August 21, 1997
Honorable Carl T Hayden
New York State Board of Regents
303 William Street
Elmira, NY 14901
Dear Chancellor Hayden:
We wish to call your attention to serious defects in the proposed New York State standards for school mathematics. These weak standards will go a long way to destroying what is left of a once excellent program in school mathematics.
Some of us gratefully recall the fine Regents programs in high school mathematics consisting in part of algebra, geometry, intermediate algebra and trigonometry, or later, of algebra, geometry, and eleventh year mathematics. Designed for average to above average students in high school mathematics, these courses provided a fine basis for subsequent courses in pre-calculus and calculus. The outstanding success of so many graduates from New York State high schools who received Regents diplomas gives eloquent testimony to the quality of those Regents programs.
With some exceptions, the mathematical content of the current Regents sequence, (Courses I, II and III), is already not as rich as that of those older sequences. Just the same, with few exceptions, the mathematical content of Courses I, II and III is better by far than that of the mathematics program outlined in The Learning Standards for Mathematics, Science, and Technology (Revised Edition, March 1996, The University of the State of New York, New York State Education Department, Albany, NY 12234). Although these standards embrace work from primary school through high school, the focus of this letter will be mainly on the high school mathematics program.
Proponents of these new standards argue that the standards, designed for all students, represent a toughening of graduation requirements for New York State students. For the very weakest students who in the past took Regents competency tests instead of the Regents exams, and for those who graduated with local diplomas, these new standards may represent a step up. But for the tens of thousands of students who today graduate after passing three Regents examinations in high school mathematics, the new The Learning Standards for Mathematics, Science and Technology (MST) represent an alarming drop in the content and quality of school mathematics.
The MST will push mastery of arithmetic, algebra and geometry to a new all time low. Factoring is de-emphasized, and treatment of the quadratic equation has apparently been postponed beyond the first two years of high school mathematics. The important technique of completing the square is not mentioned at all.
The prose itself causes one to distrust the content of the document, eg on page 22 of the MST, the discriminant of a quadratic equation is referred to as the "discriminate of a quadratic equation."
Despite their unpopularity in many educational circles, traditional word problems really did develop problem solving. These problems are barely visible in Standard 3, the mathematics component of the MST, although they make cameo appearances elsewhere in the document. So, we were pleasantly surprised to see some simple word problems on a pilot exam entitled "Mathematics A Regents Assessment Spring 1997 Pilot Questions."
The philosophy behind the curricular changes now underway in New York derives from an earlier version of the MST called the Curriculum, Instruction and Assessment Framework for Mathematics, Science and Technology (March 1994, The University of the State of New York, New York State Education Department, Albany, NY 12234). The writing there is also well seasoned with educational jargon. On page 10 of the document, one reads that a new kind of approach is needed to help students deal with the knowledge explosion. Among other things the new approach involves "decreasing the amount of content that has to be learned.." [emphasis ours] Thus, when all is said and done, a stated goal of the developers of the MST is to dumb down the requirements.
None of the trigonometric functions were mentioned by name in Standard 6, the mathematics standard of this 1994 Framework. The odd placement of trigonometry in the now published version (MST) led the late Professor Chih-Han Sah of the Mathematics Department of Stony Brook to comment: " It appears to be an after-thought when comments came in the form of 'what happened to trig'..."
But it is the subject of geometry that receives the cruelest treatment in the MST. Gone is any mention of the word "theorem" in the mathematics standard of the MST. Defenders of the MST say it is not a curriculum but a framework. But this framework will set the tone for education in school mathematics for years to come. On page 22 of the MST, one "performance indicator" says that students will have achieved mastery of some of the standards in Mathematical Reasoning provided they can "prove that an altitude of an isosceles triangle, drawn to the base, is perpendicular to that base." Since by definition an altitude of a triangle is perpendicular to its base, the author of that line is either ignorant or careless. So much for the "Tough new student standards - can WNY schools measure up?" being fed to a gullible public by the State Education Department. This headline, the title to a cover story in a special education supplement, appeared in the Sunday, August 3, 1997 edition of the Buffalo News.
We feel compelled to ask: Why the changes? While the apparent upgrading of requirements for weaker high school students is laudable, the MST would effect this by depressing the requirements for New York's better students. In newer curricular adventures, much is made of the importance of statistics. Were any statistical comparisons carried out to see whether the new program would work better than the one it appears intended to replace? It appears that no such comparisons were carried out. And this in a curriculum where statistics is cited as an important technique for quality control (MST, p 47).
The mathematics component of the MST incorporates many of the worst features of the 1989 Curriculum and Evaluation Standards for School Mathematics [NTM Standards] published by the National Council of Teachers of Mathematics. But even the NCTM Standards did not massacre geometry as thoroughly as the MST. Although the NCTM Standards initially received approval from many professional organizations - it now appears that this approval was given without much thought -, the NCTM Standards are now under heavy attack by many mathematicians: Recent issues of the Notices of the American Mathematical Society give eloquent witness of this. (1)
What many mathematicians find disconcerting is that the NCTM Standards and its cognates weaken needed preparation for the most important mathematics course that students can take: calculus. If one wants to do serious work in mathematics, physics, in some areas of chemistry and the biology of DNA, in engineering, or in economics, mastery of calculus is absolutely essential.
