Carol Findell (BU School of Education)

Glenn Stevens (BU College of Arts and Sciences)

Immersion in Mathematics

Boston University’s

Masters Degree in Mathematics for Teaching

Ryota Matsuura (BU Graduate School of Arts and

Sciences)

Sarah Sword (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/

a Wide-Ranging Partnership of Grade 5-12 Teachers, Administrators, University Educators and

Professional Mathematicians

Focus on

Mathematics(NSF/EHR-0314692)

Boston UniversityEducation Development Center,

inc and five school districts

• School-based intellectual leadership in mathematics

• Learning cultures in school settings involving – Students– Teachers– Educators– Mathematicians

Designed to develop and sustain:

Focus on Mathematics

Masters Degree in Mathematics for Teaching

Focus on MathematicsOur Approach

Depth over breadthWe work intensely on one aspect of improving education.

Focus on mathematicsEverything we do revolves around mathematics.

Capacity buildingTeachers learn to drive professional development.

Community buildingMathematicians, 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,

engage teachers in rich and ongoing mathematical experiences,

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

• Immersion Experiences of Mathematics

• Classroom Connections Seminars

• Leadership Experiences

Elements of the Program:

Masters Degree in Mathematics for Teaching

The Immersion Experience

PROMYS for Teachers

The Immersion Experience

• as a community activity

• alongside students

• as an empirical science

• as exploration

Teachers and mathematicians experiencing mathematics

• emphasis on learning• strengthening mathematical habits of mind• low threshold, high ceiling• deeply personal engagement in mathematical

ideas

Key Features

Habits of Thought

Acquiring experience- numerical experimentation- alert observationGood use of language- asking good questions- formulating conjectures- proofs and disproofsReview- identifying important ideas- formalization- looking for connectionsGeneralization- broadening applicability- questioning answers

“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

The Experience

“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• 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

• 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

• 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.

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 Coursescfindell@bu.edu

• SED ME 581 Advanced Topics in Algebra for TeachersThis 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 GroupHere 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 ExamplesExplorationThe 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 MindThe 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-leadersschapin@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 difficultiescgreenes@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 itemsOne 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.