Thursday, 12 February 2015
Response to "Proofs as bearers of mathematical knowledge"
I was looking forward to reading this article, as I hope my intended research to have proof at the forefront. Unfortunately, I was a bit underwhelmed by this article. The authors, Gila Hanna and Ed Barbeau, base their investigation on Yehuda Rav's paper "Why do we prove theorems?". In this paper, Rav states that proof should be the focus of mathematical interest, because within a proof lies the development of mathematics as a field. He contends that far too much emphasis is put on the importance of theorems (product), rather than the proof (process). While Rav focuses on the field of mathematics, Hanna and Barbeau attempt to extend his insights to mathematics education.
When I entered into a mathematics major during my undergrad, the concept of "proof" was essentially completely unfamiliar. The "proof" that I had seen in high school consisted of trigonometric identity manipulations, and most proofs I saw in calculus were either derivation type proofs, or very brief conceptual proofs of about two lines. I continually wondered why it took until my third year in university to see a rigorous mathematical proof, from which I learned a great deal; not only the final "fact", but the process in between that came in as useful later on. Throughout my teaching, I have always enjoyed presenting proofs to my students, as I feel that it gives a great deal more insight into the problem at hand, and how very simple statements, such as the Intermediate Value Theorem, have very non-trivial proofs. Proofs give rise to the importance of assumptions, something that many students tend to forget about. One of my favourite examples is that of fixing a car; if you have a specific part that only works for one specific type of car, you probably don't want to put it in a different model car; something bad might happen!
My disappointment in the article started when the authors began their "case study" of particular proofs that are apparently seen in most secondary mathematics classrooms. First of all, there is nothing I despise more than hearing from a professor "you should have seen this in high school (and/or first year)" in regards to something non-trivial. I've had this happen to me in graduate classes, and I can't express how degrading it feels in the moment. As someone relatively mathematically confident, I can't imagine what that feels like to a student who is not as confident in their mathematical ability. I felt this tone in much of the paper, there seemed to be an assumption that most students (and teachers) know these facts and their proofs inside out, when I would have to argue that most do not.
The authors consider the cases of roots of a quadratic function and the case that an angle inscribed in a semi-circle is a right angle. I have done a lot of tutoring of students working with quadratic functions, and I know from experience that 1) students have difficulty with quadratics when the coefficients are defined real numbers and 2) that this difficulty increases exponentially as soon as the coefficients are arbitrary. The authors recommend that the proof of quadratic roots be explained first by considering "nice" quadratics like x^2 - k and then moving onward to develop the idea of completing the square; a technique that the students will be able to use on quadratics in the future, as well as extend the notion to cubics and quartics.
While I do believe of the importance of conceptual proof and a lack thereof within the secondary mathematics curriculum, there are a number of issues that need to be attended to. First of all, the teachers working with such material need to be very flexible in their mathematical knowledge in order to attend to the requests of Hanna and Barbeau. Moreover, what are secondary school teachers conceptions of proof and their usefulness in the classroom? Next, the request to have substantial, conceptual proofs within the curriculum starting in secondary school might be a bit of a slap in the face to many students. If there is to be such a large curricular change in upper grade levels, shouldn't there be consideration of the importance of proof in elementary and middle school grades?
I should probably stop here. I really apologize to the people in my reading group for the length of my response. I think I could go on about this paper and the issues it brings up for hours......
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Vanessa, no need to shorten your response. It's obvious that this article struck you on a personal level, something I find rare in academic literature.
ReplyDeleteIt sounds like you and I had a similar experience in high school and with proofing, as well as with the expectation of "prerequisites", a response I know all too well. Having been a math major with an original intent on medical sciences, I took the wrong "streams" of mathematics (the biology calculus streams) as opposed to the "physics mathematics", which reviewed all sorts of concepts that I needed for my undergrad. I didn't truly find this out until one of my advisors commented on my transcript.
When it comes to proofing in the classroom, I know that during my mathematics major, there was actually a course on proofing that all majors had to take. However, the course introduced a whole host of new concepts unfamiliar prior to taking the course (namely, set theory and a vigorous look at injective, bijective, and surjective functions), and as it so happens, the course rivalled the campus' organic chemistry course as resulting in the lowest pass rate (or was it lowest average grade) on campus. It's funny, though, I actually took the organic chemistry course and excelled in it, whereas with the math proofs course I barely scraped by. I can say that my high school mathematics career did not prepare me for the rigor of proof expected in university - that is to say, there was little or no consequence for not learning the proofs presented in my prior courses (including calculus), and so I would tune out the teacher as soon as she entered into a proof. This was different than when I would cling to every word of my professors, perhaps due to my fear of failure (and the permanent mark on my transcript that this would leave). With the help of two tutors, I finally scraped by in the course.
Would proofing in elementary have helped me? Had proofs been introduced early on, perhaps I would have been used to seeing them used throughout my math learning. However, I don't know if psychologically-speaking, the human brain would be advanced enough to understand proofs until it began developing the critical-thinking portion (around the age of 12-13, I believe?). I wonder if there are any studies were persons have investigated the understanding or proofing of a concept in elementary schools. If so, would they stretch far beyond the examples done in a classroom? I would argue that the use of numbers to test a lemma is already quite advanced, but the use of variables and parameters to test it for all cases? I wonder if it would be a little too much (or, as I have found in my own learning, students may learn the tricks for proofing without understanding how they truly work). This, in addition to the lack of training in the majority of elementary school teachers would make proofing very challenging (but not impossible). One thing which would certainly need to change would be the credits of math training needed to teach mathematics in the elementary school setting. The other thing would have to be the re-introduction of proofs into the curriculum as early as grade 8, not in a non-calculus directed course in grade 11.
Wow! Vanessa, between your response and Alex's response to your response, I feel like I've read an entire separate article. . .in a good way. Unfortunately, I feel that I won't be able to contribute too much to this thread as my being an elementary school teacher, with only 1st year level university math I may be a bit over my head. That said, I am in full agreement that not enough emphasis is put on the process of solving a problem. We see this in the way we assess work. Many teachers, including myself, are guilty of heavily focussing on summative assessment. We are told time and time again to do more formative assessment. We need to assess the process, along with the product. Similarly, students should focus on the process, in this case perform proofs.
ReplyDeleteIn regards to teaching proofs in secondary, middle, and elementary school, I agree on some levels. In an elementary school setting I like the idea of using versions of proofs to possibly introduce subjects or reinforce lessons. But I think that this could only be done as a class activity, or possibly as an extension, or for a "gifted" group of children.