A discussion with my students: What works?
I engaged a few of my students in a conversation about the effectiveness of the Harkness classroom. Here is that conversation, transcribed below:
So what do you think works for you in the discussion style classroom?
Jen: I think just hearing everyone else’s ideas and all their different ways of approaching each problem is really good ‘cause if I only know one way but I’m a bit confused on it and then I learn another way to approach it that makes it a lot easier for me to do problems.
May: Yeah and I think also if I don’t get a problem on a problem set and it’s hard to see, like, different ways to do a problem and it makes it easier.
Mabel: I agree with that. You learn more from making a mistake and in this style of teaching, that’s ok and that’s actually encouraged—to just be vulnerable and just to share what you did, whether you are confident in what you did or not. I don’t feel pressured to say what is right.
Becca what works for you?
Becca: I like being able to get to see how other people did the problems, but it also kind of helps that other students are doing the problems instead of a teacher going up to the board.
Yeah, why does that work?
Becca: Because you get to hear it through the perspective of other people who are developing their language, kind of, in math and like so we’re all coming from the same place and we have different ideas and share in that, and share our language, and share our perspective on the problem.
Why do you think it helps to work through a developing language? to get to know the math through developing your language?
Jen: I think because it forces you to really think about what you’re saying and that you’re trying to articulate what you’ve written on the board especially if you’re not writing a whole proof and you’re just putting up steps of a problem, you have to like, think about what you did in a deeper way when you have to talk about it and develop your language.
Mabel: Some people are more of visual learners and everyone is a different learner in their own special way and um its really (cut off)
For visual learners or learners who don’t think they have “math brains” because "it’s not really a real thing" when you look at it through a metaphor ‘cause it helps for the common person to realize that math is not just “math”, it’s life and it’s like life in a lot of different ways…and eileen comes up with metaphors on a whim and it helps me to better understand concepts.
So like translating, it’s a way to translate the learning of math from the traditional classroom to something that is more--
Mabel: I’m used to having all of my math classes, up until now, have been you know, I come into the classroom and I am learning math language. But it feels different coming into a Harkness style geometry class because it’s like being told to me in my own language--
Mabel: And if its not, you can kind of see it better in your head through the use of metaphors and I find myself coming up with my own metaphors inspired by the metaphors that she says.
Becca: I think it also kind of like coming into the classroom, the feeling is totally different ‘cause you’re not coming in and sitting down in your own little isolated bubble of a desk and looking at the board and trying to like, figure it out on your own. And immediately there’s discussions going on, like before the class starts, about the homework last night and like, “How did you do this?” and trying to talk about it, and it’s just fun, I guess. Even before we officially start talking about it—I just like that environment.
May: Ummm I think it’s also really nice, like, going off of that, how we’re all on the same page so it’s not, like, like because we’re all learning together, it’s kind of like we’re all building off of each other rather than, like, we’re looking at the board and kind of like interpreting it how we want. So I think that’s really nice.
Mabel: I really like the backwards classroom in the way that we do it in this class because it’s very simple and we’re not going off of a complex math book. I mean we’re given a problem set and it’s consistent and we’re given problem sets all year and we can expect the length and time we are going to spend on our homework. It’s a white page and it’s got problem one, problem two, give it your best shot, it’s very straight forward and the world that we live in, simplicity is key. It’s kind of like the “mac” and “pc”—it’s like every other class is “pc” but this is “mac”.
May: I think that we said, geometry is a lot more words and algebra is more clean-cut and there is one answer and one way to get there. And here you can kind of like circumnavigate how you get around a problem and find a way of solving it and I think that’s why the discussion works because we are all branching off each other’s ideas to get to the eventual one answer.
For any of you, did you find it to be really hard in the beginning and then get used to it, and then come to like it later on in the year? Or was it something you instantly liked?
Jen: I think for me it took a bit getting used to because it was so different from everything I’ve done, and also I’ve never been really big on speaking in classes and this kind of forced that, and especially in math classes I would just kind of do on my calculator, like, write it down and that would be it. But here you have to talk about it, think about it. So yeah, I think I adjusted in the first month or so. And now it’s so helpful and, like, it’s so great.
