Paper

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

Although it features extensive (attributed) quotations from the NCTM 1989 Standards, this document's prescriptions are usually more definite and comprehensive. From the 10th grade onward, each mandate is followed by a section headed, "In addition, college-intending students will . . .," the section containing extra or more advanced material. Ohio is therefore among that minority of states that recognize in their standards that it is not wicked to offer some students opportunities not every student might desire or be able to use.

The organization is by strand and level, the levels being keyed to what are apparently state-wide "proficiency tests" given at the 9th and 12th grades, as well as some earlier standardized achievement tests at grades 4, 6, and 8. Most items are introduced with "The student will be able to . . .," and the rest is brief, e.g., "identify common shapes in the environment" (at the K level), or "identify parallel lines, perpendicular lines, and right angles in geometric figures and the environment" at the grade 4 level. The wide margins have space for clarifying comments here and there, and suggested exercises or illustrations.

The standards take less space than appears in this over-200 page book, as they are repeated, first classified by strands and then by grade level. In three pages, then, one can see what is expected by grade 5 in all subjects. Alas, one of the "subjects" in the early grades is Strand 2: "Problem-solving strategies," which contains largely vacuous demands, such as "validate and generalize solutions to problems" (p 133). Still, even in Strand 2 there may occur a valuable reminder of something not every state demands, e.g., (still at 5th grade level), "Read a problem carefully and restate it without reference to the original problem" (p. 33).

In Negative Qualities, Ohio is slightly downgraded in the category False Doctrine for its too frequent use of "explore" when naming something a student should be able to do. Anyone can explore, after all, and a teacher using these standards might well slight the direct transmission of knowledge, and demand for its exhibition, in favor of open-ended discussion, some of which is valuable in its place, of course, but the place of which should be better defined than it is in this document.

These Priority Academic Student Skills-- Mathematics employ the categories (or threads) of the NCTM 1989 Standards, except at the kindergarten level, where it wisely recognizes their inapplicability, and simply lists quite reasonably what in mathematics should be done there. It would have done well to continue in this vein, for the NCTM threads, "Problem Solving," "Number Sense," "Patterns . . .," etc., often encourage the invention of content items that make little sense in many grade levels. In the early grades, however, the items are often clearly and definitely stated, as at the grade 1 level, "Use models to construct addition facts to 10." By the 9-12 levels, however, such a statement as "Recognize the connections between trigonometry, geometry and algebra" is surely inflated, and surely untestable, as is "Understand the connections between trigonometric functions and polar coordinates, complex numbers and series" (p. 49), something that (though also vaguely stated and untestable) could only be appropriate in a calculus course, so far as "series" is concerned. (This last list of things to be "connected" also appears verbatim in the South Carolina Framework, page 144.) Vast and inappropriate generalities of this sort must leave a teacher wondering about his own competence, when it is really the author of the lines who is prescribing impossibilities.

One curiosity: The resemblance between these core skills and the NCTM Standards is reinforced by the unattributed quotation, on page 48, of a line that appears on page 150 of the NCTM Standards: "Represent situations that involve variable quantities with expressions, equations, inequalities and matrices," a strikingly awkward formulation that appears in a slightly varied and even more awkwardly augmented version in the New Jersey and New Hampshire standards.

As to Content and Reason, there is in the present document no mention of the Pythagorean theorem, or of any particular axiom or theorem connected with geometry, which is still said to be studied "from several perspectives" and whose "foundations (e.g., postulates, theorems)" are to be understood "through investigation and comparison of various geometries" (p. 48). The surrounding text does not support such ambition, even if it were possible at the high school level. Nor is the logic of algebraic procedures delineated either by prescription or by implication. The general vision of mathematics offered is lacking in both substance and coherence, and is poorly expressed as well.

