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Is it general true among math professors

  1. Jan 26, 2008 #1
    If math professors observed that their students who don't perform well in their foundation to proofs class and abstract algebra class, there not going to do well in any other math course? My math professor told me about a study of 200 students who did not do well in their analysis class who previously perform badly in their foundation to proofs class and abstract algebra class. I was in agreement with my professor in the study. But what she and the study failed to provide is explanations for why the students did not do well in the proofs class. its like she and the study were implying that improvement in writing mathematical proofs is virtually nonexistent. Its like singing or being a good basketball player. my professor seemed to be implying that writing mathematical proofs is innate, just like being a good basketball player and a good singer, you either know it or you don't. now I think its safe to say that we are all in agreement when I say that their are just some people who are just gifted in mathematics or physics , based on the way that individual's brain is structured. but I think that most mathematicians worked to become guru's in math or just to become mathmaticians. now back to my main point: What the study I initially mentioned was if failed to provide whether or not students changed their study habits or students did not show any effort to try hard. I personally think if you changed your study habits, you will improved at writing proficient mathmatical proofs. Finally , I want to know the general consensus among math professors whether they are in agreement with my professor and the study , or do all of you (math professors) think students that did bad in the proof class just were lazy or needed to change their study habits to help them improve at writing mathmatical proofs? Please I asked for the honest truth and not for any of you responders to sugar coat your replies. Thank you.
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  3. Jan 26, 2008 #2


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    i think what you should be asking is whether those students who went back and took foundations of proofs again and did well, then later did well in analysis.

    the problem i observe is that many of my students do not do this. they do poorly in foundations, then proceed on to the next class without making up this deficiency.

    it takes courage for the professor as well to flunk a student who does not know the basics, but who begs that they just need to graduate on time, when you know they really need to retake an important foundational class.
  4. Jan 26, 2008 #3
    What do the professors think is the reason for this ?
  5. Jan 26, 2008 #4
    What about students who initially take an analysis class as an audit while re-teaching themselves the basics and reteaching themselves abstract algebra throughout their summer off and then taking analysis for a grade the incoming fall semester?
  6. Jan 26, 2008 #5
    at upper level, any kind of mathematical proofs requires one very common thing- logical thinking.
    The logic of writing proofs in different area are actually quite similar which you are just given different asssumptions or axioms. In a proof writing class (or analysis class), you are given more structure and information than you are in abstract algebra class. If you have difficulty to draw information from a more structured space, how on earth do you work in a space with almost no information (say group theory).
  7. Jan 26, 2008 #6
    One of the reasons that a Intro to Proofs class is a good litmus test is that it doesn't really test knowledge. It tests, for lack of a better term, mathematical aptitude. Now it may be that students are rusty, or use the wrong language, or don't know what a contrapositive is, but the ability to understand whether something constitutes an airtight proof is difficult to "pick up" if it doesn't come naturally. Here's an example from when I took the class: after seeing the proof that the square root of two is irrational, we had to show that the square root of 12 is irrational. Some students immediately grasped the modification that needed to be made, and other students simply repeated the earlier proof by rote, without noticing the difference. It was clear that the first group actually understood the proof, while the second group simply "learned" it.
  8. Jan 26, 2008 #7
    I understand people who are born with the vocal cords to sing; I don't understand how a person can just have a knack for writing proofs. we have brains that are able too allow us to think logical. Most people would know what an irrational number is. its a non-repetitive infinite sequence of real numbers after a decimal place. Why do you think people are not able to prove that a number is irrational? if we all had a natural knack for mathematics and logical (i.e, Einstein and Godel ) we wouldn't need to go to school to develop our mathematical skills and people wouldn't need to go to college and graduate school to hone their skills at learning how to right mathematical proofs. The brain is like a sponge: it is very good absorbant for liquids. analogy: our brain is the sponge and knowledge is our liquid.
  9. Jan 26, 2008 #8
    I think the difference is in the way we see it. Most people don't have trouble writing proofs, they have trouble proving things. There's a big difference. The way you make it sound, proof-writing is no different than learning the MLA conventions for bibliographies.
  10. Jan 26, 2008 #9
    To me, you make it sound like just because people have trouble proving things, then they should not continue to try to hone their proof writing skills. Learning math is not something that innate. Learning is innate and probably people grasping mathematical proving techniques faster than those around them are probably innate. But I still think that their is room for improvement in any field that involves logical thinking.
  11. Jan 26, 2008 #10
    Zhentil, just curious, is just showing that that the root of 2 is a factor of the root of 12 enough, because you don't need to know anything about the proof to see that r6 x r2 = r12...?

    What year of undergraduate math do you usually take this Foundation to Proofs? I'm frequently questioning if I am good enough to major in math..
  12. Jan 26, 2008 #11
    Because people don't enjoy it. You can make people better at math, but you can't force everyone to become a mathematician, because that kind of dedication requires that you enjoy the subject.

    In my case, I hate poetry and analyzing literature. I also suck at it. Would I become better if I spent more time on it? You bet. But I wouldn't enjoy it and wouldn't advance as fast as someone who did enjoy it.
  13. Jan 27, 2008 #12


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    Based on your teaching experience, did you ever notice if students who retook their foundation proof classes and then later went on to higher-lvl math classes scored better grades than those which did not bother retaking?
  14. Jan 27, 2008 #13
    I never said you had to like math to become proficient at math. But fortunately , I do enjoy math. I times, with have to do things that we do not like to do in order to make ends meet. A heart surgeon may at times not like opening a person heart to remove whatever, but they have to open up the person's heart in order to save the person life.
  15. Jan 27, 2008 #14
    I'm not a prof, but I do have a bachelor's in math and I was one of these students who performed poorly early in the program.

