State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
We consider the two documents (see Appendix) together: (1) The Sunshine State Standards is filled with examples and teaching strategies as well as content advice. It is well printed and easy to use, but those parts covering grades 6-12 have been superseded by (2), Florida Course Descriptions, which describes the material by course titles rather than by "threads." The general tone of the Course Descriptions is consonant with that of the Standards; however, it features an overemphasis on the instrumental ("real-life") purposes and uses of mathematics, even in "honors" courses presumably directed at college preparatory students.
Thus, on page 3 of "Course 1206320- Geometry Honors," we see, at MA.B.3.4.1, "Solve real-world and mathematical problems involving estimates of measurements, including length, time, weight/mass, temperature, money, perimeter, area and volume, and estimate the effects of mea surement errors on calculations." This is typical, and among other things rather careless in classifying money problems under "geometry." There is indeed very little geometry in this course, from either the Euclidean or the analytic standpoint. Some other failures: On page 112, we read, "Recognizes, extends, generalizes, and creates a wide variety of patterns and relationships using symbols and objects." This is not clear or definite, and is inflated. (The associated example is actually simple, and worthy of more explicit description.) On page 50, the Performance Description MA.A.1.4.3a, "determines whether calculated numbers are rational or irrational numbers"; it is exemplified by a contrived "real-world" situation concerning when an automobile's braking distance is given by a rational or irrational number. Real-world physics has not yet achieved the means of determining whether a measurement is or is not that of an irrational number. Thus, here is an example of straining after "real-life" ends by presenting a false scientific, if not mathematical, doctrine. Finally, on page 48 Example M.A.A.1.3.3a is a multiple- choice question with all its answer choices wrong: carelessness again.
On page 1 we find that "[c]alculators and computers are essential tools for learning and doing mathematics at all grade levels." This is not so. The Georgia "Core Curriculum" is labeled "Draft," and is certainly one of the better ones; but it unfortunately avoids mention of the deductive structures of mathematics even in places where it would be natural as, for example, in "Geometry": "States and applies the triangle sum, exterior angles and polygon angle sum theorems." Why not ask for their proof, too? The proofs are not difficult, and are more enlightening than the results themselves. It is an opportunity missed. In another place concerned with geometry, the student is expected "to [use] tools such as compass and straight-edge, paper folding, tracing paper, mira or computer to construct congruent segments, angles. . . ." This reduces ruler and compass to tools like the others, unrelated to the deductive structure of the subject. As tools, they are certainly inferior to computer printouts, but this is irrelevant to the instructional value that should be their purpose. This January, 1997 draft is not yet a completed document, especially in describing the organization intended for its advanced offerings, which are numerous and rich in content, and include Advanced Placement courses in statistics as well as calculus. It is mainly the standard K-12 curriculum implied by these standards that lacks sufficient attention to Reason, overemphasizes the uses of technology, and carries the message in all courses that mathematics is preeminently for direct practical use. Almost all the demands are clear and definite, but indefiniteness appears more often than it should. In "Discrete Mathematics," the student "solves problems that relate concepts to other concepts, and to real-world applications, using tools such as calculators and computers." Probably the authors had something definite in mind here, but the words do not convey it. Furthermore, too much is expected of technology here if calculators are expected to help "relate concepts to other concepts." This is something that should be done in the mind.
Yet much that is written is definite and good. For example, at the K-5 level, "Determines the missing number or symbol in addition or subtraction number sentences" describes a necessary ability exactly, and "uses skip-counting as readiness for multiplication" pays proper attention to the pacing of the teaching of arithmetic.
In the Essential Content document (2), there appears under "Content" in the "Statistics/ Probability" thread, the heading "Probability in real-world situations"; to its right, under the corresponding Performance rubric, are three entries, of which the third is "Solve real-life problems using statistics and probability" (p. 64). This is not an enlightening amplification of the heading. Both documents are of this vague nature. In the Performance Standards document (1), on page S-17, occurs the 9- 12 instruction, "Geometric and spatial explorations include: . . . Non-Euclidean geometries, and hyperbolic and elliptical geometries." This is unbelievable where the documents do not otherwise mention axioms or proofs. And it cannot mean spherical geometry (see Note on Arkansas above), because, while spherical geometry is related to one form of non-Euclidean geometry, it is not at all hyperbolic. Very few statements in either document are definite enough to qualify as False Doctrine, or any doctrine at all.
