Presentation Slides (pdf) - Education Development Center

The next generation of

NSF-funded high school

mathematics programs.

CME Project

Promoting Mathematical Habits of

Mind in High School

Session Presenters

Al Cuoco, Bowen Kerins, Sarah Sword,

Kevin Waterman

**Center** for Mathematics **Education**, EDC

acuoco@edc.org

www.edc.org/cmeproject

www.phschool.com/cme

Opinions about the

discipline

...the future well-being of our nation and people

depends not just on how well we educate our

children generally, but on how well we educate

them in mathematics and science specifically.

John Glenn, September, 2000

The only people who need to study calculus are

people who want to be calculus teachers.

Bill Cosby, April, 1999

...and about how to learn it

I’d never use a curriculum that has worked

out examples in the student text.

Nancy McLaughlin

CME Project Teacher Advisory Board

I’d never use a curriculum that doesn’t have

worked-out examples in the student text.

Chuck Garabedian

CME Project Teacher Advisory Board

The Utility of Mathematics

Mathematics constitutes one of the most ancient

and noble intellectual traditions of humanity. It is

an enabling discipline for all of science and

technology, providing powerful tools for analytical

thought as well as the concepts and language for

precise quantitative description of the world around us.

It affords knowledge and reasoning of

extraordinary subtlety and beauty, even at the

most elementary levels.

RAND Mathematics Study Panel, 2002

CME Project

CME Project

Overview

Problem-Based,

Student-**Center**ed

Approach

Comprehensive

4-year Curriculum

CME Project Overview

Contributors

EDC’s **Center** for Mathematics

**Education**

National Advisory Board

Core Mathematical Consultants

Teacher Advisory Board

Field-Test Teachers

CME Project Overview

Fundamental Organizing Principle

The widespread utility and effectiveness

of mathematics come not just from

mastering specific skills, topics, and

techniques, but more importantly, from

developing the ways of thinking—the

habits of mind—used to create the

results.

CME Project Overview

“Traditional” course structure:

it’s familiar but different

Structured around the sequence of Algebra 1,

Geometry, Algebra 2, Precalculus

Uses a variety of instructional approaches

Focuses on particular mathematical habits

Uses examples and contexts from many fields

Organized around mathematical themes

CME Project Overview

CME Project audience:

the (large number of) teachers who…

Want the familiar course structure

Want a problem- and exploration-based

program

Want to bring activities to “closure”

Want rigor and accessibility for all

CME Project Overview

Relationship with Texas Instruments

CME Project makes

essential use of

technology:

A “function-modeling”

language (FML)

A computer algebra

system (CAS)

An interactive geometry

environment

CME Project Overview

Why CAS-Based Technology?

To make tractable and to enhance many beautiful

classical topics, historically considered too technical

for high school students.

To support experiments with algebraic expressions

and other mathematical objects in the same way that

calculators can be used to experiment with numbers.

To allow students to build computational models of

algebraic objects and structures that have no faithful

physical counterparts.

CME Project Approach

Getting Started

Example 1: Arithmetic to Algebra

Instructional Approach:

An Investigation is a collection of Lessons

The first lesson in each investigation is

Getting Started

Launched with little or no instruction

Previews upcoming ideas in simple

contexts

Provides teachers with a diagnostic tool

CME Project Approach

Getting Started

Example 1: Arithmetic to Algebra

Extending a Rule

Extending number systems

Extending operations “coherently”

CME Project Approach

Algebra 1

The multiplication table, reoriented

CME Project Approach

In-Class Experiment

Example 2: More Arithmetic to Algebra

Instructional Approach

Launches a topic, or

Provides experience before formality, or

Connects ideas

CME Project Approach

In-Class Experiment

Example 2: More Arithmetic to Algebra

Investigating Algebraic Structures

A central ingredient of algebraic thinking

Parallel structure between integers and

polynomials

CME Project Approach

Precalculus

The Polynomial Factor Game

x – 1 x 2 – 1 x 3 – 1 x 4 – 1 x 5 – 1

x 6 – 1

x 7 – 1

x 8 – 1

x 9 – 1

x 10 – 1

x 11 – 1

x 12 – 1

x 13 – 1

x 14 – 1

x 15 – 1

x 16 – 1

x 17 – 1

x 18 – 1

x 19 – 1

x 20 – 1

x 21 – 1

x 22 – 1

x 23 – 1

x 24 – 1

x 25 – 1

x 26 – 1

x 27 – 1

x 28 – 1

x 29 – 1

x 30 – 1

CME Project Approach

TI Connection

CME Project Approach

Minds in Action

Example 3: Geometry to Analysis

Instructional Approach

A mathematical conversation among students

Characters reappear throughout the program

Students begin to talk like the characters

CME Project Approach

Minds in Action

Example 3: Geometry to Analysis

Reasoning by Continuity

A central ingredient of analytic thinking

Previews ideas of calculus

CME Project Approach

Geometry

You are on a camping trip, returning

from a walk. Standing at Y, you

notice that your tent (located at T) is

on fire. You need to run to the river,

fill your bucket with water, and get to the tent. Where

should you land on the shore to minimize the total

distance of the trip?

CME Project Approach

TI Connection

CME Project Approach

Minds in Action

Geometry

Tony

Derman

Tony

Hey Derman, do you remember the

“burning tent” problem? It said...

Yes, I remember that we tried to

find the best spot to fill the bucket

in the river and minimize the total

distance you had to run.

Well, I was thinking about solving it

by looking at some “wrong”

answers…

CME Project Approach

Minds in Action

Geometry

Tony

…like this one.

Derman

Why would you do that?

