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

This Standards is very well organized: the "Outcomes" are listed and classified by subject thread and grade level early in the document, then spread out, one "outcome" to a page, followed on that page by a number of examples or exercises illustrating their import, for they are rather general in language and tone. But the illustrations are often trivial and illustrate very little, or they are carelessly written, sometimes even suggesting lessons destructive of the announced purpose. On page 74 are given two matrices named A and B, with instructions to multiply to get AB and then BA, and draw some lesson therefrom. (This is a calculator exercise.). But the desired "lesson," which appears to concern the commutative law for real numbers, according to the top of the page, cannot be had because BA does not exist, i.e., the matrices cannot be multiplied in that order. People who know matrix multiplication only from calculator exercises, even written correctly, cannot learn the main lessons to be learned from matrix multiplication in school.

A more striking example of the inappropriate emphasis on technology occurs at the 8th grade level: "Use a computer program that generates Pythagorean triples. (Note: Many text-books have BASIC language computer programs already written to do this. The student could enter the program and run it.) What are the first 10 Pythagorean triples? (i.e., the 10 Pythagorean triples with the smallest unique whole numbers)"

The reader is immediately impelled to wonder whether the triple (20,21,29) "precedes" or "follows" (9,40,41), but this is incidental. The real fault here is that using a previously written BASIC program teaches a student no more about Pythagorean triples than would the teacher's writing some of them on the blackboard. A "lesson" of this sort is one consequence of a standards document's insistence on "appropriate technology" and manipulatives at all levels. Algebra tiles, which Kansas recommends for the 10th grade, are another. The 10th grade is a time when one hopes such "training wheels" have long been left behind. Here, instead, Reason is left behind. False doctrine.

Kentucky's companion document, Transformations, is not listed in the Appendix because it is a guide to instruction rather than a standards document of the sort we find in the Core Content commented on here. The latter is organized to show, for each thread (such as "algebraic ideas"), the items labeled "concepts," "skills," and "relationships," at each of the levels--elementary (K-5), middle (6-8), and high school (9-11)--or more accurately, what is expected by the time of assessment, which will be during the 5th, 8th, and 11th grades. The list is brief and generally inadequate. As a guide to assessment it would have to be augmented by many specifics, and perhaps experienced Kentucky teachers already know them; but the purpose of a Core Content guide is not simply to be a general reminder of what one already is doing. Nor is such a guide very helpful to new teachers.

A typical "skill" listing for "algebraic ideas" at the grade 5 level is, "Students should be able to find rules for patterns, extend patterns, and create patterns" (p. 6). At grade 11, they should be able to "solve and graph a variety of equations and inequalities." It is not possible to deduce from statements like this whether the content at each grade level is rich or poor, and we have rated Content accordingly, i.e., only by as much as we have been instructed by the text. Some instructions are more definite, and this pattern should be followed elsewhere--e.g., "Students should understand arithmetic and geometric means." This is good as far as it goes, but at such places, where the Core Content is definite, it is meager.

While a vague instruction might sometimes imply more content than our scores give credit for, one also sometimes finds that the vagueness is actually impenetrable, as when students are to understand "how ratio and proportion can be used to connect mathematical ideas." Ratio and proportion are relationships that in school mathematics connect numbers, or lengths, etc., not mathematical ideas. This must be called Inflation. Again, 11th grade students are to understand "order and equivalence relations," and "how numbers in the real number system relate to each other." Were the phrasing more comprehensible and definite, these might describe some rather advanced college work. The abstract idea of "equivalence relation," for example, and any application of that idea, are found difficult by many advanced college undergraduates.

