Wednesday, February 18, 2015

Two readings on proof for discussion in today's class (and for our online participants!)

In today's class, we will take time to read and discuss excerpts from two classic and influential works about the nature of proof in mathematics & math education.

In the first, from Thinking Mathematically, John Mason discusses justifying conjectures -- that is, proving them. His 'big idea' suggestion (convince yourself, convince a friend, convince an enemy -- and then learn to be your own enemy) gives a very interesting and often quoted plain-language idea of mathematical proof. Mason refers to puzzles from elsewhere in the book, and I have included some of these at the end of the scanned
excerpt so that you can try one or two of them.

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The second, from Proofs and Refutations, is part of Imre Lakatos' influential doctoral thesis in the history of mathematics. It is heavier reading than the first, but is worth the effort for the insights it gives. Lakatos reconstructs a bit of the history of mathematics as if it were all happening in a classroom, with a Teacher and students Alpha, Beta, Gamma, Delta, etc.

The topic is Euler's Polyhedral Theorem, a lovely little bit of theory that says that, if you count the vertices, edges and faces of any polyhedron, the sum of the number of vertices is two more than the sum of the number of edges and faces: V-E+F = 2. To make sense of this theorem, it is very helpful to play around with virtual and/or actual models of polyhedra -- see previous posting. You can look at the simple, regular polyhedra (cube, tetrahedron, octahedron, dodecahedron and icosahedron -- the Dungeons & Dragons dice shapes!) and count the vertices, faces and edges to see if they work.

Lakatos shows that this seemingly-simple geometric theorem actually encountered a lot of opposition through counterexamples historically. Several kinds of responses to counterexamples are shown in this excerpt, including "the method of surrender" and "monster-barring". The main point for us reading this excerpt from Lakatos is to see that, historically, the task of creating mathematical definitions and theorems, proving theorems, and offering counterexamples was anything but clear, simple and straightforward! The idea of monster-barring (by changing our mathematical definitions to exclude "monstrous" cases) is something I have used in my secondary school math classes.

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Instructions for Keri, Vanessa, Shan and Alex (and optionally, Jubilee): Take a bit of time to read and make sense of these excerpts, then post four substantive comments each over the course of the next week in an online discussion amongst yourselves, considering the idea of proof in mathematics learning and whether it is significant, important and/or understandable for learners at a variety of levels. Is there something about proof as a way of thinking logically and critically that is important for all learners -- or not?

23 comments:

  1. I am only in limited agreement with Mason’s statement that reading other people’s proof is necessarily a good way to strengthen critical ability. I argue that it is a passive way of absorbing mathematics without being critical enough of the mathematics, much like sitting through a lecture and nodding your way through. I would argue instead that having a person refute your proof and having to defend it is a better way of learning to be critical independently. This criticism is much more in line with Lakatos’ Proofs and Refutations in which Lakatos describes a (somewhat humorous) situation where a teacher is disputing with his Greek-letter students regarding the justifiability of their proof. Student Alpha feverishly criticizes their teacher in their attempt to prove that V-E+F=2. It is at this moment that the teacher can spend some cherished times with individuals who are obviously concealing their true understanding of the problem. In any case, the idea of criticism comes with some underlying assumptions. One assumption I would pose as an argument for the group to discuss is the following:

    There exists a proof for any mathematical proposition with enough substantiality to become a lemma.

    The second observation I made in Lakatos brings some of the language used in his discussion to a second assumption. For example, “ Plausible’ or even ‘trivially true’ propositions are usually soon refuted; sophisticated, implausible conjectures, matured in criticism, might hit on the truth.” (p. 12).

    Assumption two: A lemma is only as good as its proof.

    Assumption three: There are certain kinds of truths, some more valuable than other; complex ones in mathematics, steeped in theory, are the most valuable.

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  2. John Mason in his book Thinking Mathematically demonstrated what we should do when encountering a conjecture (it is not necessarily a mathematical conjecture): (a) to convince ourselves; (b) to convince our friend; and (c) to convince our enemies. What he means by these is that we should describe the conjecture (as part (b) and explain it using formal language as proper as possible), and then see every statement as a conjecture and try to disprove it. These steps are significant in learning a mathematical concept.

    But the below is what I feel valuable:
    “Each version tends to become more abstract and formal, trying to be precise and to avoid hidden assumptions of informal language, but incidentally causing the reader to have to work harder at decoding the original insight and the sense of what is going on.” (p. 98)


    This is why the math concepts are hard to read: the insights are covered under the intricate math signs. It might be helpful to demonstrate the true meanings of math concepts by showing students their origin.