CME Project Approach

TI Connection

CME Project Approach

Minds in Action

“Students begin to talk like

the characters”

A real dialogue about

order of operations

from an Algebra 1

Field Test Site

Sara

Zoie

Sara

CME Project Approach

Minds in Action

Algebra 1

I found your number. It’s −5.9.

Wrong.

It’s right! Look.

Sara goes through a process of reversing

steps.

Sara

See?

Zoie

CME Project Approach

Minds in Action

Algebra 1

Look.

Zoie goes through steps on her calculator…

Sara

Uh-uh! You can’t do that!

You have to push

“equals” every time.

CME Project Approach

Minds in Action

Algebra 1

Zoie

Sara

Zoie

Sara

No! It’s the same.

Is it?

Well, it should be. But it’s not.

Why do you think it isn’t?

Because of “please my dear

stuff”...you know...order of

operations!

CME Project Approach

Minds in Action

Algebra 1

Zoie

Sara

Oooh. Yeah! So it’s not the same.

Eew. We sound like Tony and

Sasha.

CME Project Approach

Projects

Example 4: An Algebraic Interlude

Instructional Approach: Project

At the end of each chapter

Two types

Independent research

Develop a personal understanding of a

mathematical exposition

CME Project Approach

Projects

Example 4: An Algebraic Interlude

Patterns are non-trivial

Justifying the pattern

Seeking Patterns

CME Project Approach

Projects

Algebra 2

Number of Irreducible Factors of x n – 1

n

1

2

3

4

5

6

factors

n

7

8

9

10

11

12

factors

CME Project Approach

TI Connection

CME Project Approach

Projects

Algebra 2

Number of Irreducible Factors of x n – 1

n

1

2

3

4

5

6

factors

1

2

2

3

2

4

n

7

8

9

10

11

12

factors

2

4

3

4

2

6

Comprehensive Curriculum

Recurring Theme

Example 5: Algorithmic Thinking

Program Approach: Recurring Theme

Visited in each course

Deepened in each grade

Generalizing

A central ingredient of algebraic thinking

Emphasis on recursive thinking

Comprehensive Curriculum

Recurring Theme

Algebra 1

Find a function that matches each table:

Comprehensive Curriculum

Recurring Theme

Algebra 2

Table A: Two Models

Comprehensive Curriculum

Recurring Theme

Precalculus

CME Project Approach

TI Connection

Comprehensive Curriculum

Recurring Theme

Precalculus

g(255)

=

g(254) + 5

def of g

=

f (254) + 5

CSS

=

5(254) + 3 + 5

def of f

=

5(254 + 1) + 3

arithmetic

=

5(255) + 3

more arithmetic

=

f (255)

def of f

Comprehensive Curriculum

Recurring Theme

Precalculus

g(n)

=

g(n – 1) + 5

def of g

=

f (n – 1) + 5

CSS

=

5(n – 1) + 3 + 5

def of f

=

5n – 5 + 3 + 5

distributive prop

=

5n + 3

arithmetic

=

f (n)

def of f

Comprehensive Curriculum

Recurring Context

Example 6: Modeling with Functions

Program Approach: Recurring

Context

Visited in each course

Deepened in each grade

Building a model

Refining “actions” into coherent

processes

Chunking

Comprehensive Curriculum

Recurring Context

Algebra 1

Comprehensive Curriculum

Recurring Context

Algebra 2

Suppose you want to buy a car that costs

$10,000. You have $1000 for a down-payment.

The interest rate is 5%, and the loan must be

paid off in 3 years. If the most you can pay per

month is $350, can you afford to buy this car?

This situation leads to the questions

“How does a bank figure out the monthly

payment on a loan?”

or

“How does a bank figure out the balance

you owe at the end of each month?”

Comprehensive Curriculum

Recurring Context

Algebra 2

Take 1

What you owe at the end of the month is

what you owed at the start of the month

minus the monthly payment.

b(

n,

m)

=

# 9000

"

! b(

n $ 1, m)

$

m

if

if

n

n

=

>

0

0

Comprehensive Curriculum

Recurring Context

Algebra 2

Take 2

b(

n,

m)

=

# 9000

"

! b(

n $ 1, m)

+

.05

12

b(

n

$ 1, m)

$

m

if

if

n

n

=

>

0

0

becomes

b(

n,

m)

=

# 9000

"

!(

.05

1$

)

12

b(

n

$ 1, m)

$

m

if

if

n

n

=

>

0

0

Students can then use successive

approximation to find m so that

b(36, m) = 0

Comprehensive Curriculum

Recurring Context

Algebra 2

Let’s try it:

What’s the monthly payment on a

5% interest loan of $9000 for 36

months?

CME Project Approach

TI Connection

Comprehensive Curriculum

Recurring Context

Algebra 2

Comprehensive Curriculum

Recurring Context

Algebra 2

Comprehensive Curriculum

Recurring Context

Algebra 2

Comprehensive Curriculum

Recurring Context

Algebra 2

Comprehensive Curriculum

Recurring Context

Algebra 2

CME Project Approach

TI Connection

Comprehensive Curriculum

Recurring Context

Algebra 2/Precalculus

Comprehensive Curriculum

Recurring Context

Algebra 2/Precalculus

CME Project

CME Project

Overview

Problem-Based,

Student-**Center**ed

Approach

Comprehensive

4-year Curriculum

CME Project Availability

Dates

Available right now!

• CME Project Algebra 1 Sampler

Available in Spring 2007

• CME Project Geometry Sampler

• CME Project Technology Sampler

Available in Fall 2007

• Piloting of Algebra 1/Geometry

• CME Algebra 1 text (November 2007)

• CME Geometry text (November 2007)

Available in Spring 2008

• CME Algebra 2 text

Available in Summer 2008

• CME Precalculus text