It is possible to find a meaningful and useful item here and there. At the 5-8 level, "demonstrating a conceptual understanding and applications [sic] of proportional reasoning (e.g., determining equivalent fractions, finding a missing term of a given proportion)" is perfectly plain and definite, and suited to the level. But consider Benchmark K-4, "Data Analysis":

"Demonstrating the connection of data analysis, probability and discrete mathematics to other strands and real-life situations." It is not possible to believe this is asked of the same K-4 students, less than 10 years old, who in another part of the same standards document are "identifying and drawing lines and angles and describing their relationships to each other and to the real world," something much more rea sonable. At every level of this document, there is such a mixture of the trivial and the impossibly general or sophisticated--or opaque. Even the Glossary cannot be of real use to anybody; it defines "Coordinate Geometry" as "geometry based on the coordinate system," and "Magnitude" as "size or largeness." There is much else that is not helpful, including this: "Due to the rapid growth in technology, the amount of information available is accelerating so rapidly that teachers are no longer able to impart a complete knowledge of a subject area" (p. 1-2). Even without technology, the conclusion that more is known--about anything-- than teachers can impart was true in the time of Plato. Such platitudes do not contribute to the purpose of a standards document, and are Inflation.

Under the Content Standard, "Students understand and demonstrate that ideas are more powerful if they can be justified," is the following Performance Indicator: "[Students should understand] that proving a hypothesis false (i.e., that just one exception will do) is much easier than proving a hypothesis true (i.e., true for all possible cases)." The lesson that a single counter-example falsifies a conjecture ("conjecture" is better than "hypothesis" here) is an important one, worth teaching. However, to cast proof as a means of making an idea more "powerful," rather than merely true, is not to make a mathematical statement at all, and perhaps to make a false statement, as the history of demagoguery shows. The teaching of Reason is ill served in general in these standards, as Maine's rating for Reason indicates.

This Curriculum Framework is for mathematics and science both, and gives the mathematical reader a chance to observe another field (science) in passing. On page 4, "Understand that matter can be neither created nor destroyed" is a doctrine of great 18th century importance, and was learned by children as late as 1900 in just those terms, but it is now known to be mistaken, and its exceptions have had important 20th century implications. With mathematics, on the other hand, the Maine document has it the other way round: the (apparent) exceptions to old doctrine are being emphasized beyond their desserts. Deductive thinking, given us by ancient Greece, is here replaced by a "content standard" urging that "students use different methods of thought to justify ideas," and one performance indicator under this standard advises that students "use intuitive thinking and brainstorming." These are things we do perforce; mathematics should teach us to govern these impulses, channel them, and recognize their pitfalls. Also under the general heading of Reason is "make inductive and deductive arguments to support conjectures." This is about as much detail as the document offers.

The illustrative, mainly "real-world," examples throughout the document do not often have much mathematical content. To advertise reasoning is not the same as to teach it in contexts where its value is visible. There is not much context here to make this possible.

The most that can be said about change is that we now have some mathematical processes that were not earlier known; but the ones taught in high school have mostly been around for a long, long time, and the world still has need of people who understand them.

The Framework items of student "expectations" occupy only pages 18-23 of this 42 page pamphlet, the rest being introduction, educational philosophy, and other commentary. There are six extremely general Goals. For example, Goal 2 is "to develop an understanding of the structure of mathematical systems: concepts, properties, and processes." Each goal has four, five or six "subgoals." For example, Subgoal 2.3 is to "[u]nderstand concepts, properties, and processes of geometry," which is not much advancement in specificity. Finally, there are from two to 13 "expectancies" under each subgoal. These are also general, and not keyed to particular grade levels. Expectancy 2.3.7 is to "explore geometric ideas such as topology, analytic geometry, and transformations." This is as specific as the Framework gets, which means that it is of no practical value. This particular example was chosen to show that it can also be inflated and unrealistic: topology is no subject for school mathematics anyway, and this should be known to the person who wrote this "expectancy." Most of the Framework is of this level of generality and sometimes unreality, and while it also contains some advice about how local schools can use it in constructing curricula, it is not really a guide of the sort envisioned in the criteria for the ratings as described earlier in this report.

What value there is in the package of two documents in our possession is mainly in The High School Core Learning Goals (document (2) in the Appendix), which is more definite, but quite undemanding of content. It also has goals and expectations, e.g., under Goal 2, "Geometry, Measurement and Reasoning," Expectation 2.1 states that "the student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate." One of the "indicators" under this Expectation (2.1.3) is that "[t]he student will use transformations to move figures, create designs, and demonstrate geometric properties." Clearly this indicator is compatible with whatever geometry course the local district might want to create. Despite the vagueness, there is False Doctrine in the overemphasis on calculators and computers, and the "real-world" applications, examples of which in the Core Learning Goals overwhelm whatever intellectual content the text might attempt to imply.

