I have been following closely the controversy over the MCPS proposal to
pilot a new middle school math curriculum developed under an NSF grant.
I have a Ph.D. in mathematics and was a National Merit Scholar, a
Presidential Scholar, and an NSF Graduate Fellow. I have two children
in the MCPS, one in high school and one in middle school, but not in
the
proposed pilot. Given the serious criticisms that I have been reading,
I felt it was necessary personally to take an in-depth look at the
materials on the Connected Mathematics Project (CMP) where most of
the discussion has focused.

I have spent many hours analyzing the Guide to Connected Mathematics
produced by the CMP, the Teacher's Editions of several of the CMP
curriculum units, the National Council of Teachers of Mathematics
(NCTM) Curriculum and Evaluation Standards for School Mathematics
(1989), the
discussion draft NCTM Principles and Standards Document (1998), the
MCPS Instructional System of Mathematics (ISM) Objectives, the
California
Mathematics Academic Content Standards, my children's notes from their
middle school magnet math classes, and some of the research that
underlies the NCTM Standards and the proposed curriculum changes.
I evaluated CMP based on philosophy and goals, teaching methodologies,
use of technology, student assessment methodologies, mathematical
content,
supporting research, and evidence of improved outcomes. The conclusion
of my
evaluation is that the critics are correct. CMP is seriously flawed in
all areas.

It would take many pages for me to document all the problems I found in
CMP, but I would like to summarize some of my findings.
 
Philosophy and goals: CMP wants students to learn that math "is
arbitrary, and good solutions are arrived at by consensus among those
who are
considered expert." It wants to "show students that many mathematical
questions have more than one right answer." These are fundamentally
wrong statements that will confuse students about how formal proof is
used in
mathematics to determine truth.

Teaching methodologies: Different students will respond to different
approaches and paces. CMP, however, uses a dogmatic, one-size-fits-all
methodology that diminishes the role of the teacher in teacher-led
transmission of knowledge. CMP utilizes group cooperative learning
through student-led discovery activities and discussions where students
wander
around in hands-on investigations and "bump into the mathematics". The
activities and projects depend too heavily on manipulatives and other
concrete objects delaying the transition to symbolic abstraction that
is the essence of real math. The teacher is frequently reduced to a
group
organizer and facilitator. The discovery method, particularly group
discovery, is just too slow and inefficient to cover much material or
depth. The methodology also diminishes the time spent on math by
including too much extraneous writing, picture drawing, paper folding,
etc.. CMP sets up a straw man comparison between a traditional math
class
where, according to CMP, students learn by rote procedures that they do
not
understand and CMP where students learn to understand deep mathematical
concepts, but at the price of not learning procedures. This is a
dishonest comparison. While a traditional math class can be poor, there
are
excellent traditional math classes. The teacher makes the difference.
Parents should expect their children to learn facility with procedures
and understand the concepts behind them as well. Good teacher-led
classes
can and do achieve both.

Technology: Calculators and computers have an important role to play in
math classes, but they must be used wisely. They cannot be used as a
substitute for knowledge of procedures or how to calculate. As the NCTM
Principles and Standards Document says, "a large caveat must be issued.
enriched student understanding is possible. IF [emphasis in original]
the technology is used appropriately. Flashy. demonstrations may leave
students no more knowledgeable than before. .access to calculators does
not replace the need for students to learn and become fluent with basic
arithmetic facts. and to perform algebraic manipulations.."
Unfortunately,
CMP allows indiscriminate use of calculators ("Students will have
access to calculators at all times.") and gives no emphasis to facility
with
computational procedures or algebraic manipulations.

Student assessments: CMP utilizes fuzzy assessments that mask the math
shortcomings of students, grade them on activities not related to math,
and homogenize their grades. For example, quizzes in CMP are pair
quizzes. "Students work in pairs on the tasks. Students are able to use
their
notebooks, calculators, and any other appropriate equipment when doing
quizzes. Student pairs are given the opportunity to have their quiz
examined once by the teacher and they can then revise their work,
without penalty. The pair's first attempt . on the quiz is considered a
first
draft that is being submitted for teacher input. Following revision
based on teacher input the team can turn in the finished product." CMP
also
says, "The nature of the problems in these assessment pieces require[s]
that teachers readjust their expectation that 80% of the students will
show 80% mastery. Rather, teachers will want to use holistic scoring
techniques and multidimensional weighted grading scales that take into
account the many dimensions addressed by the test". What are some of
these "many dimensions"? Well, a lot of discussing, writing and drawing
takes
place in CMP. Students are graded on how well they "help others in the
class to understand the mathematics". Here is some of the scoring
rubric for a major project, which is to plan a park: "A total of 50
points is
possible. 23 for the scale drawing, 22 for the report, and 5 for the
letter.." Correct math calculations count for 6 of the 50 points.
Neatness, organization, grammar and spelling can be worth up to 8
points. Homework is not checked or collected until "students are given
the
opportunity to ask questions about the assignment. The teacher does not
give answers or tell how to solve the problem but, with the classes
[sic] help, tries to work with students to understand what the question
is
asking. Students have the right to revise any of their work while this
conversation is going on and not be marked down".

