A Framework for
Mathematical Knowledge for
Teaching at the Secondary Level
Conference on Knowledge of Mathematics for Teaching
at the Secondary Level
Pat Wilson pswilson@uga.edu
Kathy Heid mkh2@psu.edu
Tucson, AZ --- March 2011
Mid Atlantic Center for Mathematics Teaching and Learning
Center for Proficiency in Teaching Mathematics
Situations Project
• Classroom-based situations
• A framework for mathematical
knowledge at the secondary level
Mathematical Knowledge for Teaching
at the Secondary Level
Mathematics
Classroom
Mathematics
Knowledge for
Teaching
Created
Situations
Mathematical Knowledge for Teaching
Mathematics at the Secondary Level
What is MKTS?
• MKTS is specialized mathematical knowledge
• Knowing mathematics
• Being able to and having a tendency to use the
mathematics in appropriate circumstances
• MKTS is not pedagogical knowledge.
How is MKTS different from MKT elementary?
Examples of Situations
• Prompt
• Mathematical Foci
Circumscribing Polygons
Prompt
In a geometry
class, after a
discussion about
circumscribing
circles about
triangles, a student
asked, “Can you
circumscribe a
circle about any
polygon?”
A few mathematical foci
• Every triangle is cyclic. This fact
is core to establishing a condition
for other polygons to be cyclic.
• A convex quadrilateral in a plane
is cyclic if and only if its opposite
angles are supplementary.
• Every planar regular polygon is
cyclic. However, not every cyclic
polygon is regular.
Summing Natural Numbers
Prompt
Students found a need
to sum the numbers 1
to n. One student
offered a formula, but
was not sure if he
remembered it
correctly.
n(n +1) / 2
Students wondered if
this would always be a
natural number.
A few mathematical foci
• When n is a natural number,
n (n+1) / 2 is also a natural number.
• Strategic choices for pair-wise
grouping of numbers is critical to the
development of the general formula.
• The first n natural numbers form an
arithmetic sequence.
• Geometric arrays can lead to the
development of the formula.
Framework
Mathematical Knowledge for Teaching
• Mathematical Proficiency
• Mathematical Activity
• Mathematical Work of Teaching
Mathematical Proficiency
•
•
•
•
•
•
Conceptual Understanding
Procedural Fluency
Strategic Competence
Adaptive Reasoning
Productive Disposition
Historical and Cultural Knowledge
At secondary level
Procedural Fluency:
Knowing when and how to
apply a procedure in typical
settings
Algorithms are more complicated,
settings are somewhat varied.
Productive Disposition:
Tendency to notice and
apply mathematics in the
world around us
Mathematical applications are
verging on modeling rather than
solely recognizing or associating.
At secondary level
Adaptive Reasoning:
Adjust to changes in
assumptions and
conventions
More attention is paid to
assumptions and their
consequences. There is more
overt consideration of conventions.
Strategic competence:
Generating, evaluating, and
implementing problemsolving strategies
Strategies are evaluated for their
mathematical fidelity, rather than
solely for the viability of the
answers they generate.
Conceptual Understanding: Knowing “why”
Every triangle is cyclic.
Every regular polygon is
cyclic.
Historical and Cultural Knowledge
Awareness of how people from various times
and culture conceptualize and express
mathematical ideas
E.g., The story of Gauss and the sum of the
natural numbers from 1 to 100.
1 + 2 + 3 + … + 98 + 99 + 100
100 + 99 + 98 + … + 3 + 2 + 1
Sum = (100)(101)/2
Historical and Cultural Knowledge
E.g., The Japanese theorem for cyclic
polygons states that no matter how we
triangulate a cyclic polygon, the sum of the
inradii is constant.
Mathematical Activity
• Mathematical Noticing
• Mathematical Reasoning
• Mathematical Creating
• Integrating Strands of Mathematical
Activity
Mathematical Noticing
Structure of mathematical systems
Symbolic form
Form of an argument
Connections within and outside of
mathematics
Summing Natural Numbers
Prompt
Students found a need
to sum the numbers 1
to n. One student
offered a formula, but
was not sure if he
remembered it
correctly.
n(n +1) / 2
Students wondered if
this would always be a
natural number.
A few mathematical foci
• When n is a natural number,
n (n+1) / 2 is also a natural number.
• Strategic choices for pair-wise
grouping of numbers is critical to the
development of the general formula.
• The first n natural numbers form an
arithmetic sequence.
• Geometric arrays can lead to the
development of the formula.
Mathematical Noticing:
Structure of mathematical systems
The set of natural numbers from 1 to n is an
ordered sequence. Symmetry and the
ordered nature of the sequence allow for
rearrangements that facilitate finding a
sum. Under any rearrangement, the
cardinality and the sum of the elements
remains the same.
Mathematical Noticing:
Structure of mathematical systems
.


