State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
The following Notes are not intended as full explanations of the numerical scores tabulated in Part VII above and repeated here state-by-state. Those explanations are implicit in the numbers themselves, coupled with the description of the criteria given earlier in this report. To review all the details that entered the scores would require more space than is possible.
In particular, the quotations are not necessarily drawn from or based on the most typical or characteristic features of the document in question. On the whole, they are selected to provide, in their ensemble, a glimpse of some of the detail, here and there, that no summary or averaging procedure can provide. Since these notes mainly (though not entirely) exhibit weaknesses rather than strengths in the documents mentioned, we expect that, taken together, they provide additional insight into what, by our criteria, is the overall failure of the current efforts at writing useful state standards for school mathematics. That certain particular criticisms appear under the headings of particular states does not mean they are peculiar to the State in which they appear. Many are applicable to other states, and the objects of these criticisms tend to occur in much the same form--sometimes in exactly the same form--in the standards documents of other states.
The Content is one of the most comprehensive, and is mostly, though not always, direct in language. A typical good item: "Describe characteristics of plane and solid figures using appropriate terms. Examples: round, flat, curved, straight" (1st grade, p. 18). Too often, on the other hand, occurs something like this: "Describe, extend, and create a wide variety of numeric and geometric patterns" (5th grade, p. 46). This is not a "learning objective" at all, and even as a pedagogical device is too broad and open-ended.
There is a lot of content in the course descriptions, though including some old- fashioned items that really should be retired as unilluminating and time-wasting, such as Descartes' Rule of Signs and Synthetic Division. The document as a whole slights deductive reasoning, which is principally mentioned in the geometry courses, but not carefully outlined even there, where its use is most traditional. It is surely evidence of this lack of attention to logical structures that the authors assign this impossible task: "Use the Fundamental Theorem of Algebra to solve polynomial equations" (p. 35) which mistakes the logical status of an existence theorem. Yet the document as a whole is one of the best.
The Framework (1) is extremely vague, offering examples of lack of clarity, and also of inflation: "By strategically applying different types of logics, students will learn to recognize which type of logic is being used in different situations and respond accordingly." ("Different types of logic" is, we believe, a reference to induction and deduction, about which much is made these days, without much effect. And "respond accordingly" couldn't be vaguer.) On page 4-12, under "benchmarks" for Math Content Standard A (Content of Math), at the 16-18 year-old level, the following appears: "A student would be able to . . . explore linear equations, nonlinear equations, inequalities, absolute values, vectors and matrices." "Exploration" is not a standard, and "absolute values" is curiously misplaced in this list. Clarity has suffered here, and more than clarity.
At the grade 3-5 level, under "Problem Solving," readers are told, "Evaluate the role of various criteria in determining the optimal solution to a problem." This is not only unclear, but is Inflation. For high school: "Recognize how mathematics changes in response to changing societal needs." (We believe mathematics is eternal and unchanging. There certainly are many things that vary in response to social pressures, but the document puts it badly.) The Standards (2) partly makes up for the deficiencies in the Framework (1), especially in its avoidance of inflated language, but it outlines a program lacking in sufficient content, especially at the secondary level.
The "performance objectives" are very briefly stated, and, while the reader has to make some difficult inferences, they are apparently demanding and good. Also, Arizona lists extra objectives for honor students, a useful feature. Sometimes, however, ambition outruns the language. For example, under "Geometry 4M-P4," "PO 3: State valid conclusions using given definitions, postulates and theorems" (p. 21). This is inflated and indefinite, as none of the surrounding text asks for knowledge or skills that would give it substance. Taken seriously, so vast an instruction should be broken down into numerous graded demands, describing one or more years of algebra or geometry. (Curiously, no particular subject matter is mentioned here.) Thus, Reason is badly outlined as a thread in an otherwise comprehensive document; it is put into one corner of the curriculum, as it were. As for Inflation, here is an example of unreality at the "honors" level in high school: "Demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations." The "theory of equations" is ill-defined today, and as understood 50 years ago concerned things about real polynomials that did not really translate into "techniques" concerning transformations. Curiously enough, this standard is also found verbatim in Idaho's "algebra" strand, at the 9-12 level (see below).
The telegraphic style of this Framework may conceal a good program in many school districts, but it says so little that it cannot be of much use. "Use technology" is a sentence that occurs repeatedly and often pointlessly. Standard 5.2.9, "Use mathematical reasoning to make conjectures and to validate and justify conclusions and generalizations," is no more helpful than to say, "Use mathematics." Standard 5.2.16, "Apply algebraic processes to non-algebraic functions," is opaque. Standard 2.1.6 (grades 9-12), "Explore non-Euclidean geometries," is unreasonable where Euclidean geometry is already too little explored.
