Sometimes it feels like I’m just teaching students patterns that go in students’ short-term memory. In general, students pick up on the solution pattern if you show them enough examples. I think they’re really getting it. They say, “This is easy.” I’m glad because at least they’re not complaining about how hard math is and how they suck at it. Then I come in the next day and they can’t remember what we’ve done. I spend time showing them the pattern again and little light bulbs go on. Then I teach the new lesson, which is usually another type of problem with a slightly different solution pattern. They practice it and get decent at it. I assign them homework that is just like what we did in class. Some of them do it and some of them don’t. They come in the next day and many have forgotten how to do the problems. So I review again. Review is good, but I wonder if I’m just teaching them patterns that go in their short-term memory.
I want them to be getting the big
idea of the unit and be able to fit what they’re learning into this grand
scheme. But when I was in high school, I didn’t know how things fit together. I
learned math by recognizing patterns and mimicking problem solutions for the
most part. Sometimes I got a glimpse of how some topics were related, but
rarely saw the overall picture. It wasn’t until college when I began learning
even more advanced topics that I could look back and see how concepts in
algebra were related, for example. Many of the different patterns and
procedures I had learned were actually different perspectives of the same
concept. (Which is something I think is cool about mathematics.)
I don’t spend time teaching the
connections because I don’t think they’ll get it. I also don’t think I have
enough time to teach them. I don’t know if these are false preconceptions or if
they’re true. I also don’t know if it’s good or bad that I don’t teach the connections
very often. I used to think that you’re a bad teacher if you focus on
procedures. I still do, in a way. However, today, as I was having this thought
about whether I should be more focused on deep understanding and conceptual
understanding or on procedures, this blog post by Christopher Danielson appeared
on my NetVibes account: “Sometimes in mathematics, we need to live with new
notation before picking its meaning apart too carefully…. Patterns are powerful
tools in mathematics. Tabitha’s [my daughter’s] experience in the teens gave
her powerful intuitions for the twenties” (Christopher Danielson, http://christopherdanielson.wordpress.com).
In the comments, someone writes: “cf: John von Neumann: ‘In mathematics you
don’t understand things. You just get used to them.’ One of my favorite
thoughts about mathematics, and persistence, and playfulness.”
The perspective that learning
patterns is fundamental to learning math countered my thought about
understanding the grand scheme. I do believe that teaching connections is
important in math. But I can’t overlook the importance of the procedures
either. Sometimes you learn the procedures or notation first, and once you get
good at it, you can then start to see how it relates to other ideas. So thanks,
Christopher, for that timely thought!
You're welcome!
ReplyDeleteThis is a lovely reflective piece. As is my nature, I'd like to offer a couple of critical observations.
Please note the "sometimes" that leads that observation. Sometimes in mathematics we need to live with new notation.... Note, too, that it's about notation, not ideas. The relationship between these has surely been the subject of many dissertations, but a simple example suggests that ideas and connections can be understood before symbolism. Consider another post about Tabitha. In this one, she clearly understands something about addition without having a clue about the symbolism.
I would say that von Neumann's quote, while a useful reminder, is too strong. It is not at all the case that we are only "used to" many of the things we know in mathematics.
You wrote, I don’t spend time teaching the connections because I don’t think they’ll get it.
I guess my viewpoint is that connections is all there is to learning. All there is.
I see learners as building connections whether I teach them or not. The big question is whether we make those connections public. Once we commit to making them public, things get visibly messy. But keeping those connections bottled up and private doesn't eliminate the mess. It just keeps us teachers from dealing with that mess. Here is an example of a time when a student was making connections that weren't working.
I'd love to hear your thoughts. I'll try to remember to check back here, but feel free to shoot me a tweet or comment over in my house to continue the conversation.
Thanks for the mention and for sharing your reflections publicly. Great community of thoughtful teachers we've got here, eh?
I really appreciate you commenting on my blog. I'm kind of a math blog stalker, so I read your blog all the time, and am always impressed by the amount of thought that goes into your teaching and writing. In general, the blog writers of the mathtwitterblogosphere are pretty amazing and I am always inspired by them.
DeleteBecause I read so much, I feel like I know what good teaching is, but I suck at practicing it. I'm really new to the profession so my hope is that I get better over the years. When I was student teaching I was really fazed by what my students knew/didn't know, or at least what I thought they knew/didn't know. I relied on a lot more direct instruction than I wanted because that's the only way I knew how to get them through it, and it was also not my own classroom.
In the fist couple of classes I taught during student teaching in Algebra 2, I focused on having students make connections. For example, when teaching combining radical expressions, I had them do some problems where they had to find the greatest common factor of an expression that had no radicals in it. Some of the problems even had non-mathematical symbols just to get them to think about the grouping that takes place with GCF. Then I was going to make the connection to combining radicals. The problem is that most of them didn't remember how to find a greatest common factor. So the part about combining radicals got lost in the process. Then I showed them the more traditional way of thinking about combining radical expressions - the rule about like radicals - and they got it. They just wanted to know how to do it not why.
After that experience, I shied away from making connections. It seemed like it was only going to confuse them more and take up too much time. You're right though that ignoring making connections just because I don't know how to deal with it doesn't make things any better. It just feels a lot more comfortable doing the standard method of teaching because it's what the students expect and are used to; they're not used to it being messy. I actually like when math is kind of messy because to me that's what real mathematics is like. It's frustrating, but it's interesting and when I get it, it's exciting! I'm just afraid that students won't learn that way and that I won't be able to cover all the material that I'm supposed to in a year. I'm not really sure how I'm going to correct things in the future, except keep reading things that inspire me and try to integrate them into my teaching as I go.