The "Oregon Standards" named in the Appendix is a very brief portion of a very brief document of that name, and by itself does not serve the present purpose. However, the state sent, in answer to our letter of inquiry, the Test Specifications for Statewide Examinations at the 5th, 8th and 10th grade levels, along with sample tests at those levels and a Teachers' Support Package. Taken together, our hypothetical newcomer to Oregon has here the equivalent of the more usual form of curriculum standards for guidance; and we felt entitled to rate the results along with those of other states which provided the more usual standards or framework documents, recognizing that the materials at hand are intended for all students, and only up to the 10th grade level.

The best quality to be observed (especially in the Teacher's Support Package) is the encouragement given students to write their answers to questions in good English sentences. However, the content demanded is disappointing, and almost nothing is asked for that answers our criterion concerning Reason. The grade 10 examination booklet states (under "Measurement,") that "students will be given necessary formulas to complete problems" in the examination room (p. 6). The eligible content includes all the common figures, including circles. In other words, the basic competence demanded of Oregon high school graduates includes none of the common mensuration formulas, let alone anything of theoretical import. At the same level, "Geometry" contains an exercise in recognizing which of four drawings depicts a pair of perpendicular lines and which does not, surely an exercise in mere vocabulary, and that of a 5th grade level. The Pythagorean theorem is mentioned, but not in any significant context. Altogether, an undemanding curriculum is implied by both the very brief "standards" and the more extensive but excessively simple Test Specifications.

While the "Proposed Academic Standards" examined is a provisional document ("proposed" by the Department of Education for approval by the Governor's Advisory Commission on Academic Standards, that is, with further approvals before becoming official), the introductory commentary says that "the suggested standards contained in this report are the product . . . of more than a year of work by educators and others from all across Pennsylvania, the comments and suggestions of literally hundreds of citizens from all walks of life. . . ." The Commission did not write the document, but "the members of the Commission feel the standards being recommended here are rigorous, measurable, applicable, and understandable expectations of what students should know and be able to do." Also, "The Commission was diligent in its efforts to remove from the standards educational and professional jargon."

The result, as evidenced by a comparison between some items in this document and their correspondents in an earlier and controversial standards document of 1993, is indeed less jargon, but not, we believe, what the state hoped for. The Glossary defines "prime" so as to include "one" as a prime, and states that "1, 2, 3, 5, 7, 11, 13, 17, and 19 are prime numbers" (p. 27). "Gradient" is also incorrectly defined, and the definition of "limit" exhibits a familiar obscurity. The deficiencies in the standards themselves are congruent with these observations. The introduction states: "While mathematics is a very interesting and enjoyable field to study for its own sake, it is more appropriately used as a tool to help organize and understand information from other disciplines." We submit that this is a false dichotomy, and that the use of mathematics is appropriate everywhere.

The standards themselves are divided by grade levels: 3,5,8, and 11 marking the levels of what seem to be proposed state-wide assessments. They are also divided by threads, e.g., "Number Systems . . .," "Problem-solving . . .," "Probability . . .," etc.

The attempt to avoid jargon is palpable and often successful, so that the items are brief. Thus, the reader finds, under "Number Systems" at grade 3, "Use drawings, diagrams, or models to show the concept of fraction as part of a whole"; at grade 5, "Explain the concepts of prime and composite numbers"; at grade 8, "Use models to represent operations on positive and negative numbers"; and at grade 11, "Convert between exponential and logarithmic forms." Important vocabulary items are italicized where they appear, which is itself a guide to the intention of the phrase or sentence.

Yet jargon of a sort remains, inherent in the way the ideas are expressed and not only in the words. "Convert between exponential and logarithmic forms," once its meaning is penetrated, says no more than that the definitions of "exponent" and "logarithm" should be understood. The awkwardness of the "convert between" phrasing here derives from a long tradition of confused jargon whose usefulness was finally obliterated by the modern conception of the exponential functions and their inverses. In 1940 (already anachronistically) the schools taught that "a logarithm is an exponent. . . ." This is not actually mistaken, but it is, or was, the beginning of a tangled sentence that most high school students of that era failed to understand. The current attempt, quoted above, may be shorn of most of that tangled wording, but it still betrays the lack of change of concept. By grade 11 a student should either learn nothing at all about logarithms, or should learn about inverse functions at the same time, to be able to place the subject in its reasonable setting, a setting not recognized by the quoted item.

