Identifying Mathematical Knowledge for Teaching at the
Secondary Level (6-12)
from the Perspective of Practice
Joint NSF-CLT Conference on
Curriculum, Teaching & Mathematical Knowledge
University of Maryland
November 18, 2006
Mid Atlantic Center for Mathematics Teaching and Learning
Center for Proficiency in Teaching Mathematics
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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
investigating:
• mathematical knowledge
• ways of thinking about mathematics
that proficient secondary mathematics
teachers can use in their teaching.
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Two Assumptions
Every day teachers need to integrate their
knowledge in order to respond to students
whose real questions about mathematics do
not fall into neat compartments.
The structure and content of mathematics
requirements for a mathematics major falls
short of what is needed for secondary
teachers.
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Our work is centered on finding a balance between the
complexity and reality of practice and the goal of
creating a new vision.
We are trying to address mathematical knowledge for
teaching by grounding it in 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.
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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
• 3-6 Mathematical Foci - created from a
mathematical perspective
• Commentary - introduced and connected foci
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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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Mathematical Foci
The mathematical knowledge that teachers
could productively use at critical
mathematical junctures in their teaching.
Foci describe the mathematical knowledge that
could inform a teacher’s actions, but do not
suggest pedagogical actions.
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Some issues in creating and
identifying mathematical foci
• Straying into pedagogy
• Determining appropriate level of
mathematical detail
• Determining the appropriate level of
mathematics
• Determining what makes two foci different
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Commentary
Commentaries were added as we worked
on creating Foci for Prompts.
A commentary offers a reason for each
carefully selected focus and points out
the importance of the mathematics that
was addressed across the set of foci.
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Where are we in our work?
Collecting prompts from our experiences
in schools and teacher preparation
Creating a variety of foci that illustrate
useful mathematical knowledge
Working on characterizing and classifying
domains of knowledge that are salient in
the foci
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We need your help
We think your experiences with developing,
implementing, analyzing, studying, or assessing
curricula and the use of curricula prepare you
to:
• See events that suggest rich prompts
• Develop insights related to foci
• Provide guidance in designing our framework
• Provide helpful feedback to our process
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Sample Prompt
Three prospective teachers have 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 students first encounter what they
call the three basic functions: sine, cosine, and tangent.
They indicated in their plan they would next have students
work with the "inverse functions" apparently meaning the
secant, cosecant, and cotangent functions.
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Sample Mathematical Focus
• What is an inverse?
• The inverse is based upon set and
operation
– In this case trigonometric functions are the
set and composition of functions is the
operation.
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Brainstorming
What other foci can be developed to
explore the mathematics that underlie
this prompt? In other words, what other
key mathematical ideas would you
identify that directly relate to this
prompt?
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Other Foci We Developed
• In a Cartesian coordinate plane, a graph
of an inverse of a function is the graph
of the function reflected over the line, y
=x.
• If a function is the inverse of another
function, then the original input should
be the output of the composition.
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Commentary
• Introducing and relating the various
mathematical foci
• Providing mathematical extensions
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Analyzing the Situations
Identifying the mathematics:
Within a given situation
Across the set of situations
The goal of the analysis is to develop a
framework, based in situations arising
from practice, that characterizes
mathematical knowledge for teaching at
the secondary level
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How might this analysis proceed?
How do we go from the situations to a
framework for MKT at the secondary
level?
What might such a framework look like?
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Three Potential Lenses for
Developing a Framework
Mathematical objects, representations,
properties, and operations
Big Ideas
Mathematical activities in which teachers
engage
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Mathematical Objects, etc.
Mathematical objects
Properties of those objects
Relationships between objects
Representations of those objects
Relationships between representations
Operations on objects
Properties of those operations
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Big Ideas in Secondary Mathematics
Equivalence
Variable
Linearity
Unit of measure
Randomness
…
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Mathematical Activities in Which
Teachers Engage
Defining a mathematical object
Giving a concrete example of an
abstraction
Introducing an analogy
Explaining or justifying a procedure
Reasoning using cases
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Processes
Defining
Informal (characteristics)
Formal (minimal set)
Descriptive
Constructive
Generalizing
Recogning invariants in a
number of instances
Subclass to larger class
Symbolic working
Justifying/Proving
Non-proof justifying
(validating/refuting)
Proving
(validating/refuting)
Creating symbols
Interpreting symbols
Manipulating symbols
Connect symbols to other
representations
Reasoning from symbols
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Categories from the Situations
Mathematical Activity
Producing general case
Reasoning from def’n
Using extreme cases
Reasoning from graph
Manipulating symbols
Proof by region subdivision
Interpret relationships
transformationally
Content
Rate of change
Gaussian elimination
Linearity/non-linearity
Distributions
Mean as balance pt.
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Some issues in developing a
framework
1. Grain size
2. Direction
Given a situation, what processes are there?
Given certain processes, are they in situations?
3. General framework vs. Specific framework
4. Straying into pedagogy
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Feedback on Lenses
Preferences among the three lenses
Mathematical objects
Big ideas
Teachers’ mathematical activities
How might this differ from the perspective of a
mathematician, curriculum developer, teacher
educator, researcher, …?
Alternative lenses
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Generating Additional Prompts
NSF-funded curricula
Different mathematics
Different pedagogy
Who might best provide such prompts?
Teachers?
PD providers?
Curriculum writers?
Researchers whose data include interesting
classroom instances?
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What do we hope to get from you?
New prompts
Strategies for getting additional prompts
Who might be productive contacts?
Other strategies?
How might one use these prompts?
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