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How Do We Do It?
Teaching Mathematics to
U.S. Teachers
Jeremy Kilpatrick
University of Georgia
My experience
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Graduated in mathematics (and
science), Chaffey College, Spring 1954
Graduated in mathematics, U.C.
Berkeley, Spring 1956
Student teaching in mathematics,
Richmond High, Spring 1957
Mathematics (and science) teaching,
Garfield Jr. High, Fall 1957
No internship
More mathematics at Stanford
Richmond High
King Middle
(Garfield Jr. High)
Berkeley High
Outline:
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Who teaches?
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What mathematics?
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To whom?
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And how?
Who teaches?
Teacher developers: those who teach
mathematics to prospective or practicing teachers
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Mathematicians at two-year colleges, four-year
colleges, and universities
College and university mathematics educators
Mathematics supervisors in schools, district
offices, and state education departments
Commercial providers of professional
development in mathematics
Teacher leaders and coaches
Mathematics for prospective
teachers
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Taught by faculty in 4-year undergraduate
programs, usually in large public institutions
Two-year college faculty teach roughly 45% of
all undergraduates
Increasing numbers of extended (e.g., 5- or 6year) and alternative programs
In 1998, 28% in teacher education programs
began at the postbaccalaureate level
Newer teachers more likely to hold degrees in an
academic field rather than education
Mathematics for elementary school
teachers
●
Students likely to major in education but often
take courses in the mathematics department
Coursework in mathematics or mathematics education
Grades K–5
Number of semesters
<4
4–7
8–11
> 11
%
23
46
21
10
s.e.
1.8
2.4
2.0
1.3
Source: Horizon Research.
(2002). 2000 National Survey.
Mathematics for middle school
teachers
●
Students likely to major in education
Undergraduate major (Grades 6–8)
Mathematics
%
16
s e.
1.7
Mathematics education
Other education
Other field
10
65
9
1.4
3.1
1.9
Source: Horizon Research.
(2002). 2000 National Survey.
Mathematics for middle school
teachers
●
Most have equivalent of at least a minor in
mathematics
Coursework in mathematics or mathematics education
Grades 6–8
Number of semesters
<4
4–7
8–11
> 11
%
9
28
26
37
s.e.
2.5
3.8
3.4
3.4
Source: Horizon Research.
(2002). 2000 National Survey.
Mathematics for high school
teachers
●
Students likely to major in mathematics or
mathematics education
Undergraduate major
Mathematics
Mathematics education
Other education
Other field
%
58
22
10
10
s.e.
2.2
2.0
1.4
1.2
Source: Horizon Research.
(2002). 2000 National Survey.
Mathematics for practicing teachers
Time spent on in-service
education in mathematics
during the last 3 years
Elem.
% s.e.
Middle
% s.e.
None
15
1.8
11
3.4
7
1.4
Less than 6 hours
21
2.1
12
2.5
8
1.4
6–15 hours
33
2.0
20
2.7
18
1.8
16–35 hours
17
1.6
27
3.7
25
1.8
More than 35 hours
14
1.6
31
3.4
43
2.2
Source: Horizon Research. (2002). 2000 National Survey.
High
% s.e.
Teacher developers in
mathematics departments
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What preparation in mathematics?
What preparation in mathematics
education?
What criteria for promotion and tenure?
How to keep up with developments in
mathematics?
How to keep up with developments in
mathematics education?
What mathematics?
The Mathematics Learning Study
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K–8 teachers need to learn more
mathematics
More mathematics courses
 Not more standard mathematics courses
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Elementary school teachers
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CBMS The Mathematical Education of
Teachers (2001) calls for at least 9
semester hours of coursework on
fundamental ideas of elementary school
mathematics for teachers of grades 1–4
2000 Survey found majority of teachers in
grades K–5 have at most 7 semester-long
courses, and a quarter have fewer than 4
courses
Mathematics for elementary school
teachers
Courses taken by teachers of grades K–5
%
s.e.
