This article was very interesting to read. I agree that a lot of what we teach students is not because they will specifically use it later in life, but to develop their problem solving skills. I also agree that just teaching students formulas doesn't necessarily push them to think, rather just encourages them to memorize a process. This is exactly what Skemp brought up with his relational vs instrumental understanding argument, where there are two ways that we can understand. Lockhart says that we only really have instrumental understanding in our schools, but that we should really emphasize ways of teaching that lead students to relational understanding. Now, I do think that it is difficult (as Lockhart acknowledged) to fully switch to something like he described, but what we can do in my opinion is to try and teach with the Socratic method when we can.
One point where I am less inclined to Lockhart's argument is his thoughts about notation. Although he states that he is only disinclined to "excessive" notation, I feel like the notation that is introduced is for clarity in communication. I agree that students should be allowed to understand the subject in any way that makes sense to them, but part of math is also about communicating your thoughts to others. Using the same notation is an easy way to stay on track with what everyone is talking about. As I wasn't entirely clear on what was meant by "excessive," I'm not sure that I fully disagree with Lockhart here, but it suffices to say that I appreciate notation and I think it's part of learning any new subject. Even non-scholarly subjects have notation- for example in volleyball, you soon learn about bumping, setting, and spiking, although as a newcomer you may simply be thinking "ah yes I need to hit the ball."
Great points about the importance of notation here! And then think about the Socratic method -- both its strengths and limitations. It's a good option to have in your teaching repertoire, but I suggest that, again, it is just one of many ways to teach math well. You'll want to find your own good balance among these.
ReplyDeleteHello Minami! You’ve written a clear and well-reasoned response that shows solid understanding of both Lockhart and Skemp (1976). I like how you connect their shared concern about students being taught to memorize rather than think, and your mention of the Socratic method is a great bridge toward practical teaching—there’s real potential in that idea. Agreeing with Susan, you are finding your balance that works for you and your students.
ReplyDeleteYour comments on notation are thoughtful and grounded. You raise an important counterpoint: that shared mathematical language isn’t just about formality, it’s about communication. I love that you are seeing the value in structure without losing sight of creativity - this is powerful and absolutely something to explore further (I am thinking of you incorporating origami into your teaching).
As you move into practicum, keep exploring that tension between freedom and form. What does it look like to question students in ways that spark thinking, while still helping them communicate their ideas clearly? Watching how experienced teachers use questioning or notation as tools for understanding (rather than just correctness) could help form ways in which you can create mini-springboards in your classroom.