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Thoughts on math education - From a young college student.

  1. Aug 23, 2011 #1

    JyN

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    I am a second year student of engineering at a Canadian university and am very interested in mathematics and education. I feel like the methods of teaching math in high school and early university (I have no experience with senior university classes) are not only an inefficient use of the teachers knowledge but are also boring, stifling, and doesn’t do justice to the subject.

    If we look at English classes, students are required to be familiar with certain material before attending class. ie) reading a chapter of the book being covered. And, the majority of class time is spent discussing said material. The teacher guides the students to think about what they have read and encourages everyone to collaborate and answer some kind of question. It is essentially an exploration of the work. Personally, I find this to be a very rewarding method of learning, and only someone with extensive knowledge of the topic would be capable of guiding such a discussion.

    Now if we look at math class, the teacher dictates results to students (often skipping proofs, the mathematicians motivation, and the general concept) who copy the notes and try to essentially memorize how to apply a theorem. (This is of course somewhat of an over simplification, but that is the core of math classes in my experience). This is very boring for the student, and probably for the professor as well. Moreover, I could copy notes onto a blackboard as well. Professors with advanced degrees are essentially turned into middle-men between the text book and the student. A clearly inefficient use of the profs expertise.

    Why not model math education after English education? In my math class: students would be required to familiarize themselves with a chapter in the textbook and then come to class to discuss the material with their peers and the professor. The professor could talk to the students about the motivation for researching such a topic, explain the proof behind it, and enthusiastically answer any of the students’ curious/clarifying questions. Again – it is essentially an “exploration” of the material. Finally, I think that the most rewarding part of modeling a math class in such a manner is that the class could take part in a discussion to try to solve a very difficult problem, just like in an English class.

    I feel like this type of approach would be a much more effective way of teaching math. In my grade 12 trig class i had a seriously terrible teacher. Because of this i started to try to teach myself from the text book. And, the more i teach my self (as apposed to taking notes in class) the better i have done in class, and the greater my understanding. I recently got 99% in integral calculus, and 96% in vector calculus. I attended ~1/4 of my classes. I am not trying to brag, and i do not think i am gifted either. I spent a large portion of my studying time trying to solve the most difficult problems in the text book, and i rarely succeeded, yet i was able to achieve marks unheard of by almost anyone else in my class. How else could i have done that if it wasn't the product of a superior approach to the subject?

    I would greatly appreciate peoples thoughts on the matter, even if you only want to tell me off. Or if anyone can offer a suggestion for more acceptable place for this kind of discussion that would be incredible as well :D
     
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  3. Aug 23, 2011 #2

    Hootenanny

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    I can emphasise with you somewhat, JyN.

    I'm not sure what the US/Canadian system is like, but over here in the UK there are often dedicated courses in mathematics for non-maths majors. For example, as an undergraduate, I chose to do Mathematical Physics, which essentially meant I did a double major in Maths & Physics. Obviously, this meant I took the full Maths classes.

    On the other hand, if I had chosen to Major in Physics (or theoretical physics etc.), I wouldn't have been able to take any "true" mathematics courses. Instead, I would take courses like "Maths for Physicists" or "Mathematics for Engineering". Essentially, these are stripped down versions of the full maths courses, which contained about 50-60% of the material that Maths Majors would see. To accommodate this, most of the proofs, interesting remarks and asides would be stripped out to just leave the core "bones" of the course. Students would be expected to learn results by wrote and not expect to reproduce proofs in exams.

    I agree that this is not an effective way of teaching mathematics and is exceedingly dull, both for the students and teachers (I TA'd one of these courses earlier this year). I'm not saying that the full mathematics courses were perfect, but they were far better than these half-cooked courses. Time was allocated for discussion and "exploration", as you say. Not as much time as I would have liked, but it was still there. In the full maths courses, we were also expected to learn proofs - not just how to apply them, but the concepts behind them. Certainly in honours and graduate level course significant time was spent working on "unseen" problems and we were expected to be able to prove "unseen" theorems etc.

    As I said, the US/Canadian system is likely entirely different and this was just my own experience.
     
  4. Aug 23, 2011 #3

    chiro

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    Hey Jyn, great post and topic.

    What I've noticed is essentially the same kind of phenomena that you have described. I should note that the higher the classes, the more you are expected to do a tonne of work outside contact hours, and along with that, you are also going at a much faster pace than in junior years. Classes with research professors can go very quick in comparison to junior ones.