In short, the content implied by the entire document is less than what students can and should learn at each grade level; even if the jargon used earlier has been improved, the deficiencies once made easily visible by such language too often remain. On the other hand, the text sometimes announces a startlingly ambitious curriculum expectation, as where it asks grade 11 students for proofs by "mathematical induction." The average grade 11 student will not learn this unless thoroughly educated in other forms of mathematical reasoning and syntax at earlier stages of the curriculum, and in the context of definite subject matter, such as algebra and geometry. But the entire thread containing mathematical reasoning (p. 7) exists in isolation from the others, which show little evidence of demand for reasoning within substantive contexts. Here the Pennsylvania document exhibits what is probably the most common failing of all the standards documents this report has studied.

There must be a better compromise between the dry lecture and student-directed brainstorming than what is implied by this Framework.

This attractively packaged Framework has a small poster-sized full text of the standards folded into its inside front cover, all of which is repeated in the Framework proper, along with commentary and pedagogical notes. The notes contain a philosophical view that partially explains the vagueness of the standards themselves, but do not explain what in the way of mathematical demands upon the students they are supposed to represent.

Sample entries: Under "Algebra" (grades 5-8), "Identify and justify an appropriate representation for a given situation." Under "Reasoning" (grades K-4), "Make logical conclusions about mathematics." Under "Problem-Solving" (grades 11-12), "Use sophisticated as well as basic problem-solving approaches to investigate, understand, and develop conjectures about mathematical concepts." None of these is amplified anywhere in the document.

As philosophical background, which we count False Doctrine: On page 15 are listed features of "the traditional classroom" opposite a corresponding list for "the learning community," which Rhode Island intends shall replace it. Opposite "Teacher knows the answer" is "More than one solution may be viable and teacher may not have it in advance." Opposite "Thinking is usually theoretical and `academic'" is "Thinking involves problem solving, reasoning, and decision-making." We believe thinking involves all that is named on both sides of this apparent dichotomy and more, and deplore this denigration of theory and the academy. And while real-world problems often have no single answer, mathematical problems do.

As to the relative absence of Content in this Framework, the authors of this report are not saying that no mathematics is being taught in the Rhode Island schools, or that this Framework intends such a thing. But what is written in the "Process and Content Standards" is so vague that it is impossible to tell. In K-4, it is literally not clear at all if the student really knows how to add, subtract, multiply and divide, or just has an "appreciation" for those things. What part of these four operations shall children memorize? What algorithms are taught, and what schemes are left to their own construction? The Framework offers no answer, but the teachers across the state need one.

The Constructivist stance elaborated throughout the Framework would, if diligently followed, prevent a full curriculum from taking hold, since there is not time for the students to construct all the required knowledge for themselves. It might be that Rhode Island does not intend it so, but the vignettes contributed by teachers to illustrate ideal lessons argue that they, at least, are taking the idea seriously, so much so that one of them (at a junior high school) announced that, among other things, "Students also discovered Euler's formula, F+V-2=E." In truth, the formula is simple enough for children to understand, even if its proof is hazardous; but to say the children "discovered" it, or to have made them believe so, has its own dangers. True discovery is not taught by surreptitiously feeding answers. There must be a better compromise between the dry lecture and student-directed brain-storming than what is implied by this Framework.

The Standards is a revised version of the relevant portions of a Framework, published earlier and containing much besides content standards, some of it illuminating the intention behind the Standards, which is the principal subject of this evaluation. In the Standards are found fairly demanding curriculum indications, especially concerning numerical and algebraic concepts for the early grades (p. 22), and concerning numerical systems at the middle school level (p. 38). On page 66 ("Geometry," high school) occurs a more typical--i.e., ambiguous--demand: "Deduce properties of and relationships between figures from given assumptions by: using logical reasoning to draw conclusions about geometric figures from given assumptions." The suggested method (the only one mentioned there) uses the same words as the thing it is supposed to be a method for, and so does not add anything. But the real ambiguity is the question of the depth of investigation asked for. It could be anything from "math appreciation" to a full-scale course in Euclid. Items of this kind, which are numerous, render unclear what students are asked to accomplish, and might even render inaccurate our appraisal in this report of what "content" is being rated by our numerical scheme.

