Immersion in Mathematics
Boston University’s
Masters Degree in
Mathematics for Teaching
Carol Findell
Ryota Matsuura
(BU School of Education)
(BU Graduate School of Arts and Sciences)
Glenn Stevens
Sarah Sword
(BU College of Arts and Sciences)
(Education Development Center, Inc)
Immersion in Mathematics
Slides will be available at
http://www.focusonmath.org
and http://www2.edc.org/cme/showcase
More information available at:
http://math.bu.edu/study/mmt.html
and http://www.promys.org/pft/
Focus
on Mathematics
(NSF/EHR-0314692)
a Wide-Ranging Partnership of Grade 5-12 Teachers,
Administrators, University Educators and Professional
Mathematicians
Boston University
Education Development Center, inc
and five school districts
Focus on Mathematics
Masters Degree in
Mathematics for Teaching
Designed to develop and sustain:
• School-based intellectual leadership in mathematics
• Learning cultures in school settings involving
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Students
Teachers
Educators
Mathematicians
Focus on Mathematics
Our Approach
Depth over breadth
We work intensely on one aspect of improving education.
Focus on mathematics
Everything we do revolves around mathematics.
Capacity building
Teachers learn to drive professional development.
Community building
Mathematicians, teachers, and educators work and learn together.
Focus on Mathematics
Our Programs
The programs are designed to…

help teachers develop a profession-specific knowledge of
mathematics for teaching,
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engage teachers in rich and ongoing mathematical
experiences,
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and establish a lasting mathematical community among
mathematicians and teachers.
Focus on Mathematics
Key Questions
1.
What do we mean by a “profession-specific
knowledge of mathematics for teaching?”
2.
What do we mean by “engage teachers in rich
and ongoing mathematical experiences?”
3.
What do we mean by a “lasting community
among mathematicians and teachers?”
Focus on Mathematics
Evolving Roles of Participants and Mathematicians
 Mathematicians working with teachers as colleagues
 Sharing expertise
 Connecting to mathematics for teaching
 Increasing active involvement by teachers
 Teacher-led sessions
Focus on Mathematics
1. A Taxonomy of Mathematics for Teaching
Expert mathematics teachers…
…know mathematics as a Scholar:
They have a solid grounding in classical mathematics, including
 its major results
 its history of ideas
 its connections to pre-college mathematics
Focus on Mathematics
A Taxonomy of Mathematics for Teaching
…know mathematics as an Educator:
They understand the habits of mind that underlie major branches
of mathematics and how they develop learners, including
 algebra and arithmetic
 geometry
 analysis
Focus on Mathematics
A Taxonomy of Mathematics for Teaching
…know mathematics as a Mathematician:
They have experienced the doing of mathematics — they know
what it’s like to
• grapple with problems
• build abstractions
• develop theories
• become completely absorbed in mathematical activity for
a sustained period of time
Focus on Mathematics
A Taxonomy of Mathematics for Teaching
…know mathematics as a Teacher:
They are skilled in uses of mathematics that are specific
to the profession, including
• the ability “to think deeply about simple things”
(Arnold Ross)
• the craft of task design
• the ability to see underlying themes and
connections
• the “mining” of student ideas
Masters Degree in
Mathematics for Teaching
Elements of the Program:
•
Immersion Experiences of Mathematics
•
Classroom Connections Seminars
•
Leadership Experiences
The Immersion Experience
PROMYS for Teachers
The Immersion Experience
Teachers and mathematicians experiencing mathematics
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as a community activity
alongside students
as an empirical science
as exploration
Key Features
• emphasis on learning
• strengthening mathematical habits of mind
• low threshold, high ceiling
• deeply personal engagement in mathematical
ideas
Habits of Thought
Acquiring experience
- numerical experimentation
- alert observation
Good use of language
- asking good questions
- formulating conjectures
- proofs and disproofs
Review
- identifying important ideas
- formalization
- looking for connections
Generalization
- broadening applicability
- questioning answers
The Experience
“The first weeks of the program, I could connect to
things I knew. Even if I was frustrated one day, the next
day I'd have an epiphany - there were lots of ups and
downs. Understanding math concepts was not enough,
you had to look at things in different ways. It's not
necessarily intuitive. I learned a lot about my own
patience. Every time I felt frustrated, I realized
something that I wouldn't have realized without being
frustrated.”
FoM Middle School Teacher
“A lot of us didn't feel we were prepared
for the summer program . . .
Afterwards we felt we could do anything.”
FoM Middle School Teacher
The Mathematics
Sample Projects
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Patterns in Pascal’s triangles
Repeating decimals and other bases
Sums of Squares
Pythagorean Triples
Combinations and Partitions
Dynamics of billiards on a circular table
Stirling Numbers of the Second kind
Symmetries of cubes in higher dimensions
Applications of quaternions to geometry
The PROMYS Community
• First year participants
• 20 teachers
• 8 pre-service teachers
• 45 high school students
• Returning participants
• 8 teachers
• 20 high school students
• Counselors
• 6 graduate students
• 6 teachers (alumni)
• 15 undergraduates (for students)
• Faculty
• 5 mathematicians
• 2 math educators
Examples of
Professional Leadership
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Lead colloquia and mathematical seminars
Write and publish mathematical papers
Develop new courses for students
Lead partnership-wide seminars
Lead study groups in the schools
Lead professional development in the districts
Curriculum review and research
What lessons are to be learned?