    With that said, at least in my university, the junior classes are taken by a variety of majors as a core subject and as a result, the subject has to cover all of the basics that meet the needs of all majors taking it.

    As a result of all these requirements, you find that the mean is low for a math major, but probably average or slightly high (with regard to math content and expectations) for a "non-math" major (example chemistry major, environmental science major, and so on). So with that in mind, I don't see it surprising that math majors get frustrated in lower level courses.

    The alternative to the above is to get into a university that has specialized math pathways like say Advanced Mathematics/Science degrees with really high entry requirements. They definitely exist, but the thing is it requires a lot of work, and a lot of people that enter university need the first year to really settle in due to the fact that a lot of people have this freedom that they didn't have, and as a result need a little time to experiment, screw up, and mature so that they can pick up their game in later years. With this said, it is probably a good idea that first year is really a walk due to these factors.

    Persevere though, because as you get to top undergrad/graduate level classes, I'm fairly sure that everything will go up a gear, and then you'll have much more work and maybe look forward to a breather in place of being bored.
     
  5. Aug 23, 2011 #4

    JyN

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    I definitely see the practicality of what you are saying Chiro, but I'm not really advocating we should just make classes more difficult. For me, one of the most inspirational moments in my mathematics "career" was in grade 9 or 10 when i saw learned about the proof of the formula for the area of a triangle A=1/2bh, its just half a square.. This made me realize how simple some things could be and i really started to think about what math was really all about.

    I don't think math has to be difficult to include discussion and exploration, there is plenty to explore on the surface. I in fact think it would make it easier!

    And part of what i think would be so great about teaching math like this is not necessarily even for the sake of discovering the math itself. But more about the experience of delving deeply into something yourself. I feel like math is a great medium for learning how to think on a deeper level and to see connections between ideas. I would imagine that it's a skill that would be valuable to any profession, not just to mathematicians.
     
  6. Aug 23, 2011 #5

    chiro

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    I wholeheartedly agree with what you are advocating.

    The thing is though that most people in the western school model (for lack of a better analogy), is used to the idea that the teacher is the authority and that the student-teacher relationship is more or less a situation where the teacher is the wise one, and the student must simply absorb the wisdom from the teacher.

    If you add all the bureaucracy that happens in education, then you can see that this environment causes many people to simply think that the teacher/student relationship is the norm, even in a institution like university.

    In fact for many people, university can provide a big shock, because people have gone through their education for the majority of life learning to be so dependent on the authority which can really do disasters for independent learning and thinking.

    With regard to math in high school, there are not many truly passionate mathematicians that want to work in a high school. I myself did a practicum in a high school and I was lucky to be in a good school! With all the crap that high school teachers have to go through, it is no wonder that dedicated, motivated, and inspirational mathematicians are found in universities, corporate research labs, or in other fields where their expertise and dedication is well rewarded.

    I agree mathematics is in a lot of ways, a way of thinking. It is a language, a construction of concepts, and probably the best language in terms of the capability to accurately describe and analyze something.

    Other languages like spoken and written languages like English are powerful, but they are in no way as general or precise as mathematics. When I say the word "cat" most of us have an idea of what cat is, but it is no way clearly defined.

    If I talk about the unit circle in the complex plane, I know exactly the set of objects that this refers to and on top of that I know exactly what isn't part of that definition in the context of the complex numbers. This kind of thinking actually helps you in general communication, because it helps you formulate specific thoughts with absolute clarity. The myth that mathematicians are bad at communication is largely a myth and if you ever read a typical math paper, you'll find that the communication skills are often top notch and with regard to my comment above, I am not at all surprised.

    In spite of the above, I have found that there are good teachers out there and they do deserve due credit. Despite the restrictions placed on them, they do manage to use good examples and have a personality that allows the lecture to be that bit more enjoyable.

    Just to finish, I want to go back to your point about independent style learning. I am optimistic that it could be done, but it would require a restructuring of not only the education system, but also the values of society. Different people whether they be teachers, concerned parents, administrators (bureaucrats), and the general public all have different ideas of what education should be and this has an impact on how things get done. A lot of older people might think that education has gotten "too soft", and people like yourself and I think that it is not as interactive a process as it could be, while politicians are saying that teachers need to focus on getting said standardized test average up to x to compete with other countries. The thing is, everyone has a different idea of what the problem is, and because of this, there is no real consensus and friction tends to remain.