On the other hand, the instructions (p. 60) concerning the desired facility (grade 12 level) in manipulating algebraic expressions, and solving equations and inequalities, are well-stated and illustrated by examples, though the content appears a bit thin. Binomial theorem? Quadratic formula? Proof of either? We don't know. To add to the uncertainty, the phrase "using appropriate technology" appears repeatedly throughout the entire document, in every context. The teacher is thus deprived of necessary guidance as to the balance between what is to be internalized by the intellect and what is to be consigned to machinery.

The educational philosophy expressed in the Framework, though not a subject of numerical evaluation in this report, includes this curious observation, which might illuminate the vision of past practice in mathematics education that underlies the recommendations in the present Standards: "Mathematics should be a discipline that helps [students] to make sense of things, not a discipline that is arbitrary and devoid of meaning" (p. 11). We believe that the presumption that people, and in particular teachers, have in the past thought the latter is unwarranted.

Under False Doctrine of another sort is the instruction on page 10 of the Standards volume, concerning geometry and spatial sense for primary grades students: "Students at the primary level are not expected to know these words." It is not clear which words are in question, but the children are, in the sentence following, expected to do things with tessellations, symmetry, congruence, similarity, scale, perspective, angles, and networks; why should they be deprived of an adequate vocabulary? Grades 7 and 8, on the other hand, sometimes apparently call for too much, or too advanced, material: fractals, Fibonacci, the Golden Mean, and permutations (e.g., to calculate nPr, albeit with technology). At that level, fractals can only be "math appreciation," and nPr will never be assimilated.

The introduction tells of the purposes of standards, in particular, that they are not to be construed as a curriculum, but that they "make an effort to describe the `big ideas' or core content which should be learned by all students. . . . Content standards provide guidelines for the development of curriculum by listing important areas of content which must be addressed."

Further, "These standards compel teachers not only to look at the mathematics taught but at what is learned, and at why it is learned." And, "Checklists of concepts and skills are no longer the primary component of the mathematics curriculum." The examples of standards and the more particular "benchmarks" listing what a student should know or do at particular grade levels, to be quoted below, will indicate that these purposes are not met by the present document.

There are six standards, (here) briefly labeled algebra, geometry, measurement, numbers, functions, and statistics. Each standard is printed at the top of a new page and followed by a "Rationale," a paragraph telling of its importance and place in the curriculum, and mentioning such things as societal needs, real-life experiences, connections across the curriculum, our daily lives, etc. Finally, the "benchmarks" offer what specifics the document contains; they are all prefaced by "Students will. . . ." The quotations given below will mention the grade level intended for each. We hope the vacuity of each of them as a guide for teachers or for curriculum development will speak for itself.

• Model number relationships using a variety of strategies (3-4);

• Analyze and describe situations that involve one or more variables (9-12);

• Compare and contrast spatial relationships using geometric figures (3-4);

• Use models, manipulatives, and computer graphics software to build a strong conceptual understanding of geometry and its connection to other mathematics strands, science and art (5-8);

• Use models, manipulatives, and computer graphics software to build a strong conceptual understanding of trigonometry and its connection to other mathematics strands, science and art (9-12);

• Measure quantities indirectly using techniques of algebra, geometry, or trigonometry (9-12);

• Utilize the real number system to demonstrate problem-solving strategies (5-8);

• Determine and describe the effects of number operations on real numbers with varying magnitudes (9-12).

One more quotation before we leave: A "benchmark" on page M 11, grade level 9-12, states that "students will demonstrate the concept of limit using mathematical models." We have not been able to determine the meaning of this instruction.

