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Can you like math without liking proofs?

  1. Oct 18, 2013 #1
    (I numbered my questions- it ended up being a long post!)

    (1) I'm also wondering if anyone has any good metaphors for difference between proof and "drills" or "techniques". Maybe learning "techniques" is sort of like getting good at scales, whereas proof is actually playing songs?

    I feel that math through high school calculus is pretty much mostly learning different manipulation techniques, with the exception maybe of geometry. But those geometry proofs, often the only proof in any HS math curriculum, end up seeming kind of like an aberrance in their methodology- open-ended, no one single procedure to memorize like in say Algebra II or in finding the derivative; a lot of gruntwork for, perhaps, relatively simple payoffs.

    I have to admit, I myself often don't find some of the geometry proofs that fun- and yet they can be daunting because we aren't used to thinking that way in math, no series of steps to follow and memorize. Maybe that's partly because I'm approaching proof wrong- it can be easier if you build your way from the conclusion back on simple proofs so you don't miss overlooked steps, for example.

    But, in general, (2) how can I make proof less daunting/more interesting to others? (3) Is it possible to "like math" without liking proofs?

    (4) Why do you think it is that people who say they like math also say they "hate proofs"? Is it just because our brain's reward centers that reward us for generating correctness haven't been trained to handle those sorts of challenges throughout our primary/secondary math education?

    Finally, (5) I'm wondering how/why the US got so stuck on "math = techniques" in the first place. (6) Is it like that in other countries?
     
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  3. Oct 18, 2013 #2

    Simon Bridge

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    Can't help you there.

    Turn it into a narrative - a story. Treat math as a language.

    Yes. It is often the math formalism that makes the proofs daunting.
    Although much of what you do in maths amounts to proving relations, the formal "doing proofs" as a math exercise is more about the logic behind the math you use.

    No - I think they just don't like doing proofs.
    It's very common among physicists and engineers.

    History - I suspect it is because techniques are easier to set exams for.
    Look at any other way of doing it and ask yourself you your would set an exam that tested that sort of learning.

    Yes.
     
  4. Oct 18, 2013 #3
    >>No - I think they just don't like doing proofs.
    >It's very common among physicists and engineers.

    I don't know if that's quite true. Physicists use math to "prove" things, or to show mathematical relationships. Yes they are primarily interested in revealing the truth about something instead of just demonstrating an abstract truth of no particular real-world meaning, but I'd say proof is more similar to this process than is just doing drills. The formalism of proof can be annoying I agree.

    >the formal "doing proofs" as a math exercise is more about the logic behind the math you use.

    Of course. But conversely it can be pretty easy to lose logic in while displaying mastery of complicated procedures.

    Editing to add:

    Just went over to reddit.com/r/math and found this: http://www.reddit.com/r/math/comments/1oo1sz/worst_practices_in_math_education/

    I realized that there's a lot of what I was talking about there, particularly user knowstupidquestion's comment. I guess I feel like the distinction between more informal physics-type proofs and formal math proofs isn't too important to what I am saying. It's more about understanding the logic, the definitions, the derivations, vs just memorizing algorithms.
     
    Last edited: Oct 18, 2013
  5. Oct 18, 2013 #4
    Also I had a discrete math teacher from Czech Republic and she gave the impression she was learning basic proofs in elementary school.
     
  6. Oct 18, 2013 #5

    pwsnafu

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    Ah the legendary ε-Red Riding Hood and the Big Bad Bolzano-Weierstrass Theorem. Awesome that.
     
  7. Oct 18, 2013 #6

    SteamKing

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    Constructing a valid proof is quite different from utilizing or practicing mathematics. The 'taming' of mathematical intuition by imposing a rigid logical structure is to many akin to sucking the life out of mathematical inquiry, although it is a necessary sacrifice.

    At times, mathematics has forged ahead of those who would provide the logical rigor. For example, the calculus, in its early glory days of the late 17th to early 18th century, provided much fruitful insight into mathematics, mechanics, etc., but its logical foundations, especially for the concept of the limit, were quite shaky. It wasn't until the early 19th century that a solid logical foundation for the calculus was finally constructed.
     
