From CASE Reports, Volume 13,3

The State of Teaching and Learning of Mathematics and Science in
Connecticut’s Schools

by Robert A. Rosenbaum*

*[Editor’s Note: Robert Rosenbaum, a member of all three academies, is a Principal Investigator on the NSF grant to Project CONNSTRUCT. In addition, he is the University Professor of Mathematics and the Sciences, emeritus, Wesleyan University, and Chairman of the Project to Increase Mastery of Mathematics and Science (PIMMS), a 19-year-old statewide teacher development program with the same goals as CAE.

Several CASE members are actively involved with CAE: Andrew DeRocco, Commissioner of Higher Education, is a member of the Executive Committee of the Academy’s Board of Directors; Roger Howe, Rose Professor of Mathematics at Yale, serves on the Board of Directors; Sr. Clare Markham, professor of chemistry and dean of faculty, emerita, at St. Joseph College, and Hugh Miser are Fellows of the Academy.]

Connecticut has two academies in addition to CASE. One is the venerable Connecticut Academy of Arts and Sciences, perhaps best known to members of CASE as the publisher of a seminal paper on thermodynamics by Josiah Willard Gibbs.

The other is the CT Academy for Education in Mathematics, Science, and Technology, chartered by the Legislature in 1991 to serve as an advocate for excellence in K-12 teaching and learning of mathematics and science, including appropriate use of instructional technologies, as a catalyst for systemic change in such education, and as a coordinator of efforts to achieve the desired excellence. Called, for short, “the CT Academy for Education,” or “CAE,” this sister Academy is the operating arm of Project CONNSTRUCT, which has received large grants under the National Science Foundation’s (NSF’s) Statewide Systemic Initiatives. CAE is a major force in improving mathematics and science education.
The story of K-12 education in mathematics and science in Connecticut contains both bad news and good news. Here are a few examples:

Some bad news is that a teacher, asked by a 12th grade student for help in a “word-problem,” worked through the problem slowly with the student, arriving at 178 = 100x. “So,” said the teacher, “How much is x?” “Oh,” replied the student, “I need pencil and paper to do that.”

Some good news is that an 8th grade student, having been shown how to obtain formulas for the sum of the first n integers and of their cubes, approached the derivation of a formula for the sum of the squares of the first n integers as follows:

Student: “I thought that the desired formula ought to lie, in some sense, between the two formulas that I knew. With a little trial, I came out with

Teacher: “Well, did this prove that your formula was correct?”

Student: “No, I just showed that it worked for a few cases. So I tried the general cubic, substituted values for n = 1,2,3,4, and came out with the same formula.”

Teacher: “Well, had you then proved that your formula was correct?”

Student: “No, because I assumed that the formula is a cubic. So I then proved it by mathematical induction.”

When we realize that mathematical induction, as a method of rigorous proof, is but dimly understood by many college students, it is heartening news indeed to find an 8th grader knowing when and how to apply the method.

Some more bad news: some elementary school teachers do not have command of mathematics and science, and are uneasy about teaching these subjects. The following elementary school teacher’s announcement is both genuine and not rare:
“Class, you’ve earned a reward for good behavior today—we’ll skip math.”

The situation is even worse in science: because science is not included on the Connecticut Mastery Tests (CMT)—the tests given at the beginning of grades four, six, and eight—some teachers virtually ignore science.

A major item of good news: average scores on the mathematics portion of the CMT are going up. So are average scores on the mathematics and science portions of the Connecticut Academic Performance Test (CAPT), the tests given at grade 10. So are the average scores on the SAT. [Some data appear in the accompanying table.] Details on the upward trends in scores on CMT, CAPT, and SAT; on increased enrollments in advanced mathematics and science courses; and on Connecticut’s favorable rating in recent reports of the National Assessment of Educational Progress (NAEP) are available from the CAE web site at

How is it possible to have such dramatically different illustrative stories about education in Connecticut?

The answer is: Connecticut is a “schizoid” state, split in many ways, including education, between the economically depressed districts—all the urban, and some of the rural—and the affluent districts—largely suburban.

