The Evaluation of State Mathematics Standards: New Jersey Perspectives

Joseph G. Rosenstein

Rutgers University, New Brunswick, New Jersey

(Presented, in abbreviated form, as part of a panel on "The Evaluation of State Mathematics
Standards" at the Joint Meeting of the American Mathematical Society and the Mathematics
Association of America, San Antonio, Texas, January 16, 1999.)

I am very pleased to be here today to share with you some information about the
development of the New Jersey Mathematics Standards, and my thoughts about the evaluation of
state standards and other related matters.

Let me begin with a few words about myself. I am a professor of mathematics at Rutgers
University. In the research phase of my career, I wrote a number of articles in mathematics and
also a monograph entitled "Linear Orderings" which was published in the Academic Press Series
on Pure and Applied Mathematics. About fifteen years ago I started to work on improving
mathematics education, and for the last ten years, have devoted essentially all of my efforts on
this area.

One of my focuses has been discrete mathematics -- I direct NSF-funded teacher
enhancement projects that have involved over 1000 teachers from all K-12 grade levels in doing
discrete mathematics. One outgrowth of these projects is that we are for the first time able to
describe what topics in discrete mathematics are appropriate for children at all grade levels,
based in part on the classroom experiences of teachers in the programs. This is described in a
volume called "Discrete Mathematics in the Schools", which is likely the only volume ever co-published by AMS and NCTM. These activities have been under the co-sponsorship of two
centers at Rutgers, the Center for Mathematics, Science, and Computer Education, and the
Center for Discrete Mathematics and Theoretical Computer Science, known as "DIMACS", an
NSF-sponsored Science and Technology Center.

Another important focus of my efforts has been the New Jersey Mathematics Coalition,
of which I was a founder, and which I have served as director since its inception in 1991. The
New Jersey Mathematics Coalition is an affiliate of NASSMC, the National Alliance of State
Science and Mathematics Coalitions, which together with the MAA and the AMS Committee on
Education, is co-sponsoring today's panel discussion. The New Jersey Mathematics Coalition is
governed by a board which includes all sectors of the community -- teachers of mathematics
from elementary grades through college, mathematicians, policy makers, school and district
administrators, and members of the business community and the public.

One of the first accomplishments of the New Jersey Mathematics Coalition was obtaining
a grant from the United States Department of Education in 1992 to develop mathematics
standards and a mathematics curriculum framework for New Jersey. This grant was actually
received through the New Jersey Department of Education, as required by the proposal
guidelines. Two strengths of the proposal were, first, that the Department was in the process of
developing and adopting standards in all content areas, so that the standards developed through
the funded project would actually be adopted by the state, and, second, the Coalition Board
involved representatives of all the stakeholders in mathematics education, so that broad
involvement and support would be available in the process of developing the standards and
framework.

Members of the Coalition Board participated actively in the panel that developed the
initial version of the mathematics standards in 1993. After the completion of the draft standards
document, it was circulated widely by the New Jersey Mathematics Coalition -- to all those
involved in mathematics education in the state -- and comments were received from nearly 300
individuals.

With the completion of a draft of the standards, work began on developing a curriculum
framework based on these standards. After much discussion, it was decided that the purpose of
the mathematics curriculum framework would be to provide information and guidance to
teachers and districts in implementing the standards. Although the terms "standards" and
"frameworks" are apparently used in many different ways, in New Jersey we are very clear about
the difference between the two. "Standards" describe the goals of education, and "frameworks"
provide assistance in reaching those goals. "Standards" describe what we value, and
"frameworks" help us achieve what we value. In many states, what is called "standards and
frameworks" correspond to what we in New Jersey think of as standards alone.

New Jersey is a "home-rule" state -- each of its nearly 700 districts insists on making its
own decisions about the curriculum in its schools. The framework is not intended to be a
curriculum. It is intended to be a framework, like that of a house, which each district can use to
build its own curriculum or curricula.

As with a road-map, there are many possible routes that will get you to your goal, but
your best route obviously has to take into consideration your starting point. Thus, a framework
should provide guidance in making good choices, and not simply prescriptions of what should be
done. This is particularly true because high standards cannot be implemented overnight and
therefore require prioritization of what is to be done first.

I don't believe that a document like the New Jersey Mathematics Curriculum Framework exists in many other states.

