Seeing the Connections Karen J. Graham University of New Hampshire Neil Portnoy Stony Brook University Steve Benson and Al Cuoco Education Development Center NCTM Annual Meeting – Session 704

Agenda for Today Introduction (10 minutes) – Brief Introduction to Seeing the Connections Materials. – Motivation and Philosophy. Engage in Sample Activities as Learners (50 minutes) – Two activities - divide into working groups. – Engage in activity (don’t just read it). Follow-Up Discussion (30 minutes) –Small group discussion on “How might these activities enable “connection making”? –Brief sharing from small groups. –Discussion of additional activities, implementation issues, and resources.

Motivation I’m still not sure why I had to learn about rings and fields and other such topics to be a high school math teacher. — A veteran high school teacher

Message from the mathematics and mathematics education communities The mathematical knowledge needed by teachers at all levels is substantial, yet quite different from that required by students pursuing other mathematics-related professions.... College courses developing this knowledge should make connections between the mathematics being studied and mathematics prospective teachers will teach. — The Mathematical Education of Teachers (CBMS, 2001) Teachers need several different kinds of mathematical knowledge: – Knowledge of the whole domain – Deep, flexible knowledge about curriculum goals and about the important ideas that are central to their grade level – Knowledge about the challenges students are likely to encounter in learning these ideas – Knowledge about how students’ understanding can be assessed — Principles and Standards for School Mathematics (NCTM, 2001)

Seeing the Connections: Promoting Profound Understanding of Secondary Mathematics A collaborative curriculum project from Education Development Center University of New Hampshire Stony Brook University Funded by NSF DUE Steve Benson Karen Graham Al Cuoco Neil Portnoy

The Seeing the Connections materials Gateways to Advanced Mathematical Thinking (DUE ) Al Cuoco PI, EDC Wayne Harvey, Co-PI, EDC Making Mathematical Connections in Programs for Prospective Teachers (DUE ) Karen Graham, PI, UNH Neil Portnoy, CSU, Chico* Todd Grundmeier, UNH ‡ Making the Connections: Higher Algebra to School Mathematics (DUE ) Carole Greenes, PI, BU Al Cuoco, Co-PI, EDC Carol Findell, BU Emma Previato, BU The Seeing the Connections materials are the “offspring” of three NSF-funded proof-of-concept projects: * Now at Stony Brook University ‡ Now at California State University San Luis Obispo

Engaging in Sample Activities Engage in the activity as a learner. Think about how a future secondary teacher might engage with the activity. As a group keep track of questions and observations.

Group Discussion Questions 1) What important mathematical ideas are learned by engaging in this activity? Why would this activity be important for ALL undergraduates? Why is it important for prospective secondary teachers, in particular? Refer to specific aspects of the activity in your response. 2) In what course(s) does the activity make sense? Where does the activity fit into the course(s)? Refer to specific aspects of the activity in your response. Where could this activity lead?

OVERVIEW OF SEEING THE CONNECTIONS MATERIALS

Making Mathematical Connections in Programs for Prospective Teachers, developed a series of activities that provide prospective teachers with the opportunity to make connections between two mathematical areas (transformational geometry and liner algebra) and school and university mathematics. In addition, there is a series of 3 pedagogical activities that the prospective teachers explore within the context of the developing mathematical understandings above. These activities involve the prospective teachers in the analysis of pre-college mathematical curricula and tasks, the analysis of classroom observations conducted in middle school and/or high school classrooms, and the development, implementation, and evaluation of a class activity focused on transformational geometry. Making Mathematical Connections in Programs for Prospective Teachers

1- Isometries of the Plane Discover the four basic isometries (rotation, reflection, translation, and glide). Reinforce the place of definition in mathematics. Sharing definitions and the ensuing discourse is likely to bring out the importance of careful wording. Identify similarity transformations. Make connections between functions and geometric transformations. 2- Rotations, Reflections, Translations, and Glides Discover basic properties of various isometries. Understand definitions and invariants of each isometry. 3- Compositions Discover that the class of isometries is preserved by composition. View isometries as functions. Making Mathematical Connections in Programs for Prospective Teachers

4- Proof with Isometries Be familiar with the use of isometries in proof. Consider basic Euclidean postulates. 5- The Human Vertices Enable students to make connections (physically) between transformational geometry and linear algebra. Linear transformations are functions. Non-invertible transformations collapse R 2 to R 1 or to {0}. Sign of the determinant indicates orientation. 6- Isometries and Linear Algebra This activity is meant to bring closure to the mathematical ideas connecting transformational geometry and linear algebra by introducing the idea of a group structure. Making Mathematical Connections in Programs for Prospective Teachers

Making the Connections: Higher Algebra to School Mathematics was a proof-of-concept project, funded by the National Science Foundation (DUE ), which produced materials for use in courses for preservice mathematics teachers that make explicit connections between the mathematics they learn in college to the mathematics they will eventually teach. The content focus of this project was algebra and number theory with three main themes: Modular Arithmetic, Periods of Repeating Decimals, and The Chinese Remainder Theorem. Making the Connections: Higher Algebra to School Mathematics

Numbers, Systems, and Divisibility (prototype module) 1. Algebra as Structure 2. Modular Arithmetic 3. Making it a System 4. Decimals, Fractions, and Long Division 5. The Fundamental Theorem of Arithmetic 6. Interlude 7. Units, Orders, and Periods 8. The Chinese Remainder Theorem 9. Etude 10. Euler, Units, and Periods of Decimals 11. Irrational Numbers: An Introduction Making the Connections: Higher Algebra to School Mathematics

Gateways to Advanced Mathematical Thinking was a dual curriculum development/research project funded by the National Science Foundation (DUE ). The development component of the project built a model curriculum module for use with undergraduates, and particularly with preservice teachers, which motivates appreciation for mathematics, focuses on conceptual understanding without sacrificing formal techniques, and makes explicit connections to the high school curriculum. Topics include precalculus methods for solving optimization problems, both exactly and approximately. Gateways to Advanced Mathematical Thinking

Part1: Geometric Techniques 1.Minimizing Distance 2.Maximizing Area 3.Contour Lines Part 2: Algebraic Techniques 1.Squares are never negative 2.The Arithmetic Geometric Mean Inequality Part 3: Graphical Techniques 1.The Box Problem

Seeing the Connections materials are available online Making Mathematical Connections in Programs for Prospective Teachers Making the Connections: Higher Algebra to School Mathematics Gateways to Advanced Mathematical Thinking Copies of slides and handouts will be available at All files are in PDF or Powerpoint format Questions? Problems? Send to

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