• What is it in the structure of PROMYS that makes it possible
to “succeed” with such disparate audiences?
– the genius of Arnold Ross’s problem sets;
– the depth of the traditions and the community.
• Are these teachers “special” before they begin the program?
Undoubtedly, “yes”!
– What is special about them?
– How rare is this brand of “specialness”?
• What relationship does this have with leadership?
• How does the immersion experience affect teachers’ work in
the classroom?
• Can we replicate (generalize) key elements of the program?
Final Remarks
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The number of “special” mathematics teachers having both significant
talent and significant interest in mathematics is significantly higher than
is commonly believed.
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Helping these teachers is work that mathematicians are uniquely
prepared to do.
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The mathematical habits of thought required for excellence in teaching
are similar to those required for excellence in research.
•
Mathematicians can benefit AS MATHEMATICIANS from engagement
in issues of mathematics education.
FoM and the School of Education
• Established a new degree to focus on leadership in
mathematics education
• Created new courses to provide connections between
higher math and school mathematics
• Trained teacher-leaders to conduct needs assessments and
develop professional development courses. Provided
mentored experiences in professional development
• Conducted research on student difficulties with linear
relationships
The MMT Degree
Masters in Mathematics for Teaching
School of Education
In collaboration with
College of Arts and Sciences
Boston University
New Connections Courses
cfindell@bu.edu
• SED ME 581 Advanced Topics in Algebra for Teachers
This course focuses on how concepts developed in university level
modern algebra courses connect to and form the foundation for the
middle and high school algebra curriculum. The mathematical
structures of group, ring, integral domain, and field will be discussed.
By showing how these advanced algebraic ideas relate to school
mathematics, students will gain a deeper knowledge of the algebraic
ideas.
Examples of connections
The Parade Group
Here is an example of a group. The set of elements is the set
containing the four parade commands: left face (L), right face (R),
about face (A), and stand as you were (S). The operation is
“followed by”, which we will designate as F.
• Make a Cayley Table to show the results of each command
followed by other commands.
• Prove that the “Parade Group” really is a group. That is, show that
the group axioms hold for the four commands and the operation F
“followed by”.
New Connections Courses
• SED CT 900 Independent Study in Number Theory
Connections are made among concepts of algebra and
number theory from college level courses such as linear
algebra, abstract algebra, and number theory, and those
same concepts taught at high school and middle school.
Concepts at each level are explored.
Example of Connections
• How can modular arithmetic help you figure out if 2346 is
a perfect square?
• How can modular arithmetic help you figure out if 99416
is a perfect square?
• Explain how modular arithmetic helps you find out that x2
– 5y = 27 has no integer solutions.
• For what integer values of n does n3 = 9k + 7? How does
modular arithmetic help find the values?
New Connections Courses
• SED ME 580 Connecting Seminar:
Geometry
Focuses on how concepts developed in university level
geometry courses connect to and form the foundation for
the middle and high school geometry curriculum
Connections Examples
Exploration
The Annual Mathematics Contest presented a puzzle. The rules
and overlapping memberships caused some complications.
Here are the facts.
• Each team in the contest was represented by four students.
• Each student was simultaneously the representative of
two different teams.
• Every possible pair of teams had exactly one member
in common.
• How many teams were present at the contest?
• How many students were there altogether?
Rationale for connecting courses
In a paper written for the Mathematical Association
of America Committee on the Undergraduate
Program in Mathematics, Joan Ferrini-Mundy and
Brad Findell (2000) state that the entire set of
undergraduate mathematics courses now required
of those students preparing to teach mathematics at
the middle or high school level consists of courses
that are, at least on the surface, unrelated to the
mathematics they will teach.
Course Premises
These courses are based on the premise that
teachers not only need to understand
concepts of higher level mathematics, but
also need to know how these concepts are
manifested in high school and middle
school mathematics curricula. The courses
connect problems suitable for exploration
by middle or high school students to
problems from the college courses.
Habits of Mind
The courses help pre-service and in-service teachers refine
and expand these middle and high school concepts, and
provide experiences in posing questions that encourage
student-directed learning in the exploration of the
traditional mathematics curriculum. The courses present
strategies for developing habits of mind that motivate
students to ask questions like, “If I change the parameters
or initial conditions, how will that affect the problem and
its framework and solution?” or “How do these algebraic
and geometric ideas mesh?”
Trained Teacher-leaders
schapin@bu.edu
• The curriculum course prepares teachers to
evaluate and develop curriculum goals and
materials.
• The professional development course prepares
teachers to assess the needs of a school or district
and prepare a professional development sequence
to meet these needs.
• The field study allows teachers to present the
professional development sequence with
mentoring..
Research on student difficulties
cgreenes@bu.edu
Curriculum Review Committee
Found that all programs were aligned with Massachusetts
Frameworks
Why poor MCAS performance?
Analyzed MCAS items and student work
Focused on linearity
Discovered that student difficulties were different than
what was expected.
Assessment Tool
• The committee devised an assessment tool
Seven items
One essay, 3 short answer, 3 multiple choice
• Results: minimal understanding of linearity,
including aspects of slope, different
representations of linear relationships, and
problems that required applications of these
concepts
• More than 3000 students tested in the US and 800
more in Korea and Israel. Results were the same in
all countries.