Immersion in Mathematics
Immersion in Mathematics
a Wide-Ranging Partnership of Grade 5-12 Teachers, Administrators, University Educators and Professional Mathematicians
on Mathematics

Focus on Mathematics
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Focus on Mathematics
 Our Programs
Focus on Mathematics
 Key Questions
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Focus on Mathematics
 1.  A Taxonomy of Mathematics for Teaching
Focus on Mathematics
 A Taxonomy of Mathematics for Teaching
Focus on Mathematics
 A Taxonomy of Mathematics for Teaching
Focus on Mathematics
 A Taxonomy of Mathematics for Teaching
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The Immersion Experience
PROMYS for Teachers

The Immersion Experience
as a community activity
alongside students
as an empirical science
as exploration

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“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
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Sample Projects
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
6 graduate students
6 teachers (alumni)
15 undergraduates (for students)
5 mathematicians
2 math educators

Examples of
Professional Leadership
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

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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
The number of “special” mathematics teachers having both significant talent and significant interest in mathematics is significantly higher than is commonly believed.
Helping these teachers is work that mathematicians are uniquely prepared to do.
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.

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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
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
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
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
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.