Infusing Connections into Core Courses for Future Secondary Teachers

Steve Benson Al CuocoEducation Development Center

Karen GrahamUniversity of New Hampshire

Neil PortnoyStony Brook University

MAA Minicourse #3; MAA MathfestAlbuquerque, NM; August 4 - 7, 2005

Agenda for today

Introduction Philosophy Brief introduction to Seeing the Connections materials

Engage in sample activities as learners Two activities - divide into working groups Engage in activity (don’t just read it)

Discussion Brief sharing between groups Knowledge of mathematics for teaching How might these activities enable “connection making”?

Agenda for Saturday

Thursday’s questionnaire

“Homework” questions

Adapting existing materials

Lawrence (MA) high school study group

Adoption and implementation

Final thoughts, next steps

Motivation

I’m still not sure why I had to learn about rings and fieldsand other such topics to be a high school math teacher.

— A veteran high school teacher

Over the past 15 years, two refrains have echoed through the discourse about teachers’ knowledge of mathematics: (1) that U.S. teachers mathematical knowledge is weak(2) that the mathematical knowledge needed for teaching is different from that needed by mathematicians. — Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education (RAND, 2001)

The mathematical knowledge needed by teachers at all levels is substantial, yet quite different from that required by students pursuing other mathematics-related professions. . . . Collegecourses developing this knowledge should make connections between the mathematics being studied and mathematics prospective teachers will teach. — The Mathematical Education of Teachers (CBMS, 2001)

Messages from the mathematics community

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

Messages from the mathematics community

— Principles and Standards for School Mathematics (NCTM, 2001)

Responses to the call for connections

Ways to Think About Mathematics: Activities and Investigations forGrade 6-12 Teachers; Benson, Addington, Arshavsky, Cuoco, Goldenberg, Karnowski; Corwin Press, 2004.http://www2.edc.org/wttam

Mathematical Connections: A Companion for Teachers and Others;Cuoco, MAA, 2005Mathematics for High School Teachers - An Advanced Perspective; Usiskin, Peressini, Marchisotto, Stanley; Prentice Hall, 2003.

Seeing the Connections: Promoting Profound Understanding of Secondary Mathematics; Benson, Cuoco, Graham, Greenes, Grundmeier, Portnoy (in preparation)

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-0231342

http://www2.edc.org/connect

Steve Benson sbenson@edc.org Karen Graham karen.graham@unh.eduAl Cuoco acuoco@edc.org Neil Portnoy nportnoy@math.sunysb.edu

The Seeing the Connections materials

Gateways to Advanced MathematicalThinking (DUE-9450731) Al Cuoco PI, EDC Wayne Harvey, Co-PI, EDC

Making Mathematical Connections in Programs for Prospective Teachers (DUE-9981029) Karen Graham, PI, UNH Neil Portnoy, CSU, Chico* Todd Grundmeier, UNH‡

Making the Connections: Higher Algebra to School Mathematics (DUE-9950722) 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 a sample activity

Engage in the activity as a learner, yourself

Think about how a future secondary teacher might engage with the activity, as well

Keep track of questions and observations (share them with your working group, as well as the whole group)

“Homework” questions

Think about the activity you worked on Thursday.

• What important mathematical ideas are learned from engaging in this activity?

• Why is this activity important for prospective secondary teachers? For other students?

• In what course(s) would this activity fit?

PROBLEMFind the line that passes through the point (3,4) that cuts off the smallest area in the first quadrant.

Solve the problem any way you can and, whether you come up with an exact or approximate solution, pay attention to the process and methods you use in solving it. Record some of your thinking about how to solve the problem, and any insights you gained by thinking about the process/methods used. Make sure to include the different approaches you tried, and which directions or methods failed to help, which seemed most helpful, and why.

Adapting Existing Materials

Audio Works makes an excellent CD player in its Dexter, Maine facility.(The facility was formerly occupied by a shoe manufacturer.) A thintriangular piece of gold must be manufactured to cover a 3 cm by 4 cmrectangular integrated circuit. This gold triangle shields the other circuitryfrom the radio frequency waves emitted by the digital-to-analog converterintegrated circuit. The two components must be positioned so that onecorner of the integrated circuit is on the hypotenuse of the triangle, as shownin the picture.

What is the size of the right triangle that should be manufactured tominimize the costs of the gold radio frequency shield?

Integratedcircuit 3 cm

4 cm

Adapting Existing Materials

Knowledge of Mathematics for Teaching

What undergraduate mathematics is important for secondary teachers to understand?

How might an understanding of that mathematics help someone be a better teacher?

What topics in secondary mathematics provide seeds for the study of undergraduate mathematics?

How might the study of that undergraduate topic be designed?

Knowledge of Mathematics for Teaching

Not everything a teacher needs to know ends up on the chalkboard.— Mark Saul

The ability “to think deeply about simple things” (A. Ross) What’s really behind the geometry of multiplying complex numbers?

The ability to create activities that uncover central habits of mind What do 53/2 and 5 mean?

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The ability to see underlying connections and themes

Connections

Linear Algebra brings coherence to secondary geometry

Number Theory sheds light on what otherwise seem like curiosities in arithmetic

Abstract Algebra provides the tools needed to transition from arithmetic with integers to arithmetic in other systems.

Analysis provides a framework for separating the substance from the clutter in precalculus

Mathematical Statistics has the potential for helping teachers integrate statistics and data analysis into the rest of their program

Knowledge of Mathematics for Teaching (cont’d)

The ability to see underlying connections and themes

Themes

Algebra: extension, representation, decomposition

Analysis: extension by continuity, completion

Number Theory: reduction, localization

Knowledge of Mathematics for Teaching (cont’d)

The “mining” of student ideas

The class was using calculators and estimation to get decimalapproximations to . One student, Marla, looked at how you do out long multiplication and realized that none of thesedecimals would ever work because if you square a finite (non-integer) decimal, there’ll be a digit to the right of the decimalpoint, so you can’t ever get an integer. So, Marla had the startof a proof that can’t be represented by a terminating decimal.But where does she go from here?

— Adapted from “A Dialogue About Teaching” in What’s Happening in Math Class? (Teachers College Press, 1996).

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Knowledge of Mathematics for Teaching (cont’d)

The Seeing the Connections project is producing curriculum modules for use in mathematics courses that help preservice teachers develop a knowledge of mathematics for teaching.

The StC curriculum will help secondary teachers develop important mathematical knowledge and skills required in their future careers:

designing effective lessons emphasizing certain ideas over others connecting ideas across the grades understanding germs of insight in students' questions placing precollege topics in the broader mathematical landscape.

The project staff, combining extensive expertise in curriculum development, undergraduate and secondary teaching, teacher preparation and professional development, and education research, will create and make widely available (in paper and electronic formats) a library of materials that can be used in a wide range of preservice and inservice environments.

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 R2 to R1 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-9950722), 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 RepeatingDecimals, 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 9450731). 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

Gateways to Advanced Mathematical Thinking

Part1: Geometric Techniques 1. Minimizing Distance2. Maximizing Area3. Contour Lines

Part 2: Algebraic Techniques 1. Squares are never negative2. 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 Teachershttp://www2.edc.org/connect/mathconnlink.html

Making the Connections: Higher Algebra to School Mathematicshttp://www2.edc.org/connect/connectionslink.html

Gateways to Advanced Mathematical Thinkinghttp://www2.edc.org/connect/gatewayslink.html

Copies of slides and handouts will be available athttp://www2.edc.org/cme/showcase.html

All files are in PDF or Powerpoint formatQuestions? Problems? Send email to sbenson@edc.org