The Framework for grades K-8 is a 1996 document while the Curriculum Framework for grades 9-12 is dated 1991. Tennessee states that the latter is now under revision. Since they are so different in every way, obviously written by people with vastly different purposes, we are giving Tennessee two separate sets of ratings. Averaging them does not honestly reflect our estimate of either.

The K-8 Framework begins with four "Process Standards": Problem Solving, Communication, Reasoning, and Connections, each incorporated into each of seven "Content Standards": "Number Sense and Number Theory"; "Estimation"; "Measurement and Computation"; "Patterns"; "Functions and Algebraic Thinking"; "Statistics and Probability"; and "Spacial Sense and Geometric Concepts." Then there is a set of five learning "expectations," and a Glossary for terms that appear in the document.

Some idea of the educational philosophy underlying the entire Framework may be had from the Glossary entries. The authors warn that these definitions are not complete, or designed for student use, but that "the definitions given are restricted to only the common mathematical meaning." It is further explained that "the terms and definitions included in this Glossary were produced by the Mathematics Curriculum Framework Committee for Grades K-8." Here are a few of these definitions:

Algebraic thinking--thinking skills which are developed by working with problems which require students to describe, extend, analyze, and create a variety of oral, visual, and physical patterns (such as ones based on color, shape, number, sounds) from real life and other subjects such as literature and music.

Equation--two mathematical expressions joined by an equals sign.

Model--(verb) to show or illustrate a concept or problem by using physical objects with manipulations of these objects; to use simpler or more familiar objects and situations to explain a new concept, to solve a given problem, or to demonstrate understanding of a concept.

The first of these definitions ("Algebraic thinking") shows the marks of the deplorable current fashion emphasizing the primacy of "real-life" applications of mathematics to such a degree that mathematics is deliberately confused with the world it seeks to model. Except for "thinking skills," a nebulous phrase in itself, there is absolutely no reference here to anything remotely mathematical, symbolic, or deductive, let alone algebraic.

The definition of "equation" is amusing; it is a definition of what an equation looks like on the printed page, perhaps, but seems not to understand that an equation is a certain sort of statement, a sentence written in English. The sad state of public understanding of mathematics can be traced to the early acquisition of such "definitions," which can only lead to meaningless manipulation of symbols, if that.

The definition of "model" shows a misunderstanding of the role of mathematics in modeling phenomena; the given definition totally reverses the process, betraying the notion in the mind of the writer that the mathematics is the difficult thing, and the real world the simpler, by which the mathematics is to be elucidated. This may be true for very young children, whose understanding of arithmetic must of course be rooted in experience, but suggests that the authors of these Frameworks wish to convey an understanding of mathematics and its relation to the real world that stops at that point. Certainly, they are ignoring the current use of "model" in mathematics: as a verb it refers to the use of mathematics to picture real phenomena, while as a noun it refers to the mathematical structure that serves in place of the reality. This definition has it backwards.

A study of this Glossary does not cause one to anticipate much mathematical content in the Framework itself, for all that it lists an enormous number of rubrics by which to classify what happens in grades K-8. There are also some goals: "Learning to value mathematics"; "Becoming confident in one's own ability"; "Becoming a mathematical problem solver"; "Learning to communicate mathematically"; and "Learning to reason mathematically." Yet with all these rubrics, there turn out to be only 12 pages of non-repeated text material in this 46 page Framework. These convenient pages order the items by grade level, with the mathematical categories listed within each level. Here is the place, if any, to give substance to the philosophy and categories of the surrounding texts. We are willing to tolerate vagueness in a title or category description, provided the text ultimately makes plain what the generalization describes. In the K-8 Framework, however, most generalizations are not later made more specific.

A typical "learning expectation" from the K-8 document concerns "Patterns, Functions and Algebraic Thinking" for the Grade 6-8 level: "Describe, extend, analyze, and create a wide variety of patterns in numbers, shape, and data using a variety of appropriate materials, including manipulatives and technology."

