I am currently pursing a graduate degree in Pure Math at Bowling Green State University.
I won’t go into a long story of how or why like I usually do when I overblog. Just wanted to share a few thoughts.
Something I have noticed that I really like but I did not experience as much as an undergradute studying mathematics and secondary education is how my fellow students and I tackle problems together.
We use the phrase “I’m not convinced,”.
When I say this to a fellow graduate student or someone says it to me, we generally mean one of two things:
- I don’t understand your argument. Can you explain it again? Convince me.
- I agree with your conclusion, but I don’t know if your argument is sufficient. Convince me.
I think both of the meanings for this phrase are incredibly useful. The first allows us to express a lack of knowledge without saying “I don’t understand” or “I don’t know.” This may sound like math students being overly concerned about saving face, and I can’t say that it isn’t, but we are all generally secure in our mathematical abilities. If we admit that we don’t know, it is okay. It is not really a reflection on our intelligence or ability or personhood.
On the other hand, to our own students who may not have that confidence and who view their mathematical knowledge as not independent from their intelligence and worthiness as a person, this phrase is great. It gives them a way to express a lack of understanding that relates back to the communication of mathematics. “I don’t understand what you are saying,” not “I don’t understand this” (And the subtext in the latter option of course being “And I never will so why try?” or “Because I am not smart enough” or “Because I am not a math person” etc.).
The second option is great for my fellow graduate students and myself because it gives us a tool to challenge each other to be better mathematicians by making better arguments without telling someone their argument isn’t good. (It may have even been good, but was missing a minor detail here or there.) It says “I’m not sure if that is how you argue that. Can you walk me through your thought process” rather than “Man, you suck. That isn’t how you prove that.” This friendly challenge to make a better argument makes us better mathematicians together not only because we increase our understanding, but it also our skill at communicating that understanding gets better.
This second meaning is also completely applicable to our own students. The phrase “I’m not convinced” not only give students a way to communicate mathematics to each other in a positive way, it inherently shifts the focus of the conversation to how to best express mathematical ideas instead of how a certain person understands those ideas.
I think this phrase encourages students to practice the Communication and Reasoning and Proof Mathematical Process Standards much better than a “I don’t know” or a “I don’t understand”.
In the future I will be encouraging my students to say “I’m not convinced” whenever they want to say the other things.