Assessing the
Mathematics Knowledge
of Teachers
UNIVERSITY OF LOUISVILLE
William S. Bush
University of Louisville
North Carolina Association of Mathematics Teacher Educators
Chapel Hill, North Carolina
April 21, 2007
Purpose of Session
To analyze the types of mathematics
knowledge necessary for teaching
 To review assessment items that assess
these types of knowledge
 To discuss strategies for helping teachers
build these types of knowledge

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University of Louisville
Think, Pair, Share
What kinds of mathematics
knowledge do teachers of
mathematics need to do their
work effectively?
Discuss in groups of three or
four.
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University of Louisville
What Others Say

Lee Shulman (Educational Researcher,
February 1986)
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Content Knowledge
– Subject matter content knowledge
– Pedagogical content knowledge
– Curricular knowledge
4
University of Louisville
Subject Matter Content Knowledge
The amount and organization of knowledge per
se in the mind of the teacher. . . Teachers must
not only be capable of defining the accepted
truths in a domain. They must also be able to
explain what a particular proposition is deemed
warranted, why it is worth knowing, and how it
relates to other propositions, both within the
discipline and without, both in theory and in
practice.
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University of Louisville
Pedagogical Content Knowledge
Includes, for the most regularly taught topics in
one’s subject area, the most useful forms of
representation of those ideas, the most powerful
analogies, illustrations, examples, explanations,
and demonstrations . . . the ways of
representing and formulating the subject that
makes it comprehensible to others . . . what
makes the learning of topics easy or difficult, the
conceptions and preconceptions students of
different ages and backgrounds bring with them
to the learning of those most frequently taught
topics and lessons.
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University of Louisville
Curricular Knowledge
The full range of programs designed for
the teaching of particular subjects and
topics at a given level, the variety of
instructional materials available in relation
to these programs, and the set of
characteristics that serve as both
indications and contraindications for the
use of particular curriculum or program
materials in particular circumstances.
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University of Louisville
What Others Say (Continued)
Ball and Others—Univ. of Michigan
Mathematics knowledge for teaching requires knowing in
detail the topics and ideas that are fundamental to the
school curriculum, and beyond. This detail involves a kind of
unpacking that is often difficult to produce. . . It requires
knowing how mathematical topics are connected, and how
particular ideas anticipate later ones . . . [It] also involves
using tools and skills for reasoning about mathematical
ideas, representations, and solutions, as well as knowing
what constitutes adequate proof. . . [It] also requires
fluency and care with mathematical language and notation.
. . [It] also depends on a broad familiarity with applications
of mathematics.
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University of Louisville
Mathematics Knowledge about
Instruction
Design mathematically accurate explanations
 Use mathematically appropriate and comprehensible
definitions
 Represent ideas using a variety of models
 Respond productively to mathematical questions
 Explain mathematical terms and concepts
 Use representations accurately in the classroom
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Use appropriate technology
Help students abstract/generalize mathematical ideas
Assist students in building mathematical structures
University of Louisville
Mathematics Knowledge About
Student Learning
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Pose good mathematical questions and problems that
are productive for student learning
Interpret and make mathematical and pedagogical
judgments about students ideas and work
Assess students’ mathematics learning
Interpret student statements/solutions and provide
remedies
Provide students examples of mathematical concepts,
algorithms, or proofs
Know when student reasoning is valid and assist them
with errors in reasoning
University of Louisville
Mathematics Knowledge About
Curricula
Make judgments about the mathematical
quality of instructional materials and
modifying as necessary
 Judge and correct textbook treatments of
particular topics
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Connect mathematical ideas within and
across other mathematical topics
University of Louisville
What the Research Says
Hill, H. C., Rowan, B., & Ball, D. L. (2005, Summer). Effects of teachers’
mathematical knowledge for teaching on student achievement.
American Educational Research Journal, 42(2), 371-406.
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Teacher and student data collected in 115 elementary
schools in 2001-02 through 2003-04.
Mathematics achievement data and parent interview data
collected on 1st and 3rd grade students.
Teacher questionnaires, including mathematics knowledge
for teaching, collected on teachers
Teachers’ mathematical knowledge for teaching positively
predicted student gains in mathematics.
University of Louisville
Think, Pair, Share
Review the following sets of
assessment items.
Which type of mathematics
knowledge for teaching is
being assessed?
 What underlying mathematics
knowledge is needed to
respond to the item?

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University of Louisville
Questions for Discussion
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Where can preservice and practicing
teachers learn these types of
mathematical knowledge?

What strategies can we use to help
teachers gain this knowledge?
University of Louisville
Closing Comments
Mathematics knowledge for teaching is based on
an underlying knowledge of mathematics.
 Tasks that focus on or assess mathematics
knowledge for teachers:

 can uncover teachers’ misconceptions about
mathematics.
 are inherently more relevant and motivating for
teachers.
 Can and should be used in methods and mathematics
courses for teachers.
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University of Louisville
For More Information about
Teacher Assessments

University of Louisville DTAMS
William S. Bush [email protected]
Web http://louisville.edu/edu/crmstd/

University of Michigan Teacher Assessments
Heather Hill [email protected]
Web http://www.sii.soe.umich.edu/
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University of Louisville