Cathy Carroll

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Developing Mathematical Knowledge for Teaching

• Cathy Carroll Judy Mumme
• WestEd

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Purposes for Doing Mathematics in PD

• Some commonly identified purposes
• To have a common experience for a group of people
• To prepare for using with students
• An opportunity to increase teacher content
  knowledge of math
• Begin to address particular math issue(s)
• Opportunity to reconstruct ones notion of what
  it means to do mathematics, nature of
  mathematical activity
• Identify certain kinds of pedagogy that support
  learning in particular situations
• How do these resonate with your experience?

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Session Overview

• Consider a purpose of developing teachers
  mathematical knowledge for teaching (with a focus
  on specialized content knowledge) through
• Work on a mathematics task
• Watch a video
• Consider next steps from the video to work on MKT
  with teachers

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Doing Mathematics in PD

• Pairs (1 minute each)
• What supports you in doing math in PD? What gets
  in the way?
• Small group (5 minutes)
• Share what things are supportive and what are
  not. (Note these may be different for each
  person)

With this in mind, it is important to respect
individual differences and provide everyone equal
access in group work
Pairs/Small Group
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Logos
Assume the pattern continues to grow in the same
manner. Find a rule or formula to determine the
number of tiles in a figure of any size.
What are the different ways that the geometric
model can be decomposed and how can those ways be
connected to symbolic expressions? How do
those different expressions represent the growth
of this model?
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Considering the Task

• What are some expressions you came up with to
  represent the growth (un-simplified versions)?

Whole Group
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Considering the Task

• Take each of the expressions and see if you can
  figure out how that group was thinking about how
  the model was growing
• What is the relationship between these
  expressions and the logos model?

Small Group
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Considering the Task

• How did you map the expressions to the logos
  model?

Whole Group
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Considering Purpose

• What potential mathematical ideas could teachers
  work on through the use of this task?
• Why would those be good goals for teachers?
• Would you do that with students?
• How might ones purposes be similar/different
  there?

Whole Group
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Mathematical Knowledge for Teaching (MKT)

• Frame knowledge entailed by the work of
  teaching
• Knowledge used or needed in practice
• What do we mean by knowledge?
• Mathematical knowledge, skills, habits of mind
• What do we mean by the work of teaching?
• The activities in which teachers engage, and the
  responsibilities they have, to teach mathematics,
  both inside and outside of the classroom

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Mathematical Knowledge for Teaching
Pedagogical Content Knowledge
Subject Matter Knowledge
Knowledge of Content and Students (KCS)
Common Content Knowledge (CCK)
Specialized Content Knowledge (SCK)
Knowledge of curriculum
Knowledge at the mathematical horizon
Knowledge of Content and Teaching (KCT)
Ball, D. L., Thames, M. H., Phelps, G. (2008).
Content knowledge for teaching What makes it
special? Journal for Teacher Education, 59(5),
389-407.
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Common Content Knowledge (CCK)
Mentally find the answer to 92-56
92-56 36
Knowing how to estimate that it is greater than
30 and less than 40
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Specialized Content Knowledge (SCK)
92 - 56

• What meanings of subtraction might each
  represent?
• Do these methods always work?
• If so, why do they work?

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Pedagogical Content Knowledge
92 - 56

• Knowledge of content and students (KCS)
• Knowing which approaches students are likely to
  use
• Knowing which ways of decomposing the numbers are
  likely to lead to confusion (e.g. rounding the
  two numbers in different directions - 92 to 90
  and 56 to 60

• Knowledge of content and teaching (KCT)
• Choosing numbers which invite particular
  approaches or stumbles
• Choosing contexts or models to illustrate
  different approaches
• In a whole-class, choosing which approaches or
  methods you want to pursue and in what order

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Tasks of Teaching Mathematics That Require SCK

• Unpacking and decomposing mathematical ideas
• Explaining and guiding explanation
• Using mathematical language and notation
• Generating examples
• Making mathematical practices explicit
• Choosing and using representations
• Comparing the affordances of different
  representations or methods
• Analyzing and interpreting alternative solutions
• Analyzing errors
• Interpreting and evaluating alternative solutions
  and thinking
• Analyzing mathematical treatments in textbooks

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SCK in Logos

• What SCK might it be possible to develop using
  the Logos task?

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SCK in Logos

• How did our framing of the task focus your
  attention on SCK?
• In what ways did our discussion of the task
  enable consideration of SCK?

Our framing of the task What are the
different ways that the geometric model can be
decomposed and how can those ways be connected to
symbolic expressions? How do those different
expressions represent the growth of this model?
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Pause Point

• What issues does this session have you thinking
  about now?
• How are those related to your practice?

