The document discusses a presentation about the nature of teacher knowledge, specifically specialized content knowledge (SCK) which is knowledge used for teaching but not directly taught to students. It describes research assessing SCK among different groups like prospective teachers and more experienced teachers, finding that SCK increases with experience. The presentation also discusses goals for further research on developing teachers' SCK, especially regarding integers and students' reasoning about them.

1. The Nature of Knowledge for
Teaching and Implications for
Research and Practice
Yopp Distinguished Speaker Series
University of North Carolina, Greensboro
September 8, 2014
Dr. Randy Philipp
San Diego State University
RPhilipp@mail.sdsu.edu

2. Thank You!
UNCG Mathematics Education Group in the
School of Education
The Graduate Students
James D. and Johanna F. Yopp

3. Presentation Plan
• Knowledge for Teaching and Knowledge
for Teaching a Particular Subject
• Assessing Specialized Content Knowledge
• Rethinking Specialized Content Knowledge
in a Particular Domain
• Discussion

4. Presentation Plan
• Knowledge for Teaching and Knowledge
for Teaching a Particular Subject
• Assessing Specialized Content Knowledge
• Rethinking Specialized Content Knowledge
in a Particular Domain
• Discussion

5. What Knowledge of Mathematics Do
Teachers Need?
Example, Javier, Grade 5
At the time of this
interview, Javier had
been in the United
States about one year,
and he did not speak
English before coming
to this country.
(Javier, VC #6, 0:00 - 1:10)

6. One Representation of Javier’s Thinking
6 × 12
= (5 × 12) + (1 × 12)
= [(1
2
× 10) × 12] + 12
= [1
2
× (10 × 12)] + 12
= [1
2
× (120)] + 12
= 60 + 12
= 72

7. One Representation of Javier’s Thinking
6 × 12
= (5 × 12) + (1 × 12) (Distributive prop. of x over +)
= [(1
2
× 10) × 12] + 12 (Substitution property)
= [1
2
× (10 × 12)] + 12 (Associative property of x)
= [1
2
× (120)] + 12
= 60 + 12
= 72

8. One Representation of Javier’s Thinking
6 × 12
= (5 × 12) + (1 × 12) (Distributive prop. of x over +)
= [(1
2
× 10) × 12] + 12 (Substitution property)
= [1
2
× (10 × 12)] + 12 (Associative property of x)
= [1
2
× (120)] + 12
= 60 + 12
= 72
Place value

9. Unpacking The Knowledge Demands
What is the nature/category/classification of the
knowledge required to…
…solve 12 x 6 procedurally?
…solve 12 x 6 using number sense?
…understand Javier’s (and other students’) reasoning?
…think to ask “How did you know that 12 x 5 is 60?”
…think of a productive follow-up question to pose to
Javier after he solved this task?
…situate the mathematical issues embedded in Javier’s
thinking in terms of the mathematics that has come
before and how these ideas might unfold in future
mathematics courses?

11. Knowledge
for
Teaching
Ball,
Hill,
&
Bass,
2005;
Hill,
Sleep,
Lewis,
&
Ball,
2007
Common
Content
Knowledge—the
knowledge
teachers
are
responsible
for
developing
in
students
Evaluate
and
understand
the
meaning
of
12
÷
3.
Specialized
Content
Knowledge—knowledge
that
is
used
in
teaching,
but
not
directly
taught
to
students
Write
a
real-­‐life
story
problem
that
could
be
represented
by
the
expression
12
÷
3.
Pedagogical
Content
Knowledge
(Shulman,
1986)—the
ways
of
represenCng
and
formulaCng
the
subject
that
make
it
comprehensible
to
others
including
knowledge
of
how
students
think,
know,
and
learn.
How
might
children
reason
about
this
task?

12. Unpacking The Knowledge Demands
What is the nature/category/
classification of the knowledge
required to…
…solve 12 x 6 procedurally?
…solve 12 x 6 using number sense?
…understand Javier’s (and other
students’) reasoning?
…think to ask “How did you know
that 12 x 5 is 60?”
CCK
CCK, SCK
CCK, SCK, PCK
CCK, SCK, PCK

13. Consider an example of SCK in a field
other than mathematics
Common
Content
Knowledge—the
knowledge
teachers
are
responsible
for
developing
in
students
Specialized
Content
Knowledge—knowledge
that
is
used
in
teaching,
but
not
directly
taught
to
students
Pedagogical
Content
Knowledge
(Shulman,
1986)—the
ways
of
represenCng
and
formulaCng
the
subject
that
make
it
comprehensible
to
others
including
knowledge
of
how
students
think,
know,
and
learn.
Is this knowledge taught? If so, where? If not, why not?
Is this knowledge assessed by researchers? How?

