1
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Overview
1. The “problem” of teacher knowledge
2. Knowing mathematics for teaching
3. What do we know about the role of
mathematical knowledge in teaching?
4. Teacher education for better
mathematics teaching
2
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3
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Clarifying the Problem
Teachers’ knowledge of mathematics
and their ability to use it in teaching
4
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• What’s the evidence?
– Anecdotes
– Research
• What does it tell us and what is not well
understood?
– Lack of a system
– This is a societal problem, widespread
5
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• Require more mathematics for certification
– More mathematics courses
– A major (or minor) in mathematics
– Mathematics test
• Recruit mathematically trained people into teaching
– Engineers, accountants, mathematicians, …
• Fund mathematically focused professional development
But -- these approaches assume we know what
mathematical knowledge and skill is needed for
high-quality teaching.
6
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Clarifying the Problem
Teachers’ knowledge of mathematics
and their ability to use it in teaching
How MUCH mathematics do teachers need to know?
But: What mathematical knowledge and skill
does high quality teaching require?
7
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8
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49
X 25
9
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(a)
49
x 25
(b)
49
x 25
405
108
225
100
1485
325
(c)
49
x 25
1250
25
1275
10
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Put these numbers in order:
.7
.2 3.4
.05
11
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A.
.5
7
.01
11.4
B.
.60
2.53
3.12
.45
C.
.6
4.25
.565
2.5
D. These lists are all equally good for assessing whether
students understand how to order decimal numbers.
12
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Simplify:
150
13
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(a)
(b)
54
(c)
156
128
(d) These examples all work equally well.
14
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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
15
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The mathematical knowledge and skill
expected of any well-educated adult
Teachers need to be able to:
•Recognize wrong answers
•Spot inaccurate definitions in textbooks
•Use notation correctly
•Do the work assigned to students
Example Items
What is 11/4 ÷ 1/2 ?
What number is halfway
between 1.1 and 1.11?
16
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The mathematical knowledge and skill needed by
teachers in their work and beyond that expected of
any well-educated adult
Teachers need to be able to:
•Analyze errors and evaluate alternative ideas
•Give mathematical explanations and use mathematical
representations
•Be explicit about mathematical language and practices
17
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Which of these students is using a method
that could be used to multiply any two whole
numbers?
Student A
35
x 25
1 25
+75
8 75
Student B
35
x2 5
1 75
+ 70 0
875
Student C
35
x 25
25
1 50
1 00
+ 6 00
875
18
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Combines knowledge of mathematics with
knowledge of students or knowledge of teaching
Teachers need to be able to:
Teachers need to be able to:
•Anticipate student errors and
common misconceptions
•Interpret student thinking
•Predict what students are likely to
do with specific tasks and what they
will find interesting or challenging
•Sequence content for instruction
•Recognize instructional pros and
cons of different representations
•Size up mathematical issues in
responding to students’ novel
approaches
19
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Ms.Violeta was looking carefully at her students’ papers, and she
saw the following responses to the problem:
8 + 4 = ___ + 5
i) 12
ii) 17
iii) Can’t do it
iv) 1
Which of the following is the most likely explanation of the
difficulty the students are having? (Mark ONE answer.)
a) They do not know their basic addition facts.
b) They cannot do multi-step problems.
c) They do not know that addition is commutative.
d) They do not understand the meaning of the equals sign.
20
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To introduce the idea of grouping by tens and ones with
young learners, which of the following materials or tools
would be most appropriate? (Choose ONE.)
a. A number line
b. Plastic counting chips
c. Pennies and dimes
d. Straws and rubber bands
e. Any of these would be equally appropriate for
introducing the idea of grouping by tens and ones.
21
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Common
Recognize
incorrect
answers
Specialized
Analyze
errors
Students
Know
common
errors
Teaching
Know
what to
do next
307
- 168
307
- 168
307
- 168
307
- 168
261
169
261
261
22
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• Questionnaire consisting of 30 items (scale reliability .88)
• Model: Student Terra Nova gains predicted by:
– Student descriptors (family SES, absence rate)
– Teacher characteristics (math methods/content, content
knowledge)
• Teacher content knowledge significant
– Small effect (< 1/10 standard deviation): 2 - 3 weeks of
instruction
– But student SES is also about the same size effect on
achievement
(Hill,Rowan, and Ball, AERJ, 2005)
23
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• Mathematical knowledge for teaching is the knowledge
needed for the work of teaching.