(In fact, "non-Euclidean geometry" in today's high schools, where it is mentioned at all, sometimes designates spherical geometry, which is quite Euclidean. If this is what is meant, it should be stated; if this is not what is meant, it is inflation, since what the history of mathematics calls "Non-Euclidean Geometry" is any of a number of sophisticated axiomatic systems that differ from the Euclidean system only in postulating alternatives to the famous parallel postulate.)
The best feature is the document's relative lack of outright False Doctrine, but even in its brevity there is a great deal of inflation, e.g., "Visualize algebra as a bridge between arithmetic and higher level mathematics" (p. 9). This is neither clear nor definite nor testable, nor yet an item of content nor of reasoning. Yet it is labeled a "learning expectation."
The Standards is a scant 37 pages in length, and is classified by grade level from K-7 and then by subject headings (the subjects are strangely called "disciplines"), from Algebra I (in Grade 8) through courses which prepare the student for AP Calculus and AP Statistics. The writing is always terse and to the point.
At the start of each grade (K-7), the expectations for that grade are summarized in a hundred words or so, permitting a rapid and accurate overview of the whole. The details follow by rubric, the same for all grade levels:
i Number Sense
Although naming a rubric "Algebra and Functions" is stretching things at the lower levels, there is in general no undue multiplicity of rubrics, e.g. a strand labeled "Calculus" which at least one other state mysteriously included in its framework right down to the first grade. "Mathematical reasoning" is the only one here whose presence might be questioned, for its demands are of a rather general nature, and some could be considered "inflation" by the authors of this report if they were not thoroughly exemplified in the content standards of the other rubrics.
Each rubric is headed by one or more general admonitions which would also tend to be labeled not "Definite" or "Testable" were it not that their subheadings explain exactly what is meant. Under "Algebra and Functions," grade 3, for example, students are to "...represent simple functional relationships" (which is vague to say the least), but then the provincial teacher imagined in the Criteria is immediately told what that means in terms of content, e.g. "solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the per unit cost)."
Again, under "Mathematical Reasoning," grade 3, we find that "Students use strategies, skills and concepts in finding solutions." So often admonitions of this sort are simply left hanging as empty exhortations, but here follow six specifications, e.g., "express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work." A tall order, perhaps, but conveying (in passing) another important point: In speaking of "the" solution, the phrasing insists that mathematical problems have a single solution. Here the "reform" philosophy of what its opponents sometimes have called "fuzzy math" is firmly rejected.
In grade 4, instead of reading "Investigate the relation between the area and the perimeter of a rectangle" (a popular, though confusing, entry in many state standards, and in any case "investigate" is not a content standard), we find (1.2) "recognize that the rectangles having the same area can have different perimeters," and (1.3) "understand that the same number can be the perimeter of different rectangles, each having a different area." It is also refreshing to observe, in grade 4 "Number Sense" and in the Glossary, that mathematical educators in California know what prime numbers are and tell us carefully. (cf. Pennsylvania, below.) Scientific calculators are mandated for the first time in grade 6, but only for good reason, and after the essential properties of the real number system have been assimilated through hand calculations. (Japan introduces these electronic aids in grade 5, while most American states demand their use from the beginning.) By the end of grade 5, California students are to be able to do long division with multiple-digit divisors and to represent negative integers, decimals, fractions, and mixed numbers on a number line. By the end of grade 7 they graph functions, use the Pythagorean theorem, evaluate algebraic expressions, organize statistical data, and in general are prepared for high school algebra and geometry.
The years 8-12 are described by subject: Algebra I and II, Geometry, Probability and Statistics, Trigonometry, Linear Algebra, Mathematical Analysis, and as the "Advanced Placement" subjects of statistics and calculus. The Standards is not prescriptive in its pedagogy. How the teacher, or the textbook, goes about the job is left to the discretion of the teacher or school district. A table is provided suggesting placement of this material by year, so that integrated curricula are possible by the same standards as the subjects would demand when taken in the form of courses. It is clear that a course-by-course program would place Algebra I in the 8th grade, Geometry in the 9th, and Algebra II in the 10th, completing the state-mandated curriculum for graduation.
Algebra I names 25 items, some necessarily mechanical and some with welcome attention to logical structure, including knowing the quadratic formula and its proof. "Practical" rate problems, work problems, and percent mixture problems are to be studied as well as the (impractical?) Galilean formulas for the motion of a particle under the force of gravity.
In Geometry, students must be able to prove the Pythagorean theorem and much else, including proofs by contradiction, and classical Euclidean theorems on circles, chords, and inscribed angles. "Geometry" also includes the basic trigonometry of solving triangles, and the properties of rigid motions in the plane and space.
