Seeing the Quality of Mathematical Knowledge in Instruction

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Seeing the Quality of Mathematical Knowledge in
Instruction
Learning Mathematics for Teaching
ProjectUniversity of Michigan
Deborah L. Ball, Hyman Bass, Merrie Blunk, Katie
Brach, Charalambos Charalambous, Carolyn Dean,
Seán Delaney, Imani Masters Goffney, Heather
Hill, Jennifer Lewis, Geoffrey Phelps, Laurie
Sleep, Mark H. Thames, and Deborah Zopf

• TNE Observation Protocol Meeting
• Academy for Educational Development
• University of Wisconsin-Milwaukee October
  17-19, 2007

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Mathematical Knowledge for Teaching

• Essential for high-quality mathematics teaching
• Learned in a variety of settings school,
  university, teaching
• Includes common content knowledge
• Comprises other domains of mathematics content
  particular to the work of teaching

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An example of MKT

• Common content knowledge
• Can you simplify these radicals?
• v48
• v63
• v99
• v150

• Mathematical knowledge for teaching
• Mr. Squire is designing a lesson on simplifying
  radical expressions. He wants to pick one example
  from the previous days homework problems to
  review at the beginning of todays class. His
  goal is to select a problem that will lead to a
  good discussion about different solution
  strategies for simplifying radicals.
• Which of the following is best for setting up a
  good discussion about strategies for simplifying
  radical expressions?
• v48 v63 v99 v150

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Why does MKT matter?

• MKT is associated with childrens achievement
  therefore we want to know
• What are the domains of MKT?
• How do teachers acquire MKT?
• What is the relationship between instructional
  practices and MKT?
• What professional development experiences
  contribute to MKT?

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Measuring MKT

• Paper-and-pencil multiple-choice items
• Develops the construct of MKT
• Provides data about professional development

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Validating MKT measures

• Does achievement on paper-and-pencil measures
  matter in instruction? We videotaped lessons of a
  subset of teachers to validate.
• 10 teachers
• 9 videotaped lessons each, over one year
• Interviews
• If the MKT measures are valid, then teachers
  performance in practice would correspond to their
  achievement on the paper-and-pencil test.

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Coding Videotapes of Mathematics Teaching

• Videocodes cover these domains
• Mathematical content and instructional format
• Teachers use of mathematical language,
  representations, explanations
• Teachers mathematical interaction with student
  productions
• Teachers use of mathematics to teach equitably
• Videocodes are neutral regarding teaching style
  (reform-oriented, didactic, etc.)

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Sample Representations
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An excerpt of code

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A teacher models

• Red pies represent negative numbers
• Green pies represent positive numbers

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Coding Glossary

• To code, first decide whether the mathematical
  element is present (P) or not present (NP). If
  present, then
• Mark appropriate (A) if the teachers use of the
  element was, for the most part, mathematically
  accurate and appropriateit did not distort the
  mathematics.
• Mark inappropriate (I) if the teachers use of
  the element distorted the mathematics or was
  inappropriate for the grade level.
• If not present, then
• Mark appropriate (A) if absence of the element
  seems appropriate.
• Mark inappropriate (I) if absence of the element
  seems problematici.e., the element should have
  happened.

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Use of videocodes

• Our findings
• inter-rater reliability of .7 to .85
• teachers clinical practice scores and test
  scores correlation of .75 or higher
• Others potential use of codes would require
• Shared understandings of the codes
• Users with MKT

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Continuing challenges

• How do we appraise teaching for equity?
• Some essential qualities difficult to nail down
  (e.g., scaffolding)
• High qualifications for coders
• Gap between quality teaching as envisioned by the
  field and quality teaching that produces results
  in schools