Mathematics education (math. for elem. tchrs.)
94
1.0
College algebra/trigonometry/elem. functions
44
2.2
Probability and statistics
36
2.1
Applications of mathematics/problem solving
21
1.6
Geometry for elementary/middle school teachers
21
1.4
Calculus
13
1.5
Source: Horizon Research. (2002). 2000 National Survey.
Middle school teachers
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CBMS The Mathematical Education of Teachers
(2001) calls for 21 semester hours of
mathematics, including at least 12 on
fundamental ideas of school mathematics for
teachers of grades 5–8
According to 2000 Survey, NCTM recommends
coursework in abstract algebra, geometry,
calculus, probability and statistics,
applications/problem solving, and history of
mathematics
2000 Survey found nearly two-thirds of teachers
in grades 6–8 have taken 8 or more semesterlong mathematics courses
Mathematics for middle school
teachers
Selected courses taken by teachers of grades 6–8
%
s.e.
Mathematics for middle school teachers
45
3.5
College algebra/trigonometry/elem. functions
66
3.5
Probability and statistics
56
4.1
Applications of mathematics/problem solving
27
2.6
Geometry for elementary/middle school teachers
36
3.2
Geometry
47
3.9
Calculus
43
3.1
Linear algebra
28
3.1
Abstract algebra
22
2.3
History of mathematics
16
2.1
Source: Horizon Research. (2002). 2000 National Survey.
Mathematics for middle school
teachers
Number of “recommended” courses
%
s.e.
None
24
3.0
1–2 courses
37
3.8
3–4 courses
28
2.8
5–6 courses
11
1.4
Source: Horizon Research. (2002). 2000 National Survey.
High school teachers
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CBMS The Mathematical Education of
Teachers (2001) calls for the equivalent of an
undergraduate major in mathematics (9–12)
But “future high school teachers need to
know more and somewhat different
mathematics than mathematics departments
have previously provided to teachers”
Recommends a 6-hour capstone course to
connect college with high school mathematics
Mathematics for high school
teachers
Selected courses taken by teachers of grades 9–12
%
s.e.
College algebra/trigonometry/elem. functions
80
1.5
Calculus
96
0.8
Geometry
83
1.3
Linear algebra
82
1.7
Abstract algebra
65
2.0
Probability and statistics
86
1.7
Applications of mathematics/problem solving
37
1.8
Geometry for elementary/middle school teachers
17
1.7
History of mathematics
41
2.0
Source: Horizon Research. (2002). 2000 National Survey.
Mathematics for high school
teachers
Number of “recommended” courses
%
s.e.
None
1
0.7
1–2 courses
10
1.4
3–4 courses
48
2.1
5–6 courses
40 2.0
Source: Horizon Research. (2002). 2000 National Survey.
Mathematical knowledge for
teaching
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An application of mathematics to the
practice of teaching
The mathematics that is imperative/useful/
important for teachers to know
Just as the school mathematics curriculum
is a selection from all that could be taught,
so is the curriculum of mathematics for
teaching
To whom?
Elementary school teachers
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93% of K–5 teachers are female; 90% are
White; 58% are over 40; 29% have taught
more than 20 years; 42% have a master’s
degree
In mathematics, 54% of K–5 teachers
consider themselves very well qualified,
45% adequately qualified, and only 1% not
well qualified
Source: Horizon Research.
(2002). 2000 National Survey.
Middle school teachers
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72% of grades 6–8 teachers are female;
85% are White; 52% are over 40; 29%
have taught more than 20 years; 44% have
a master’s degree
Perceived qualifications to teach subjects (in %)
Subject
Not well
Adequate
Very well
Computation
0
8
92
Numeration and number theory
1
21
78
Algebra
9
33
57
Statistics
37
43
20
Source: Horizon Research.
(2002). 2000 National Survey.
