Summary Commentary

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

Supporting Student Learning

Sometimes I'm not sure whether the students are learning more from the classroom experience, or whether I am. Perhaps both.

Within this Teaching Event and this Curriculum Unit, I had it hammered home that students will learn in their own good time. A teacher can facilitate. A teacher can make materials available, accessible, and interesting (or not). A teacher can even check in with individual students to acknowledge issues they bring with them from outside the classroom. Nevertheless, in the end, it's the student who makes the necessary connections to other topics, the student who decides to spend the necessary time, the student who constructs his/her own understanding.

In view of that, I've learned--and it has been a continuing lesson, not a fireworks epiphany--that the most effective method I've seen of supporting student

I've also learned that asking students to produce feedback or reflective commentary not only helps them learn metacognition as a way to regulat their own learning, but it's a not-so-sneakily effective way for me to learn from their experiences, which might otherwise go unspoken. When students feel comfortable enough to express the things that do and don't work for them in the classroom, it improves my ability to adjust instruction accordingly.

A strong starting-point is important so students have some incentive to persevere even through less successful activities. The activities they experienced through GIVING DIRECTIONS and NINE MEN'S MORRIS, for example, captured students' interest enough that they tolerated the overlong and less-well-thought-out TREASURE HUNT. Students let me know some of the flaws ("too long" and "too complicated") in the three-class-long activity, and even though by then it was too late to backtrack, the fact that I assured them that I'd know better "next time" seemed to reassure them.

I found flexibility to be the key. Given that strong starting-point and a willingness to change whatever is feasible and necessary, most things will work themselves out. I do not expect perfection on the first try. It is also important to convey both that fallibility and that flexibility to students, as it's their learning experience at stake.

Students' Progress in Learning

In the feedback phase of creating potential test questions for the final, Student #1 commented on another student's question, "Give three examples of when you would use Cartesian and polar coordinates," as follows: "I would love to see this question on the final. It pays more attention to common knowledge and on how well I understand the uses for the things I have learned. If I didn't know what it was for it would be useless. Right?" I couldn't say it any better myself.

From the beginning, when I introduced polar coordinates, to conversions as shown in the LETTER TO ZAP (in particular the Student Learning Samples), I saw a lot of increased comfort and familiarity as students acclimated to the strange things I was asking them to do. The number of errors in their work declined as they were given time to check with each other and get used to what they were doing. Though they may not remember specific formulae, I am confident that next year, they will be able to use polar coordinates, discuss some of their uses, and remember some part of their relationship to Cartesian coordinates.

Even more importantly, above and beyond the specific topics, they will have learned that one area of mathematics has real-world relevance and one way of using estimation as a tool. They will have learned to think about why they are learning a particular topic, no matter how many times they rolled their eyes when I asked, "Why do you think I'm giving you this activity to do?" Despite the rolled eyes, they began to answer.

Ten years from now, if a particular student doesn't remember how to compute theta using arctangent, I'm not going to lose any sleep over it (unless, of course, they happen to be in some critical position where it's vitally important that they know how to do so). On the other hand, if a particular students considers math to be useful or interesting enough to continue pursuing for some part of the next ten years, I shall consider myself to accomplish some small part of the purpose of education. It is not that the particulars of a topic (such as coordinate systems) are unimportant, but that it makes far more sense for students to study particular topics, which are necessary in particular contexts, with the parallel goal of learning how to learn mathematics effectively.

Changes for Next Time

Next time, I would especially like to build in more metacognitive exercises like the "What Did I Learn About Grading?" poster exercise. "Why are we doing this" is an important question for students to answer for themselves in general; it also helps them monitor their own learning processes. When I ask a student asking me for help, "What part is confusing?" and they can answer me, they have already stepped a long way toward resolving their confusion.

Specifically, questions such as "What do you think you will learn in this activity?" before the activity begins, and "What was the purpose of this activity?" after it is done, might prompt students more explicitly to think about their learning. I think it would also be important to ensure that "I don't know" or "I'm not sure" is an acceptable answer when students are new to such exercises. Of course, moderation is also important. I scaled back on "quickwrites" earlier in the year when students complained they were a little too frequent. Moderation in all things, I suppose.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 8 June 2002