School-based intellectual leadership in mathematics Focus on Mathematics Secondary Teacher Preparation in Mathematics Glenn Stevens Boston University Robert Weldin Lawrence High School

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… 3help teachers develop a profession-specific knowledge of mathematics for teaching, 3engage teachers in rich and ongoing mathematical experiences, 3and establish a lasting mathematical community among mathematicians and teachers.

Focus on Mathematics Mathematical Community 1.In the Schools --- study groups of teachers and mathematicians doing mathematics together 2.At the University --- courses and degree programs for pre- and in-service teachers 3.Across the Partnership --- five school districts, three universities, and growing

Focus on Mathematics Evolving Roles of Teachers 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

an Immersion Experience a Research Experience 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 deeply personal engagement in mathematical ideas Key Features

The Community First year participants 20 in-service 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

Culture of Exploration Immersion in new ideas Necessity for open communication Acquiring taste for hard problems The central role of experience –empirical basis of mathematical knowledge –personal experience as guide for new explorations Learning good judgement in recognizing significant ideas Sharing ideas with others –in writing –in seminars Questioning answers Low threshold -- high ceiling

Habits of Mind 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

Beliefs about the Nature of Mathematics Mathematics is natural –The empirical nature of mathematics –People do mathematics naturally Mathematics exists independent of us –We can perform experiments, explore, and investigate –We can test ideas and decide for ourselves Experience precedes formality –“Meaning” is determined by experience –Definitions and theorems are capstones –Language is a tool for coming to terms with experience Mathematics is the study of structure –Operations, order –Shape –Continuity –Transformation Mathematics is the art of figuring things out

“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

Bob Weldin’s Slides

FOM Study Groups at LHS Lawrence High School is an urban school with over 3,000 students. –About 83% qualify for free or reduced lunch –About 83% are classified as “First Language not English”, and 23% are “Limited English Proficient”. –Anecdotally, LHS has a high percentage of transient students –LHS is ranked last in the Commonwealth on the 10 th grade Math “MCAS” test.

FOM Study Groups at LHS LHS moved to a new campus this year –Over 500,000 square feet of instructional space –Reduced overcrowding from 200% of designed capacity to about 90% –Divided into six separate “small schools” –Math class sizes still remain large for 9 th and 10 th grade – close to 30 students per class

FOM Study Groups at LHS Between the six schools, LHS has about 35 math teachers, up from about 28 when the FOM study groups formed in About one-third of our math teachers are “career-changers”. Over half have 5 or fewer years of experience in teaching. With the split into six schools, FOM group meetings are the only time we see math teachers from the other schools

FOM Study Groups at LHS Our study group currently meets on a biweekly basis throughout the school year. Average attendance is about teachers with nearly half of the math teachers participating on a fairly regular basis. Al Cuoco has been with us from the start and we have also worked with Bob Devaney and Marvin Freedman from BU and Ken Levasseur, Marvin Stick and Kiwi Graham- Eagle from UM-Lowell

FOM Study Groups at LHS At our organizational meeting at the beginning of the year, and at a few other points of the year we talk about what topics interest us and decide what problems or areas to work on for the next several meetings. Our professor partners usually take the lead by facilitating the meetings and bringing interesting problems, but teachers sometimes fill that role.

FOM Study Groups at LHS Some of the topics we have worked on… –Pythagorean triples –Fractal geometry –Patterns of repeating decimals –Divisibility Tests –Using CAS (N-Spire) to test conjectures about polynomials (finding the factors of x n – 1 and other polynomials) –Combinatorial proofs

FOM Study Groups at LHS Some of the topics we have worked on… –Pascal’s Triangle patterns –Farey Sequences and Continued Fractions –Cryptography and code-breaking –Geometry problems from Hadamard’s classic text Lessons in Elementary Geometry (currently being translated into English) –Fundamental Theorem of Arithmetic –Euclid’s Algorithm

FOM Study Groups at LHS First, we have fun doing math together. Sometimes we work on problems that connect to the math we teach in our classes. Sometimes we just work on what seems interesting. We always learn something. It makes us think about ways to give our students similar experiences.

Promys/MMT at BU The Summer program: Total Immersion – all day, 5 days a week for 6 weeks. “Thrown into the deep-end of the pool” The answer to every question is a question Guess what: we CAN swim. Learning to work together and to work alone. The curious spontaneity of small group formation.

Promys/MMT at BU Between the summers –Abstract Algebra – taught in a more traditional manner –Math for special needs students – diversification of lessons –Development of Math Curriculum – where we’ve been, where we are going –Professional Development for Teachers – taking what we’ve learned back to other teachers.

Promys/MMT at BU The second summer: the forest begins to emerge from the trees Learning to do mathematical research: observation, conjecture, experiment, prove Presenting your results: It helps to have a great audience. Bringing Promys into your lesson plans. The third summer: Geometry and Symmetry and more lesson planning.