It is expected that you will share the connections that you see between your high school classes and the material in the program during the class and you should be talking and asking about how what you teach in your classes is related to the class material.
If you don't bring this up in classes you will find that the professors teaching may not bring it up either. In my case its because I assume it is obvious a lot of the time (and probably for most people it isn't so obivous).
Mike,
ReplyDeleteI think a 3.0 crypto class would be amazing for this program. The formal and the reading class were the most fun I had during my undergrad. I also think a lot of people would enjoy it since it puts a different spin on recreational math, which I'm guessing most of the people in the program enjoy.
Alice,
I like your suggestion for a financial math course. I would definitely be interested in taking that one, but mostly for personal reasons. Entertainment math would also be an interesting one.
Louis,
I want to clarify: my thinking that this was a math ed degree was MY issue; I don't think that that is necessarily the impression that most people had. I don't believe that that was really ever the case. I'm enjoying it the way it is now, and do not regret that I am in it. It is up to us to find the 'direct relevance to our teaching', given that we are all teaching different grades and different levels.
Been great discussion and I wish more people would have been a part of it. This will be my final posting in this discussion as I have so many things to do (enjoyed the March Break a bit too much :-| ).
ReplyDeleteMike -- it wasn't me that you had an argument with last year. I didn't take it with the lens of thinking how I could apply it to my teaching. In fact, I do believe that mathematics helps develop logical and abstract reasoning -- not everything we do has to have applications to the real-world; it may, but as I said, when we learn the mathematics concepts, it's not the applications that are at the forefront -- it's the procedure or skill being learned.
We will need to agree to disagree that everything in the programme has relevance to teaching. If I agreed with this, then I would be very disappointed with the programme -- maybe it is my fault for not trying to find relevance of what is taught to teaching. If that is the intention of the programme, great; but for me, the implemenation of the programme meant doing mathematics often for the sake of it -- it's good stuff for the brain. Although I don't feel there's any direct impact on my teaching, there may be indirect impact that I don't see at the moment. I've solidified this through the wonderful discussion of the past little while.
You should know that during the past 2 years, I have applied to 2 leadership positions and on both occasions, I mentioned that I am working on a Master of Arts degree for mathematics teachers -- one of the interviewers mentioned that it is "not a real math degree" -- and in another position, I was told, "You're getting a second masters degree in mathematics education." My intention was to demonstrate a strong mathematics background but that didn't come through.
PS: On the application form, it does say that there may be an interview with the coordinator of the programme. Does this ever happen? I think it could be a good thing to ensure people that accept the offer of admission have a clear understanding what the intentions/purpose of the programme are. I'm still not clear (I'm slow...) but I've come to the intentions/purpose that will work for me.
Go out and enjoy the March Break everyone :-)
Louis, Finally we agree on something...we are going to have to agree to disagree. What you call "indirect" and "direct" relevance to teaching I think is reversed from my view.
ReplyDeleteDirect relevance to teaching to me would be absorbing more math and become an expert to a degree where one is able to make the links and express to your students that there is math everywhere. Ideally you should understand where mathematics comes from so that you are able to make the decisions about what should be in the curriculum so that your students are prepared when they get into university.
Indirect relevance to me is scholarly research on the practice of teaching mathematics.
....po-tay-to/po-tah-to.
One of the problems with the way that you have come through this program. You didn't take Algebra, Analysis, Computation or some of the other classes. I hope that you get as much as you can out of the Modeling class. I think it is an exciting offering.
Here are some references where generating functions are clearly indicated to be appropriate as a topic in advanced high school level mathematics:
ReplyDeletehttp://mathforum.org/library/drmath/view/51564.html
http://en.wikibooks.org/wiki/High_School_Mathematics_Extensions/Counting_and_Generating_functions
http://www.jstor.org/pss/3026852
http://www.imomath.com/tekstkut/genf_mn.pdf
Here is a list of references on the Ask Dr. Math website on high school level questions related to Diophantine equations:
http://mathforum.org/library/drmath/sets/select/dm_diophantine.html
Here is a really good diophantine equation exercise set appropriate for high school (or MA in Math for Teachers):
http://www.math.harvard.edu/~knill/seminars/diophant/diophant.pdf