Seeing Mathematical Connections in Courses for Teachers and Other Mathematics Majors

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Seeing Mathematical Connections in Courses for
Teachers (and OtherMathematics Majors)
Steve Benson Al Cuoco Education Development
Center
Karen Graham University of New Hampshire
Neil Portnoy Stony Brook University
PMET WorkshopTuscaloosa, AL May 28, 2005
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Motivation Im 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
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Messages from the mathematics community
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)
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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

Principles and Standards for School Mathematics
(NCTM, 2001)
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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. Mathematical
Connections A Companion for Teachers and
OthersCuoco, MAA (in press) Mathematics 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)
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Many of the materials used in todays activity
were adapted from Ways to Think About
Mathematics Activities and Investigations for
Grade 6-12 Teachers, available from Corwin Press.
A Facilitators Guide and Supplementary CD
(including solutions and additional activities)
are also available.
More information athttp//www2.edc.org/wttam
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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
Steve Benson sbenson_at_edc.org Karen Graham
karen.graham_at_unh.edu Al Cuoco acuoco_at_edc.org
Neil Portnoy nportnoy_at_math.sunysb.edu
http//www2.edc.org/connect
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The Seeing the Connections materials
The Seeing the Connections materials are the
offspring of three NSF-funded proof-of-concept
projects
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
Gateways to Advanced Mathematical Thinking
(DUE-9450731) Al Cuoco PI, EDC Wayne Harvey,
Co-PI, EDC
Now at Stony Brook University Now at
California State University San Luis Obispo
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Engaging in sample activities
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)
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Homework questions 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?
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Additional questions
Think about the activities you worked on this
morning. 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. 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?
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Altering problems
PROBLEM Find 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.
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• 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.

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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) Whats 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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Knowledge of Mathematics for Teaching (contd)

• 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

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Knowledge of Mathematics for Teaching (contd)

• The ability to see underlying connections and
  themes
• Themes
• Algebra extension, representation,
  decomposition
• Analysis extension by continuity, completion
• Number Theory reduction, localization

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Knowledge of Mathematics for Teaching (contd)

• The mining of student ideas
• The class was using calculators and estimation to
  get decimal
• approximations to . One student, Marla,
  looked at how you do out long multiplication and
  realized that none of these
• decimals would ever work because if you square a
  finite (non-integer) decimal, therell be a
  digit to the right of the decimal
• point, so you cant ever get an integer. So,
  Marla had the start
• of a proof that cant be represented by a
  terminating decimal.
• But where does she go from here?
• Adapted from A Dialogue About Teaching in
  Whats
• Happening in Math Class? (Teachers College
  Press, 1996).

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Making Mathematical Connections in Programs for
Prospective Teachers
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.
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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.

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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.

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Making the Connections Higher Algebra to School
Mathematics
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.
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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
• Etude
• 10. Euler, Units, and Periods of Decimals
• 11. Irrational Numbers An Introduction

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Gateways to Advanced Mathematical Thinking
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.
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Gateways to Advanced Mathematical Thinking

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

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Seeing the Connections materials are available
online
Making Mathematical Connections in Programs for
Prospective Teachers http//www2.edc.org/connect/m
athconnlink.html Making the Connections Higher
Algebra to School Mathematics http//www2.edc.org/
connect/connectionslink.html Gateways to
Advanced Mathematical Thinking http//www2.edc.org
/connect/gatewayslink.html Copies of slides and
handouts will be available at http//www2.edc.org/
cme/showcase.html All files are in PDF or
Powerpoint format Questions? Problems? Send
email to sbenson_at_edc.org