Malcolm Swan, on how to begin a lesson:
Every lesson should begin by getting [students] to articulate something about what they already understand or know about something or their initial ideas. So you try and uncover where they’re starting from and make that explicit. And then when they start working on an activity, you try to confront them with things that really make them stop.
And it might be that you can do this by sitting kids together if they’ve got opposing points of views. So you get conflict between students as well as within. So you get the conflict which comes within, when you say, “I believe this, but I get that and they don’t agree.” Or you get conflict between students when they just have fundamental disagreements, when there’s a really nice mathematical argument going on. And they really do want to know and have it resolved. And the teacher’s role is to try to build a bit of tension, if you like, to try and get them to reason their way through it.
And I find the more students reason and engage like that then they can get quite emotional. But when they get through it, they remember the stuff really well. So it’s worth it.
ScottNovember 7, 2014 - 11:25 am -
That gels so well with Fawn’s post from the other day: http://fawnnguyen.com/driving-them-nuts/
Her kids were obviously itching:
>“This problem is making me crazy!” Or, “I won’t be able to think about anything else until I get this!”
I like also that Fawn’s problem is on the “pure” math side of the spectrum. It demonstrates that “real world application” is not necessary for kids interest. (here’s the problem: http://fawnnguyen.com/wp-content/uploads/2014/11/missing-area2.png )
Math for math’s sake! Math for the art of math! Math for the love of math!
Is it Fawn setting certain sociomathematical norms in her classroom that allows this? Probably to some degree. But I think also she chose or constructed this problem well too: it has an inherent appearance of simplicity and yet allows for complex exploration. “complex scratching” ?
I think also it is evident from the student work that the the problem allowed so many points of entry to begin the solution, and yet the solution itself initially lay outside the student’s understanding. Very Vygotsky-ian (“zone of proximal development”)
So if we merge with Swan’s thoughts here: put the itch just at the edge of students’ reach and the students will grow to scratch it.
NickNovember 7, 2014 - 8:43 pm -
It is a border line strategy. Some kids may not appreciate the conflictual part of the conversations. More that that, these days kids are though to agree to disagree. It’s easier and safer. I see this strategy more for a debate class …
Melissa C.November 18, 2014 - 6:43 pm -
I partially agree with starting class this way. I think it’s a great idea to kick off class with a small debate or a hook of sorts to engage students, but sometimes I think people might become really emotional and stirred up about their feelings on a particular topic.
I’m a student teacher in a science classroom and we have just finished our unit on evolution. Granted, nobody came up to me afterward and proclaimed what I was teaching was wrong or went against their beliefs, but I can easily see it turning into a religion vs science debate.
That being said, I think setting up that itch is terrific. Some of the more boring topics in biology, like teaching biochemistry, can be fun if I can find the right itch for my students.
AdrianNovember 19, 2014 - 4:31 am -
When students don’t appreciate a cognitive conflict, it tends to be because they have not been exposed much to the experience. I have often found this to be the case, especially at the start of the year.
The trouble is, when they leave school they are going to be faced with cognitive conflict at every turn and it is imperative that we, as educators, prepare them for this.
I would also encourage you that I have found as you provide more itching, the scratching becomes oh so much more satisfying! They will respond as your give them more opportunities to problem solve using their innate intuition and creativity. We need our Math classrooms to be a debate class.
JayantNovember 20, 2014 - 1:48 pm -
Challenging students is always a good idea. The metacognitive load of just the content is easily balanced by using some hands on exercises and that along with flipping the learning can help setup the “itch” and the form there it is a matter of involving them in the hand on activities, easily donein math classes.
CoryDecember 10, 2014 - 2:14 pm -
I’ve been thinking about how a “mathematical argument” takes shape and what differentiates a “good” one from a “bad” one. I’ve observed and participated in ones where two students are opinionated and are very attached to those opinions, which I would be inclined to call a “bad” argument. How can the teacher accommodate fast, confident thinkers and slower, insecure thinkers within one group?
I’m wondering if it is necessary to preface any classroom that will incorporate heated discussions, especially with younger kids, with suggestions on ways to contribute and make a positive environment. For example, if you think you know the answer, allow space for others to think about it. Or how to react if you see a blatant mistake made by another — that this mistake should not weigh into how we view this person’s contributions.
In terms of the itch metaphor, I really like it. The only problem I have is that I might find a particular puzzle exciting, the question is, will they? Sometimes my enthusiasm can be enough, other times not.
Dan MeyerDecember 10, 2014 - 6:50 pm -
Nothing really to add here, Cory. I think you nail a big issue around mathematical argumentation in whole-class instruction.
I’d only add to your last paragraph that nothing guarantees student interest. All I want is to understand the kinds of task properties that correlate with student interest.