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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 CLT’s to
Identify and Characterize the
Mathematical Knowledge for Teaching
at the Secondary Level
The Center for
Proficiency in Teaching
Mathematics
Mid Atlantic Center for
Mathematics Teaching
and Learning
Focus: The preparation of
those who teach
mathematics to teachers
Focus: The preparation of
mathematics 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:
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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 student’s 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 teacher’s 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
1
f the reciprocal of f.
If people think about the “inverse of sine” as sin-1, they
1
might use sin(x) 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 2 and 3 and got 5 .
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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
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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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