    I don't think all schools have it. I know Michigan at Ann Arbor didn't. But I think it's normally a second year course.

    I think this is really good advice. It would be a good study to conduct before we concluded that mathematics is something "innate."

    At Michigan, Linear Algebra is the first course introducing students to proofs. I wound up failing the first time I took the course (with Prof. Griess I think). I gradually did better in the pure math courses (B+ Abstract Algebra, B Analysis, A- Topology), but I didn't move on until I passed Linear Algebra.

    The main reason I did poorly was that I didn't show up enough to lecture and didn't go ask for help when I should have. The only way to do better was to do more work (a lot more), get involved in study groups, and get help from the prof on the questions I couldn't grasp. I'm kind of happy Griess failed me then. It was what I needed to resolve to do better. If he didn't, then I probably would have left school with a sub 3.0 gpa.
    Last edited: Jan 27, 2008
  16. Jan 27, 2008 #15
    Then you may conclude that it holds because an irrational times an irrational is irrational (which is not true in general). The easier way is to note that the square root of twelve is two times the square root of three, and the proof that the square root of three is irrational is identical to the the proof for the square root of two.

    I took the course when I was a sophomore. If you've had linear algebra, it's not bad, since lin alg is usually proof-based. If you've only had calculus and ODE, then it might be very different.
  17. Jan 27, 2008 #16
    I think you're absolutely right, and it probably wasn't smart for the professor to reveal that he made pre-judgments about his students' aptitude based on that one course. I'm just saying that I understand the sentiment on his part. I've seen studies that compare mathematical aptitude to being a musician or a chess grandmaster. Certainly, if you practice, you will get better at chess, and certainly, the grandmaster practiced a lot to get there. And you can certainly practice enough to play a serviceable game and enjoy yourself. But practice alone will not make someone a chess master.
  18. Jan 27, 2008 #17
    I totally agree. Some person said that to be greate at something, its 90% practice and 10 % genius. I think it was edison. i don't want to be the next newton, guru, or einstein , I just want to be proficient in mathmatics thats all. Not everyone who receives good grades in math will become world renown mathmaticians. I think what makes you a great mathmatician , is how many original discoveries you made in your respective field.what you contribute to Mathematics Not just mastering the mathematical techniques that makes you a great proof writer.
  19. Jan 27, 2008 #18


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    i do not know any stats on the value of retaking. in my own case it was more like that of the sensation, i did better later when i begN WORKING HARDER, GOING TO CLASS,...
  20. Jan 29, 2008 #19
    I recently heard an interview with a mathematician turn philosopher. He makes a dichotomy between abstracted (not abstract) mathematics and symbolic mathematics specifically to answer these types of questions. The truth, as he "reports" it, is that human beings are almost all naturally talented at abstracted mathematics as this ability is inseparably linked to our language ability and fundamentally tied to the same area of our brain that has necessarily had to evolve over the past 10000 years. An example of abstracted mathematics is the following: when you were in grade school, invariably, your teacher would hand out a sheet of 50 or so arithmetic problems of the same nature (say all empathizing substraction). You work on the first few with great strain--i.e., for 13-5 you first take 1 away from 12 and then take 1 away from 11 and then 1 from 10 and so on until you get to 5. After working quite hard on the first five problems you have an epiphany: you can think of 13-5 as the distance between 5 and 13 which immediately you know is 7. The rest of the problems becomes easy. It seems that when bombarded by these types of problems (for instance, managing a cash register) we can preform quite quickly by going to some abstracted intuition about what is going on.

    Symbolic mathematics on the other hand is not this way. For me it is hard to actually understand what the nature of doing mathematics is really like (mostly becomes of my own limited study). But, it is really like learn a language as well as learning to be able to think of a bundle of ideas as just one concept (only separating one or two bundles at any given time to prove the validity of very complicated sentences). Symbolic mathematics is not something everyone is good at (it serves very little evolutionary purpose). In fact, only a small percent of the population are good at it mostly because it takes years and years to learn as well as great discipline. As I remember it, my introduction to proofs textbook said this, in a manner of words, point blank by stating that only a small percentage of the people who read this book will become mathematicians.

    I have four inexperienced suggestions on how to get people to do bette in a intro to proof course:

    (1) State the fact that to write proofs well and do well in the class you must work hard and practice as well as realize that this type of mathematics will be, on the whole, wildly different than most of their previous courses.

    (2) Continually reiterate and display the power, beauty, and genius of the axiomatic method which would be the foundation for any of these intro to proofs courses.

    (3) Discuss historical figures in mathematics briefly as well as a brief flavor of their research and the work it inspired.

    (4) Teach to start from examples (as general topology was the first axiomatic subject I studied, I had, and still have, a very hard time doing this--but for most people (if not all) it is crucial). The reason for this is that examples are exactly that which gives rise to abstracted mathematics which in turn gives rise to intuitions which in turn helps separate the necessary pieces of the proverbially bundle in orderly fashion.

    Of course, the real distress of anyone interested in mathematics education is the existential question of how much real help any of this is within a culture that shapes people to not really care all that much in the first place. I guess the only thing the instructor can do is fail or give relatively bad marks to such students which I am sure gives rise to a whole other dilemma altogether.
    Last edited: Jan 29, 2008
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