Between 1993 and 1995, the adoption of standards moved to the back-burner in the
Department of Education, with the arrival of a new Commissioner of Education, then the election
of a new Governor, and then, of course, the appointment of a new Commissioner of Education.
In the meantime, however, the New Jersey Mathematics Coalition involved over 100 people in
different aspects of the development of a preliminary version of the curriculum framework, and,
as called for in the project proposal, the Coalition established teams in thirty district clusters,
which used the framework as a guide to implement the standards in their schools. All of these
efforts led to many comments about the draft version of the standards and the preliminary
version of the framework. The involvement of an extensive and diverse group of people in the
development of the standards and framework turned out to be very significant, because when the
Governor and the Department of Education returned to the development of standards in 1995, we
had in place not only a set of standards and a preliminary framework, but also a community-based model for how both standards and frameworks can be developed, and widespread
involvement in and approval for the recommendations of the standards.

New panels were set up by the New Jersey Department of Education in 1995 to review
and modify the earlier drafts of the standards, and comments were solicited very widely from
both educators and the public. Standards were adopted in all content areas in May 1996 by the
New Jersey State Board of Education. Once it became clear that the mathematics standards that
we had generated would likely be adopted by the Board without major changes, we proceeded to
develop the revised version of the New Jersey Mathematics Curriculum Framework. This was
completed in December 1996 and displayed to the legislature in the Governor's state of the state
address the following month. Given the positive reception accorded the mathematics curriculum
framework, the New Jersey Department of Education embarked on the development of
frameworks in all other content areas; moreover, it adopted our development model by recruiting
for each content area an appropriate agency outside of the state department of education to serve
in a leadership role.

The New Jersey Department of Education has taken the standards-based approach beyond
the development of standards and frameworks. Indeed, since the standards reflect what we value
in mathematics education, our statewide tests should assess whether students are achieving the
goals of the standards, and we should be encouraging teachers to become involved in standards-based professional development activities. The state assessment program includes the
development of new standards-based assessments in all content areas at grades 4, 8, and 11, and a
state Professional Teaching Standards Board has been appointed to ensure that districts
implement standards-based professional development in response to newly adopted state
regulations.

A few words about the structure of the New Jersey Mathematics Standards and the New
Jersey Mathematics Curriculum Framework. There are altogether 16 standards, each consisting
of a short declarative statement, a longer "descriptive statement", and three sets of "cumulative
progress indicators" which describe what students should know and be able to do at the end of
the 4^{th}, 8^{th}, and 12^{th} grades. Roughly speaking, the curriculum framework has a chapter devoted
to each standard. The chapter begins with a K-12 overview of the standard, which provides
district personnel with a perspective on how they might implement the standard across grade
levels. After that, each chapter contains five sections, directed to teachers at the K-2, 3-4, 5-6, 7-8, and secondary grade levels; each section includes an overview of the standard at those grade-levels and activities which would help the teacher implement each individual cumulative
progress indicator at those grade levels. The framework was distributed in loose-leaf format so
that teachers could copy sections that were appropriate to their grade level; the framework is also
available on the Web (http://dimacs.rutgers.edu/nj_math_coalition/framework.html) and on CD-Rom (with the science framework).

I should add that I played a leadership role in the development and adoption of the
standards and the framework. To my knowledge, New Jersey was the only state where a
mathematician was involved so intimately in the development of state standards.

People often ask how New Jersey has avoided the controversies that have arisen in
California. Let me offer three reasons for this. First, as I have already emphasized, the process
of developing the standards and framework was inclusive. All were invited to participate, all
views were shared, all views were discussed. Second, common sense prevailed over ideology.
On issues such as use of calculators, instructional methods, grouping of students, we tended to
make recommendations which took into consideration the valid and sensible arguments of both
sides. The preface of the framework states: "The recommendations provided here are very
specific. Yet, it is not intended that they be implemented dogmatically; different situations call
for different responses and different strategies. In education, as in other areas, there is a tendency
to swing from one extreme to another. We hope that educators will utilize their common sense,
judgment, and experience in finding appropriate ways of using the recommendations in this
Framework to inform their decision-making." Finally, in California, education decisions are
made in a way that invites confrontation and politicization. On one important spectrum, "state
mandate" vs. "home rule", New Jersey is at the opposite end from California. In California, it is
determined at the state level which textbooks may be used in the state; in New Jersey, each of
our 700 districts decides for itself. Large-scale decisions like that invite protest and
politicization. Ballot referendums have the same effect. In California, programs can be initiated
or abolished by referendum, so California will now swing from an all-ESL state to a no-ESL
state, instead of allowing local conditions to determine the kinds of programs are most suitable.
To summarize, I believe that New Jersey has not experienced the "math wars" because we have
involved all stakeholders in the process of developing the standards and framework, we have
used common sense in drafting our documents, and, serendipitously, we have avoided the
politicization of mathematics education.

