Yesterday afternoon I worked for an hour with the student consultants at our university's Writing Center, helping them to understand what mathematical writing might look like, preparing them to work with students in the quantitative sciences who might come in with writing assignments from their quant courses. I was happy with the conversations we had together, and delighted by the students' energy and enthusiasm.
I shared with the consultants one precalc student's response to the midterm question in which I asked the students to describe the most meaningful learning outcome they've achieved so far this semester (see here for one response, not the one I shared at the writing center), and promised that I'd share a few more.
Let me throw a few more out there, all with a common theme. Several students, including the three quoted below, indicated experiencing the realization that mastering math is more than just memorizing formulas, and that if you stop trying to memorize every last formula but rather try to break each down and understand its inner workings, you gain immeasurably through your effort. That understanding is strengthened if math is put in a contextual matrix, placed alongside other disciplines so that its relevance becomes more apparent.
Katarina had this to say about her personal revelations:
This course is unlike any other math class I've taken. We actually discuss math in English at a level that I believe everyone in the class can understand....Instead of memorizing formulas, we play them out on the board, multiple ways, so that it is almost illogical not to understand their function and there is no need to actually memorize them....We write our answers in paragraph form, truly explaining the reason for doing them and our end result....It is so unusual to me to have math and other subjects (such as writing) overlap. My initial response was to avoid it but now I'm beginning to embrace the idea. After all, isn't UNCA's liberal arts program all about integrating many subjects in order to have a broader education?
Thomasina (who took a year off from high school before coming back to college) had this to say, upping the ante by acknowledging not just understanding, but enjoyment, and even aesthetic appreciation:
When I graduated high school, I decided that school was pointless....Read, memorize, regurgitate. That is all I was ever taught. But to understand? To break something down to its very core and build it back up, seeing every piece as they’re placed together to form a whole again. It's actually beautiful. I never got what you meant when you’d say math is beautiful, but now I get it. To have the ability to look at something complex and make it simple and tangible - it's art. And it's not just with math. It applies to everything: decisions, work, other people. Everything is a complex formula waiting to be taken apart and understood, and then put back together in a way that makes sense and feels right.
Kurt reports an experience similar to that of Katarina and Thomasina:
I've found more interest and enjoyment in something that I previously found tedious and boring and have found that I actually can relate math (including calculus) to the real world and my everyday life in ways I'd not before considered or imagined....I see this insight as far more valuable to me as a person than any one mathematical concept on its own. This is the kind of insight that changes people's lives, gives the potentially brilliant scientist a view into the potential locked up inside, or even just changes a fundamental attitude or a paradigmatic shift in thinking altogether....It's a great feeling to realize suddenly 'hey, I GET this!' and even better to find 'hey, I actually LIKE this!' I truly wish there were more classes like this one which, if not persuading one to major and work in the field, to at least open one’s eyes to the possibilities.
If a precalc course can move students to wax this rhapsodically about their learning, how much can students get from still deeper and more meaningful courses? It's up to those of us who teach to make our courses as relevant as we can.