Capturing Growth in Teacher Mathematical Knowledge

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Capturing Growth in Teacher Mathematical Knowledge
An Inquiry into Elementary and Middle School
Teacher Understanding of Algebraic Reasoning and
Relationships

• The Association of Mathematics Teacher Educators
• Eleventh Annual Conference
• 26 January 2007
• Dr. DeAnn Huinker, Lee Ann Pruske Melissa
  Hedges
• The Milwaukee Mathematics Partnership
• University of Wisconsin - Milwaukee
• www.mmp.uwm.edu

This material is based upon work supported by the
National Science Foundation Grant No.
EHR-0314898.
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Session Goals

• Contribute to the discussions around defining and
  measuring the specialized mathematical knowledge
  needed for teaching.
• Share and examine performance assessments that
  look more closely at growth in the mathematical
  knowledge targeted on algebra.

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What distinguishes mathematical knowledge from
the specialized knowledge needed for teaching
mathematics?
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Common vs. Specialized Mathematical Knowledge

• Encompasses
• Common knowledge of mathematics that any
  well-educated adult should have.
• Specialized to the work of teaching and that
  only teachers need to know.
• Source Ball, D.L. Bass, H. (2005). Who knows
  mathematics well enough to teach third grade?
  American Educator.

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Some interesting dilemmas
Mathematical Knowledge for Teaching (MKT)

• Why do we move the decimal point when we
  multiply decimals by ten?
• Is zero even or odd?
• For fractions, why is 0/12 0 and 12/0
  undefined?
• How is 7 x 0 different from 0 x 7?
• 35 x 25 ? (30 x 20) (5 x 5) Why?
• Is a rectangle a square or is a square a
  rectangle? Why?

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Capturing Growth in Teacher Mathematical Knowledge
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Setting

• Content Strand Algebraic Reasoning and
  Relationships
• Pretest September 2005
• School Year Monthly sessions (20 hours)
• Posttest June 2006
• 120 Classroom teachers Kindergarten - Eighth
  Grade

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Algebraic Relationships
Expressions, Equations, and Inequalities
Generalized Properties
Sub-skill Areas
a x b b x a
Patterns, Relations, and Functions
??? 25? 37
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Items

• Measure mathematics that teachers use in
  teaching, not just what they teach.
• Orient the items around problems or tasks that
  all teachers might face in teaching math.
• MMP performance assessments to give insight into
  depth of teacher knowledge developed around
  monthly seminars.

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Teacher Growth in Mathematical Knowledge for
Teaching (MKT)
Gain 0.296 t 5.584 p 0.000
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Complete the following

• A) Draw a sketch of a rectangle to represent the
  problem 46 x 37. Partition and label the
  rectangle to show the four partial products.
• B) Make connections from your partial product
  strategy (in part A) to the traditional
  multiplication algorithm, explaining how they are
  related.
• C) Make connections from your partial products
  strategy (Part A) to the problem (4x 6) (3x
  6), explaining how they are related.

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Reflect and Discuss

• What is the pure mathematical knowledge you
  employed while completing this task?
• What mathematical knowledge embedded in this task
  might be accessed during the teaching of this
  concept?
• Is this knowledge the same?

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Performance Assessment

• Gain additional insights into our teachers
  abilities to
• Make solid connections between the area model of
  multiplication and the distributive property.
• Understand and explain connections between the
  standard algorithm and use of the distributive
  property for multiplication.
• Generalize use of the distributive property.

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Examining Teacher Work

• As you reflect on teacher work samples consider
  the following
• Is the mathematics correct? Are mathematical
  symbols used with care?
• Are the connections between representations
  clear?
• Are explanations mathematically correct and
  understandable?

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Performance Activity Results

• 16 (9/56) proficient, good explanations and
  connections.
• 50 (28/56) getting there, good procedural
  skills, limited explanations.
• 34 (19/56) did not accurately or completely
  solve the tasks.

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Next steps
Next Steps . . .

• Do teachers scores predict that they teach with
  mathematical skill, or that their students learn
  more, or better?
• How might we connect teachers scores to student
  achievement data?
• More open-ended items to show reasoning

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Knowing mathematics for teaching includes
knowing and being able to do the mathematics that
we would want any competent adult to know. But
knowing mathematics for teaching also requires
more, and this more is not merely skill in
teaching the material.

• Ball, D.L. (2003). What mathematical
  knowledge is needed for teaching mathematics?
  Secretarys Summit on Mathematics, U.S.
  Department of Education, February 6, 2003
  Washington, D.C. Available at http//www.ed.gov/in
  its/mathscience.

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Mathematical knowledge for teaching must be
serviceable for the mathematical work that
teaching entails, for offering clear
explanations, to posing good problems to
students, to mapping across alternative models,
to examining instructional materials with a keen
and critical mathematical eye, to modifying or
correcting inaccurate or incorrect expositions.

• Ball, D.L. (2003). What mathematical knowledge
  is needed for teaching mathematics? prepared for
  the Secretarys Summit on Mathematics, U.S.
  Department of Education, February 6, 2003
  Washington, D.C. Available at http//www.ed.gov/in
  its/mathscience. (p. 8)

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Knowing Mathematics for Teaching

• Demands depth and detail that goes well beyond
  what is needed to carry out the algorithm
• Use instructional materials wisely
• Assess student progress
• Make sound judgment about presentation, emphasis,
  and sequencing often fluently and with little
  time
• Size up a typical wrong answer
• Offer clear mathematical explanations
• Use mathematical symbols with care
• Possess a specialized fluency with math language
• Pose good problems and tasks
• Introduce representations that highlight
  mathematical meaning of selected tasks