Transcript www.teachersdg.org

Developing Mathematical
Knowledge for Teaching
Cathy Carroll & Judy Mumme
WestEd
1
Purposes for Doing Mathematics
in PD

Some commonly identified purposes…
•
•
•
•
•
•

To have a common experience for a group of people
To prepare for using with students
An opportunity to increase teacher content knowledge of math
Begin to address particular math issue(s)
Opportunity to reconstruct one’s notion of what it means to do
mathematics, nature of mathematical activity
Identify certain kinds of pedagogy that support learning in
particular situations
How do these resonate with your experience?
2
Session Overview

Consider a purpose of developing teachers’
mathematical knowledge for teaching (with a
focus on specialized content knowledge)
through
 Work
on a mathematics task
 Watch a video
 Consider next steps from the video to work on MKT
with teachers
3
Doing Mathematics in PD
Pairs (1 minute each)
 What supports you in doing math in PD? What gets in
the way?
Small group (5 minutes)
 Share what things are supportive and what are not.
(Note: these may be different for each person)
With this in mind, it is important to respect individual
differences and provide everyone equal access in group work
Pairs/Small Group
4
Logos
Assume the pattern continues to grow in the same manner.
Find a rule or formula to determine the number of tiles in a
figure of any size.
Size 1
Size 2
Size 3
Size 4
• What are the different ways that the geometric
model can be decomposed and how can those
ways be connected to symbolic expressions?
• How do those different expressions represent the
growth of this model?
5
Considering the Task

What are some expressions you came up with to
represent the growth (un-simplified versions)?
Whole Group
6
Considering the Task

Take each of the expressions and see if you can
figure out how that group was thinking about
how the model was growing
 What
is the relationship between these expressions
and the logos model?
Small Group
7
Considering the Task

How did you map the expressions to the logos
model?
Whole Group
8
Considering Purpose

What potential mathematical ideas could
teachers work on through the use of this task?
 Why

would those be good goals for teachers?
Would you do that with students?
 How
might one’s purposes be similar/different
there?
Whole Group
9
Mathematical Knowledge for
Teaching (MKT)

Frame: knowledge “entailed by the work of
teaching”


What do we mean by “knowledge”?


Knowledge used or needed in practice
Mathematical knowledge, skills, habits of mind
What do we mean by the “work of teaching”?

The activities in which teachers engage, and the
responsibilities they have, to teach mathematics, both inside
and outside of the classroom
10
Mathematical Knowledge for
Teaching
Subject Matter Knowledge
Common
Content
Knowledge
(CCK)
Knowledge at
the
mathematical
horizon
Specialized
Content
Knowledge
(SCK)
Pedagogical Content Knowledge
Knowledge of
Content and
Students (KCS)
Knowledge of
Content and
Teaching (KCT)
Knowledge
of
curriculum
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it
special? Journal for Teacher Education, 59(5), 389-407.
11
Common Content Knowledge
(CCK)
Mentally find the answer to 92-56:
92-56 = 36
Knowing how to estimate that it is
greater than 30 and less than 40
12
Specialized Content Knowledge
(SCK)
92 - 56
56 = 50 + 6
92 – 50 = 42
42 - 6 = 36
92 = 80 + 12
80 – 50 = 30
12 – 6 = 6
30 + 6 = 36
+30
+4
56
56 = 52 + 4
92 – 52 = 40
40 – 4 = 36
60
92 = 32 + 60
(32 + 60) – 56
32 + (60 – 56)
32 + 4 = 36
+2
90 92
What meanings of subtraction might each represent?
Do these methods always work?
If so, why do they work?
13
Pedagogical Content Knowledge
92 - 56

Knowledge of content
and students (KCS)


Knowing which approaches
students are likely to use
Knowing which ways of
decomposing the numbers
are likely to lead to
confusion (e.g. rounding
the two numbers in
different directions - 92
to 90 and 56 to 60

Knowledge of content
and teaching (KCT)



Choosing numbers which
invite particular
approaches or stumbles
Choosing contexts or
models to illustrate
different approaches
In a whole-class, choosing
which approaches or
methods you want to
pursue and in what order
14
Tasks of Teaching Mathematics
That Require SCK











Unpacking and decomposing mathematical ideas
Explaining and guiding explanation
Using mathematical language and notation
Generating examples
Making mathematical practices explicit
Choosing and using representations
Comparing the affordances of different representations or methods
Analyzing and interpreting alternative solutions
Analyzing errors
Interpreting and evaluating alternative solutions and thinking
Analyzing mathematical treatments in textbooks
15
SCK in Logos

What SCK might it be possible to develop using the
Logos task?
16
SCK in Logos

How did our framing of the task focus your attention on
SCK?
Our framing of the task:
• What are the different ways that the geometric model
can be decomposed and how can those ways be
connected to symbolic expressions?
• How do those different expressions represent the growth
of this model?