The two documents have the ambition to provide a philosophical background for curriculum and teaching, but do not end by saying anything that can be used, and by occasionally saying what is not so, as when the author of the Core Learning Goals makes this statement: "With the change in technologies, the mathematical processes change" (p. 1). How can this be, when on page 15 is pictured a house with a patio, inviting the student to find some optimal dimensions? This problem and its solution have not changed in 5000 years; though we now have a more convenient notation than did the ancients, the mathematical processes are the same. The most that can be said about change is that we now have some mathematical processes that were not earlier known; but the ones taught in high school have mostly been around for a long, long time, and the world still has need of people who understand them.

The Framework is nicely laid out and printed, with excellent photographs as well as diagrams, but it is pedagogically and mathematically incoherent, and lacking in content. It says that manipulatives should not be confined to the early grades, and indeed it illustrates their use at all levels, but manipulatives are poor preparation for proofs "by mathematical induction" in the 9th and 10th grades (p. 78). Any college mathematics teacher will testify to the difficulty of teaching proofs by induction to college mathematics students, and the idea that this should occur in 10th grade is unrealistic, suggesting that "mathematical induction" has been confused with the rather vague notion of "inductive reasoning" in the mind of the author. Furthermore, the placing of mathematical induction under the heading of "Geometry and Measurement" is also incoherent.

At grade level 11-12, under "Geometry and Spatial Sense," we have this incoherent Learning Standard: "Use vectors, phase shift, maxima, minima, inflection points, and precise mathematical descriptions of symmetries to locate and describe objects in their orientation" (p. 81). The other standard in this section is "deduce properties of, and relationships between, figures from given assumptions," which is coherent but vague. Opposite these two, under "Example of Student Learning," is a picture of a parallel pair of Plexiglas rectangles held in place by some bolts, and the instruction to dip this into a soap solution to illustrate Steiner points for networks. This model has nothing to do with the two standards, and, except for its "popular science" appeal, has no educational value at this level of mathematical instruction.

At grades 5-8, students are to "engage in problem-solving, communicating, reasoning, and connecting to . . . develop and explain the concept of the Pythagorean Theorem." This kind of verbiage is inflated; the thing is a theorem, not a concept of a theorem, and "communicating" and "connecting" etc., are indicative of the sort of time-wasting richly repeated in other parts of the Framework, where "Examples of Student Learning" are given. All this in a document that doesn't ask for the essential skills of factoring of polynomials, or the quadratic formula, or the binomial theorem. Page 28 contains this: "Move away from the notion that basics must be mastered before proceeding to higher-level mathematics." For students of ordinary ability this is the road to "mathematics appreciation," not mathematics. Indeed, "mathematics appreciation" seems to be the main theme of the Framework which talks of Fibonacci numbers and Nautilus shells, soap films and Steiner points, fractals, notions whose mathematical content is generally beyond the school mathematics level, and which can only serve as amusements. The document as a whole is not serviceable for the purposes outlined in the introduction to the criteria, as given earlier in this report.

The educational philosophy given in the introductory pages of the Model Content Standards is clear and definite: These standards are to be consistent with the "constructivist" view of the educational process, "firmly grounded in the work of John Dewey. . . . In other words, it is no longer sufficient to simply know mathematical facts; learners must be able to understand the concepts behind them and to be able to apply them to problems and situations in the real world." However, it is not possible to determine from the standards as written what are the "mathematical facts" desired, insufficient though they might be, let alone the way to "understand the concepts." The items are vague, and are more exhortations than standards, or are incoherent.

Under Content Standard 9, concerning number systems, at the High School level, students are to "develop an understanding of irrational, real, and complex numbers." There is something mathematically offbeat about the progression, "irrational, real, and complex," where the historical construction runs "rational, real, and complex"; and a mathematician's unease increases two pages later, where students are to "develop an understanding of the real and complex number systems and of the properties of special numbers including i, e, and conjugates." Conjugates? Again, this is an incoherent list, collected without thought of the pedagogical sequence it is intended to suggest, and perhaps without understanding of the non-parallel qualities of that particular listing.