At this point, it would be appropriate to review the ratings that the New Jersey
Mathematics Standards received from each of the three groups that reviewed state standards.
Interestingly, our scores were all different -- an "A", a "C", and a "D".

In the rankings of the Council for Basic Education, New Jersey was the only state to
receive an "A" for the rigor of its mathematics standards. That means that our standards, when
compared with each of the 81 benchmarks in the C.B.E. mathematics framework, received an
average score of at least 3.7 out of 4. I don't have a copy of the item-by-item report; however,
my calculations indicate that we must have received a score of 4 on at least 60 of the 81 items,
indicating that with respect to those 60 benchmarks, the New Jersey Standards "require every
student to learn all of the essential concepts and skills as defined by the framework benchmark at
or above the level of sophistication specified." It should be noted that two of the New Jersey
Mathematics Standards were not referenced at all in the C.B.E. benchmarks -- the areas of
discrete mathematics and underpinnings of calculus, in both of which the New Jersey standards
go far beyond what is expected in other states. So I will continue to state my "unbiased" opinion
that New Jersey's Mathematics Standards are the strongest in the nation.

Since my comments about the other two reports will take me in a different direction, I
want to point out that the information provided by the C.B.E. report, unlike that of the other
reports, will indeed be useful when it comes to the five-year revision of our mathematics
standards that is due to begin this year. That is because the C.B.E. report provides concrete
information about how our standards correspond to a set of independently generated and
approved benchmarks; the other two reports only provide general information.

Can the C.B.E. benchmarks be improved? Certainly. And that would be a very
worthwhile effort. A broadly approved set of benchmarks would be very valuable to every state
that is reviewing its standards, and would undoubtedly influence most states to move in the
direction of those benchmarks.

In the rankings of the 1997 report of the American Federation of Teachers, New Jersey's
mathematics standards received a grade of "D". The only statement made in its report was that
"the math standards overemphasize skills without adequate grounding in content knowledge". In
a telephone discussion I learned that this comment was based on our placing at the beginning of
our list of standards those which dealt with problem-solving, reasoning, communications, and
connections. I examined the standards of states which received an "A" from the AFT, and found
no evidence that our standards were in any way weaker than theirs. (The framework was
apparently not taken into consideration since it had no official status.) In the 1998 AFT report,
recently released, no ratings are given; however, this time the New Jersey Mathematics Standards
receives a check mark because "the new Test Specifications strengthen the math standards
significantly by clarifying the specific content and skills that are absent in the standards". For
example, they note that, according to the test specifications, eighth graders should be able to
"find equivalent forms of fractions, decimals, and percents", apparently preferring that
formulation to the more general formulation in the standards, which say that all students should
"investigate the relationships among fractions, decimals, and percents, and use all of them
appropriately". Of course, the committee which developed the test specifications based them on
the standards, so it appears that the AFT's previous ranking was based less on what the standards
said, but on the format that was used.

Finally, in the rankings of the Fordham Foundation, New Jersey's mathematics standards
received a ranking of "C". In this study, that was a good ranking -- indeed, only 12 states
received "A" or "B", and New Jersey's score was clearly the highest of those getting a grade of
"C", so we placed 13 out of the 46 states rated. Curiously, the New Jersey mathematics
standards received high marks at the elementary and middle school levels, but a very low mark at
the secondary level, and a comment about the "falling off of Content in the high school years".
This is particularly striking because New Jersey's mathematics standards includes benchmarks
under probability and statistics, discrete mathematics, and underpinnings of calculus that likely
go far beyond what is expected in every other state. That was apparently ignored, however, but
the report did cite that "no proof of the Pythagorean theorem is demanded". (At the 8^{th} grade
level, the standards say that "all students should understand and apply the Pythagorean
theorem".)