    As an example, you should ask a few people if they think mathematics is important in education, and ask them why. You would be surprised of how many parents say "yes, absolutely" for the first question, but then for the second give a really vague or non-informative answer for the second question.
     
  7. Aug 23, 2011 #6
    I hear you JyN, I could quote many of the things you say.
    During the years, I was able to find out only unsatisfactory and few ideas, but first I think that the school in western countries (and maybe everywhere) is modeled on old standards which are very slow to change. Today we have plenty of resources to study and get information: internet, TV, books, libraries are almost everywhere. Unfortunately school was conceived when even books were a scarcity, so the only way to collect knowledge was to listen the professor talking and copy down the knowledge.
    Another issue is that maybe you and I are able to learn from a book without any human interaction. It seems like in general people learn much more when they are in a classroom with a teacher instead that in front of a book. I never understood why but it has to be like that.
     
  8. Aug 23, 2011 #7
    Hey JyN - I just finished my bachelor's in math and found that my experience was very similar to yours for many of my classes. The professor would stand in front of the board and regurgitate the material in the book for the period of the class, everyone would take notes and then class would let out. For many of these classes I would skip every day that didn't involve tests, quizzes, or handing in homework, and I would often get better grades than my classmates who attended every day.

    In upper level classes the issue abated somewhat - we were expected to learn much of the material independently, and a fair amount of time in class was reserved for questions and discussion. This was unquestionably a more effective method of learning.

    It is worth mentioning that at the summer camp for high school kids participating in the international mathematical Olympiad, the approach to teaching is more like an upper level math class. Only a short amount of time is reserved for introducing a concept, and the rest of the "lecture" is devoted to working on and discussing problems.

    I think the problem isn't necessarily that educators don't know how to improve math classes - surely they must realize how dull and redundant their methods are - but that there is a feeling that the general population isn't able to absorb the material independently. This is ridiculous of course, because if a student can absorb material from notes directly based off the textbook, they can absorb material from the textbook itself. But there is also an expectation from many students that professors will teach them exactly the contents of their textbooks, because they have been trained to believe that they cannot possibly learn anything without a teacher - even if the teacher is completely redundant! To me this is the most insidious part of it - there is no expectation, from educators or the students themselves, that students will be able to learn anything on their own. That attitude has to somehow be reversed if we are to see any positive changes in education.
     
  9. Aug 23, 2011 #8

    Stephen Tashi

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    That would be fine if all students were talented and diligent enough to do that, but they aren't. (Not all are talented enough to do that in English class either.) Your prescription for educational reform won't work for schools that must teach a population of students who have very diverse capabilities.
     
  10. Aug 23, 2011 #9
    I can understand different students have different skills in math or whatsoever, but when it comes to diligence, then I start to complain.
    I can only speak of my school career until 19 yo, I never attended university because my only parent could not afford it with other 4 children.
    When I was in high school I found very annoying and boring following lectures because as it is already evident from this 3d, classes consist of re-writing books which are already written, by listening to a professor which has learnt them by heart.
    And then there were the many times where we would do very easy exercises in class, because maybe half of the class didn't understand, or better, they pretend to do so in order not to add new material.
    I don't claim that school has to leave behind students with difficulties, but after all it was my time, my years.
    I spent 13 years in school doing what I probably could have done in 9 years or so. A lot of time throw away. Then in the end the hoax: I could not afford the engineering course for economic reasons while almost all my classmates did it !
    School sucks and professor suck. Until under age, school is basically a park for children, and then a place which pretends to instill a few concept in the minds of boys who cannot care less.
    (Nothing to personal with you, this is just a random vent)
     
  11. Aug 23, 2011 #10
    Well, the average moron doesn't know what a square root is. They just answer "yes absolutely" as a jerk reaction, because somewhere in their minds math is linked with "culture" and education.
     
  12. Aug 25, 2011 #11
    It really depends on what you are trying to do.

    It's very boring for *you*. I think it's also very boring for most of the people that are reading this forum. However, it gets the job done for most people who aren't particularly interested in math and who have no interest in science and engineering.

    What will happen if you try to do something other than memorization with people that don't have good math skills to begin with, is that they will totally freeze, and they won't end up learning anything. If you give them a cookbook, and have them follow the recipe, they will end up learning something, and that something might come in useful for the types of problems that they end up doing in their day to day lives.

