Paper

State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan

Page viii states this directive: "The mathematics/science learner will demonstrate mathematical and scientific literacy in a global environment." At page D-5 (high school geometry) occurs the "measurable performance" demand: "Use the deductive nature of geometry to solve problems." This demand is, as is the style of this Framework, followed by an entry under the follow-up rubric, "A closer look," urging real-life experiences to "enhance the understanding of modeling as a problem-solving tool." Finally, there are the "sample investigations" to close out this instruction: The first is a treasure hunt, and the second concerns the decorating and furnishing of a room. All told (and "global environment" or not), none of this helps "demonstrate mathematical and scientific literacy." The idea that deduction, allegedly asked for in this "measurable process," is a mental process is lost in all these "experiences." Another example occurs on page D-11: "Technology . . . must be used to build the basic notion of a function." Here the false doctrine is more urgent. Technology is no longer an aid, but a necessity, or so it is said. But the basic notion of a function was understood well before the technology required here was invented. Indeed, reasoning about functions helped develop technology, not the other way round.

This Framework is easy to handle and read, for it spends a bare four pages explaining what it is about, and then goes about it. The "Societal goals" are excellent, e.g., "All students will develop strong mathematical problem solving and reasoning abilities" (p. 3). Its page on "How students learn mathematics" quite properly emphasizes students' engagement with the material, and does not consider "real-life" applications the only path to knowledge. However, it repeatedly prescribes manipulatives for students at levels that should be beyond that sort of thing (False Doctrine), as at "Spatial Sense," on page 19, where the "proficiency standard" for End of Grade 10 says, "Use manipulatives, and/or coordinate geometry to explain properties of transformations. . . ." At this level, geometry should be presented conceptually, and indeed logically, rather than as a continuation of the wooden block constructions of kindergarten.

By the end of grade 6, this 30 page document uses three lines under "Number Operations" to ask students to "demonstrate an understanding that when dividing two whole numbers that are greater than one, the quotient will be smaller than the dividend." Such a detail is jarring in a list which, a few lines earlier, rather more grandly asks 3rd graders to "explain the relationship among the four basic operations."

Another curiosity, unrelated to our numerical scores under the criteria listed above, occurs at algebra level 7-12, where students are to, "[m]odel and solve problems that involve varying quantities with variables, expressions, equations, inequalities, absolute values, vectors, and matrices" (p. 25). This sentence is a bit awkward, and would attract the attention of any mathematician by its inclusion of the misfit phrase "absolute values," which is simply not parallel with the other items named in a list already too-long. The word "with" is also strange here, but apparently means "using," as in New Jersey's Standard 4.13 for 12th grade algebra, which reads, "Model and solve problems that involve varying quantities using variables, expressions, equations, inequalities, absolute values, vectors and matrices," which (by what cannot be a mere coincidence) repeats the same list in the same order. Believing there must be a common origin to this rather strange listing, the authors went to the NCTM 1989 Standards, which is the acknowledged model for so many of the states' standards, and found on page 150 a similar, if simpler sentiment: "Represent situations that involve varying quantities with expressions, equations, inequalities, and matrices." The NCTM list, however, omits the misfit phrase "absolute values" and the word "vectors" that both New Hampshire and New Jersey chose to include at identical points in their identical listings.

A teacher who follows the outer implications of the Framework's many suggestions could very well prepare a student to bypass the first year at MIT, while others, using other textbooks equally consistent with these standards, would be able to avoid much that should be mandatory in any school system.

The Framework (document [2], see Appendix) is a very ambitious and very long commentary on the Content Standards (document [1], which appeared a little earlier). A teacher who follows the outer implications of the Framework's many suggestions could very well prepare a student to bypass the first year at MIT, while others, using other textbooks equally consistent with these standards, would be able to avoid much that should be mandatory in any school system. More than most states, New Jersey seeks to embed mathematics in a cultural frame of reference, which is a worthy effort in the hands of well-educated teachers; but the classical content of mathematics, and its backbone of deductive reasoning, without which no amount of cultural framework can really be understood, are often slighted in these standards.

No proof of the Pythagorean theorem is demanded, yet the student is expected to "explore applications of other geometries in real-world contexts" (Standard 7, grades 912). While this is not explained in the standards, the Framework gives as an exam ple of "other geometries" the thoroughly Euclidean geometry of figures drawn on a sphere. (There really are other geometries than the Euclidean, such as projective geometry; and the discovery of hyperbolic geometry in the 19th century was a milestone in the history of mathematics.) Without a good background in the Euclidean deductive system, or of more analytic geometry than any high school is likely to offer, such "non-Euclidean" studies are more "math appreciation" than mathematics.