How do you feel like it works…so essentially you do your work on your own at night, though I know that some people team up and do their work together, which I think is really cool. Um but for the most part, you do it on your own, then we come to class and we talk it out, and we see different ways of doing it. How does that effect your ability to do assessments on your own? How do the discussions inform your work on an assessment where you know that you’re going to be graded and it’s going to "count"?
Mabel: I mean, I’m not usually a very good test taker, and I mean, I don’t know if I’ll every be, but when I am taking a test for geometry, the conversations kind of play back in my head instead of what did I do for homework or what is x plus whatever…the conversations we've had on a perpendicular bisector will come up in my head and again, it’s just simplified, um especially for someone like me, I need that imagery and I remember the imagery and it’s like when you’re studying for a test and you can connect a picture with a word and you make that link, and when you have that visual link, it makes it easier to remember when you have to remember it.
Becca: Um for me I guess similarly, I can hear the voices of my classmates when I am taking an assessment and when I can’t get through something in how I would do it, I picture like “what would June do in this situation” and kind of enter into other people’s modes of thinking to, like, get through an assessment sometimes.
You really do that?
Becca: Um a lot of the time though, most of the time though, I’d say I don’t do that, because all the practice we have and all the talking we do in class, that kind of sticks with you because you’re struggling through in class—it’s not “This is how you do it,” it’s “How do you do this?” and the students have to figure it out, and by figuring it out yourself, I think it makes it easier to remember. Like, certain techniques and even the things you should be memorizing, the theorems--like, they’re all provable! So there really is no memorization and because it’s all one thing, and I just think it’s cool.
Jen: I have kind of a photographic memory so when we do problems in class, and there are like multiple different ways of doing it up on the board, when I can look back on tests and think, “here are the different ways I can do this,” and I can find out which one works best for the problem and which one works best for me, which I wouldn’t be able to do if we weren’t discussing it together in class.
Yup. Excellent. Do you somehow prefer this environment for learning math? Like if all things created equal, if it was constructed and scaffolded correctly, would you prefer to be in this style or this environment or no? You can say not, I would not be offended at all by it.
Jen: Yeah, so this is a lot less intimidating for me to talk about my work and to discuss, like um, to discuss different problems and that helps a lot when you’re not scared to talk about what you did and you think it might be wrong. Whereas in algebra sometimes I would have a question but I would feel a bit scared to ask and I think that was more because of the way the class was set up and how we weren’t always discussing things, and it was like “Ok this is the answer. Can we move on?”
May: I think in this structure it’s kind of easier to, I guess, like, level out because I know that in algebra we had a lot of different speeds that people worked at and so some people would be far ahead and reading the next chapter versus some people need another review and so we would spend another class period on it when there are people who can move on. I feel like in this set up, it’s a lot easier to pace ourselves so I really like that.
Mabel: Piggybacking your point, um, I feel like some people at first struggle with this class because they are faster or they grasp the concepts more quickly than the person sitting next to them but I feel as we, as a class, adapt to the new environment and the learning style, the people who it comes more naturally to, they become, like, kind of the teachers. They become the enlighteners—they enlighten the rest of us. ‘Cause I know there are some very strong students in my class and you know, not everyone is going to be at the same level mentally, it’s just--you know, we’re all human. But if you look at it in a way where they illuminate the problem for everyone else and it’s one of your peers--it’s not just a teacher. I went to public school and I sat in the conventional style and my teacher would just tell it like it was and then you would go and you would sit and you would do the quiz. But you know this way, the teacher isn’t the only teacher.
Anything else you want to add?
Becca: What was the last question you asked again?
(student reiterated question)
Becca: Oh, I would prefer this style classroom. And that’s me.
“The essence of metaphor is understanding and experiencing one kind of thing in terms of another.”
-George Lakoff and Mark Johnson, Metaphors We Live By