Some argue that most students embark on careers that do not require calculus. But since most high school students have no idea what career they will eventually pursue, the kind of curricular degradation being promoted by the MST will effectively foreclose to otherwise talented college bound students those career opportunities requiring mathematical competence. Moreover arithmetic, algebra, geometry, and the rudiments of trigonometry are often needed even for those not planning careers in the above disciplines. The fourth year now added to the MST, while welcome, amounts to too little of the necessary mathematics too late. Are equally trivial mathematics curricula being adopted in Japan, France, or the Peoples Republic of China? We think not.
Reformers say that computer algebra systems or calculators are able to execute operations of arithmetic or traditional algebra. Why not eliminate [NCTM Standards, p.8] or de-emphasize [NCTM Standards, p. 21] the teaching of long division too? Those wanting to promote a curriculum of equity and ease will find that the MST of New York State does not disappoint them. The NCTM Standards, a leading voice in the campaign for lower standards, is now taken by New York as its principle guide: New York's teachers can find a useful summary of the NCTM Standards in the so called Resource Guide. The guide describes the NCTM outline as "a useful reference as districts align their curriculum with State standards." And of course in the 94 Framework the NCTM Standards is singled out as one of the documents motivating the current changes in New York (Framework, p 7)
We support the intelligent use of computers and calculators in school mathematics. For example, the lovely theorem of Euclidian geometry "The medians of a triangle are concurrent" comes to life with the use of software like Cabri-Geometry or The Geometer's Sketchpad. By moving the vertices of the triangle independently, the student can really see the universality that is the content of the theorem. This kind of experiment represents a lovely combination of twentieth century technology with the mathematics of ancient Greece.
But Cabri-Geometry will never prove the Theorem, or be able to show the intersection's most telling property, that it lies at the triangle's centroid. Mathematics is more than arresting pictures. Excessive reliance on computers and calculators at the expense of the traditional basics will only have the effect of creating a new learning disability among New York students: calculator assisted mathematical incompetence.
In addition to the changes in mathematical content made in the MST, there appears to be a decided change in the way student mastery of mathematics will be tested in high school. It appears that Regents exams given each year in Courses I, II and III will eventually be replaced by one exam to be taken usually at the end of the sophomore year.The latest version of this exam called "Mathematics A Regents Assessment Spring 1997 Pilot Questions" was mentioned briefly above. While far better than the 1996 pilot, this exam follows closely the MST outline, and inevitably omits important topics covered in the Course I and II Regents Exams. As a crib-sheet, moreover, a generous list of mensuration formulas are provided for the standard geometric figures, also formulas for the slope of a line, two forms for the equation of a line, the formula for the distance between two planar points, the midpoint formula, trigonometric formulas, the Pythagorean Theorem, and distance = rate x time. With all of this help one is forced to ask: what will the better students be required to know?
Not very much.
After dumbing down the mathematical content and the testing of that content, the designers of New York's MST offer some really novel scoring guides. The interested reader may wish to follow up on this aspect of the MST by reading the document itself. But that reader will also want to get a copy of the 1994 Framework to learn the educational theory behind this flawed endeavor.
Although this letter primarily concerns the high school mathematics framework and standards, we would be remiss if we failed to indicate that the horrors done to high school mathematics in the MST are also present, in somewhat diluted form, in the earlier mathematics recommended for elementary and middle school. And - in passing- the Science component of the MST [mathematics, science and technology] is equally flawed and superficial. Regents Exams in Physics, Chemistry and Biology might very well become a thing of the past if the reformers get their way.According to one document from SED, a Science Regents will be required of all students for the Freshman class of 2003. The current science component of the MST is perhaps best summed up by the words of our colleague Ralph Raimi of the University of Rochester who describes it as : interdisciplinary baloney."
Where will these reforms lead? The interested New Yorker can actually read what had happened in a similar case in California in an article appearing in the June-July and August 1997 issues of the Notices of the American Mathematical Society. The article's title? : "The Math Wars California Battles It Out Over Mathematics Education."
The time has come for the Regents of New York to take immediate and drastic change in New York State educational bureaucracy. Curricular content in important areas like mathematics, physics, chemistry an biology cannot intelligently be dictated entirely by people with terminal degrees in education. Even at the most elementary levels, the subjects of mathematics, physics, chemistry and biology should each be represented by several seasoned professionals with doctorates in the disciplines in question. There should also be several well-trained and experienced practicing teachers representing each of the various disciplines. And just as no one shoe size fits all, there should be several challenging mathematics curricula to accommodate students of different ability.
Finally, it is absolutely vital that more mathematicians and scientists take active interest in what is happening in this nations' schools. We should hope that the Regents of the State of New York take some practical measures towards enlisting them rather than alienating them.
(1) Articles and letters from Notices of the American Mathematical Society are on the World Wide Web at one of the following locations:
http://www.ams.org or http://e-math.ams.org
Herbert A. Hauptman
Hauptman-Woodward Medical Research Institute
SUNY College at Fredonia
University of California at Berkeley
Michael J. Cowen
John Bascom Professor of Mathematics
University of Wisconsin-Madison
Richard H. Escobales, Jr., Ph.D.
Professor of Mathematics
Canisius College, Buffalo, NY 14208
SUNY at Buffalo
Ronald I. Rothenberg
Queens College of CUNY
Encs: Selected Papers
Cc: Honorable Members of the Regents
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