Mathematical Induction is introduced in Algebra II, which material is apparently intended for the 10th grade or earlier. Experience will have to show if this and certain other ambitious demands are really possible, or should rather be left for the following years and a volunteer audience. Other topics are either preparation for college work or are (these days) college-level work themselves (de Moivre's Theorem, the Binomial Theorem, Conic Sections with foci and eccentricities, etc.). We are not told how much of this program must be taken by all California students, or whether the courses named Trigonometry and Statistics are intended as options.
There is no mention of "shop math," "finite math," or "business math" (etc.), directed at the non-college bound student, or remedial courses for older students who have failed earlier. Perhaps the Mathematics Framework (see below), required by California law to be revised for 1999, will address these issues. The authors of this report did not downgrade California (or any other States) for this sort of omission, when the core curriculum is adequate and well stated. We do, however, expect that California will in due course find it advisable to add something appropriate to its "elective" curriculum at the high school level, as does, for example, Tennessee, which offers a branching of alternate tracks.
If teachers and textbooks can be found to carry it through properly, this Standards outlines a program that is intellectually coherent and as practical for the non-scientific citizen as for the future engineer. Whatever of "real-world" application school mathematics can have, is found here, set upon a solid basis of necessary understanding and skill. Initial reaction to the adoption of this document included a widespread apprehension that this "return to basics" represented an anti-intellectual stance: rote memorization of pointless routines instead of true understanding of the concepts of mathematics. The opposite is true. One can no more use mathematical "concepts" without a grounding in fact and experience, and indeed memorization and drill, than one can play a Beethoven sonata without exercise in scales and arpeggios.
There is always a danger that intellectually challenging material, be it in music, literature, or mathematics, will in the hands of ignorant teachers, or bowdlerized textbooks, become reduced to pointless drills. The history of American school mathematics in the 20th century has largely been a chronicle of conceding defeat in advance, teaching too little on the grounds that trying for more will fail. California is to be commended for taking up the challenge head-on, and announcing its intention in the clearest terms in its Content Standards.
It is the more curious, then, that the adoption of this Standards has been attended by an extraordinarily bitter public debate centering on their characterization as a reactionary document discouraging, rather than demanding, the "real understanding" of mathematics. An earlier version of this Standards had been composed by a special Commission on Standards which worked through most of 1997 on standards for English and mathematics, to be approved by the State Board of Education. The appointment of the Commission was itself extraordinary, and the consequence of public dissatisfaction with current teaching of core academic subjects.
California by law publishes a Framework for mathematics instruction every seven years. Past Frameworks included standards of the sort under review here, along with pedagogical and administrative information, and the most recent was published in 1992 in the midst of enthusiasm, on the part of the school administrators, for the point of view represented nationally by the NCTM Standards of 1989, and often called "reform." (The "reform" trend had been visible in California even earlier.) Two foci of opposition soon appeared: "HOLD" in Palo Alto and "Mathematically Correct" in San Diego, both citizens groups (including mathematicians and engineers) publishing web pages designed to persuade readers that the "reform" represented by the 1992 Framework and its progeny, should be discarded in favor of something usually (though simplistically) called "traditional." In particular, they pointed to what they said were deteriorating scores of California children on national tests.
In a word, the anti-reform camp claimed "Johnny can't add," but instead spends his school time measuring playgrounds and talking it over with his classmates; and he uses a calculator when asked to multiply 17 by 10. Thus the new, ad hoc Commission on Standards was appointed, quite apart from the legally mandated Framework committee, to--in effect--adjudicate this controversy. The Commission, a citizens' commission not intended to be expert in mathematics (new standards for other core subjects were also part of its charge), took advice from experts of its own choosing and ended sharply divided. It voted by a large majority in favor of a document that it submitted to the State Board of Education on October 1.
The Board, which has final say, heard much public testimony, including opposition expressed by mathematicians, and rejected the draft. The Board used that draft, however, as a beginning for the very substantial revision it ultimately approved in December of 1997. That revision, which is the Standards reviewed here, was mainly prepared by a group of mathematicians at Stanford University, and its publication has generated more public controversy than anything seen earlier. Apart from segments of the public, two groups of professionals are now in contention: the mathematics education community, or a vocal part of it, against the mathematicians' community, or a vocal part of that.
(In the meantime, the Board has the task of reconciling the mathematics standards implicit in the new Framework, expected to be published in 1998, with the revised Standards under review here.)
Newspaper reports of the controversy make it apparent that those opposed to the Board's revisions, and who wish the Board to return to the document submitted to them by the Commission in October, include the California Superintendent of Public Instruction and at least one high-ranking official of the National Science Foundation. This party portrays this revised Standards as a return to the failures of past years, a document devoted to the mindless, pointless manipulation of outdated algorithms. The authors of this report believe such a characterization is mistaken, and that the mathematicians who participated in the final revision had no such intention, and their product no such result--except as poor teaching might make it so. It is to the better mathematical education of teachers that California (and the rest of us) must look for improvement of result.