14. Presentation Plan
• Knowledge for Teaching and Knowledge
for Teaching a Particular Subject
• Assessing Specialized Content Knowledge
• Rethinking Specialized Content Knowledge
in a Particular Domain
• Discussion

15. Principal Investigators
Randy Philipp, PI
Vicki Jacobs, co-PI
Faculty Associates
Lisa Lamb, Jessica Pierson
Research Associate
Bonnie Schappelle
Project Coordinators
Candace Cabral
Graduate Students
John (Zig) Siegfried
Others
Chris Macias-Papierniak,
Courtney White
Funded by the National Science Foundation, ESI-0455785

16. Participant Groups (N=129 with 30+ per group)
PSTs, Prospective Teachers
Undergraduates enrolled in a
first mathematics-for-teachers content course
________________________________________________________________________________________________________________________
IPs, Initial Participants
0 years of sustained professional
development
APs, Advancing Participants
2 years of sustained
professional development
ETLs, Emerging Teacher Leaders
At least 4 years of
sustained professional development and some
leadership activities
_______________________________________________
SMSs, Strong Mathematics Students - Graduate or
advanced undergrads taking advanced math courses
K–3 Teachers

17. Ones Task
Andrew Task

18. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew ?
Ones ?

19. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew 1.48 ?
Ones 1.58 ?

20. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew ? 1.48 2.36
Ones ? 1.58 2.49

21. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew 1.67 1.48 2.36 ?
Ones 0.31 1.58 2.49

22. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew 1.67 1.48 2.36 2.55
Ones 0.31 1.58 2.49

23. Two ETL’s solution to Andrew Task
1) Because he made 63 into 65 so that he could solve
the problem. He got 40 and he subtracted the 2 in
which he had added to simplify the problem.
2) 5 = 3 + 2 and if you only have 3 and you're
subtracting 5, you can take away the 3 but you still have
two more to take away, hence the -2.
63
– 23
40
– 2
38
63
- 25
-2
40
38

24. SMS’s solution to Andrew Task
Explain why Andrew’s strategy makes mathematical
sense.
He makes the problem simpler by subtracting 20 from 60
and 5 from 3 and adding the results.
Please solve 432 – 162 = ☐ by applying Andrew’s
reasoning.
432
−162
0
− 30
300
270
63
- 25
-2
40
38

25. The Land of Specialized Content
Knowledge
The$Land$of$Specialized$Content$Knowledge$
The$Land$of$SCK$
A$Mathema2cal$Path$to$the$
Land$of$SCK$
A$Path$Through$Children’s$
Mathema2cal$Thinking$to$the$
Land$of$SCK$

26. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew 1.67 1.48 2.36 2.55
Ones 0.31 1.58 2.49 ?

27. Group Means by Task
(0–4 scale)
PST IP AP/ETL SMS
Andrew 1.67 1.48 2.36 2.55
Ones 0.31 1.58 2.49 0.94

28. Presentation Plan
• Knowledge for Teaching and Knowledge
for Teaching a Particular Subject
• Assessing Specialized Content Knowledge
• Rethinking Specialized Content
Knowledge in a Particular Domain
• Discussion

29. Project Z: Mapping Developmental Trajectories
of Students’ Conceptions of Integers
• Lisa Lamb, Jessica Bishop, & Randolph Philipp, Principal
Investigators
• Ian Whitacre, Faculty Researcher
• Spencer Bagley, Casey Hawthorne, Graduate Students
• Bonnie Schappelle, Mindy Lewis, Candace Cabral, Project
researchers
• Kelly Humphrey, Jenn Cumiskey, Danielle Kessler,
Undergraduate Student Assistants
Funded by the National Science Foundation, DRL-0918780

30. Solve each of the following and think about how you
reasoned. If you have time, solve another way.
1) 3 – 5 = ___
2) -6 – -2 = ___
3) - 2 + ___= 4
4) ___+ -2 = -10

31. So, Why Negative Numbers?
Even secondary-school students who can
successfully operate with negatives have
trouble explaining.
KCC Montage, High School, 2:35

32. So, Why Negative Numbers?
Many middle-school stud
ents do not understand
what they are doing with negatives.
Valentin, Grade 7, 1:25

33. So, Why Negative Numbers?
Many young children hol
d informal knowledge
about negatives on which instruction might be
based.
Rosie, 1st grade, ___ + 5 = 3, 1:23

34. One Last Reason…
• Negative numbers comprise (almost) half of the
reals!

35. Why Study Negative Numbers?
• The literature that exists tends to either point
out student difficulties, or offer purported
instructional paths.
• Too little literature documents students’
informal understandings.
• Can we connect the goals of integer
instruction to something other than
procedures? And if so, what might that be?