• A kind of content knowledge distinct from that
required by other mathematically-intensive professions
• Importance of building this theory IN practice
• Possible to write valid and reliable survey measures of
MKT
• Credibility of this approach with teachers
• This kind of testing is professional
24
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Employed Persons by Occupation (2003, in 1,000s)
3,681
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0
3,632
3,343
2,449
1,884
1,842
1,811
1,639
1,617
1,362
1,313
952
819
)
al
av
st
gi
-n
lo
(n
on
ho
yc
ite
ct
s
ps
d
an
ns
ar
ch
ia
ic
ys
ph
180
s
ns
eo
rg
su
la
l
ia
oc
/s
rs
lo
se
un
co
ye
w
or
w
ty
ll
(a
s
er
ne
gi
en
rs
rs
ke
s)
pe
es
ut
ec
ex
f
ie
ch
s
nt
ta
un
co
iv
to
di
au
d
an
he
e
m
ho
ac
rs
s
de
th
al
ai
w
d
an
rs
te
ai
w
ai
ss
tre
ta
en
es
pr
re
s
le
sa
es
es
rs
nu
d
re
ste
gi
re
tiv
es
rs
ne
ea
cl
e
om
,h
ja
ni
to
rs
se
,m
cr
ai
et
ds
ar
ie
s/a
dm
te
in
ac
he
as
rs
sts
185
Source: Bureau of Labor
25
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Distribution of TE Programs by Size
500
450
444
median
350
300
250
226
200
133
150
88
9
0+
96
95
0-
91
3
2
0
9
9
088
0-
83
1
87
9
2
92
3
1
84
71
0-
68
64
0-
67
9
9
63
9
60
0-
59
9
56
0-
55
9
52
0-
51
9
048
44
0-
47
9
9
43
0-
40
36
0-
39
9
9
35
32
028
0-
31
9
9
024
020
27
23
9
9
0-
19
15
16
12
0-
19
9
-1
-7
80
39
40
2
0
9
4
0-
7
80
8
9
10
8
79
11
0
0-
20
17
75
30
76
31
9
40
0-
46
50
72
100
0-
Number of Programs
400
Size of Program
26
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Te
ch
no
m
la
Ka
az
o
Al o C
b
io olle
lo
Co gic n C ge
ol
nc al
or Un leg
e
di
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Hi
Un rsi
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La
dr ale rsi
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en
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Fe eig vet lleg
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Al ive y
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U
of
ge
D ni
M
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r
si
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Sp
Un erc
rin
y
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g Hop ver
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si
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r
No
Un lleg
rt
U M ive e
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- D rsi
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M Ca ear ty
lv
ic
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hi
i
ga n C o r n
ol
n
Aq Un leg
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iv
ui
Sa
na er
gi
si
s
na
t
Co y
w
lle
Va
lle UM UM ge
y
St -An Flin
O ate n A t
a
W
Un r b
Ce a y k l a
o
nt ne nd iver r
ra
S t Un sit
lM
y
a
iv
M
e
ic te
i
W Gra chig hig Un rsit
es
n
an ive y
a
te d V n
r
al Sta Uni sity
Ea rn
le
ve
M
t
st
e
y
er ich
St Un rsit
n
i
M gan ate ive y
ic
rs
U
S
hi
ga tat niv ity
e
er
n
St Un sit
at
y
iv
e
e
Un rsi
t
y
iv
er
si
ty
n
ga
hi
ic
M
MTTC-passers (2000-2001)
900
200
100
837 837
800
700
600
500
495
400
300
95
69 92
41 51 54 54 65
13 20 22 27 27 30 36
157
114 133
164
192 204 214
258
305
349
533
567
612
378
0
27
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Series1
• Inappropriate subject matter preparation
• Inadequate preparation for diversity of U.S.
classrooms
• Lack of focus on practice
• No professional system for training,
licensure, certification, ongoing professional
education
28
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Key elements
• Practice-based (focused on core practices, study of artifacts and
records, use of a variety of school settings as labs)
• Knowledge of academic subjects for teaching
• Skills for working to address inequities
• Interpersonal and relational skills (diversity, families, colleagues)
• Preparation to manage and change school and policy
environments
• New strategies for recruitment, admissions requirements
• Assessment of performance throughout the program
• Programs for early career professionals
29
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