Supporting Productive Whole Class Discussions

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Supporting Productive Whole Class Discussions

• Paul Cobb
• Vanderbilt University

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Overview

• Classroom social and sociomathematical norms
• The norm of what counts as an acceptable
  mathematical argument
• Calculational and conceptual discourse
• Reflective shifts in classroom discourse

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Organization of Classroom Activities

• Initial discussion during which the teacher
  introduces instructional activities
• Students work on the instructional activities
• Concluding whole class discussion of students
  solutions and interpretations

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Organization of Classroom Activities

• Teachers goal To achieve a mathematical agenda
  by building on students contributions

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The Swing of the Pendulum

• Student-centered approaches
• Celebrate students discoveries and methods as
  ends in themselves
• Teacher-centered approaches
• Focus on conveying mathematical ideas and
  procedures to students

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Transcending This Dichotomy

• Keep one eye on the mathematical horizon and the
  other on students current understandings,
  concerns, and interests
• (Deborah Ball, 1993)

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Making Sense of Classrooms

• What do students have to know and do to be
  effective?
• What obligations do they have to fulfill?

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Making Sense of Classrooms

• It is just a class. Most classes teach then they
  give you class work then homework. She the
  teacher goes over the homework. Then she goes
  over new stuff. Then we start on homework. And
  then it is time to go.

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Classroom Norms

• Classroom social norms -- general classroom
  obligations
• Explain and justify solutions
• Attempt to make sense of explanations given by
  others
• Indicate understanding and non-understanding
• Ask clarifying questions
• Question alternatives when conflicts in
  interpretations have become apparent

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Classroom Norms

• Sociomathematical norms -- specifically
  mathematical obligations
• What counts as a different mathematical solution
• What counts as an efficient mathematical solution
• What counts as a sophisticated mathematical
  solution
• What counts as an acceptable mathematical
  explanation

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Classroom Norms

• You talk about your way, or you add something to
  someone else's way. You can't just say that you
  agree or you disagree. Mrs. M the teacher makes
  you explain it. You have to ask questions about
  things that you don't understand.
• You have to do a good job explaining how you
  looked at the problem. That's important since
  you didn't talk with everybody else when you were
  doing the problem.

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Establishing Classroom Norms

• Scaffolding and holding students accountable
• Indicate understanding and non-understanding
• Ask clarifying questions

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Equity In Students Access To Significant
Mathematical ideas

• All students are able to participate
  substantially in classroom activities
• All students see reason and purpose to engage in
  classroom activities
• Students view classroom activities as worthy of
  their engagement

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Equity In Students Access To Significant
Mathematical ideas

• Differing norms of participation, language, and
  communication
• Potential conflicts with the norms that the
  teacher seeks to establish in the classroom
• Explicit negotiation of classroom norms is a
  critical aspect of equitable instructional
  practice

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What Counts as an Acceptable Mathematical
Explanation

• Calculational explanation
• Explain the process of arriving at a result or
  answer
• Conceptual explanation
• Also explain the reasons for this process process

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Calculational and Conceptual Discourse

• Chris and Juan have 12 candies altogether. Chris
  has 8 candies. How many candies does Juan have?

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Calculational and Conceptual Discourse

• A directory of 62 pages has 45 names per page.
  How many names are in the directory?

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Calculational and Conceptual Discourse

• Initial activities for linear measurement
• Counting the first step

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Calculational and Conceptual Discourse
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Calculational and Conceptual Discourse
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Calculational and Conceptual Discourse

• Casey And I was saying, see like theres seven
  green that last longer.
• Teacher OK, the greens are the Always Ready, so
  lets make sure we keep up with which set is
  which, OK.
• Casey OK, the Always Ready are more consistent
  with the seven right there, and then seven of the
  Tough ones are like further back, I just saying
  cause like seven out of ten of the greens were
  the longest, and like ...

