Developing a Framework for Mathematical Knowledge for Teaching at the Secondary Level

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Developing a Framework for Mathematical Knowledge
for Teaching at the Secondary Level

• The Association of Mathematics Teacher Educators
  (AMTE)? Eleventh Annual Conference
• Irvine, CA
• January 27, 2007
• Mid Atlantic Center for Mathematics Teaching and
  Learning
• Center for Proficiency in Teaching Mathematics

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Situations Research Group

• Glen Blume
• Brad Findell
• M. Kathleen Heid
• Jeremy Kilpatrick
• Jim Wilson
• Pat Wilson
• Rose Mary Zbiek

• Bob Allen
• Sarah Donaldson
• Ryan Fox
• Heather Godine
• Shiv Karunakaran
• Evan McClintock
• Eileen Murray
• Pawel Nazarewicz
• Erik Tillema

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Collaboration of Two CLTs to Identify and
Characterize the Mathematical Knowledge for
Teaching at the Secondary Level

• Mid Atlantic Center for Mathematics Teaching and
  Learning
• Focus The preparation of mathematics teachers

• The Center for Proficiency in Teaching
  Mathematics
• Focus The preparation of those who teach
  mathematics to teachers

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Our collaborative work is addressing

• the mathematical knowledge
• the ways of thinking about mathematics
• that proficient secondary mathematics teachers
  understand.

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The Problems

• How to get teachers acquainted with secondary
  mathematics in ways that are useful in their
  teaching.
• How to help secondary mathematics teachers
  connect collegiate mathematics with the
  mathematics of practice

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Grounding our Work in Practice

• We are drawing from events that have been
  witnessed in practice.
• Practice has many faces, including but not
  limited to classroom work with students.
• Situations come from and inform practice, making
  this mathematics for practice.

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Working toward a Framework

• We would like to build a framework of
  Mathematical Knowledge for Teaching at the
  Secondary level.
• The framework could be used to guide
• Research
• Curriculum in mathematics courses for teachers
• Curriculum in mathematics education courses
• Design of field experiences
• Assessment

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Situations

• We are in the process of writing a set of
  practice-based situations that will help us to
  identify mathematical knowledge for teaching at
  the secondary level.
• Each Situation consists of
• Prompt - generated from practice
• Commentaries - providing rationale and extension
• Mathematical Foci - created from a mathematical
  perspective

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Prompts

• A prompt describes an opportunity for teaching
  mathematics
• E.g., a students question, an error, an
  extension of an idea, the intersection of two
  ideas, or an ambiguous idea.
• A teacher who is proficient can recognize this
  opportunity and build upon it.

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Commentaries

• The first commentary offers a rationale for each
  focus and emphasizes the importance of the
  mathematics that is addressed in the foci.
• The second commentary offers mathematical
  extensions and deals with connections across foci
  and with other topics.

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Mathematical Foci

• The mathematical knowledge that teachers could
  productively use at critical mathematical
  junctures in their teaching.
• Foci describe the mathematical knowledge that
  might inform a teachers actions, but they do not
  describe or suggest specific pedagogical actions.

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Example of a Situation Inverse Trig Functions

• Prompt
• Three prospective teachers planned a unit of
  trigonometry as part of their work in a methods
  course on the teaching and learning of secondary
  mathematics. They developed a plan in which high
  school students first encounter what they called
  the three basic trig functions sine, cosine,
  and tangent. The prospective teachers indicated
  in their plan that students next would work with
  the inverse functions, identified as secant,
  cosecant, and cotangent.

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Example of a Situation Inverse Trig Functions

• Commentary
• The problem seems centered on knowing about the
  entity of inverse. Connections can be made to
  the notion of inverse from abstract algebra.
  When we think about inverses, we need to think
  about the operation and the elements on which the
  operation is defined. The selection of foci is
  made to emphasize the difference between an
  inverse for the operations of multiplication and
  composition of functions. The foci contrast how
  the multiplicative inverse invalidates the
  properties for an inverse element for the
  operation of composition. The contrasts will be
  illustrated in a variety of approaches
  graphical, numerical, and verbal.

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Example of a Situation Inverse Trig Functions

• Mathematical Focus 1 What does it mean to be an
  inverse?
• The problem seems centered on knowing about the
  mathematical entity of inverse. An inverse
  requires two elements the operation and the
  elements on which the operation is defined.
  csc(x) is an inverse of sin(x), but not an
  inverse function for sin(x). For any value of x
  such that csc(x) ? 0, the number csc(x) is the
  multiplicative inverse for the number, sin(x)
  multiplication is the operation in this case and
  values of the sin and csc functions are the
  elements on which the operation is defined. Since
  we are looking for an inverse function, the
  operation is composition and functions are the
  elements on which the operation is defined.