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BREAK

• Well start back promptly at 1030

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We are looking at this
We are here
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Context

• Group of 27 high school and middle school
  teachers from 6 districts
• Session 7 of of an ongoing series--8 Saturday
  sessions over the school year
• Small groups have worked on the logos task and
  posted charts with their solutions
• 3 groups have shared their work, and Shamshir is
  asked to talk about his poster
• Mike is the PD leader

We drop in here
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Caveat

• Mike and the teachers are offering us a gift of
  allowing us to carefully examine a real instance
  of practice. We are examining their practice, not
  critiquing them.

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Mike
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Viewing the Video

• We will watch the clip one time, then use it as a
  jumping off spot for connecting to practice, with
  a focus on helping teachers develop SCK

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Frame for Viewing

• What approaches are teachers sharing?

Suggestion Use transcript to think about issues
after viewing the video
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What approaches are teachers sharing?
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Small Group Discussion

• What approaches were teachers sharing?

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Whole Group Discussion

• How did the approaches that were shared relate to
  the approaches we came up with?

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Connecting to Practice

• Assume a goal of developing teachers
  understandings of how various ways of decomposing
  or transforming the model relate to different
  symbolic expressions--possibly including how they
  show aspects of quadratic growth (i.e., square,
  linear, and constant components)
• How would you frame the next steps in sharing?
• What solutions would you select to pursue to help
  develop teachers SCK? Why? How do they relate
  to your goal?
• How would you sequence and connect these to
  achieve your goal with teachers?
• How would you highlight aspects of SCK?
• What ideas did you consider in your process?
  What issues arose for you in this task?

Small Group
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Connecting to Practice

• What ideas did you consider in planning?
• What issues arose for you?

Whole Group
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Reflecting on the Experience

• What are you thinking about now with regard to
  developing SCK with teachers in PD?

Small Group/Whole Group
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A High Leverage Purpose

• Were placing our bets that a focus on developing
  mathematical knowledge for teaching (specifically
  specialized content knowledge) will better enable
  teachers to be effective in the classroom

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Logos
Assume the pattern continues to grow in the same
manner. Find a rule or formula to determine the
number of tiles in a figure of any size.
What are the different ways that the geometric
model can be decomposed and how can those ways be
connected to symbolic expressions? How do
those different expressions represent the growth
of this model?
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The Mathematical Task Framework (MTF)

Tasks as enacted by teacher and students
Tasks as they appear in curricular materials
Tasks as set up by teachers
Student learning

• Smith Stein (1998)
• Stein, Smith, Henningsen Silver (2000)

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The Mathematical Task Framework Adapted to PD

Tasks as enacted by PD leader teachers
Tasks as they appear originally
Tasks as set up by PD leader
Teacher learning
Kazemi 2009
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Important Features of SCK Task Design

• Unpacks, makes explicit, and develops a flexible
  understanding of mathematical ideas that are
  central to the school curriculum
• Opens opportunities to build connections among
  mathematical ideas
• Provokes a stumble due to a superficial
  understanding of an idea
• Lends itself to alternative/multiple
  representations and solution methods
• Provides opportunities to engage in mathematical
  practices central to teaching (explaining,
  representing, using mathematical language,
  analyzing equivalences, proving, proof analysis,
  posing questions, writing on the board)

Suzuka, K., Sleep, L., Ball, D. L., Bass, H.,
Lewis, J., Thames, M. (in press). Designing and
using tasks to teach mathematical knowledge for
teaching. Association of Mathematics Teacher
Educators (AMTE) Monograph.
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Challenges of Teaching SCK

• Staying focused on the mathematics, and not on
  how to teach the math
• Keeping the problems focused on SCK and not just
  sliding to Knowledge of Content Students or
  Curriculum
• Unpacking the mathematics sufficiently and
  convincingly helping teachers see what there is
  to learn and do
• Making visible the connections to the kinds of
  mathematical thinking, judgment, reasoning one
  has to do in teaching

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Enactment

• What are key questions and moves that leaders can
  use to keep a task focused on developing SCK?
• Asking teachers to
• Explain their solutions to others
• Make correspondences between solutions and/or
  representations
• Explain someone elses thinking
• Figure out what might be confusing/difficult for
  someone else about the problem
• Having teachers
• Explain what is/was confusing to them
• Ask questions to become clearer about their
  colleagues solutions
• Providing opportunities to talk mathematics and
  write on the board
• Narrating how something a teacher does/says
  relates to or is a skill used in teaching

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To Learn More About

• Learning to Lead Mathematics Professional
  Development
• Leadership Development Seminars
• Contact Information
• Email
• ccarroll_at_wested.org
• jmumme_at_wested.org
• Web
• WestEd.org/LLMPD
• WestEd.org/PLMPD