In what ways did our discussion of the task enable
consideration of SCK?
17
Pause Point

What issues does this session have you
thinking about now?
 How
are those related to your practice?
18
BREAK
We’ll start back promptly at 10:30
19
20
Context


We drop
in here



Group of 27 high school and middle school
teachers from 6 districts
Session 7 of of an ongoing series--8 Saturday
sessions over the school year
Small groups have worked on the logos task and
posted charts with their solutions
3 groups have shared their work, and Shamshir is
asked to talk about his poster
Mike is the PD leader
21
Caveat

Mike and the teachers are offering us a gift of
allowing us to carefully examine a real
instance of practice. We are examining their
practice, not critiquing them.
22
Mike
Viewing the Video

We will watch the clip one time, then use it as
a jumping off spot for connecting to practice,
with a focus on helping teachers develop SCK
24
Frame for Viewing

What approaches are teachers sharing?
Suggestion: Use transcript to think about issues after
viewing the video
25
What approaches are teachers sharing?
Small Group Discussion

What approaches were teachers sharing?
27
Whole Group Discussion

How did the approaches that were shared
relate to the approaches we came up with?
28
Connecting to Practice

Assume a goal of developing teachers’ understandings of
how various ways of decomposing or transforming the
model relate to different symbolic expressions--possibly
including how they show aspects of quadratic growth (i.e.,
square, linear, and constant components)





How would you frame the next steps in sharing?
What solutions would you select to pursue to help develop
teachers’ SCK? Why? How do they relate to your goal?
How would you sequence and connect these to achieve your goal
with teachers?
How would you highlight aspects of SCK?
What ideas did you consider in your process? What issues
arose for you in this task?
Small Group
29
Connecting to Practice


What ideas did you consider in planning?
What issues arose for you?
Whole Group
30
Reflecting on the Experience

What are you thinking about now with regard
to developing SCK with teachers in PD?
Small Group/Whole Group
31
A High Leverage Purpose

We’re placing our bets that a focus on
developing mathematical knowledge for
teaching (specifically specialized content
knowledge) will better enable teachers to
be effective in the classroom
32
Logos
Assume the pattern continues to grow in the same manner.
Find a rule or formula to determine the number of tiles in a
figure of any size.
Size 1
Size 2
Size 3
Size 4
• What are the different ways that the geometric
model can be decomposed and how can those
ways be connected to symbolic expressions?
• How do those different expressions represent the
growth of this model?
33
The Mathematical Task Framework
(MTF)*
Tasks as
they appear
in curricular
materials
Tasks as
set up by
teachers
Tasks as
enacted by
teacher and
students
Student
learning
* Smith & Stein (1998)
Stein, Smith, Henningsen & Silver (2000)
34
The Mathematical Task Framework
Adapted to PD*
Tasks as
they appear
originally
Tasks as
set up by
PD leader
Tasks as
enacted by
PD leader
& teachers
Teacher
learning
* Kazemi 2009
35
Important Features of SCK Task
Design*





Unpacks, makes explicit, and develops a flexible understanding of
mathematical ideas that are central to the school curriculum
Opens opportunities to build connections among mathematical
ideas
Provokes a stumble due to a superficial “understanding” of an idea
Lends itself to alternative/multiple representations and solution
methods
Provides opportunities to engage in mathematical practices central
to teaching (explaining, representing, using mathematical
language, analyzing equivalences, proving, proof analysis, posing
questions, writing on the board)
* Suzuka, K., Sleep, L., Ball, D. L., Bass, H., Lewis, J., & Thames, M. (in press). Designing and using
tasks to teach mathematical knowledge for teaching. Association of Mathematics Teacher Educators
(AMTE) Monograph.
36
Challenges of Teaching SCK




Staying focused on the mathematics, and not on how to
teach the math
Keeping the problems focused on SCK and not just
sliding to Knowledge of Content & Students or
Curriculum
Unpacking the mathematics sufficiently and
convincingly helping teachers see what there is to learn
and do
Making visible the connections to the kinds of
mathematical thinking, judgment, reasoning one has to
do in teaching
37
37
Enactment

What are key questions and moves that leaders can
use to keep a task focused on developing SCK?
Asking teachers to
 Explain their solutions to others
 Make correspondences between solutions and/or representations
 Explain someone else’s thinking
 Figure out what might be confusing/difficult for someone else
about the problem
 Having teachers
 Explain what is/was confusing to them
 Ask questions to become clearer about their colleagues’ solutions
 Providing opportunities to “talk mathematics” and write on the board
 Narrating how something a teacher does/says relates to or is a skill
used in teaching

38
To Learn More About…


Learning to Lead Mathematics Professional
Development
Leadership Development Seminars
 Contact Information

Email:
[email protected]
 [email protected]


Web:
WestEd.org/LLMPD
 WestEd.org/PLMPD

39