## What We Do

### Mathematics Courses

Foundations of Mathematics

The Foundations course was a "hands on" elementary introduction to the basic material of mathematics: Working with definitions and proofs, sets, functions, number systems, symmetries, and examples of algebraic structures like fields and groups. The exposition was guided by the search for a generalization of the quadratic formula, and the content also served as a supporting outline to the introduction to Galois theory described in the text Field Theory and its Classical Problems by Charles Hadlock. Students worked on guided problem sets in class, supported by minimal lecturing. The only prerequisites were a knowledge of high school mathematics and a willingness to think about problems that may not have an obvious answer. Specific topics covered included basic operations on sets, definitions and examples of functions, properties of natural numbers, rational numbers, irrational numbers, proofs by induction, proofs by contradiction, compass and straightedge constructions, polynomials, subfields of real and complex numbers, symmetries of regular polygons, finite groups, and n­-dimensional space.

Views of Geometry

In this class, we covered topics in geometric theory that are motivated by the NY State middle and high school curricula. We began with a review of basic logic and the axioms of planar Euclidean geometry. Then, we focussed our attention on developing several common geometric notions: congruence, similarity, rigid motions, and symmetry. Though proofs were a part of the course, a stronger emphasis was placed on developing geometric intuition. For this, we used the free and easy­-to­-use geometry software called GeoGebra.

The seminars focused on the role of the teacher leader, advocating for the Common Core Standards in mathematics, using protocols to facilitate inquiry groups, and strengthening content through problem solving. Every session modeled innovative pedagogy in areas such as motivation, group work, questioning techniques, and formative assessment.

Topics for the seminars included:

• Common Core State Standards
• The Math Leader’s Domain of responsibility
• Making the Case for Change
• Building sensible, sense­making mathematics
• Insights into recognizing and overcoming obstacles to change
• Issues related to looking at student work
• The rationale for using protocols
• Implementing protocols to look at student work, i.e the Tuning Protocol
• Implementing protocols to look at practice, i.e. The Consultancy Protocol
• Performance-­based assessments
• Teaching for conceptual understanding
• Problem­-solving

Each class began with a problem of the day to explore content and pedagogy. As a result of our three years’ experience with Cohort 1 and discussions of content and pedagogy, we decided to support and complement the mathematics content courses with Cohort 2 through the use of problems of the day, which parallelled the content that is being taught in the mathematics courses.

### On­-Site Consulting

During the fall 2013 and spring 2014 semesters, 30 MTTI teachers were visited in their schools by an MTTI teacher­-consultant. Each teacher was visited monthly over the course of each semester. Most of the support provided was related to the Common Core. Consultants shared outlines for unit plans, procedures for creating units, websites for tasks, and information relating to state exams. outlines and slides for common core professional development being conducting on the chancellors day and keeping all teacher leaders up to date with relevant information from the state and city education departments. Several teacher consultants also coached and prepared their participants for presentations that they conducted as part of the MTTI.

### Spring Workshop Series

These workshops were MTTI’s major professional development outreach effort of the year. Facilitated and conducted by MTTI participants themselves, the two free series of workshops, one for high school teachers and one for middle school, consisted of three weekly sessions and focused on fostering student engagement.These workshops offered strategies, activities and information on resources to motivate and engage students, to boost their achievement not only on assessments but beyond, and to help them see mathematics as something worthwhile and beautiful. The emphasis throughout was on useful and practical ideas and information, which teachers could put to use in their classrooms immediately.

Topics

High School:
Solving Problems Systematically
Art and Mathematics
Looking at Area and Perimeter

Middle School:
Engaging the Struggling Youngster
Theoretical vs. Experimental Probability
Proportional Relationships