  8. Oct 18, 2013 #7
    Yeah I think we have different definitions of shaky. Also please see my edit above.
     
  9. Oct 18, 2013 #8

    Simon Bridge

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    ... oh sure, as part of our jobs we are often called upon to do stuff we don't like. We still don't like it.

    And I will concede that there exist a number of physicists and engineers greater than zero and less than the population who do like proofs. However, people in math courses who like proofs tend to become mathematicians rather than physicists or engineers. OTOH: I am extrapolating from personal observation over 30 years teaching sciences and mathematics so it is not a scientific conclusion.

    Personally I wouldn't get worked up about students not liking some aspect of the course material.
    Every course has some drudgery in it - ask chemists about titrations in their junior year. How many enjoyed them?
     
  10. Oct 18, 2013 #9
    Yeah I think there are a couple misunderstandings going on here. When I say "proof", I would include physics-type derivations or demonstrations of relationships in there, and contrast it against the sort of rote computation-given-certain-prompts that many high school math students do. Also, although I don't have a background in physics, I have observed evolutionary theorists at work (pretty much the same approach), and I have to say that demonstrating the mathematical relations is the heart of what they do! Regardless of how I define or group things as "proof" I don't see how you can say that physicists don't like to do what amounts to doing physics.

    Again, I'm not talking about the difference between physicists leaving out the boring steps in favor of clear presentation of the intuition and logic, and mathemeticians insisting on formalism in mathematics. I'm highlighting the tendency of high school math students to conceive of math as a series of computational techniques to be memorized, and recapitulated when given a certain prompt. Perhaps if you haven't been in high school for a while or haven't worked with such students you may not remember what it's like. If so I again suggest reading the comments in the link.

    I just don't see how one can be creative in math without creative deployment of definitions and logic, whether it be couched in the language of proof or some other more informal approach. This is what's lacking in high school math. When it isn't, often the complaint is it's "too hard" or "I know the material but the tests are weird." And I admit that I tend towards non-formalist intuition/logic in my approach to math- but nonetheless there is a useful creativity and emphasis on knowing definitions inside and out in proof that can be absent elsewhere in lower-level math courses. But I'm only a tutor without a formal background in physics or math so I could be off base here.

    I have to agree with pwsnafu here. this is a kinda trite response.
     
  11. Oct 18, 2013 #10

    Mark44

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    Proofs involve logic, and don't necessarily have to be about math. There's an old example in logic that goes like this:
    All men are mortal.
    Archimedes is a man.
    ∴Archimedes is mortal

    This is something that you can show with Venn diagrams.

    Kind of a lame example, but a better example might be something that at first blush seems plausible, but falls apart under a bit of scrutiny.

    Examples that involve pictures with little or no text could be interesting to your students. Here's a geometry puzzle that they might enjoy, as the two images appear to contain exactly the same shapes, but are obviously different.- http://www.folj.com/puzzles/easy.htm
    I found that by doing a web search for logic puzzles.

    There are a number of "tricks" that make the operator appear to be a mind reader. I've run these on a number of kids, and they all seemed to like them.

    1. Have them think of a three-digit number, say XYZ (letters represent actual digits).
    Ask them to rearrange the digits, as for example, YZX.
    Ask them to subtract the smaller from the larger.
    Ask them to tell you any two of the digits of the difference, provided that if a 0 digit or a 9 digit is present, they need to tell you that one.
    Tell them the missing digit.

    If they are mathematically sophisticated enough, ask them to explain why this works.

    2. Ask them to enter any 3-digit number in their calculator, say XYZ.
    Ask them to enter a number with the 3-digit group repeated, as in XYZXYZ.
    Ask them to divide the number by 7. Tell them the answer will be a whole number.
    Ask them to now divide what they got in the previous step by 11. Tell them the answer will again be a whole number.
    Finally, ask them to divide the result of the previous step by 13. They will probably be surprised that they ended up with the first 3-digit number they started with.

    As before, ask them to explain why this works.
     
  12. Oct 18, 2013 #11

    Stephen Tashi

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    Doing proofs well requires that you have a good (and legalistic) understanding of what you are doing and also that your are able to express this understanding clearly. Some good chess players, bridge players and athletes understand what they are doing, but can't explain it or teach it well. To do proofs you must have the skills of both a player and a coach.