What disturbed me about the Fordham report was the bias of the authors. One bias was
their focus on the words "proof" and "prove". In their brief discussion of the New Jersey
standards, the authors note that "the classical content of mathematics, and its backbone of
deductive reasoning ... , are often slighted in these standards", and later they note disparagingly
that in our discussion of the evaluation of the sums of finite and infinite series, we omitted the
word "prove". The rare use of the words "prove" and "proof" was a conscious decision on our
part. Outside of our own circles, those words evoke the almost universal response "two-column
proofs in geometry, the kind I had to memorize in high school, ugh!" Our goal was not to
rehabilitate those words, but, rather, to get students to reason. We insisted throughout the
standards that all students at all grade levels be able to understand, to explain, to demonstrate,
and to justify all of the steps that they carried out. That was not good enough for the authors of
the Fordham Report; we didn't "demand" proofs.

A second bias is their aversion to "real world" terminology. We challenged teachers to
motivate their students through the many connections between mathematics and the real world,
between mathematics and their world. That is one reason we gave greater focus, at the secondary
level and earlier, to the areas of probability and statistics, discrete mathematics, and the concepts
of calculus. Here is what the authors say about this: "There is visible in these documents a
currently fashionable ideology concerning the nature of mathematics that is destructive of its
proper teaching. That is, mathematics is today widely regarded (in the schools) as something
that must be presented as usable, "practical", and applicable to "real-world" problems at every
stage of schooling, rather than as an intellectual adventure." They go on to note that despite its
successes in modeling reality, mathematics is to be seen as a "deductive system".

Let me ask you how many of your first-year liberal arts students would respond
positively to the idea of mathematics as an intellectual adventure, or to the idea of mathematics
as a deductive system. Then imagine how many students in an ordinary high school would
respond positively to mathematics were it presented in that way. Then imagine how students in
an urban high school would respond. Now all of us here became mathematicians because we
loved the intellectual adventure, because we loved playing with equations. But that's not how
the rest of the population appreciates mathematics. Students in the United States want to know
"what it's good for", and the only way we will get them to appreciate mathematics is through its
applications.

In a way the biases of the Fordham Report crystallize what "the math wars" are all about.
It should be understood that there are two major problems for mathematics education in this
country. One problem is to provide a good background in mathematics to all students, so that
they will be able to find jobs in our increasingly global, technological, and information-based
economy, and so that our nation will have enough skilled human resources to meet employment
needs. Let's refer to this as the 80% problem. The second problem is to train the highly
qualified personnel that will replace us -- the mathematicians, computer scientists, scientists,
engineers. Let's refer to this as the 15% problem. Both problems are very important. Let me repeat that -- both problems are very important.

The standards movement primarily addresses the 80% problem. It seeks to improve the
mathematical knowledge and competency of the student population as a whole. Its focus is not
on dealing with the 15% problem. Many of those opposed to the standards movement are
primarily concerned with the 15% problem -- in part because of their concerns about future
scientists and mathematicians, in part because of concerns about their own children's future, in
part because of excesses of the standards movement (where, as always, dogma tries to drive out
common sense), and in part because some of those who support the standards movement do not
recognize that there is a 15% problem at all.

Unfortunately, the solutions offered by those opposed to the standards movement may be
solutions to the 15% problem but do not address the 80% problem. For example, an "intellectual
adventure" or "deductive reasoning" approach to mathematics may help solve the 15% problem;
but it will be entirely useless in dealing with the 80% problem.

We must reject the two radical perspectives that fuel the "math wars" -- the one which
says, in effect, that we should forget about the 15% problem and focus entirely on the 80%
problem, and the one which says, in effect, that we should forget about the 80% problem and
focus entirely on the 15% problem. Each problem needs its own set of solutions, and trying to
impose a 15% solution on the 80% problem is counter-productive.

The common sense perspective is to combine a "standards" approach to dealing with the 80% problem with a "beyond the standards" approach to dealing with the 15% problem. That's what our focus should be. Both problems can be addressed, and must be addressed, by our society. And the mathematical community should be supporting efforts which work toward addressing both problems.