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  3. And one thing I am confused is on page 101. What iterates does Mason mean on page 85? the following example may be based on that question.

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  4. While reading these articles, I continually thought to myself how useful these would have been if I had read them in the early years of my undergraduate degree. I was initially so blind on what it meant to “prove” something, and these chapters would have been very insightful at the time.

    In Proofs and Refutations, I found Lakatos’ mention of definitions particularly interesting. Mathematical definitions are a rare beast and I struggle to think of anything similar. Mathematical objects are different from other “objects” in the sense that we can apply a number of axioms to them and this will totally define the object. For if we try to define table, we could say it has four legs and a solid top. But by doing this, we have removed pedestal tables. Ok, so we can say a table is an object with a support and a solid top. But is then a barstool a table? When we look at the definition for a group on the other hand, we have a number of axioms which we can check for a specific “potential group” and then determine that the “potential group” is, or is not, a group.

    The rigidity of mathematical definitions is something that I suspect, many students are unfamiliar with. Lakatos’ mention of numerous definitions of “polyhedron” is an interesting point, and a point I have seen many students struggle with. To consider the underlying definition of the object you’re considering is invaluable during mathematical proof. If you are wanting to find a counterexample for an implication resulting from continuous functions, you have to make sure that the function you use in your counterexample is in fact continuous. This is something I have seen many students overlook in my teaching experience.

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  5. Alex,
    In response to your comment about viewing other people's proofs as not being as effective in developing critical ability, I partially agree. Reading and refuting other's proofs is certainly a good exercise and one learns a great deal on how to be critical in the proof writing process. However, there is a benefit in seeing elegant, concise, beautifully written proofs. In a way, this helps one to be critical of their own proofs, when they are able to compare it to ones they have seen that are possibly a bit more elegant. I remember when I was writing my master's thesis, my second reader mentioned that there was a very nice 5 line proof (taking advantage of some very heavy machinery, of course) of proposition I was using. Although I considered changing the proof in my thesis, I opted not to. Why? Because the way I had proven it originally, was the way that I understood it. This thesis was for me, and I felt a great deal of attachment to the way I proved the statements.

    So there is benefit in seeing elegant proofs in that it teaches you to be critical of how you write your proofs; you gain the ability to judge what information is extraneous or what you might do to make the statement a bit more concise. The problem is that in most undergraduate classes, these are the only proofs that students see. Students walk into their classes, their professors lecture, and then they present the most elegant way of proving a particular theorem/lemma/proposition. Students are rarely exposed to the beaten path that most mathematicians take before getting to the elegant proof. It's a bit overwhelming at times, and at least for me, caused a great deal of stress when I first started writing proofs. Students need to know that it is ok to go down a beaten path; there's a great deal of discovery along the way!

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  6. Hi Vanessa,
    I totally agree that reading different proofs may help students understand a theory through different perspectives, especially when these proofs are arranged in an order that shows how the theory has been developped. Most textbooks introduce their theorems in this order; but perhaps because of efficiency and printing costs, each theorem are introduced in the most elegant ways.

    Clearly, this efficiency increases the difficulty of understanding the theorem. Even though students are directly exposed to the "best" way, they lose the opportunity of thinking it through different perspectives. I think the value of a theorem is fully dependent on the way people use it to explain/interpret a phenomenon; having no chance to see these wordy/"wrong"/inefficient explanations/proofs leads to a potencial loss to the theorem itself.

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    1. Shan, are you saying that there is value to seeing the wordy/"wrong"/inefficient explanations/proofs? The excerpt from Lakatos' book is a perfect example of the wordy/"wrong"/inefficient explanations/proofs that proves to in fact be very efficient and valuable for the students (Alpha, Delta, Gamma, etc.) in exploring different mathematical concepts and ideas in relation is polygons and polyhedrons. It's as if the act of "talking in circles" about the proof that the students participated in actually strengthened their understandings and explorations. This is something that I think teachers often times do not do enough of in their classroom due to things such as time constraints and standardized tests. However, having students explore a concept/theorem/proof/etc. by trying to come up with counterexamples and their own definitions might lead to great mathematical learning.

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    2. It is true that these seemingly wordy explanations are helpful to those who carefully read them. In fact, if people carefully read those proofs, they can actually learn more than they might have expected. The thing here is that, unless necessary, people normally do not WANT to read these proofs.