    In any activity there are routine matters that can be practiced and drilled. Players with the highest level of skill can do more than the routine type of play. The same is true of proofs. Discussions about teaching proofs generally refer to teaching average students how to do routine proofs.
     
    Last edited: Oct 18, 2013
  13. Oct 18, 2013 #12

    Simon Bridge

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    Spoken like someone who has not done physics.

    Physics is an empirical science .
    Mathematics is a language used to describe the physical relationships that we see or suspect may be present.

    We do not prove our relations mathematically - but, empirically, try to disprove them.
    A relationship that is proveably true in math may not be valid in Nature.
    So we don't think in terms of proving things the same way a maths course may.

    Oversimplifying for a bit: A math proof involves determining the truth of a statement a-priori ... which can be done for analytic statements. But empiricists have to deal with somatic statements. The truth of such a statement cannot be known a-priori so empiricists don't get to prove such statements. There's an enormous body of literature on this.

    But the discussion has turned up something that should interest you - clearly people who say stuff like: "I hate doing all these proofs" are not talking or thinking about proofs the same way that you do.

    What you are doing, in effect, is trying to define the problem away by pointing out that stuff those people like to do are actually a kind-of proff (what they say they don't like) so how can they say that they don't like proofs? (Rhetorical question: Are you saying they are lying? Misinformed? What are you trying to say here?)

    You are best advised to investigate which subset of "proofs" they associate with the term "doing proofs". Then you should be able to discover an effective strategy to help them over that dislike.

    i.e. you need to understand how your students are using (possibly "misusing") the language before you start looking for inconsistencies in what they say.
     
    Last edited: Oct 18, 2013
  14. Oct 19, 2013 #13

    pwsnafu

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    :confused: I'm not saying Simon's response is trite at all. I do think ε-Red Rding Hood is an awesome story. That being said not all proofs can be turned into something like.
     
  15. Oct 19, 2013 #14

    Simon Bridge

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    @pwnsnafu: +1

    Technically all math expressions tell a story - the trick is to find the narrative that fits the audience.

    i.e. I had to teach probability and stats at an all-girls school ... the book I had to use did all it's examples in terms of sport - particularly rugby and cricket: boys sports right? They figured it didn't matter - women's equality and math is math but the class was bored.

    I changed it and since they were seniors and it was a hot-(off-)topic in class that month I reworded all the problems in terms of a male-review show that had just hit town. Sudden interest - but the math was exactly the same.

    (I did get into trouble BTW.)

    Context is everything.

    At a college level - faced with the proof that $$\sum_{n=1}^{N}n = \frac{1}{2}N(N+1)$$ ... which is normally done in typically boring fashion I came across the story of a kid being punished by a math teacher... instead of doing lines, he had to add up all the numbers from 1 to 1000.

    Before producing the statement to be proved I told the story:
    He didn't want to spend all that time so he looked for a shortcut.

    There is a standard shortcut which involves reversing the order of the sum and adding the corresponding terms, and seeing the pattern. But that is not how people discover things. Instead I asked the students if they could do 1+2+3+4+5+6+7+8+9=x, what is x?

    Because they hadn't just come down from the trees they could do that quickly x=45 ... how did they do that? Well they just have to pair the 10's compliments: there's four of them for 40, with a 5 left over unpaired for 45. Bear in mind here: they are telling me this.

    The more you can get the class to tell you, the less work you have to do.

    Now the challenge becomes to exploit that method for sums to more than 9.
    For 1000, can they add up the 1000's compliments with 500 left over?

    OK - what about if the number is round ... say it is 467?
    Then express for any sum up to N and simplify.

    When they'd done all that I could show them the "elegant" proof (either myself or a student who already had it could show the others).

    This sort of lesson is more dynamic and the exploration is usually more entertaining than dry proofs.
    The students ended up producing the statement "to be proved" and they already had a proof which they had worked out themselves.

    OTOH: people who like doing dry proofs will hate it.
    It can also take a while longer than the institution will allow so some judgement is called for here.
    Do this sort of thing early, though, and doing the dry proofs later is less daunting.

    You can't win.
     