A report only on average scores (as though Connecticut were homogeneous!) is as misleading as the statement of the fellow, sitting on a hot stove with his feet in a bucket of ice water, “On average, my temperature is pretty comfortable.”

Groups involved in and responsible for formal education—the Connecticut State Department of Education (CSDE), the
Connecticut Association of Boards of Education, the various associations of school administrators, the teachers’ unions, CAE, professional associations like the Connecticut Science Teachers Association (CSTA) and the Associated Teachers of Mathematics in Connecticut (ATOMIC), and many more—consider the disparity between the affluent and the economically depressed districts to be unacceptable, and are committed to excellent education for all. Despite that commitment, the goal is not easy to achieve, for several reasons.

For one thing, societal problems impinge on what schools can accomplish. Mobility is high. In some urban schools, none of the students in a class early in September are still in that school the following June. Moreover, absenteeism is a major problem, occasioned by various circumstances such as the lack of new clothes; the early morning departure of parents for work, leaving students at home with no “push” to get to school; and the breakdown of day-care arrangements, requiring a high-school student to stay home to take care of a young sibling. Some teachers find that 25% to 30% of their students are absent, day after day—but not always the same students. It is fatuous to hold teachers accountable in such circumstances—even Mark Hopkins couldn’t teach if his student didn’t show up at the other end of the log.

The societal problems noted in the previous paragraph are principally associated with poverty. But there is another problem, not related to poverty, that also affects education. Many people recognize education for its credentialing value, but they don’t esteem education for its own sake, nor do they appreciate that real education requires sustained hard work. Education is not generally assigned high value, even by parents who have had much schooling. In affluent communities, parents often take their children out of school for a family winter vacation of skiing or snorkeling. And many high school teachers report that they don’t give juniors and seniors as much homework as they consider desirable because those students hold jobs after school—to pay the operating costs of the cars their parents have given them.

Of the many issues involved in mathematics and science education, there is space here to address only the most significant and urgent:

1. NCTM and NRC Standards. The National Council of Teachers of Mathematics, along with other organizations, published a set of volumes containing suggestions and recommendations about Curriculum (1989), Instruction (1991), and Assessment (1995). Awareness of these NCTM Standards has become widespread in the education community, many new teaching materials are aligned with them, and they have become a guiding principle for NSF’s Directorate of Education and Human resources (EHR).

There are analogous Standards in the sciences, developed by the National Research Council (NRC) and other organizations.

Having undergone several years of scrutiny and testing in the courts of professional and public opinion, the NCTM Standards are currently undergoing revision. Mathematicians and parents (especially mathematical parents) have not adopted the NCTM Standards with the unanimity displayed by NSF’s EHR.*

Indeed, discussion has been so heated and sometimes acrimonious, on the Internet and elsewhere, that CAE has conducted two conferences—one, in June 1997, devoted to increasing areas of agreement about reform of mathematics education, and the second, in June 1998, on appropriate uses of instructional technologies to enhance the teaching and learning of mathematics.

One of the criticisms of the NCTM Standards and of the teaching in our schools is that students, these days, don’t learn the “basics.” But scores on the CMTs show that, whatever the case may be in other states, Connecticut students are not deficient in the basics. For example, on the 1997 mathematics 6th grade CMT, 93% of students achieved mastery on multiplication and division facts. Faced with the numbers 7 and 9 in a significant problem, the vast majority of students immediately know the value of their product. The difficulty is that many don’t know whether multiplication is the appropriate operation for the problem at hand.

2. Individual Differences; High Expectations. All of us know that individuals have different abilities, motivations, learning styles, and rates of development. The Standards (see above)

*Similarly, the NRC Standards have not met with universal approval in their current form, with sharp criticism centering on the Standards’ alleged emphasis on process at the expense of facts.
devote attention to ways that these differences can be accommodated in a classroom.

Crucial elements in all education are expectations about student achievement—expectations of parents and teachers, and of students themselves. To reach our goal of excellent education for all, we must hold high expectations for all students, provide support for all students to perform up to their potential, and anticipate that they will strive for the highest achievement of which they are capable. Expectations, high or low, commonly turn into self-fulfilling prophecies.