The phrases, "a variety of . . ." and "a wide variety of . . .," as emphatic modifiers of perfectly understandable plurals, are frequent enough in many standards documents, but more frequent in this K-8 Framework than usual. They add nothing to the statement but words; they are inflation.

Coming to the level Grades 3-5, now, under "Number Sense and Number Theory," we find that students should

• Articulate and model the relationship between fractions and decimal notation for a rational number;

• Select a notation and justify its appropriateness for a particular situation;

• Demonstrate an understanding of decimals by extending whole number place value concepts;

• Use geometric models to illustrate properties of whole numbers and fractions and their operations.

Here the second expectation is opaque to the outsider, including the professional mathematician. The fourth expectation is indefinite enough to be the basis for either a good lesson or a poor one. The first and third expectations are appropriate but repetitious; one cannot do the first without the understanding demanded in the third.

The 9-12 Framework is a document of comparable length, though without all the rubrics and repetitions of the other; hence it contains more. It outlines one sequence, "Arithmetic 9" (remedial), "Applied Mathematics I" (also remedial), and "Applied Mathematics II" (partly remedial) as evidently designed for the non-college bound students; another, consisting of "Algebra I," "Algebra II," and "Unified Geometry," is the standard pre-college curriculum that has traditionally occupied grades 9, 10, and 11 in many parts of the country.

There are descriptions of yet other courses ("Pre-algebra," for example) which fit into alternate tracks, all of which are outlined in such a way that students show ing unusual proficiency, or difficulty, have a key to an appropriate next step in a different track, much as one would find in a college catalogue. As a whole, the 9-12 document spends very little space on "overarching" category listings. Algebra is algebra, geometry geometry and trigonometry trigonometry. There is no talk of "patterns," no "spatial sense and geometric concepts," and no "mathematics as communication."

For example, under Content for "Algebra II" we have a section on the complex numbers, with the direction, "Simplify complex fractions involving complex numbers." This is something that can hardly be done by people who have scanted computation with ordinary real fractions in favor of decimal representation and machines. The instruction is clearly stated and substantial in content, unlike so much in the K-8 listings. Other typical 9-12 demands, from various courses and levels are

• Construct a line parallel to a given line;

• Transform expressions with rational exponents into simplest radical form and conversely;

• Solve simple exponential and logarithmic equations;

• Classify a system of equations in three variables as consistent, inconsistent, dependent, [or] independent;

• Use the Gaussian elimination method.

Not everything old-fashioned is good, though, and certain truly pointless old topics appear here and there in Tennessee's 9-12 Framework. "Synthetic division" for example, is a lavender-scented relic of forgotten ballrooms and will surely not survive the revision now under way. Descartes' Rule of Signs is another fossil which is seldom accompanied by an understandable rationale in school instruction, and at best gives insights of little value. (And despite all the changes of the past 50 years, Tennessee is not the only state to retain these two topics.) Also, there are some important things missing in the present 9-12 curriculum because they have always been missing; precedent is not always a good guide.

Thus, in our Table of Ratings in Section VII we omit grading the K-8 Framework for "Content" at the "Secondary" level, and we omit grading the 9-12 Framework for "Content" at the two earlier levels, assigning "n" for "no grade" in those places. Then we average what is left, giving four totals for each document as if it were a state unto itself. At the end there is the grade of C, corresponding to an overall, though misleading, "average" for all of Tennessee.

This document is particularly lucid and easy to read, though our copy lacks page numbers. It takes the curriculum grade-by-grade for K-8, and by course titles thereafter, a logical choice, and one which avoids the fragmentation of the later curriculum so common in "integrated" high school mathematics courses. The specifications of what is demanded at each level and each subject heading are introduced with useful summaries telling the reader where the document is and where it is going; and then the specific items are indeed specific, as they should be. The content is better than average, with the important omission of almost everything having to do with mathematical reasoning. There are other omissions concerning content, sometimes requiring the reader to infer it--for example, that the quadratic formula is to be proved and not just used from memory is inferred from the mention of "completing the square" as a method of solving a quadratic equation. In pre-calculus there appear to be no trigonometric identities, and the binomial theorem gets only brief mention; in these cases we infer less rather than more.