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Calculational and Conceptual Discourse

• Ken Good point.
• Janice I understand.
• Teacher You understand? OK Janice, Im not sure
  I do, so could you say it for me?
• Janice Shes saying that out of ten of the
  batteries that lasted the longest, seven of them
  are green, and thats the most number, so the
  Always Ready batteries are better because more of
  those batteries lasted longer.

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Calculational and Conceptual Discourse
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Calculational and Conceptual Discourse

• Teacher So maybe, Casey, you can explain to us
  why you chose 10, that would be really helpful.
• Casey Alright, because theres ten of the Always
  Ready and theres ten of the Tough Cell, theres
  20, and half of 20 is ten.
• Teacher And why would it be helpful for us to
  know about the top ten, why did you choose that,
  why did you choose ten instead of twelve?
• Casey Because I was trying to go with the half.

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Calculational and Conceptual Discourse
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Calculational and Conceptual Discourse

• Brad See, theres still green ones Always
  Ready behind 80, but all of the Tough Cell is
  above 80. I would rather have a consistent
  battery that I know will get me over 80 hours
  than one that you just try to guess.
• Teacher Why were you picking 80?
• Brad Because most of the Tough Cell batteries
  are all over 80.

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Calculational and Conceptual Discourse

• Jennifer Even though seven of the ten longest
  lasting batteries are Always Ready ones, the two
  lowest are also Always Ready and if you were
  using those batteries for something important
  then you might end up with one of those bad
  batteries.
• Barry The other thing is that I think you also
  need to know something about that or whatever
  youre using them the batteries for.
• Teacher You bet.

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Calculational and Conceptual Discourse

• S I knew what they the other students did so I
  didn't want them to tell me what they were doing,
  but what were they thinking, yeah, what was your
  intention.

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Calculational and Conceptual Discourse

• S You can't just talk about your conclusion
  because that doesn't let anybody know why you did
  things.
• I Is that important?
• S If you don't talk about what you were thinking
  about then we don't know if it all is okay we
  can't figure out if it is a good point.

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Calculational and Conceptual Discourse

• Gives students access to each others thinking
• Supports situation-specific imagery that
  facilitates problem solving
• Brings significant mathematical ideas to the fore
  as a focus of discussion

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Reflective Shifts in Discourse

• Reflection viewed as a critical aspect of
  mathematical learning
• Learning as problem solving-- students reflect
  when they experience perturbations
• Teachers role limited to posing tasks that are
  genuinely problematic for students and that give
  rise to perturbations

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Reflective Shifts in Discourse

• What is said and done in action subsequently
  becomes an explicit focus of discussion and
  analysis

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Reflective Shifts in Discourse

• First-grade classroom
• Instructional intent Flexible partitioning of
  small quantities (e.g., five as four and one,
  three and two, etc.)
• Instructional activity Five monkeys in two trees

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Reflective Shifts in Discourse
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Reflective Shifts in Discourse

• Anna I think that three could be in the little
  tree and two could be in the big tree.
• Teacher OK, three could be in the little tree,
  two could be in the big tree writes 32 between
  the trees. So, still 3 and 2 but they are in
  different trees this time three in the little
  one and two in the big one. Linda, you have
  another way?
• Linda Five could be in the big one.

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Reflective Shifts in Discourse

• Teacher OK, five could be in the big one writes
  5 and then how many would be in the little one?
• Linda Zero.
• Teacher Writes 0. Another way? Another way
  Jan?
• Jan Four could be in the little tree, one in the
  big tree.

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Reflective Shifts in Discourse
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Reflective Shifts in Discourse

• Teacher Are there more ways? Elizabeth.
• Elizabeth I dont think there are more ways.
• Teacher You dont think so? Why not?
• Elizabeth Because all the ways that they can be.