      John Borwein in Experimental Mathematician lists so many proofs, but I have to say that I did not read all of them, especially those explained with computer programming language. Clearly, I can simply say that those proofs are not the things I can easily understand; although the potential gain is high, the cost of understanding them is high as well, i.e., I need to spend time learning a language that I may never use again, and so on. Is this somehow similar to what students may feel in our math class? It is true that teaching mathematical proof may provide great benefits to students, but it is also true that most people consider mathematical proofs "uselss" in daily life. Is this the reason our curricula do not emphasize mathematical proof as before?

      If this is true, then to what degree should we teach, or understand, mathematical proof? Or similarly, other subjects?

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    3. We have returned to the "usefulness of mathematics", something that has really been bugging me as of late. Why does math need to be useful? People study poetry and don't complain that it might not be "useful" to some people. Studying it is considered "good for you." In fact, in the early days of the university, mathematics was considered a classical training of the mind, which helped to build the critical thinking of students. So, we could consider mathematical proof to be "useful" if we put it in this context. Of course, should we tell our students that this is why we do proofs? Or, should we leave it as math for the sake of math and not give a "use" to it? I suppose the difference between the "usefulness" of functions and the "usefulness" of proofs is that we are putting usefulness in the category of general knowledge that can be used in all types of thinking. I'm unsure....

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    4. I think time is more precious now than it ever has been. I'm impatient with studying things that aren't useful. I would much rather study analytical writing and essay development than study poetry, because I can find and read poetry on my own time. I think the discussions around literacy are an entirely different subject, but what relates them for me is that I learn things because they could be useful, not because I have the time to learn them for my own enjoyment (hence, a used ukelele I bought last summer lies quiet in a bag behind my bed).

      Actually, a counterexample is this Master's degree. I don't need it, but I want to be a better teacher. I can argue, however, that it is still useful.

      I do wonder if knowing proofs is about empowerment as well. I mean, rather than taking something for granted on your own, the power to demonstrate to yourself that something is true is quite different. It is, I think, the assessment of this takes away from the enjoyment of doing the proof (at least, it did in my case). Meanwhile, I know that if I fail a Master's class, etc, that it won't keep me from working as a teacher.

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  7. Vanessa, your comment about the beaten path is interesting, because I had several professors who would struggle with a proof in front of a class, and then would present a bumpy (yet rigorous) proof. Somehow, I always felt that it was safer for a professor to go down a "beaten path" (safer, how? I couldn't tell you) because they had enough of a foundation to do so, to experiment, and to be wrong. I see this a lot in the students I teach, and I still haven't found a way to help them improve their confidence.

    Shan, what do you mean by "potential loss to the theorem itself"? Do you mean the theorem will be misinterpreted/misunderstood? If so, is there any benefit to that? Can a mathematics classroom be designed around misinterpretations and correcting these, as some of the problems in Mason's book would cause?

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    1. Alex, it is interesting how you mention that several of your professors would struggle with a proof in front of a class leading to a bumpy proof, and that you felt as if it were safer for a professor to go down a "beaten path" rather than you trying to go down that "beaten path" yourself. I think this is something that is very true for students, no matter their age or ability. It seems as if it is okay for the "professional" to stammer through an explanation or a proof, but students always feel as if they have to do it correctly every time, like they cannot make a mistake, when going through the same process. Is this because students are marked on their work or because they are not confident with the knowledge they have on a specific topic? I'm not sure why this is, but I too agree that as a student, I felt more comfortable with my professor stammering through a proof rather than struggling through it myself.

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    2. I suspect that the feeling has to do with confidence in the topic - I'm similar in that sense. One of the teachers in a staffroom where I'm working commented that this time of the year is the time you "check your notes three times, because you don't want to look like a jerk when you make a mistake on the board". In addition to lack of confidence, I wonder if an over-glorification of self-described "math doers" is also partly to blame. I can imagine someone saying something along the lines of, "Well, if my prof/teacher can't do it, how will I be able to do it?" I would sooner lean toward over-glorification than low self-confidence, because with low self-confidence, I might argue that you don't have an "audience" to criticize your mistakes. At that point, though, I imagine most students criticize themselves sufficiently.