    1) That's why high school teachers (and in practice a lot of teachers in undergraduate courses) don't have advanced degrees, and

    2) In my experience, professors with advanced math degrees aren't particularly good at teaching basic math.

    This gets to another point in that one reason that math is taught in the way that it is, is that you don't have a lot of people with advanced math degrees that are also great teachers. One thing about cookbook methods is that it greatly reduces the amount of skill and expense necessary to teach math.

    Because you'll find that in 90% of the situations, that the students will end up clueless about what to do. At that point you *could* tell all the students that haven't done their homework to leave, but you haven't taught them anything, whereas if you go through "cook book math" they would have learned something.

    This will work if you have the right students and the right teachers. MIT has been restructuring its physics curriculum along this sort of model, but not everyone has the math skills or the motivation of an MIT frosh.

    That's a wonderful approach for a motivated student that is gifted in math. Most people aren't motivated students that are gifted in math. One problem with class discussions on math is that some people are much, much better at math than other people, and if you have open class discussion, then what ends up happening is that the people at the bottom how have math anxiety will end up with even more math anxiety.

    What I've found in teaching people basic Algebra I is that you have to give some pep talks. We both know that you aren't going understand quantum field theory, but if you follow these step-by-step instructions, I can show you how to solve a linear equation, and I can give you some ways that you can take that knowledge back to the office and be a more efficient worker.

    For you and me, yes. For 95% of the people in the world, no. And remember, part of the reason that the school system really caters to the 95% of the people in the world, is that you and I can survive an awful math class. Most people can't.
     
  13. Aug 25, 2011 #12
    The Eastern school model is even worse at this. In China, English classes are taught pretty much like math classes. You read the passage, give the expected answer about what the passage means. If you have any of your opinions, you keep them to yourself because you are the student and the teacher is the teacher.

    One huge problem that Chinese students have in adjusting to the United States is being able to cope with seminar classes. Writing an essay about what you think or reading a book and then talking about what you think of it is just not part of the Chinese academic system.

    In part this has to do with the political system. If I'm walking on a street in Beijing, and some stranger asks me what I really think about the Communist Party, I'm sure as hell not going to give any original answers to that question. I'll tell my wife and friends what I think. But I'm not talking to strangers, and I'm sure as hell not going to write anything down in anything that I think anyone other than the people I trust are going to read.

    And you'll find that it's just not China. If someone I don't know at work asks me what I really think of the CEO, I'm not going to immediately answer that question.

    But that "crap" is part of the job. It's a 100x easier to teach if you have bright, motivated students. The trouble is that people that are not bright and non-motivated also need education.

    And at some point, you have to wonder if you should change societies values. For example, the people that I taught at UoP. They really have no interest in the beauty and wonder of math and physics. They are there because they think that if they learn this stuff, they'll make more money. Now we could get into a long philosophical discussion about whether this is a good thing or not, but ultimately, I just teach algebra and I can't and probably shouldn't tell people how to run their lives.
     
  14. Aug 25, 2011 #13
    Further, the assumption behind this thread is that bright/motivated students can effectively teach themselves. Given that, there's no need for a teacher to structure a curriculum for those students since they can obtain good results on their own.
     
  15. Aug 25, 2011 #14
    These threads are always very interesting to me in order to learn about the differences between academic systems. I am from Europe and I was tought proofs, mathematical motivations and concepts from day 1 at the university but also in the 2-3 final years in high school.

    This is more than 20 years ago and I feel things have changed a bit - nowadays there are lectures called mathematics for physicists. Back then all sciences and engineering students needed to take exactly the same algebra and calculus lectures in the first year as the math students. Since in my country there were no entry exams or other barriers these mathematics lectures constituted the 'weed out classes'. You did not need to prepare before lectures but in order to solve problems based on these lectures you needed to put in a lot of additional time immediately after the lectures not to loose track and to turn these rather abstract concepts into problem solving skills.

    Some other guys in this forum from European countries have also confirmed that math education in Europe is still more 'proof and concept based'.

    I really enjoyed this style of teaching, but the majority of colleagues considered it 'too theoretical' - they might have been happier with more examples and demonstrations of how to solve problems and less proofs.
     
  16. Aug 25, 2011 #15

    mathwonk

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    It sounds to me as if you were in the wrong class. Usually you have some responsibility for that yourself, i.e. taking an easy class, or one that is non honors for fear of a more difficult honors class. Maybe not in your case, I don't know.

