This document discusses the differences between mathematical knowledge for teaching and other specialized knowledge. It begins by asking critical questions about what mathematics is important for teaching and how it differs from mathematics used in other fields. It then defines mathematical knowledge for teaching as having two components: common mathematical knowledge that all should know, and specialized knowledge for teaching mathematics, such as representing ideas, providing explanations, and appraising unusual student solutions. The document provides examples of these different types of knowledge and knowledge used in pedagogical situations versus pure content knowledge. It concludes by introducing a diagram that further categorizes the specialized knowledge needed for teaching mathematics.

1. Selecting, Designing & Using
Mathematical Tasks in the K-5 Classroom

2. Critical Question…
o What mathematics matters for
What
teaching?
Distinguishes o Is it different from mathematics
Mathematical knowledge as known and used by
mathematicians, engineers, biologists,
Knowledge
candy sellers? If so, how can we
From The
characterize what makes it different?
Specialized o What of that mathematics is “content,”
Knowledge
and what is more properly called by
other names such as “practices,”
Needed For
“sensibilities,” or “orientations”?
Teaching
Mathematics? o What is the difference between “pure”
content knowledge and knowledge used
in pedagogical situations?

3. Mathematical Knowledge for
Teaching
Common mathematical knowledge
 Number halfway between 1.1 and 1.2
Specialized knowledge for teaching mathematics
 Representing mathematical ideas and operations
 Providing explanations for mathematical ideas and
procedures
 Appraising unusual student methods, claims, or
solutions

4. Examining Mathematical
Knowledge
Solve:
25 x 35 = ?

5. Appraising Unusual Student Solutions
Student A
35
x 25
125
+750 _
875
Student B
35
x 25
175
+700 _
875
Student C
35
x 25
25
150
100
+600
875
Which of these students is using a method that could
be used to multiply any two whole numbers?

6. Reflect and Discuss
o What did you do to solve the original
problem of 25 x 35?
o What did you do to solve the task
concerning the varied student solutions?
o What was different in the way you thought
mathematically about each task, and what
you needed to know?

7. Subject Matter
Knowledge
Common
Content
Knowledg
e (CCK)
Knowledge
at the
mathematica
l horizon
Pedagogical Content Knowledge
Specialize
d Content
Knowledg
e (SCK)
Knowledge
of Content
and Students
(KCS)
Knowledge of
Content and
Teaching (KCT)
Knowledg
e of
Curriculu
m
Diagram created by Deborah Ball & colleagues, University of Michigan