3. Teacher Preparation. There are 15 public and private colleges and universities in the state that prepare students for teacher certification by CSDE. Most of them are seriously re-examining their programs for prospective mathematics and science teachers, to increase their effectiveness. To aid in the process, CAE has a Task Force on Higher Education, which is striving to deal with the following matters, among others:

Moreover, while instructional technologies can be powerful aids to teaching and learning, mere access to hardware and software is not sufficient. Hardware and software must be supplemented by extensive guidance in how best to use these aids in the classroom and laboratory. Professional development programs (see #6 on page 9) provide this guidance for inservice teachers; faculty of teacher-preparation programs should do the same for future teachers.

Unless our colleges and universities strengthen the preparation of prospective teachers, we will forever be playing “catch-up,” with ongoing professional development programs devoted solely to addressing deficiencies in teachers’ backgrounds.

4. The Roles of Reading and Writing in the Learning of Mathematics and Science. An apparent paradox: students’ scores on the SAT (Verbal) are better predictors of college success in mathematics and science than are scores on the SAT (Mathematical). Considering what students are called upon to do in a classroom or laboratory, we should not be astonished by this fact; and the Standards emphasize the importance of reading with understanding, and of facility with written and oral expression.

Unfortunately, some teachers of mathematics and science undervalue the learning of reading and writing, to the detriment of their students’ progress. CSDE’s consultants (Steven Leinwand and Mari Muri in mathematics; Steven Weinberg in science) are working hard to rectify the balance.

5. Remediation. Many students do not have good study habits. Some students don’t learn well, but are passed along to the next grade, anyhow. Few students internalize what they learn, relating new material to what they have previously studied. As a result of such factors, our educational system is permeated with remediation at all levels.

It is probably unrealistic to expect to obviate the need for all remediation, but the vast amounts needed in high schools and two- and four-year colleges incur costs of human and financial resources that society can ill afford to pay. Some schools use imaginative ways to address remediation, successfully rescuing students before they fall too far behind; but in many schools and colleges the problem remains serious.

6. Professional Development. Members of CASE well know that mathematics and science are continually developing and evolving, with the changes in the disciplines requiring that teachers reconsider what they teach and how they teach it. Thus, every teacher, no matter how well prepared initially, needs continuing professional development to remain current, just as a physician does.

Some school districts provide their own staff development to their teachers and administrators. In addition, there are numerous professional development providers around the state, including the mathematics and science consultants of CSDE; the Regional Educational Service Centers (RESC’s); some of the colleges, universities, and museums; SMARTCenter at Sacred Heart University; Talcott Mountain Science Center; and the Project to Increase Mastery of Mathematics and Science (PIMMS) at Wesleyan. For example, each summer PIMMS provides substantial programs for about 750 teachers in grades K-14, with emphasis on subject-matter content and effective teaching strategies. Attention is also paid to leadership development so that PIMMS Fellows generate a multiplier effect through the workshops that they offer to colleagues in their own and neighboring schools. For further information, go to

To maintain certification, a teacher must earn nine Continuing Education Units (CEUs) over the course of each five-year period. This represents 90 hours of attendance at professional development programs—a useful step, but not really adequate to the need. In particular, many teachers require extensive and
sustained guidance in appropriate uses of instructional technologies.

7. The Connecticut Academy for Education. To give readers a feel for the activities of CAE, we note here some of its undertakings, in addition to those already mentioned. There is not sufficient space to do more than sketch even these few examples. To learn more, go to

In 1991, Connecticut was in the first cohort of 10 states to receive grants from NSF’s Statewide Systemic Initiative; and, at the end of five years, it was one of only two to receive a renewal grant. This bespeaks NSF’s recognition of CAE’s substantial contributions to the improvement of mathematics and science education—contributions made by many people, including the Academy’s dedicated Executive Director and staff, Board of Directors, and Fellows. Further help is always welcome.

Steve Leinwand says, “The overall situation is good, but not good enough. Things are getting better, but lots more is needed to raise the floor and close the gap.” It is reasonable to hope that, through the efforts of the large number of Connecticut citizens committed to excellent education for all, we will eventually reach our goal.

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