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Example of a Situation Inverse Trig Functions

• Mathematical Focus 2 Are these three functions
• really inverses of sine, cosine, and secant?
• Suppose cosecant and sine are inverse functions.
• A reflection of the graph of y  csc(x) in the
  line
• y  x would be the graph of y  sin(x). Figure 1
  shows, on one coordinate system graphs of the
  sine function, the line given by y  x, the
  cosecant function, and the reflection in y  x of
  the cosecant function. Because the reflection and
  the sine function graph do not coincide, sine and
  cosecant are not inverse functions.
• The reflection in the line given by y  x of one
  function and the graph of an inverse function
  coincide because the domain and range of a
  function are the range and domain, respectively,
  of the inverse function.

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Example of a Situation Inverse Trig Functions

• Mathematical Focus 3 For what mathematical
  reason might one think the latter three functions
  are inverses of the former three functions?
• The notation f -1 is often used to show the
  inverse of f in function notation.
• When working with rational numbers, f -1 is used
  to represent
• the reciprocal of f.
• If people think about the inverse of sine as
  sin-1, they
• might use to represent the inverse of
  sine.

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Samples of Prompts for the MAC-CPTM Situations
Project
1. Adding Radicals A mathematics teacher, Mr.
Fernandez, is bothered by his ninth grade algebra
students responses to a recent quiz on radicals,
specifically a question about square roots in
which the students added and and got
.
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2. Exponents In an Algebra II class, the teacher
wrote the following on the board xm . xn x5 .
The students had justt finished reviewing the
rules for exponents. The teacher asked the
students to make a list of values for m and n
that made the statement true. After a few
minutes, one student asked, Can we write them
all down? I keep thinking of more.
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Mathematical Lenses

• Mathematical Objects
• Big Mathematical Ideas
• Mathematical Activities of Teachers

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Mathematical Lens Mathematical Objects

• A mathematical-objects approach
• Centers on mathematical objects, properties of
  those objects, representations of those objects,
  operations on those objects, and relationships
  among objects
• Starts with school curriculum and
• Addresses the larger mathematical structure of
  school mathematics.

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Mathematical Lens Big Mathematical Ideas

• A big-mathematical-ideas approach
• Centers on big ideas or overarching themes in
  secondary school mathematics
• Examples ideas about equivalence, variable,
  linearity, unit of measure, randomness
• Begins with a mix of curriculum content and
  practice and uses each to inform the other and
• Accounts for overarching mathematical ideas that
  cut across curricular boundaries and carry into
  collegiate mathematics while staying connected to
  practice.

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Mathematical Lens Mathematical Activities of
Teachers

• A mathematical-activities approach
• Partitions or structures the range of
  mathematical activities in which teachers engage
• Examples defining a mathematical object, giving
  a concrete example of an abstraction, formulating
  a problem, introducing an analogy, or explaining
  or justifying a procedure.
• May also draw on the mathematical processes that
  cut across areas of school mathematics.

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Looking at the Inverse Trig Function Mathematical
Foci through an Object Lens

• Focus 1 Inverse
• Focus 2 Relationship between graphs of inverse
  functions
• Focus 3 A conventional symbolic representation
  of the inverse of f is f -1. The exponent or
  superscript -1 has several different meanings,
  not all of which are related to inverse in the
  same way.

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Looking at the Inverse Trig Function Mathematical
Foci through an Big Ideas Lens

• Focus 1 Two elements of a set are inverses under
  a given binary operation defined on that set when
  the two elements used with the operation in
  either order yield the identity element of the
  set.
• Focus 2 Equivalent Functions/ Domain and Range
  Two functions are equivalent only if they have
  the same domain and the same range.
• Focus 3 The same mathematical notation can
  represent related but different mathematical
  objects.

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Looking at the Inverse Trig Function Mathematical
Foci through an Activities Lens

• Focus 1 Appealing to definition to refute a
  claim
• Focus 2 Using a different representation to
  explain a relationship
• Focus 3 Explaining a convention

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The Problems

• How to get teachers acquainted with secondary
  mathematics in ways that are useful in their
  teaching.
• How to help secondary mathematics teachers
  connect collegiate math with the math of practice

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Engaging with the Mathematical Lenses

• Big Ideas Lens
• U GA course on Secondary Mathematics from an
  Advanced Standpoint
• PSU course based on ideas of function highlights
  notion of equivalence, system, variable.
• Mathematical Activities Lens
• PSU course for teachers based on Situations
• U GA course with major component on Situations

INSERT U GA Examples
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The Mathematics of Your Courses

• Where do you see objects, big ideas, and
  mathematical activities in your courses?
• Which mathematical lenses (these or others) or
  combination of mathematical lenses influence your
  courses?

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What insights can you now offer regarding the
problems we posed?

• How to get teachers acquainted with secondary
  mathematics in ways that are useful in their
  teaching.
• How to help secondary mathematics teachers
  connect collegiate math with the math of practice

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This presentation is based upon work supported by
the Center for Proficiency in Teaching
Mathematics and the National Science Foundation
under Grant No. 0119790 and the Mid-Atlantic
Center for Mathematics Teaching and Learning
under Grant Nos. 0083429 and 0426253 . Any
opinions, findings, and conclusions or
recommendations expressed in this presentation
are those of the presenter(s) and do not
necessarily reflect the views of the National
Science Foundation.
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