36. Ways of Reasoning
Students who have negative numbers in their
numeric domains typically approach integer tasks
using one of the following five ways of reasoning:
Order-based reasoning
Analogically based reasoning
Formal mathematical reasoning
Computational reasoning

37. Ways of Reasoning
Order-based
Analogically based
Formal mathematical
Computational
RandyLogNec6, Grade 1, 1:11-1:43
–2 + 5 = __

38. Ways of Reasoning
Order-based
Analogically (Analogy- based)
Formal mathematical
Computational
Roland, Grade 4, 0–0:49
–5 + –1

39. Ways of Reasoning
Order-based
Analogically based
Formal mathematical
Computational
Roland, Grade 4, 0–0:48, -5 - -3
-5 – -3

40. Teachers’ Knowledge About Integers
To begin to make sense of how teachers think
about integers, we interviewed 10 seventh-grade
teachers to determine their
understanding of integers and their
perspectives about teaching integers and
about students’ thinking. We posed integer
tasks, we asked them about their teaching,
and we showed them video clips of children
solving open number sentences.

41. Research Questions
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways of
reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that they
use or that students use?
5) What are the seventh-grade teachers’ goals
for integer instruction?

42. Research Questions
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways of
reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that they
use or that students use?
5) What are the seventh-grade teachers’ goals
for integer instruction?
Yes, very
successfully.

43. Research Questions
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways of
reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that they
use or that students use?
5) What are the seventh-grade teachers’ goals
for integer instruction?
Yes, very
successfully.
They invoke ways
of reasoning.

44. Research Questions
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways
of reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that they
use or that students use?
5) What are the seventh-grade teachers’ goals
for integer instruction?
Yes, very
successfully.
They invoke ways
of reasoning.
This is mixed.

45. Research Questions
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways of
reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that
they use or that students use?
5) What are the seventh-grade teachers’ goals
for integer instruction?
Yes, very
successfully.
They invoke ways
of reasoning.
This is mixed.
Not at all.

46. Research Questions
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways of
reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that they
use or that students use?
5) What are the seventh-grade teachers’
goals for integer instruction?
Yes, very
successfully.
They invoke ways
of reasoning.
This is mixed.
Not at all.
Almost entirely
procedural/rule-based.

47. Brief Examples
1) Do 7th-grade teachers answer integer tasks
correctly, and if so, how successfully?
2) Do 7th-grade teachers invoke ways of
reasoning, or rely instead upon procedures?
3) Do 7th-grade teachers recognize ways of
reasoning in students?
4) How explicitly do 7th-grade teachers
articulate the kinds of reasoning that they
use or that students use?
5) What are the seventh-grade teachers’ goals
for integer instruction?
Yes, very
successfully.
They invoke ways
of reasoning.
This is mixed.
Not at all.
Almost entirely
procedural/rule-based.

48. 2) Do 7th-grade teachers invoke ways of reasoning, or
rely instead upon procedures?
Yes, they do.
Teachers invoked ways of reasoning. Even with the result
unknown sentence, -3 + 6 = !, all but one invoked reasoning.
Example: -3 – ! = 2
Raymond: “So I am thinking about the number
line…So I am starting somewhere… and what
do I do to end up at positive two? I am moving
one, two, three, four, five––five units to the
right.”
“I am moving the opposite direction, so I
would write down negative five here.”
Order-based
reasoning
Formal
reasoning

49. 3) Do 7th-grade teachers recognize ways of reasoning
in students?
This is mixed.
Roland, -5 – (-3)
Jessica:
“He seems to have a solid understanding
that adding negatives to negatives gets you
further away. So in his mind, he is saying,
so subtracting, that must bring me closer.”
Raymond: “I think he got confused. There’s no
context involved…I don’t quite understand
him when he used the opposite. Opposite of
what? …He said, “minus minus. I don’t
believe that he knows the meaning of
“minus minus.”

50. Three Implications for Teacher
Integer Study
• Revise the Specialized Content Knowledge
About Integers

51. Three Implications for Teacher
Integer Study
• Revise the Specialized Common Content
Knowledge About Integers

52. Three Implications for Teacher
Integer Study
• Revise the Specialized Common Content
Knowledge About Integers
• Consider the Challenge in Teachers’
Adopting New Goals for Integer Instruction

53. Three Implications for Teacher
Integer Study
• Revise the Specialized Common Content
Knowledge About Integers
• Consider the Challenge in Teachers’
Adopting New Goals for Integer Instruction
• Stop Seeking the Holy Grail for Integer
Instruction: There is No One Best
Approach or Model for Teaching Integers

54. The relationship among three types of
knowledge for mathematics teaching.

55. Thank you.
Discussion