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-Centered 

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


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.


“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 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


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


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 


Launched with little or no instruction 

Previews upcoming ideas in simple 

contexts 

Provides teachers with a diagnostic tool



Example 1: Arithmetic to Algebra 

Extending a Rule 

Extending number systems 

Extending operations “coherently”


Algebra 1 

The multiplication table, reoriented


In-Class Experiment 

Example 2: More Arithmetic to Algebra 

Instructional Approach 

Launches a topic, or 

Provides experience before formality, or 

Connects ideas


In-Class Experiment 

Example 2: More Arithmetic to Algebra 

Investigating Algebraic Structures 

A central ingredient of algebraic thinking 

Parallel structure between integers and 

polynomials


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


TI Connection


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



Example 3: Geometry to Analysis 

Reasoning by Continuity 

A central ingredient of analytic thinking 

Previews ideas of calculus


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?


TI Connection



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…



Geometry 

Tony 

…like this one. 

Derman 

Why would you do that?


TI Connection



“Students begin to talk like 

the characters” 

A real dialogue about 

order of operations 

from an Algebra 1 

Field Test Site

Sara 

Zoie 

Sara 



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 



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.



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!



Algebra 1 

Zoie 

Sara 

Oooh. Yeah! So it’s not the same. 

Eew. We sound like Tony and 

Sasha.


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


Projects 

Example 4: An Algebraic Interlude 

Patterns are non-trivial 

Justifying the pattern 

Seeking Patterns


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


TI Connection


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



Algebra 1 

Find a function that matches each table:



Algebra 2 

Table A: Two Models



Precalculus


TI Connection



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



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


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



Algebra 1



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?”



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



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



Algebra 2 

Let’s try it: 

What’s the monthly payment on a 

5% interest loan of $9000 for 36 

months?


TI Connection



Algebra 2



Algebra 2



Algebra 2



Algebra 2



Algebra 2


TI Connection



Algebra 2/Precalculus



Algebra 2/Precalculus

CME Project 

CME Project 

Overview 

Problem-Based, 

Student-Centered 

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

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