While the word "reasoning" appears frequently, it most often refers to making connections between the real world and its mathematical models, rather than to the logical connection between mathematical statements. Thus, "uses pictures or models to demonstrate the Pythagorean theorem," and "use the Pythagorean Theorem to solve real-life problems," ("Geometry," 8th grade) are not followed up by a demand for its proof, or even its placement in an organized mathematical system--not even where axiomatic systems come under discussion in grades 9-12. There the course description leaves unclear what sorts of proofs, if any, will be produced or learned. The "multiplicity of approaches" to geometry here does not outline a coherent course of study. For example, "In a variety of ways, the student develops, applies and justifies triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples" gives a confused message. Pythagorean triples are of algebraic interest and teach little about geometry, and ". . . develops, applies, and justifies triangle similarity relationships, such as . . . Pythagorean triples" doesn't quite make sense.

The geometry course, little as its description implies as to content, goes so far as to suggest that non-Euclidean geometries are to be studied; this won't do, when even Euclidean geometry gets such short shrift. Texas is not the only state whose standards make such a recommendation (see Notes for Arkansas and Hawaii above).

The organization of this document emphasizes the continuation of strands through several grade levels to such a degree that particular instructions are sometimes mistakenly placed. Thus Standard 5000, is intended for kindergarten, 5010 for grade 1, 5020 for grade 2, etc. For each grade level there are strands, e.g. Strand 12 having to do with fractions and decimals. Thus 5030-12 names an instruction for the 3rd grade level concerning fractions and decimals, and 5000-12 the same but for kindergarten. The following instruction might be appropriate for 5040- 12, and is given there: "Develop concepts of fractions, mixed numbers, and decimals." But the same sentence appears also at 5000-12, 5010-12, 5020-12, and 5030-12, making it appear that decimal fractions are a feature of kindergarten instruction and then repeated each year for the next four years in identical terms. It is careless of the writers to appear to suggest this.

Each strand, at each grade level, begins with very general "objectives," which cannot be of much value--e.g., 5040-07 begins with "Develop meaning for the operations by modeling and discussing a rich variety of problem situations." (This is 4th grade level, for the strand concerned with what ordinary people call the arithmetic of whole numbers.) However, after the objectives come the "Skills and Strategies" section, offering particulars. In the case of 5040-07, one of them is "Demonstrate through the use of manipulatives that multiplication and division are inverse operations . . . (3X4=12; 12/4=3)," which could not be clearer or more definite. This is general throughout the document, and a good program is defined. Even Reason is well treated from time to time, as in 5350-14, "Recognize the applications of field properties in solving equations and inequalities," a recognition not often enough referred to in most states' standards.

On the negative side, the document is entirely too long and repetitious, the same instruction occurring in the same words again and again, so that the reader will be unable to decide the proper placement of skills, or unable to see the desired progression of skill as students progress through the grades. This is a species of Inflation, and another is the curious jargon that appears repeatedly, as if the processes of arithmetic were more mysterious than they are. In 5350-14 it is asked that students "solve problems with real numbers using Venn diagrams" when all that is meant (we believe) is that students should recognize the inclusion relations between, say, the set of rationals and the set of reals. The word "strategy" turns into a quite unacceptable technical term in 5010-07: "Recognize and employ the strategy that division by zero is undefined. (You do not divide by zero.)" The parenthetical command carries the message, after all, without any strategy. The following item in the same "skills and strategies" section says, "Recognize and employ the strategy that when zero is a factor, the product is zero." This is no more or less a fundamental fact of multiplication than that 3X4=12, and it certainly is not a "strategy." Inflation again. A careful pruning of about half the words, or maybe two-thirds of them, in these core standards would make a much better program description, for the scattershot listings do harbor very good curriculum choices.