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Reflective Shifts in Discourse

• Initial focus of classroom discourse Generating
  possible ways the monkeys could be in the two
  trees
• First reflective shift Determining whether there
  are more possibilities by checking the table
  empirically

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Reflective Shifts in Discourse

• Second (teacher initiated) shift
• Teacher Is there a way that we could be sure and
  know that weve gotten all the ways?
• Demonstrating that there were no further
  possibilities by identifying patterns in the table

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Reflective Shifts in Discourse

• Jordan See, if you had four in this big tree
  and one in this small tree in here, and one in
  this big tree and four in this small tree,
  couldnt be that no more. If you had five in this
  big tree and none in thissmall tree you could
  do one more. But youve already got it right here
  points to 50. And if you get two And if you
  get two in this small tree and three in that
  big tree, but you cant do that because three
  in this small one and two in that big
  onethere is no more ways, I guess.

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Reflective Shifts in Discourse

• Teacher What Jordan said is that you can look at
  the numbers and there are only a certainthere
  are only certain ways you can make five.
• Mark I know if you already had two up there and
  then both ways, you cannot do it no more.

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Reflective Shifts in Discourse

• Initially, the teacher and children generated
  possible ways in which the monkeys could be in
  two trees
• Subsequently, the teacher and children
  collectively constituted the (symbolically
  recorded) results of prior activity as an
  explicit focus of discussion and discerned
  structural patterns

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Reflective Shifts in Discourse

• The identification of patterns is central to
  mathematical learning
• How can we know for sure questions are natural
  precursors to mathematical proof
• Students learn to ask mathematical questions

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Reflective Shifts in Discourse

• S Now we know the terms like mean, median,
  range, and when you would want to use those
  terms.
• I You said you know what average is. How is it
  different from before? Average is something you
  talked about before the class in other classes?
• S I learned when to use it to describe the data.

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Reflective Shifts in Discourse

• I When should you use it?
• SMost of the time I don't use the average. I
  like using the range. I use the range when the
  points are spread out. If the points are around
  in a really small area you probably want to use
  the median since that would be a better way to
  let someone know about the points.

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Reflective Shifts in Discourse

• Mathematical ideas as tools
• Discussing particular mathematical ideas while
  explaining specific solutions
• Discussing when particular mathematical ideas
  prove to be useful
• Identify patterns in (collective) activity of
  using ideas

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Reflective Shifts in Discourse

• Students do not just happen to spontaneous
  reflect at the same time
• Opportunities for the students to reflect on and
  identify patterns activity arise as they
  participate in and contribute to the reflective
  shifts in discourse
• Central role for the teacher in supporting
  reflection on collective as well as individual
  activity
• Wisdom and judgment

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Agency Revisited

• Disciplinary agency Involves applying
  established mathematical methods
• Conceptual agency Involves choosing mathematical
  methods and developing meanings and relations
  between concepts and principles

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Equity in Students Access to Significant
Mathematical Ideas

• All students are able to participate
  substantially in classroom activities
• All students see reason and purpose to engage in
  classroom activities
• Students view classroom activities as worthy of
  their engagement

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Equity in Students Access to Significant
Mathematical Ideas

• Structural significance Attaining entry to
  college and high-status careers, and gaining
  approval at home and in social networks
• (John DAmato, 1993)

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Equity in Students Access to Significant
Mathematical Ideas

• Many students either do not see themselves going
  to college, or hold activist stances, or have
  more pressing daily concerns (e.g., housing,
  safety, healthcare), or do not believe that hard
  work and effort will be rewarded in terms of
  future educational and economic opportunities
• (Rochelle Gutierrez, 2004)

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Equity in Students Access to Significant
Mathematical Ideas

• Students access to a structural rationale varies
  as a consequence of family history, race or
  ethnic history, class structure, and caste
  structure within society

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Equity in Access to Significant Mathematical Ideas

• Situational significance Gaining access to
  experiences of mastery and accomplishment, and
  maintaining relationships with peers
• Students view classroom activities as worthy of
  their engagement in their own right

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Equity in Access to Significant Mathematical Ideas

• Failure to give all students access to a
  situational rationale for learning mathematics
  results inequities in motivation
• Access to a situational rationale requires that
  have the opportunity to express conceptual agency
  as well as disciplinary agency in the classroom

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