The emphasis on "real-world" applications and activities is extreme, and one must often guess at the intellectual content implied. In other places, there are admirable instructions-- e.g., "Describe and apply procedures for finding the sum of a finite arithmetic series and for finite and infinite geometric series," which would be even more admirable if the word "prove" had appeared somewhere in that sentence. There is some attention to Reason in the New Jersey standards, but not enough, which is one concomitant of the falling off of Content in the high school years.

One good thing about these standards is the forthright statement at the outset that "proficiency in English is of the highest importance." This is immediately qualified, however, by supporting "the use of the student's primary, or home language, as appropriate . . . while the student acquires proficiency in English." There is not much to object to here, but there is also not much guidance, as to when and how much and how.

Similarly, Standard 1 ("Problem solving") states that "students will use manipulatives, calculators, computers, and other tools, as appropriate, in order to strengthen mathematical thinking, understanding, and power to build on foundational concepts." This instruction is given verbatim for each of the levels K-4, 5- 8, and 9-12. But the evasive "as appropriate" renders this recommendation nugatory, just as in the earlier recommendation concerning bilingual education. Teachers are looking to the state's standards exactly to find out what is appropriate and when; it is not for the state to leave the essence of its recommendations undefined. Instructions so heavily qualified as to be interpretable at will must count as Inflation. And if the interpretation is the use of manipulatives in high school, or calculators at K-4 when arithmetic begins, it is False Doctrine.

The language of the Standards is sometimes garbled by excessive devotion to generalities. Standard 2 asks that students "use mathematics in communication" where the most likely interpretation, as evidenced in the associated benchmarks, is that "use communication in mathematics" is closer to the intention. The NCTM 1989 Standards has a thread called "Mathematics as Communication," which is probably closer yet. A similar inversion takes place in Standard 3, which states that "students will understand and use Mathematics in Reasoning," apparently a misprint for ". . . use reasoning in mathematics," since at least two of the associated benchmarks concern the use of reason in obtaining mathematical results.

Even so, the specifics are vague, here and throughout the document. Under "Number systems and number theory," for example, appears the line, "develop and analyze algorithms." Specifications this general cannot serve as a guide to teaching or curriculum. More general yet, indeed breathtaking, is this one, under Standard 4, for grade Level 5-8: "Students will apply mathematical thinking and modeling to solve problems in other curriculum areas such as employability, health education, social studies, visual and performing arts, physical education, language arts, and science; and describe the role of mathematics in our culture and society." Among other things, this particular skill is hardly testable.

The best feature of this booklet, in which only 16 pages are devoted to mathematics, is the absence of inflated verbiage and unrealistic expectations, and the absence of downright bad advice. However, the brevity of the mathematics section of the "mathematics, science, and technology" standards permits extremely varying interpretations as to content. On page 26 it asks for proofs by mathematical induction, for example, a difficult topic at the high school level, and it asks for the analysis of infinite sequences and series. These are quite demanding and specific. Yet at the intermediate level the requirement that students "develop appropriate proficiency with facts and algorithms" (this comes under mathematical "operations" as a rubric) permits too much leeway. Sometimes only a teacher already experienced in what New York means by certain terminology will catch the drift of a too- condensed demand--e.g., at high school level, "Students . . . model and solve problems that involve absolute value, vectors, and matrices" (p. 27). Any mathematician would be puzzled at this juxtaposition of vectors and absolute values. And, in the same list of performance standards, "represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations" seems to imply a lot of content, but it does so somewhat vaguely, in view of the phrase "represent problem situations," which does not quite say what it probably means. Does it mean "solve problems," and if so, what sorts of problems? Here is a place where there is white space on the page, in which illustrative exercises would be valuable; but in general, the illustrative examples are few, and most of them are on the whole much less demanding than any reasonable interpretation of the text.

There is very little to complain of in this generally excellent document, from which other states could learn.

There is very little to complain of in this generally excellent document, from which other states could learn. It is carefully done. It has over 500 pages, but they are occupied with practical information of exactly the sort the new teacher in town would need to know. Only pages 13-61 are required to outline the content, but it is done briskly, by topic and grade level. Most of the rest of the book reiterates the content standards and couples them with "sample measures," which are test questions or classroom exercises. These can serve a teacher either as class material or, most important, as clarification of the import of the standards themselves.

