Discoverd or Invented...or Both?

Having math as the minor to my physics major, I never progressed deep enough into my mathematical studies to get to the meaty philosophical issues we are faced with in this week’s readings. However, I have often pondered on my own whether mathematics was invented or discovered, so I am excited to finally gain some real insight from scholars within the field. As I progress through my Masters studies I am becoming less and less comfortable with the objectivist view that reality is “out there” as an external entity. Consequently I intuitively reject the romantic nature of universal mathematical truths discussed in the first reading. True, we do see evidence of mathematics in physical phenomena such as elliptical orbits and fractal shapes in leaves. But is this really evidence that we are slowly “discovering” a universal truth? Or have we simply “invented” a working conceptualization of such phenomena? I feel the latter is more aligned with my developing philosophy of education.

I found the Reuben Hersh interview, perhaps my favorite of the three readings, to be quite fascinating. I definitely had an “Aha!” moment when he declares that things don’t necessarily have to be either internal or external, but can exist as social beings, or as he states, “part of human consciousness.” Here I am struggling with this discovery vs. invention issue, all the while never considering the possibility of it being both somehow. It’s a strange concept to wrap your head around, but I find the easiest way is to use the parallel of money, an example used by Hersh. Money is only real because we as humans say it is real. We can look at mathematics in the same way as only being useful because we say it is useful. The whole issue is still kind of fuzzy, but one thing I think I can commit to is that mathematics is a human activity, and would not exist without us. I also found his conception of good math teaching very valuable and in alignment with my currently held ideals. A teaching style based around interaction and communication sounds appealing. Starting with examples, a question instead of an answer, is a very practical piece of advice as well.

Brent Davis offers up some rich food for thought in our final reading for the week. What if we “are not converging onto a totalized knowledge of the universe?” What if mathematics is a tool that allows us to perceive the tiny percentage of possibilities that make it to consciousness? Mathematics has allowed us to conceptualize so much more than our ancestors of two hundred years ago. But whether these conceptualizations are snippets of universal truth depends on your philosophy. I am reminded of the example given in ED 6100 to illustrate the point that a theory need not be real. No one has ever seen an electron, but it is widely accepted that they exist. The theory serves its purpose until a better theory comes along. Tomorrow we may uncover evidence that the whole idea of electrons is no longer accurate. That said, can everything we perceive as scientific and mathematical truth potentially change at the drop of a hat?

My Math Autobiography

I have very little in the way of distinct memories of my early mathematics education. I can recall glimpses of a workbook here and a test there, but I have no recollection of any one specific moment of mathematical wonder. What I do remember is that I did indeed enjoy math and quickly developed a reputation among my fellow students as well as the teachers for being a strong math student. I remember taking a pre-test for one of the math units in the third grade and scoring high enough that I was not required to participate in the classes for that particular unit. Instead I would leave class and go to another room all by myself and complete word problems from a higher grade level that the teacher had given me, which I often completed with ease and promptly returned to her for confirmation. One day I remember the teacher, perhaps bordering on annoyance that she couldn't keep me going in word problems, professed out loud "Ok Ryan, if you can do this problem then you're a genius." Now, to a bunch of third-graders you can imagine this was a fairly big deal. So I went back to my little room and went to work. After a few failed attempts and a hint or two from the teacher, I finally figured out the answer to the problem. And from that moment on, I was the kid who was good at math.

My next real memories of my math education come in the fifth grade. I was still very good in the subject but had began proclaiming a disdain for it altogether. Looking back this was probably a social ploy to fit in and be cool, since most other students truly did dislike math. I've actually taught with my fifth grade teacher and asked her about this very topic, and she also believes that I secretly did like math but was unwilling to admit it. Oh the things we do to fit in.

I remained among the top of my class on up through junior high, but upon entering Level I my always high math marks began to plummet like an '08 recession. I had floated through most of my early schooling without ever having to put in much effort, and quickly learned the hard way that this was not going to fly anymore. I still took advanced math in Level I and II but only got 66 and 63 respectively in the courses. At the end of Level II, my teacher Mr. Sheppard recommended me for the academic stream, which to me was a real eye-opener. By the time I got to Level III I had came to my senses, and ended up getting 77 in the final advanced math course of high school. I have to give at least partial credit to Mr. Sheppard for keeping me on the right path and always showing enthusiasm and love for the subject. He made me want to succeed. At the time I remember a school board official (who was also my friend's father) referring to Mr. Sheppard as the best math teacher in the province, and over time I've truly understood why. Mr. Sheppard is retired now and working at the project desk at Kent, having the time of his life helping people figure out their design problems. To me that sounds like fun! For real! And I guess I ave him to thank for that.


My main teachable is in fact not math, but physics. Now I always tell students "physics is a combination of math and thinking", but really they are two different things. In reality, a more accurate statement would probably be "you use math to do physics." In any case the two are closely linked. But I decided to focus on math education in my Masters program to better balance my teaching abilities in the subjects. I have never actually taught a math course on my own in my teaching career. Two years ago as a Numeracy Support Teacher, I co-taught ninth grade math with a teacher who had been teaching the course for years, an extremely valuable experience. We were able to do a lot of things most teachers only hear about in PD sessions. But overall, my experience in teaching math is nominal, a zygote of what I hope it will become.


I think there are two things that would help me as a math teacher at this point. The first is to sufficiently review some of the specific methods and concepts I learned four years ago in my B.Ed.'s math methods course. The second is to teach more math! There's no learning like learning from experience. In my teaching I focus more on the students than the content, and try to really figure out what works for the individual. The more experience I can get in utilizing the vast array of available methods and strategies in the classroom, the better I can hone in on what tends to work well.