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    3. I had a professor that would often spend a good portion of the class working through some proofs through the text that we had difficulty on. Most of the time, he came in completely blind and unprepared. I remember a number of times where he was unable to complete the proof that we were having trouble with. Of course, he would send us an email a day later with a proof, but it was very reassuring that sometimes the magical proof is hard to see for a very experienced mathematician! I think that being very, very prepared for lectures is part of this. We have a very specific way that we are thinking of presenting the material to the point that it may make the material seem easier than it actually is. Perhaps leaving space for "free work" within lectures and lessons might be a good idea?

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    4. I would've loved free time in a lecture. I knew a girl lecturing at..... I think at UVic who would separate people into groups in-lecture and get them to work on problems together? I remember my professor would haphazardly ask students to work together to solve a problem, but there was always this air of competition that existed (and that never was addressed by the professor) that nobody really began to talking to each other until the end of term. I guess that's what happens in a biology-calculus course in a lecture hall full of prospective med students.

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    5. *my professor in my first year differential calculus course.

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  8. When reading John Mason's book on Thinking Mathematically, the first examples I thought related to his ideas of convince yourself, convince a friend, then convince an enemy were not mathematical at all! I thought of using this concept in something like writing research proposals and/or research papers. When I took the research methodology course last term we went through a similar process of convincing ourselves that our research question was good. We then worked with a peer on fixing our research question. Lastly, we got feed back from our professor about our research question in the form of a grade. In this case the peer played the role of a friend and the professor played the role of the enemy.
    It is interesting to me that my first thought on this article was to use the information/concepts on something that was not mathematical. This relates to Whiteley's article I read for this upcoming week on teaching geometry because a point is made in that article about how math helps students think abstractly, and this type of thinking can be used on non-math related things, such as writing critically.

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    1. A professor played the form of an enemy? Ha! I find that amusing. I wonder if my students feel that way about me. :) (I'm teasing).

      I'll note, the argument about doing math because the skills are useful for something else isn't new to the situation; my follow-up to that question is, why do we use geometry to build those skills? I think finding flaws in arguments in a debate club, for instance, involves the same sort of skills and engages people in creating a structure for arguments and logic, etc - skills available to persons through geometry and geometric proofs. There are other aspects of the article where, yes, I can see why Whiteley would support the teaching of geometry (I can't think of a more efficient way of modelling molecules, for instance). Developing skills that can be attained outside of a subject (and, I would argue, would only properly develop with a solid enough foundation in mathematics in the first place) does not suffice as justification.

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    2. Hmm...I did not think about getting the same benefits from partaking in a debate club as you would from geometry, but this is all through a proofs stand point. However, the act of visualizing objects and thinking about their properties, whether you are actually proving those properties or not provides great benefits for students, especially those who are more visual learners or who want to go into something like the trades. While I do think that things like geometrical proofs and memorizing lots of postulates and axioms can be very overbearing and not beneficial for students at a certain point, I do believe that geometry also brings some positive aspects to the table for math education.

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    3. I do sometimes question why we put mathematical thought as so separate from other "types of thought". It's true that the process of convincing in mathematics is very similar to other research, but there is a slight variation. If I listen to someone's education argument, I might be "convinced" that what they are telling me is valuable, but these arguments are related to specific cases and are being generalized. Mathematics on the other hand, covers the general case; to look at a specific example does not mean that what holds there holds for every example. Perhaps this is one of the large differences between mathematical thinking and "other types" of thinking? The idea of looking at ALL possible cases might be a somewhat overwhelming concept for many students.

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    4. I think it's because many people assume (or express) that mathematical thought is only akin to certain types of people and is a characteristic with which you're born rather than a skill you develop. People in higher education have enough experience to know that this difference does not exist (I think? I hope....)

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  9. It is always fun to try games mentioned in books. Yesturday in 555 class we tried the Eureka sequence mentioned in John Mason's Thinking Mathematically. The process was a little bit more difficult than what I had thought to be. If the rule was simple, people might guess the answer quite fast; otherwise they would take extremely long time to figure out the rule. A team even tried a rule like 3n^2+2. There are two common strategies used by all the teams. The first was to test the next value of the given sequence, and the second is to exclude all the testified incorrect answer. People were able to guess all the rules at the end.

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    1. I liked that idea, Shan. I also had a different understanding of what the game was, so it was nice to see it in action. Reminds me of what a teacher at Argyle school does - she starts every class with a puzzle. In honours classes, students are responsible to produce the puzzles themselves, with a rotating order. The answers/work is not for marks, but their reflection on their metacognition is.

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