Grades 5-6 were plainly written by a different team than the others, and are much better than all the rest. We count grade 6 as part of "middle school" for scoring purposes here; hence, this part of Content receives a better grade than the rest, which is inadequate on all counts. The total organization of the Content Guide and Framework generates some careless repetitions-- for instance, fractions and decimal calculations are prescribed in nearly identical terms at all grades from K through 4. Surely not "decimals" in kindergarten? And one "performance objective" at 1st grade level is that "[a]ll students will reflect on and clarify their thinking about mathematical ideas and situations." This is not testable.
Of the 9-12 level the reader is told, "Every mathematics course in Grades 9-12 will address objectives from each of the fourteen standards included in this framework. In order to accomplish this, tedious computations and graphical representations by pencil and paper and pencil drill must be deemphasized to the point that the use of technology (calculators, computers, etc.) will be used to perform these tasks at all levels of mathematics." Yet Idaho is one of the few states to include "geometries" [plural] in its high school curriculum, asking all students to "develop an understanding of an axiomatic system through investigating and comparing various geometries." The unreality of this prescription is discussed above, where Arkansas and Hawaii make a similar demand. "Geometries" cannot be understood in the absence of the understanding of at least one example, preferably the Euclidean, something Idaho does not sufficiently offer. To speak in such grand terms, of "geometries," is Inflation. Under "Algebra" this standard appears: "demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations," and under "functions," "recognize that a variety of problem situations can be modeled by the same type of function." The first of these statements is incoherent. (It also occurs verbatim in one of Arizona's "honors" performance objectives.) The second, which catches one's attention with the awkward and obscure phrase, "the same type of function," also occurs verbatim on page M-13 of the District of Columbia's Framework.
Under Goal 7, dealing with measurement, at 7.C.4a, is an example of something clear and good: "Make indirect measurements, including heights and distances, using proportions (e.g., finding the height of a tower by its shadow)" is a plain and direct entry relevant to the Goal. Indefinite or obscure entries are much more numerous, however. For example, under "Number Sense," for late high school, the reader finds, "Determine the level of accuracy needed for computations involving measurement and irrational numbers" (p. 19). Again, 6.A.4, for early high school, states, "Identify and apply the associative, commutative, distributive and identity properties of real numbers, including special numbers such as pi and square roots"; this is quite incoherent, apart from the indefiniteness of "apply." "Calculators and computers" are apparently mandated in grades K-4 for teaching "number sense," among other things (Standard C, see p. 18), though the itemization is obscure on this point. This is either lack of clarity or False Doctrine. Certainly the standard algorithms for the decimal system are not prescribed. Here is excess verbiage: "Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space." This can only mean "Learn Euclidean geometry," and indeed the document does call for the traditional two-column Euclidean proof format at 9.4.4c. But 9.4.4c is hard to take seriously in that it is confined to a single item among 18 such, the rest regarding geometry as mainly a practical and empirical study; and there is not enough time left to do all that is implied by "two-column proofs." There is, in general, a paucity of demand for deductive reasoning in other parts of the curriculum, too, though a wise teacher might read some such demand into the rather indefinite instructions that are given. The Learning Standards are avowedly intended for all students, but appear to be a compromise: overambitious for some and neglectful of others, and vague enough to permit many sorts of teaching and many depths of curriculum.
The writing in this Mathematics Proficiency Guide is very good, direct and plain, and only sometimes indefinite, as when phrases like "in a variety of ways" are used. The content, however, suffers from an unusually heavy commitment to "real-life" applications as the rationale for all lessons. On page 207, "Constructive learning should be incorporated into almost every mathematics lesson" apparently requires the state to omit all too many mathematics lessons, as our ratings for Content indicate. The logical structures of algebra and geometry are among the omissions.
Even the announced commitment to "real-world" examples is sometimes belied by a predisposition to theory, as when a table of heights attained by an upwards thrown baseball turns out to be a theoretical set of heights attained by the Galilean formula for falling objects, and not a set of measurements as advertised.
There is also mathematical error, as in the "topology lesson" on page 154: "Give students a piece of paper with a circle the size of a dime. Cut out the circle. Can a quarter be made to pass through this hole without tearing the paper?" This is not a lesson in topology. Topology is, of course, not a K-12 subject, but if the word is mentioned it ought to be mentioned correctly. On the same page, "The formation of a sea shell depicts many golden ratios which are intertwined" is very badly stated. There is only one golden ratio, not that this fact is the only error in the quoted statement. However, the fact that the author here cannot or will not distinguish between a mathematical abstraction (the golden ratio) and its applications or exemplifications is characteristic of the view of mathematics inculcated by the entire Proficiency Guide. These are but a few examples of the plethora of "activities" and "real-world" applications that take up most of the space in this long document, in which actual demands for mathematical content take but a small space.