Math content: The CMP content is not without merit; it does contain
some good math. However, its content is less than the ISM Objectives,
particularly the revisions in ISM proposed by Nancy Metz. The CMP
content fares even worse when compared to the more demanding California
Standards. Many MCPS magnet children currently take Algebra I in 8th or

even 7th grade. A worthy improvement would be to have on-track students
take
Algebra I in 8th grade. Here is what CMP says is not in CMP that is in
the traditional Algebra I course: " emphasis on manipulating symbolic
expressions, such as multiplying and factoring polynomials; operations
on algebraic functions; formal solutions of linear systems in 2 or more
variables; formal study of direct and inverse variation; radicals and
simplification of radicals; operations on polynomials other than linear
polynomials; and completion of the square and the quadratic formula."
CMP divides its math vocabulary into terms that are essential and terms
that are nonessential for understanding the unit and future units. Here
are
some terms that CMP says are nonessential in the two dimensional
measurement and two dimensional geometry units: "diagonal, equilateral
triangle, interior angle, isosceles triangle, trapezoid, base, height,
length, width, and perpendicular." About their two dimensional geometry
unit, CMP says, "we do not stress the fact that all squares are
rectangles and that all rectangles are parallelograms. Furthermore, we
use a few special names for types of quadrilaterals (square, rectangle,
parallelogram), but not the arcane vocabulary (obtuse, acute, scalene)
often used to sort triangles."

Supporting research: Most persons will be surprised to learn how little
support research provides for proposed innovations in math curricula
such as CMP. A very good and candid article, "Relationships between
Research
and the NCTM Standards," by James Hiebart appears on the NCTM Web site.
The research is also much more cautious and nuanced in its findings
than programs like CMP are in their dogmatic approaches.

Evidence of CMP effectiveness: Studies designed to evaluate CMP are
self-serving evaluations that define effectiveness in terms of CMP's
values. However, CMP admits "some CMP students may not do as well on
parts of standardized tests assessing computational skills.." The
phrase
"computational skills" includes algebraic manipulations as well as
arithmetic. Not to worry, CMP believes the "trade-off" is worth it.
Anyway, "calculators are available to perform computation with larger
numbers." Most parents should not be satisfied with such a trade-off.
It is possible, in fact necessary, to have both good computational
skills
and mathematical understanding.

Superintendent Weast has evidently ordered a review of the pilot. This
is good, but it must not be a superficial review and whitewash. MCPS
administrators must be open and share information with parents. MCPS
needs to do more than give lip service to parental involvement. There
needs
to more of a partnership.

I would like to offer some positive alternatives and not just criticize
implementing programs like CMP. Although it is true that traditional
math classes often are not as good as they should be, there are no
faddish
shortcuts to producing the excellent math classes that all children
need and deserve. The solution lies in several areas.

Students' responsibility is to learn, not, as in CMP, to teach.
Students must be held to high expectations, which include both a
facility
with procedures and an understanding of the concepts behind them.

The curriculum must be strengthened. The standards should be more like
California's. The American math curriculum is too often "a mile wide
and an inch deep." Math courses should cover fewer topics but cover
them in
more depth. On-track students should be prepared for Algebra I in 8th
grade.
 
 

Technology should be used properly to illustrate concepts that
otherwise could not be demonstrated or to remove the tedium of
intricate
calculation after the procedure is understood.

Student assessments must be real and rigorous. Program, school, and
teacher effectiveness must be measured by meaningful, objective tests
and statistics. However, as statistical quality control expert Dr. W.
Edwards Deming pointed out, you cannot inspect quality into a system.
Dr.
Deming also warned the dangers of misinterpreting random variation in
statistical measures as meaningful changes.

The key to excellent math classes is the teacher. All of us remember a
class that inspired us because of a charismatic teacher. MCPS must have
good teachers with a good command of the subject. MCPS does have some
excellent math teachers in middle school; my children have had some of
them. MCPS needs to identify these master teachers, their curriculum
content, and methods. MCPS must train other teachers to be more like
them. This takes money, which is in short supply. MCPS, however, should
not
be seduced by NSF grant money to train teachers in a program that will
diminish the role of the teacher and hurt the children.