A typical example of a standard is Content Standard 3: "Students develop spatial sense, identify characteristics and define shapes, identify properties and describe relationships among shapes." Below the standard are specifications, some for elementary, some for middle school, and some for high school. Here is one, for middle school: "Generalize the characteristics of shapes and apply their generalizations to classes of shapes." People who do not understand mathematics might be intimidated by the technical words "generalize" and "classes of shapes" into believing this sentence was written in English, or (alternatively) into believing that mathematics is not really supposed to be written in English, but either deduction would be incorrect.

The principal faults of the entire document are its brevity and its vagueness--a standards document cannot afford both-- which render it an unsuitable guide to instruction or choice of curriculum.

This Curriculum Structure is well written and rich enough in content to serve its purpose, though it is sprinkled throughout with an excessive number of references to "real-world" applications, and is correspondingly short on instruction concerning the structures of mathematics (i.e., relative to our criterion, Reason). Page 51 contains this dangerous doctrine, though one observed by most other states without explicit mention: "Even though proof remains an important component in the geometric [sic] course, a shift from the traditional two-column deductive proof removes proof as the primary focus of the course to one in which the student provides informal arguments either orally or in writing." More happy, however, is the fact that Mississippi even mentions that proof (deductive proof) is important at all.

The exaggerated devotion to "real-life" applications sometimes leads to inflated claims, or verbiage, e.g., "The student will understand the role of application of matrices in connection with conic sections in describing real-life phenomena" (p. 65). Apart from the fact that no high school student is in a position to relate the mathematical properties of the conic sections with real-life phenomena, and that the use of matrices in their analysis is quite an advanced branch of algebra, this particular instruction is found under the goal concerned with statistics and probability, with which it has no discernable connection. (The conic sections are indeed the paths of comets and planets, but analyses of this depth cannot be done without some knowledge of calculus, which is not expected here, and is not particularly aided by the use of matrices anyway.)

Another contrived "real-life" connection is the following: "Solve problems involving factors, multiples, prime and composite numbers; include concepts of common factors and multiples and prime factorization (expressed using exponents); include real life applications of these concepts" (p. 33). The entire instruction is admirable except for the last clause, which is impossible.

By making some inferences that might or might not be consistent with the intent of the rather unclear wording of this entire document, one can deduce that Missouri intends a fair amount of mathematical content, which gives this Framework a better score for Content than under the other criteria. The document is well organized, not only by strand (e.g., "Data analysis, Probability and Statistics") and grade level (K-4, 5-8, and 9-12), but by "What all students should know," "What all students should be able to do," and "Sample learning activities." But these carefully constructed categories are not often filled with valuable information.

Under "Mathematical Systems and Number Theory," Grades 9-12, all students should be able to "select and apply appropriate technology as a problem-solving tool to achieve understanding of the logic of algebraic and geometric procedures" (p. 64). However, the Framework does not mention axioms and theorems with proofs as another way (i.e., in addition to "technology") of achieving this understanding. There is at this point a reference to NCTM Standard 14 which, upon examination, shows no mention of technology in this connection. The implied technology apparently includes "algebra tiles," which are mentioned on page 64, too, though they are better described as manipulatives. This is False Doctrine by our criteria, and an avoidance of the sort of mathematical instruction that should be implied by the title "Mathematical systems and number theory" that heads the page.

Other instructions on the same page are less mischievous because less definite: "Compare and contrast the real number system and its various subsystems," and "extend understanding and application of number theory concepts." Such language characterizes the Framework; here is another example taken almost at random: "Experiences should be such that students use discovery-oriented, inquiry-based and problem-centered approaches to investigate and understand mathematics" (p. 9). One can hardly select textbooks on the basis of this sort of instruction without already knowing from other sources what is to be taught.

The major part of the cited document concerns institutional matters: accreditation, teacher education, and general goals (and visions) for education in science and mathematics. The "Mathematics Curriculum Standards" occupy page 57 and part of page 58, and are mostly too general to be of use--e.g., "include the study of trigonometry." Brief and general as they are, the listed items exhibit a disinclination to ask that anything be overtly taught, suggesting instead "explorations" and "experiences," as in the instruction for grades 5-8 to "include explorations of algebraic concepts and processes."