S
S
1
n
1  n 
2S  n  1
2S  n n  1
n n  1
S
SS 
2


2
n  1


3
n  2 
 ... 
 ... 
n
1

2  n  1
n  1

3  n  2 
n  1
 ... 
n  1
n  1


 ... 
Mathematical Noticing:
Structure of mathematical systems
.
n(n  1)
2 sum  n(n  1)  sum 
2
Mathematical Noticing:
Structure of mathematical systems
.
n
1
n(n  1)
sum 
 n
2 2
2
2
Mathematical Noticing:
Structure of mathematical systems
.
n 1
n(n  1)
sum =
gn 
2
2
Mathematical Reasoning
• Justifying/Proving
• Reasoning when conjecturing and
generalizing
• Constraining and extending
Mathematical Reasoning
Constraining and extending
We have looked at the sum of the first n
natural numbers.
What is the sum of the squares of the first
n natural numbers?
What is the sum of the kth powers of the
first n natural numbers?
Circumscribing Polygons
Prompt
In a geometry
class, after a
discussion about
circumscribing
circles about
triangles, a student
asked, “Can you
circumscribe a
circle about any
polygon?”
A few mathematical foci
• Every triangle is cyclic. This fact
is core to establishing a condition
for other polygons to be cyclic.
• A convex quadrilateral in a plane
is cyclic if and only if its opposite
angles are supplementary.
• Every planar regular polygon is
cyclic. However, not every cyclic
polygon is regular.
Mathematical Reasoning
Constraining and extending
Regular polygons are cyclic.
What are the conditions under
which non-regular polygons are
cyclic?
Mathematical Creating
• Representing
• Defining
• Modifying/transforming/manipul
ating
Mathematical Creating :
Representing
.
n 1
n(n  1)
sum =
gn 
2
2
Mathematical Creating :
Representing
.
n
1
n(n  1)
sum 
 n
2 2
2
2
Mathematical Work of
Teaching
Teachers of secondary mathematics need
the mathematical knowledge to be able
to:
• know and do mathematics themselves
• facilitate their students’ development
of mathematical knowledge
Mathematical Work of Teaching
• Analyze mathematical ideas
• Access and understand the
mathematical thinking of students
• Know and use the curriculum
• Assess the mathematical proficiency of
learners
• Reflect on the mathematics of practice
Analyze mathematical ideas
Mathematics is dense, often succinct,
elegant, but teachers need to see
behind the elegant product.
Teachers need mathematical
knowledge that helps them:
• Probe; pull apart; unpack; dissect
• Recognize conventions vs. core ideas
• Understand the structure of mathematics
Analyze mathematical ideas
Probe; pull apart; unpack; dissect
What is an inverse?
(operation & domain required)
Types of inverses (additive, multiplicative, function)
Conventions:
notation (-1)
operation for inverse function
Strategies and rationale for finding inverse
What is a function? (Need to restrict the domain)
Access and understand the
mathematical thinking of students
Students often express ideas in vague
ways with imprecise terms.
Teachers need mathematical
knowledge that helps them:
• Find key mathematical ideas within the
students’ thinking;
• See important mathematics within
misconceptions; and
• Understand the role of mathematical rigor.
Access and understand the
mathematical thinking of students
Find key mathematical ideas within the
students’ thinking
Students found a need
to sum the numbers 1
to n.
n(n +1) / 2
Students wondered if
this would always be a
natural number.
Odd number/2 ≠ integer
Why is n(n+1) always even?
•If n is even, n+1 is odd
•If n is odd, n +1 is even
•Even x odd = even
Why are even numbers 2n? Odd
numbers 2n +1?
What about (n2 + n)/2?
Know and use the curriculum
Curricula help to organize mathematical
ideas.
Teachers need mathematical knowledge
that helps them:
• Understand the mathematical scope and
sequence in the curricula so it can be used
or modified (know the flow);
• Know how concepts are related or build on
each other; and
• Provide rationales for mathematical ideas.
Assess the mathematical knowledge
of learners
Assessing mathematical knowledge is
uncovering understanding as well as
misconception.
Teachers need mathematical
knowledge that helps them:
• Assess more than procedural knowledge;
• Identify the key concepts within problems or
exercises; and
• Ask formative and summative questions.
Assess the mathematical knowledge
of learners
Assess more than procedural knowledge
There are 49 even numbers from 1 to 99.
5
49
The mean of 48 of these numbers is
.
12
Which even number was not included in the
calculating of this mean?
From Taiwan’s May 2009 Basic Competency Test, as reported by Lo &
Tsai in MTMS, March 2011.
Reflect on the mathematics of practice
Revisiting mathematical ideas after a lesson or unit
can provide new mathematical insights.
Teachers need mathematical knowledge that helps
them:
• Recognize assignments that hide, distort or illuminate
the mathematics;
• Understand cultural factors that enhance or detract
from the mathematics. (e.g., vocabulary, contexts or
problems); and
• Recognize implicit mathematical ideas that need
explanation? (e.g.,domain, convention, orientation of
drawing)
Reflect on the mathematics of practice
Recognize implicit mathematical ideas that need
explanation? (e.g.,domain, convention, orientation)
Can a linear equation
have neither an x-intercept
nor a y-intercept?
A teacher asked her
students to sketch the
graph of
f (x)  x
y
A student
responded, “That’s
impossible! You can’t take
the square root of a
negative number!”
x
Framework
Mathematical Knowledge for Teaching
• Mathematical Proficiency
• Mathematical Activity
• Mathematical Work of Teaching
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.
Question 1
What is the role of mathematical actions
in mathematical knowledge for teaching
at the secondary level and how can this
be built into their formal mathematical
background?
Question 2
How could we develop and measure
mathematical knowledge that enables
the work of teaching?
Questions
What is the role of mathematical actions in
mathematical knowledge for teaching at the
secondary level and how can this be built into
their formal mathematical background?
How could we develop and measure
mathematical knowledge that enables the
work of teaching?