    Last edited: Oct 19, 2013
  16. Oct 19, 2013 #15
    I'm south African, we use a European style. Yes it is the same. In America do they show you proofs as to why things are the way they are in high school? We don't, I hate that. I think a lot more people would be into math if they fundamentally understood it (for the less axiomatic sections in mathematics)
     
  17. Oct 21, 2013 #16
    Regardless, it is a trite response.
     
  18. Oct 21, 2013 #17
    As I told you, I am not a physicist. However, I am a scientist by training. I do have an understanding of how math is deployed in science.

    Just curious, what is your degree in physics?

    Yes, and to best model a natural process, we have to understand why mathematical objects work the way they do, how they can be understood in terms of simpler mathematical objects, and if a given mathematical model fits empirical data, what that means about the essential parts of how the process work. Of course, modeling is an iterative process between the mathematical and empirical.

    Very good. However, one still needs to know how mathematical objects function and "why" in terms of simpler objects. A good model in physics or other mathematical science does this implicitly. A good model can use the mathematical properties of its constituents to illuminate the essential variables in a natural process. This same process is used in math proofs.


    Yeah so again I'm not trying to get into a debate about the difference between empirical science and mathematics. That's not the distinction I am asking about. I may have implied so in my original comment, but then I clarified myself.


    Yep, some have contributed to the discussion and for that I am thankful.

    Actually, I'd say that it's you that's trying to define the problem away. I clarified my original meaning. You should go read that and not tell me what I mean.
     
  19. Oct 21, 2013 #18
    Thanks for this post.
     
  20. Oct 21, 2013 #19

    Simon Bridge

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    What would that tell you? :)

    I don't like to make a big deal about it because: I am a time travelling anthropologist from the Antares cluster and hold transcendental doctorates in psychology (which is a branch of physics in the future) and quantum meta-dynamics.

    In this interference zone I pretend to postgrad college degrees in physics and education - with undergrad qualifications in law and engineering. I seem to be getting away with it ... I have to explain some of my more advanced knowledge away by citing extended experience as a professional in secondary and tertiary education as well as in academic and private research. But so what?

    I could be lying :)

    Yes - you told us in the very section I quoted back at you when I replied.

    You speak (write) like one too :) Very passionate and sure that you are right.

    It can help but it is not essential - people use stuff they don't understand all the time.

    Maybe I missed that - please provide the post number.
    Reviewing... still don't see. Maybe there was a miscommunication - that is common in international forums.

    I was responding to the post I quoted from ... which was pretty recent. If I treat the "redefinitions" I see in that post as "clarifications then I need to rephrase:

    Your clarification suggests that you have been taking a broader view of "proof" than the people you have been frustrated at. If you take a narrower view of "proof" meaning the dry math exercises students are often given to do, then you can see how people can like maths and not like proofs can't you?

    All the clarification has changed is the words used - instead I'd have to clarify earlier assertions as saying, in light of your clarification, that people like some proofs and not others.

    Similarly - your original question in post #1 would get reworded to ask: "is it possible to like some proofs and not others?" (or something - you try) which kinda answers itself doesn't it?

    You are off base. Probably not due to a lack of formal background though.

    People are messy and complicated and disorganized. Your mistake is in attempting to apply stuff like logic when you are talking about what makes sense for human behavior and attitudes.

    I suspect that you have tended to get good grades in the math you are tutoring?

    No worries. It dawned on me that people did not realize what I was talking about with "narrative" etc. so I needed an example. Hopefully the reply does not seem quite as "trite" now.

    What level are you tutoring at?

    Do you have a formal background in education - i.e. have you trained as a tutor?
     
    Last edited: Oct 21, 2013
  21. Oct 21, 2013 #20
    (I don't like quoting long posts)
    You can like math without proofs. You can be a mathematician without liking proofs. You can be a lot of things without liking proofs.

    However, there is an important note to be made here.

    You cannot do mathematics without proofs.

    This is a very important thing to understand before going into rigorous mathematics. Let's say it again.

    You cannot do mathematics without proofs.

    Mathematics is a beautiful thing. In order to truly do mathematics, in my opinion, one must look at it rigorously. Otherwise, it isn't really math. [\rant]

    I googled this. I'm dying of laughter. :rofl:
     
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