The best feature of the Content standards is the listing of topics additional to those demanded of all students. Under False Doctrine is the prescription of calculators and computers for whole number arithmetic at the K-4 level. The entire text takes up only 16 pages, but then is followed by an Index occupying three pages, surely inflation of a sort. The text is often vague, indefinite, or careless--thus at page 12 the reader finds this phrase: "solving real-world problems with informal use of combinations and permutations"; and in the Glossary, "Algebraic Methods" are defined as "the use of symbols to represent numbers and signs to represent their relationships" (p. 22). As to the former, why "informal"? And by the Glossary definition, algebraic methods could be exemplified by "2+3=5," since 2,3, and 5 are symbols and + and = are signs representing their relationships. But "algebra" means something more than symbolism. On page 14, this carelessly placed item for grades 5-8 occurs: "Solve problems using coordinate geometry." Taken seriously, this is something done in 12th grade pre-calculus courses, or in college, unless "problem" means something quite trivial. Such carelessness must be counted False Doctrine.
This August 7, 1997 document is marked "Second Draft," but will require expansion and definiteness of reference to be of any value. Reason is pushed to one corner of the "Geometry" page--at the 9-12 level, and among five other items-- where readers are told, "Develop an understanding of an axiomatic system through geometric investigations, making conjectures, formulating arguments and constructing proofs." This recapitulation of the thousand-year long development of ancient Greek mathematics is unlikely in a high school, and should be replaced by a realistic set of standards for deductive geometry, if that is what is meant. As written it is too ambitious. On the other hand, the following, from level 5-8, is too childish: "Use real-life experiences, physical materials and technology to construct meanings for the whole numbers. . . ." By grade 5 the abstract notion of a whole number should long have been in mind, without blocks and calculators.
The Framework is too brief to contain much False Doctrine or Inflation. Still, standard 4, "ratios, proportions and percents," runs through all grades; under it, at grades 5-8, we find the following: "use dimensional analysis to identify and find equivalent rates"; and this, at grades 9-12: "Use dimensional analysis and equivalent rates to solve problems." Not much progress there. The entire page, of which the two quoted items form about 20 percent, is inflated by the strain of finding enough words to justify the existence of standard 4 altogether, for it surely does not deserve equal billing with (say) standard 9, "Algebra and Functions"--and even Standard 9 omits geometric series, the quadratic formula, and the binomial theorem.
The 1995 text is brief, though augmented by pedagogical suggestions, and it covers K-10 only, as a guide to statewide testing at the grade 10 level. An "Appendix A" has been added to the state's Web site, giving an indication, all too brief, of what will be expected at grades 11 and 12, and including a welcome announcement that the "logical framework" of algebra, and deductive proofs in general, will form an important part of the program. However, this part is not yet as fully elaborated as the main text covering K-10. Here, where the standards themselves are vague, which is often, the suggested instructional activities generally illustrate a very minimal content. Projects, keeping journals, reporting to the class, and "real-life" applications are too often emphasized above intellectual content. The constructivist stance leads to instructions concerning student "exploration," sometimes producing mystifying open-ended demands, e.g., "Examine the relative effect of operations on rational numbers" ("Number Sense," grades 6-8). Relative to what? Which operations? Why rational numbers?
Where items are less vague, they can describe a valuable curriculum--e.g., for "Geometry" at grades 6-8, "Use a compass and straight edge as tools for basic geometric constructions" (p. 51). If this were accompanied by a lesson in the reasoning behind the validity of some such constructions, it would prepare for the logical analyses indicated for the grade 11- 12 levels. But the very use of the word "tools" for the ruler-and-compass backbone of Euclid's geometry generates a misunderstanding of that ancient branch of mathematics. There is very little of Reason threaded through the performance indicators.
There is here an occasional good idea, but such ideas are few. On page M-13 (grade 11 level), the reader discovers this: "Illustrate that a variety of problem situations can be modeled by the same type of function." As a summary of what can be learned from several years of progress in algebra, this request is exemplary, but the rest of the document simply does not present or even outline such a program. (Idaho--see below--posts the same demand, verbatim but for the word "recognize" in place of "illustrate.") On page M-9 occurs this example of Inflation: "Through the use of technology, students can experi ence a richer set of algebra experiences that allow them to investigate algebraic models at a conceptual level through representations in terms of graphs, tables, polynomials and matrices." Richer than what? Algebraic models of what? Without definition, such sentences do not guide instruction. And "matrices" at the 11th grade level are a disproportionate demand, where algebra does not include the quadratic equation or binomial theorem. While the D.C. document doesn't contain much under the heading Negative Qualities, it also contains very little that is positive.
District of Columbia