Immersion in Mathematics Boston University s Masters Degree in

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: //www 2. 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 – – 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… 3 help teachers develop a profession-specific knowledge of mathematics for teaching, 3 engage teachers in rich and ongoing mathematical experiences, 3 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 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 • • 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. ” Fo. M Middle School Teacher

“A lot of us didn't feel we were prepared for the summer program. . . Afterwards we felt we could do anything. ” Fo. M 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.

Fo. M 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 x 2 – 5 y = 27 has no integer solutions. • For what integer values of n does n 3 = 9 k + 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.