Not everything clear and definite is necessarily good, however. On page 55, competency goal 5.2, "Use synthetic division to divide a polynomial by a linear binomial," betrays a rather old-fashioned origin for this Algebra II curriculum. Synthetic division is a superb example of an unnecessary algorithm, one which, unlike the "long division" algorithm for the 5th grade in the hands of a good teacher, does not offer any insight into the nature of the process. Such time-wasting is unusual in this curriculum, however.

There is nothing dated about 7.11: "Explore complex numbers as solutions to quadratic equations" (p. 55). Filled in with good teaching, the instruction is excellent (though "explore" ought not to be called a "competency goal"), but will this rather general "exploration" lead to learning? On page 472 this goal is repeated, with some "sample measures," among them, "Find a quadratic equation that has these roots: 5+2i and 5-2i"; and, "solve the quadratic equation . . ." [here an example requiring the quadratic formula is given]. However, here as in all too many other places, the inner structures of mathematics are ignored in favor of a telegraphic crispness. It would seem exemplary to say, "Here is the equation; find the roots," and then, "Here are the roots; find the equation." But there is more that can and should be said: What if the given roots are not complex conjugates of each other? Does the sum of the roots have the same connection to the coefficients in the complex case as the real? What good is a quadratic equation?

It is hard to introduce such considerations into a list of contents, and most states have gone overboard trying to do so, generally to the disadvantage of the contents themselves. Yet it can be done, and the places where North Carolina's scores are less than 4 are indications of such minor failures.

Reason, for example, gets a good workout in the geometry course, but somewhat less in the algebra and trigonometry sections, where it could have been injected with little extra effort. The excessive number of threads in the elementary grades leads to a diffusion of content. And in the "sample measures" sections that follow the content listings, a disproportionate amount of space is devoted to the 9-12 segment of the curriculum. Elementary teachers, who are generally not specialists in mathematics, need more guidance, year-for-year, than those in the more advanced years.

The organization is excellent, each page containing one standard (e.g., "Number Theory" or "Geometric Concepts") and one of the three grade levels K-4, 5-8, or 9- 12. Then "Benchmarks/Performance standards" give some particulars, and are followed by examples of "specific" knowledge that support them. Finally, there are "performance activities," exercises or classroom projects, to illustrate the knowledge to be acquired, and how.

It is to be expected that the standards are quite general, but the benchmarks are often not much more particular, while the "examples of specific knowledge" are usually lists of vocabulary items with little indication of how much is to be known concerning each. For example, under "Data Analysis" at the 5-8 level, two of the benchmarks are "evaluate arguments that are based on statistical claims" and "display and use measures of central tendency and measures of variability." As written, these could as easily be placed at the 9-12 level, or in college, and indeed some of them do reappear at 9-12. The reader has little idea what progression in knowledge is contemplated as the student grows older, i.e., what is written is not sufficiently definite and is not testable.

Finally, the associated activities often are trivial, and not up to the intellectual level suggested by the benchmarks, both here and elsewhere. In the present case, one of them reads as follows: "Students work with partners to measure one another's height in centimeters. These heights are recorded on the board. Each student is to find the mean, mode, range, and median of the heights." As this could be done with any set of numbers, nothing was gained by the time consuming use of teams and yardsticks. (In the following activity, by the way, where predictions are called for at a somewhat higher intellectual level, the data set is hypothetical.)

Much of the document is good, but the intellectual level deteriorates in the higher grades. There are brave words-- e.g., in 9- 12 algebra all the elementary functions, all of them, are to be "investigated," but it is not clear how deeply or to what end. One cannot determine from these standards whether students learn the binomial theorem or the proof of the quadratic formula or Pythagorean theorem. The geometry course in particular shows little logical structure or coherence. Yet "Fermi problems" are mandated at page 35, "discovering fractals" at page 29, and "coding theory" at page 31. "Fractals" and such things as "conjecture" are defined in the Glossary, but Fermi problems and coding theory are not. (The Glossary contains some careless errors, too.) At the grades 5- 8 level students are to "solve problems using coordinate geometry" (p. 21). This is not believable, unless by "problem" is meant something like plotting a point whose coordinates are given. Some of the performance activities that illustrate this section on geometry require the use of "multilink cubes," marshmallows, tooth-picks and a "geoboard."