    I myself have almost never been in a boring university level math class, and i have sat in many of them since 1960, at Harvard, Brown, Utah, UNC, UGA, Oklahoma State, often ones in which I already knew the material. In almost every instance the professor has said something that I did not know before, or presented it in a way that I found illustrative and that either helped me understand more or suggested how I could better teach that material myself.

    One of my favorite ways to learn is to sit in on a class aimed at less sophisticated students than myself and listen to a lecture I would ostensibly not need. I almost always gain something.

    I say almost always because just once, in over 40 years, I attended a lecture for graduate students at Berkeley by a famous researcher in which he did just what you say, regurgitated the definitions of basic objects, with no motivation or insight drawn from his own understanding, and no examples. I did not learn a single thing. It was so boring I looked around for a while to see why the students (I was a professor) were putting up with this awful instruction, and then I slipped out early.

    But in 40 years that is the only such case I can remember. Maybe you chose your professors (or your school) poorly. Mind you there were some really bad classes even at Harvard, but I did my homework on those professors first and avoided them.

    My teachers at Harvard always presented not the book's version of the material but their own higher level version. Trying to emulate them, when I teach a class I not only do not present the material exactly as it is in the book, I try to create some new take on the material that I myself have never known before, so that each class I teach has some content that it did not have the year before.

    This summer e.g. I made up a computation of the volume of a 4 dimensional ball that Archimedes could have done, to present to a class on Euclidean and Archimedean geometry.

    Everyone knows that Archimedes used Cavalieri's principle to deduce the volume of a 3 ball by subtracting the volume of a cone from that of a cylinder, by noting that the slice areas of the ball and the cone add up to those of the cylinder, hence so do the volumes.

    I realized that since one sweeps out a 4 ball by revolving half a 3-ball around a plane in 4 space, that one can compute the volume of the 4 ball by revolving the cylinder and the cone and subtracting the volumes they sweep out. Moreover Archimedes could have made this calculation without calculus since he knew the centers of gravity of the cylinder and the cone.

    In fact his calculations of volume were based on work, which is really a 4 dimensional calculation anyway involving three dimensional solids moving in another dimension. Hence Archimedes was really deducing results about 3 dimensional volume from understanding 4 dimensional volume!

    Of course my classes were often derided bitterly by students who did not want them to contain more than the usual ones as that made them seem harder. So look for more challenging classes and do not comp[lain when they are more challenging to succeed in.

    Good luck.
     
    Last edited: Aug 25, 2011
  17. Aug 25, 2011 #16

    chiro

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    That is a very open issue and one that is very difficult to answer.

    Everyone has their own viewpoint about what is important, and its really not hard to see why scientists (and those who follow science) think science is important, and sportspeople (and those who follow sports) is important.

    You have mentioned an interesting point, in that you are talking about things that give a good "return on investment" or in your words "things that will make you money".

    Money is a very powerful element in how society is structured and how it functions, and the financial system plays a big role in the "how" and the "what" of what gets done. (You've pointed this out in other prior posts).

    As a consequence, the people with power can actually just control the flow of money and in a way create an environment where things with the most funding get more attention, and those with less funding get less attention.

    As a consequence of this, it might be interesting to ask exactly who is responsible for this and question their motives to see if it really is in the public's best interests. Personally I think they follow the same idea of "getting a return on investment" just like the rest of public does.

    It would be interesting to see a world without money, and I do think at some stage in the future this could happen (this kind of thing has been happening in tribal communities for many years), but for now we have the system that we have.
     
  18. Aug 25, 2011 #17
    You are a redditor :biggrin:!
     
  19. Aug 26, 2011 #18
    It's an assumption that I think is true. If you have someone bright and motivated, the important thing that a teacher can provide is a good environment and moral support.

    Not sure this is what I'm saying. What I am trying to say is that if you have bright motivated students, you can have teachers that are totally incompetent at teaching, and you won't have a huge disaster.
     
  20. Aug 26, 2011 #19
    There are lots of societies in which status isn't based on money. Curiously physics and math is one of them. There is a definite pecking order among physicists and mathematicians, and it's not how much money that you make that determines social status.
     
  21. Aug 26, 2011 #20
    As I understand it, the OP is advocating for increased self-learning in standard math curricula. The argument doesn't make sense to me b/c the kind of student who would benefit from such a change is already learning in the status quo, whether via self-study or traditional means.
     
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