Agree to Disagree

Creating a classroom environment where students publicly disagree with one another is often a challenging endeavor. During this class, my class of Integrated Math 1 students were finding the rate of change (ROC) from graphs, tables, and two points. Students had been introduced to slope triangles and the slope formula, but the majority of the work had been solo as students worked on finding the ROCs. In this extended warm up, I asked students to evaluate each “student’s” claim.

Activity adapted from colleague Thach D.

This is an instance of some fictional students making mathematical claims. I gave this task because I wanted students to continue working on critiquing arguments while being removed from the person itself. I have noticed that my students seem to avoid constructing critiques about students in class because they do not want to be the one to “hurt” someone’s feelings.

First, I had students write down their ideas independently for Damon, Mari, and Anika’s claims using the sentence frame. After about 2 minutes of writing independently, I asked students to share with their elbow partners as a “turn and learn.” 

As I walked around, some pairs were still reluctant to share with their elbow partners (about 3 of these pairs also had little written down. I think they were struggling to form an opinion). For these pairs, I asked more directed questions such as, “Do you see a rise of 1 and a run of 2 anywhere on the graph?” Then, that was usually enough for students to try and evaluate Mari’s and Anika’s claims.

I then asked for volunteers to share for the whole class. For Damon’s claim, one student said, “Damon’s wrong because he switched the 1 and the 2. He has run over rise, but it should be 2 over 1.” 

With this, I jumped in and continued to clarify that we have been defining slope as rise over run. I wonder if I could have let students expand further. For Mari’s claim, another student said, “I agree with Mari because there’s a rise of 2 and a run of 1.” The students who shared were both volunteers who typically share often. However, I believe using “fictional” students helped students feel more comfortable in critiquing the displayed mathematical claims. The multiple representations (graph, words, equation) also opened access for more students to evaluate the claims.

One comment on “Agree to Disagree

  1. Sharing this in case it is helpful!!

    In 2008 Dr. Ilana Horn published a study that asked the question: How can classroom discourse be organized to support mathematical disagreements that (a) are intellectually productive, and (b) minimize social discomfort? She ends up calling this kind of discourse “accountable argumentation.”

    Paraphrase: The norms, expectations, language, and roles aspects of accountable argumentation mediate the tension usually associated with disagreement in American culture by framing disagreement as collective rather than as personally threatening.

    The following are all direct quotes from the paper:

    Accountable Accountable argumentation is a participation structure embedded in whole class discussion that organizes the public disagreements between students and provides interactional resources for clear mathematical reasoning and the production of mathematical generalizations.

    Accountable argumentation is organizationally distinct from whole class discussions because, although it is comprised of public talk, (a) turn-taking is managed by the students engaged in accountable argumentation, not by the teacher; (b) it places students at the interactional and often physical center of the classroom; and (c) it permits students freer movement and greater access to classroom resources such as the chalkboard.

    Accountable argumentation also provides interactional roles to students to support their mathematical positioning during disagreements, as well as communicating and supporting the expectation that students justify these positions to their classmates.

    Accountable argumentation has the following “norms”:
    1. Accountable argumentation uses terms from the mathematical and academic registers (e.g. “proof,” “conjecture”)…. In other words, because of their impersonal nature, using words from the mathematical registrar might alleviate some of the personal feelings that could arise during conflict.
    2. Discussions have a slow and measured pace… This normative pace provides interactional support for participants to think through and substantiate their positions.
    3. Disagreements are important and may not (and need not) be resolved… the common discomforts associated with everyday disagreements are alleviated by the normalization and the containment in academic language of these disputes (norms 1 and 3).”

    The expectations listed below are constantly communicated during interactions.
    1. Attend to contributions in a whole-class discussion.
    2. Have a justified position in a discussion.
    3. Act on or defend a position in a discussion.
    4. Respectfully respond to other’s positions in a discussion.
    5. Revise a position in light of new questions or convincing evidence.

    The following roles are available to participants engaged in accountable argumentation:
    1. Principal of a controversy: A person held accountable for a position that others disagree with or question
    2. Dissenter: A person who takes an opposing position to the principal’s position
    3. Ally: A person (or thing) who supports the principal’s position

    During disagreements, participants may take up any of these first three roles. Oftentimes, they blend them with one or more of the following:
    4. Questioner: A person who asks questions, especially to the principal, perhaps because of confusion or an uncertain stance
    5. Reasoner: A person who provides an exposition of reasoning
    6. Listener: A person who listens to the arguments
    7. Norm-maintainer: A person who explicitly evokes norms during conversation
    8. Clarifier: A person who clarifies or summarizes another participant’s statement’ (login with your CPM account to view)

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