NCTM
2001 Annual Conference Orlando, FL Session 703.1 
April
6, 2001,
Friday, 12:00 noon  1 P.M. at Room
224F, Orange County Convention Center, Orlando, FL
Kristen Herbert
Wayland Middle School, Wayland, MA 

Kenneth J. Najjar
Maynard High School, Maynard, MA 



Scott Kellett
DoverSherborn Middle School, Dover, MA 

Andrew S.C. Chen
MIT/ EduTron, Winchester, MA 



Dr. George L. Johnston
EduTron, Acton, MA glj@zone.ultranet.com 


NCTM
2001 Annual Conference Orlando, FL Session 703.1 
CLEAR Math
How Does It Work?
How Do We Use It? Does It Help?
Wayland (1 Computer in Classroom)
Maynard (20 Computers in Lab)
DoverSherborn (24 Computers in Lab)
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
What
is CLEAR Math?
Computerbased
Learning Environment with Advanced Remediation
CLEAR Math is designed by MIT scientists and engineers for middle and high
school use. It is being developed by expert groups of teachers and math faculty
and a continuing collaboration between the EduTron team (www.EduTron.com) and college and school
teachers, students, and administrators.
CLEAR
Math is a learning resource for Prealgebra, Algebra I, and Algebra II. It
combines active concept explorations, dynamic tutorials, interactive
stepbystep practices, applications, and testing in an integrated, intelligent
environment. Its imbedded intelligence assesses student competence on a detailed
level, reports its findings to the stateoftheart classroom management system,
and remediates students on the fly. Its competence and tirelessness in the areas
of concept explorations, tutorials, and practices shift the effective
teacher/student ratio closer to 1:1. Furthermore, it thereby frees teachers to
engage in important class and individual instruction.
CLEAR
Math has been proven to be very effective in improving students’ Algebra
performance.
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
Wayland
Middle School’s Experience with CLEAR Math
Hi Andrew!
We just finished up Chapter 10 of CLEAR Math in my
modified algebra course. The
students performed well on the final test.
As you know, we used one computer hooked up to the
overhead to use the program. Although
this method of presentation does not take advantage of all of the capabilities
of the program, it worked pretty well. The lessons and examples were at an
appropriate level for these students. Also, I appreciate the fact that there
aren't a lot of glitzy graphics to take students' attention away from the
content.
One student each day worked the computer.
First we would listen to the "lesson" and students would take
notes. Students asked the student who was working the computer to
"pause" when it was needed. Then
we would do the practice exercises and the activities.
Each student would work the problem at their seat individually or with a
partner. Then I would call on one
student to answer the question. If
the rest of the class generally agreed with the response, we would enter it as
the response and check it with the computer.
This worked well.
The one issue with using the program as the base
of our math program for these students is homework. I wanted to use homework to
reinforce the concepts we covered in class.
However, I could not use our regular algebra text (UCSMP
"Algebra") for this purpose because the concepts are not grouped in
the same manner. I decided to
photocopy pages from Barbara Poole's "Introductory Algebra" text as
homework. Although the concepts are
not introduced in the same order, the examples and problems have a similar
approach and depth so that there was a seamless transition from working with
CLEAR Math in class and doing the homework.
For our next topic in this class, we have returned
to our algebra text. However, I am planning on using the program again after our
February break. I will call you to
invite you in at that time. I think
the students would really like to meet you in person because they have heard so
much about you! Actually, now that
I am thinking about it, if you had some time to come and talk with them before
February break, I think they would love to give you feedback in person!
Thanks!
Kristen Herbert
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
….20Compuer
Lab….
Dear fellow teachers:
I
have been using CLEAR Math at Maynard High School for more than two years.
I have used it as a teaching tool and for the purpose of review for
classes. The students find the immediate response given by the
software helpful. Also, the visual
lessons that the software provides add to the clarity of mathematical concepts
that I, as a teacher, provide with the lessons in the classroom. On the mathematical concepts that the software has available,
I have found that the more students master the concepts and achieve a deeper
understanding, as well as, retain the information better.
The
student reaction to CLEAR Math is very positive.
They like the idea of moving at their own pace and being able to go back
over lessons as they wish. Students
have found the software easy to run and need little time to learn how to move
through it. Even students that do not like computers find that CLEAR Math
is interesting and a very helpful addition to classroom lessons and texts.
I
like to use CLEAR Math to introduce concepts and then I can add to the depth, as
I deem necessary depending on the capability of the class.
Also, I find that using as a review of concepts taught in a prior course
has refreshed the students’ knowledge quicker, clearer, and in less time than
I could do in the classroom. It
allows me to get around to those students that may have more difficulty, to give
them more individual attention.
Feel
free to contact me if you have questions on CLEAR Math.
Ken
Najjar
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
….24Compuer
Lab….
Math Connections
Course
Description
(Semester
course meeting alternate days)
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
…One
PC in Classroom…
Kristen
Herbert
Kristen_Herbert@wayland.k12.ma.us
(508) 6556670
Wayland Middle School, Wayland, MA
Hi Dr. Chen!
Thanks for your inquiry.
We are using the program in my period 4 modified algebra class. I am
projecting it on to a large screen. I