High school teachers
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55% of grades 9–12 teachers are female;
91% are White; 59% are over 40; 34%
have taught more than 20 years; 51% have
a master’s degree
Perceived qualifications to teach subjects (in %)
Subject
Not well
Adequate
Very well
Algebra
0
5
94
Geometry and spatial sense
4
26
70
Statistics
23
51
26
Calculus
39
37
25
Source: Horizon Research.
(2002). 2000 National Survey.
High school teachers
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The more undergraduate mathematics that high
school teachers have studied, the better the
performance of their students (effect is small and
may decrease beyond 5 courses; larger for
teaching advanced than remedial courses)
“Whether a degree in mathematics is better than
a degree in mathematics education … remains
disputable”
Students of teachers certified in mathematics do
better than students of uncertified teachers
Sources: Floden & Meniketti; Wilson & Youngs. (2005). In Cochran-Smith &
Zeichner (Eds.), Studying Teacher Education. AERA Report
Ingersoll, R. M. (2003, September). Out-of-field teaching and the limits of teacher
policy. Center for the Study of Teaching and Policy.
Ingersoll, R. M. (2003, September). Out-of-field teaching and the limits of teacher
policy. Center for the Study of Teaching and Policy.
And how?
Pólya’s principles of teaching
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Active learning: The ideas should be born
in the students’ mind and the teacher
should act only as midwife
Best motivation: Pay attention to the
choice, formulation, and presentation of a
worthwhile task
Consecutive phases: Learning begins with
action and perception, proceeds to words
and concepts, and ends with ideas
Pólya on teaching
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Mathematics consists of information
and know-how
“Nobody can give away what he [or
she] has not got.”
Teachers need “experience in
independent (‘creative’) work on the
appropriate level in the form of a
problem-solving seminar or in any
other suitable form.”
Teaching teachers mathematics
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In a genetic approach, the learner
rediscovers, retracing the major steps in the
path followed by the original discoverers
Otto Toeplitz (1963/2007) The calculus:
A genetic approach (MAA & U. Chicago Press)
Prospective teachers need a synoptic view of
mathematics
Just as prospective high school teachers
need a capstone mathematics sequence, so
do teachers of other grades
Felix Klein
(18491925)
Elementary Mathematics From
an Advanced Standpoint
Vol. 1: Arithmetic Algebra
Analysis (Dover, 1932/2004)
Vol. 2: Geometry (Dover,
1939/2004)
Felix Klein
Purpose: “to take into account, in university
instruction, the needs of the school teacher”
“My task will always be to show you the mutual
connection between problems in the various
fields, a thing which is not brought out sufficiently
in the usual lecture course, and more especially to
emphasize the relation of these problems to those
of school mathematics”
Real goal of your academic study: “to draw (in ample
measure) from the great body of knowledge there
put before you a living stimulus for your teaching”
Jens Høyrup
Historian of mathematics
and philosopher of science
Roskilde University
Denmark
Extended episodes from history:
Scribal computation in Mesopotamia
Axiomatics in Greece
Merged traditions during Latin Middle
Ages
Relations between
• Development of mathematics
• Character of the mathematical
discourse
• Institutional setting of mathematics
teaching (mainly adults)
Jens Høyrup
In Measure, Number, and Weight
SUNY Press, 1994
Mathematics is a reasoned
discourse
It is the product of
communication by argument
Not only is teaching the vehicle by
which mathematical knowledge
and skill are transmitted to the
next generation, but also
Mathematics is constituted
through teaching
Hans Freudenthal
Mathematician and
mathematics educator
Utrecht University
The Netherlands
(19051990)
Freudenthal on mathematics
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Mathematics starting and
staying in reality
UCSMP International Conference on
Mathematics Education, Chicago, March 1985
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Mathematics starting and staying
within common sense
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Revisiting Mathematics Education, Kluwer, 1991
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Mathematics starting and staying in
teaching
Starting, as noted by Høyrup
 Staying, because teaching preserves
mathematics
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Therefore, those who teach
mathematics are keeping it alive
And those who teach teachers
mathematics are keeping it alive for
future generations