I have been writing a bit about the problem with "right" answers in the mathematics classroom lately. I think it is a big concern. Upon further reflection, I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:

"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students."

There is some great reading about how this relates to "mathematics for all" and teaching for social justice. Building this community is extremely difficult. Students have developed expectations about what learning and teaching mathematics "should" look like. As a result, if we are to promote this type of student discourse it becomes necessary to renegotiate the "didactic contract" (Brousseau, 1986). This "contract" both explicitly and implicitly outlines the role of both teacher and student, the expectation for classroom discourse, and, as a result, the locus of authority.

This type of discourse simply isn't going to take place if "mathematics" is practiced as "solving" a bunch of related problems (read: 1-30 odd). Instead, find the big idea, pick one rich question to lead an investigation of that idea, and then let the students sort it out. You are likely to see students doing mathematics. To return to Schoenfeld,

"This <is> their mathematics. They <have> ownership of it, not only in the motivational sense, but in the deep epistemological sense that characterizes the true mathematical knowing and understanding possessed by mathematicians."

 


Comments

Robert Gordon
04/08/2012 12:40pm

Bryan,

Thank you for this post and your ability to articulate the realities of teaching you face and sharing your theories to change these realities. Its funny that I, too, was just writing today about my similar experiences with students lack of "authority," or confidence, with their answers to math problems.

"As it stands now, students' ways of thinking are always subject to some greater authority." This unfortunate circumstance is entrenched in our students once they make it to our high school classrooms. In my first student teaching experience, I was shocked by the number of students whom need validity/assurance of their work. Maybe this is because they aren't truly doing their own work, but trying to emulate the work that the teacher just wrote on the board, or that a PowerPoint just told them to write down when they are given a problem of that manner.

Am I on the right page? Have students been told that they are wrong (and ultimately been made to feel dumb) so many times that they are afraid to use their imagination and give their own 2-cents into the classroom dialogue, perpetuating the problem we speak of?

I read your 3 Keys to Get Students "Doing Mathematics" and would love to read some of the lessons you've written that will suffice in a standards-driven classroom of which I am currently student teaching.

Cheers,
-Robert Gordon
rpg.cbad@gmail.com

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04/08/2012 7:20pm

Hi Robert...
Yes, I think you are on the right page and I agree with much of what you write here. I also hear your statement about providing more concrete examples of the type of mathematics teaching and learning that I often write about. I have some hesitations about this. One is that I think that lessons and instruction should be designed in response to a specific student "community." The other is that this type of instruction is usually most successful when certain norms have been established (didactic contract). However, with that said, I have been thinking about ways of sharing lessons without them serving as "cookie-cutter" approaches to be brought into the classroom. Thanks for your comments and thoughts!

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08/14/2013 8:43pm

I came upon your website when doing a search for: "Breaking the didactic contract" (believe it not there were only 3 links.)

I'm interested in your strategies and techniques. Could we collaborate over a inquiry based problem regarding systems of equations?

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Bryan S Meyer
08/20/2013 9:48pm

Thanks for visiting. My email is under the "About Me" section if you would like to collaborate or share ideas.

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Jamie Sneddon
11/10/2013 11:26pm

Great post Bryan, thanks.

The 'right' answer is a terrible, insidious thing. It's a hidden, secret thing, which the teacher knows, and the students are trying to guess.

I like the approach you suggest in your other post, where open-ended questions are used (rather than closed, single answer questions). Students will only get more comfortable with open-ended questions with practice; when they find it hard, they need to be exposed to it more. That's a difficult place for teachers to be in, but so worth it when the students start working mathematically instead of robotically.

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