did change the desks around in my room so that all the students could see the
screen and the board at the same time.
The students love the program!
Everyday, I have the students fill out evaluation sheets.
Here are some of the comments (direct quotes from students):
I have written copies of all
the evaluations. I can show them to
you when I see you next. Obviously,
some of the suggestions are problems caused by the speed of our machines not
your program.
Thanks so much!
Kristen
Since January 2000, Kristen has used her Massachusetts DOE Lighthouse Grant to build up a Math Lab with more than 20 computers running CLEAR Math.
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
Students:
“I love it! It catches my mistakes on the fly!”
"I get to practice as much or as little as I want. There are lots of problems."
“It never gets tired of watching over my shoulder.”
“Again, the magical combination of sound, animation and interactivity helps a great deal.”
Teachers:
“This is not fluff. This is meat!”
"MathCad gives the answers instantaneously.
CLEAR Math guides students' work through the how and why, and step by step.
Most importantly, it goes at the student's own pace."
“I call this true interactivity. Why? In MathBlaster, the interactivity (zap, move, pick) has very little to do with math itself. In CLEAR Math, all the items you directly interact with are mathematical objects – equations, lines, points, mathematical words, shapes and terms.”
Students:
"It's a lot clearer than reading a
textbook."
"It's clearer, easier to understand, than a
textbook.”
"The combination of voice + visuals +
stepbystep animation really works!"
“I can proceed at my own pace.”
Teacher:
"I can't do blackboard presentations this
good. There's no way I can use as many colors of chalk, manipulate an eraser
as skillfully, or handwave as expertly. I can't make the words and symbols
fly over the board. I don't have the energy for endless selective or
complete repetition."
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
Other
Teacher Comments
"I
believe CLEAR Math will be extremely useful for classroom and for selfpaced
settings. I am intrigued by how much detailed thinking, planning and programming
were put into this thing. If you ever need me to speak up in your future
marketing campaign, I will be more than willing to do it. This is because I
REALLY believe it will do great things for the teaching communities!"
"This
is wonderful! I witnessed students being engaged in math activities in CLEAR
Math for long periods of time. CLEAR Math is almost the perfect learning tool 
something one only dreamed of."
"I
believe most students will find it easy to use, they would learn better because
their attention has been sparked and they can go over and over at own
pace."
"Diagnosing
students as they progress is the best feature…I also liked the content and the
user interaction."
"It
is flexible, has a variety of formats. ... The voice instruction is very
clear."
"It
is clearly written and the transferring between screens is great!"
"CLEAR
Math represents a novel and revolutionary way in harnessing technology to ease
the teaching and learning process. I am totally impressed by what is being
offered."
"I
am a visual learner, and I just learned from the dynamic example you showed... I
wish we had this program when I was learning Algebra…"
"This
is by far the best Algebra Software I have seen. I would require all of my
students to use it, not just the students requiring additional attention."
"I
can see CLEAR Math increase learning efficiency for first time learners as well
as remedial students… our assistant superintendent definitely should see it
soon."
"This
may be the optimum mix of (NCTM and traditional) material… Personally I am
biased, I prefer to see mostly openended and not precanned content. Your stuff
is good  it does allow student manipulation … it also has multiple
presentations… It contains many facets and components of useful stuff. I like
it. Although I have no economic interests in it, it has the potential for
becoming a successful product."
"All
of my students will be able to use and learn from an environment like CLEAR
Math. The exploration parts that offer students a chance to construct their own
knowledge is very good. The lessons are well done, in fact, I think our teachers
will also learn from them…"
"It
looks like an exciting product. I have not seen any system quite like it. Does
it run on Macs?"
NCTM
2001 Annual Conference Orlando, FL Session 703.1 
CLEAR Math Table of Contents
0.
Arithmetic Review
1.01
Whole Numbers
1.02
Fractions
1.03
Decimals
1.04
Percents
1.
Whole Numbers
2.
Fractions
3.
Decimals
4.
Practical Geometry
5.
Ratio and Proportion
6.
Percent
7.
U.S. Units of Measure
8.
Basic Statistics
9.
Signed Numbers
10.
Variable Expressions
10.01
Properties of Real Numbers
10.02
Evaluating variable expressions
10.03
Simplifying variable expressions
10.04
Translating Expressions
11.
Linear Equations in One Variable
11.01
Solutions to Equations
11.02
Equations: x + a = b
11.03
Equations: ax = b
11.04
Equations: ax + b = c
11.05
Equations: MultiStep
11.06
Equations with Fractions
11.07
Percent Equations
11.08
Translating Words to Algebra
11.09
Absolute Value Equations
12.
Basic Inequalities
13.
Exponents
14.
Polynomial Operations
15.
Polynomial Factoring
15.01
Common Factors
15.02
Factoring Trinomials, a = 1
15.03
Factoring by Grouping
15.04
More on Trinomials
15.05
Special Factoring
15.06
Factoring Strategy
15.07
Solving Quadratic Equations
16.
Rational Expressions
17.
Linear Equations in Two Variables
17.01
Cartesian Coordinate System
17.02
Tables and Graphs
17.03
Slope
17.04
Linear Equations
18.
Linear Inequalities
19.
Systems of Linear Equations
19.01
Graphical Method
19.02
Elimination Method
19.03
Substitution Method
19.04
Graphing Calculator Methods
19.05
Graphing Linear Inequalities
19.06
Solution Space
20.
3x3 Systems of Linear Equations
21.
Roots and Radicals
22.
Rational Exponents
23.
Complex Numbers
24.
Quadratic Equations
25.
Parabolas
26.
Conic Sections
27.
Functions
28.
Matrices
29.
Exponential and Logarithmic Functions
30.
Probability
31.
Statistics
32.
Sequences and Series