State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
Only three pages are given to mathematics, yet the generalities, while often indefinite, portray an adequate curriculum up to the high school level. The high school descriptions, however, permit too many interpretations. The high school student is to "understand the basic structures of number systems," for example--possibly a large order and possibly not, depending on what is meant. On the other hand, arithmetic at K-4 and at 5-8 are reasonably outlined, and include negative numbers, all rational numbers, and the mention of non-repeating decimals. The entire document is too brief to give much detail, yet some particular indicators outline a strong primary school program. Negative numbers, for example, are introduced in level K-4. The weakest thread is "Problem-Solving and Reasoning," especially at the high school level, where the text is impossibly general, and indeed inflated, e.g., "Work to extend specific results and generalize from them; and gather evidence for conjectures and formulate proofs for them; understand the difference between supporting examples and proof." That last clause is excellent; it is a pity that its points of reference are not hinted at. One does not make proofs; one makes proofs about things, and those things should be named in a sufficient standards document.
This standards document is so well written, definite, and free of jargon and inflation, that one is disappointed at the modesty of its expectations. Despite repeated disclaimers that "technology shall not be regarded as a substitute for a student's . . . proficiency in basic computations," there is excessive emphasis on technology throughout, e.g., under Trigonometry": "Graphing utilities . . . provide a powerful tool for solving/verifying trigonometric equations and inequalities" (p. 22). If "solving/verifying" is related to "proving," this is not good advice. If only verification is wanted, it is probably easier to ask the teacher. On page 10, at grade 4, standard 4.8 prescribes calculators for products of two three-digit numbers, at grade 5 any divisor of more than two digits will call for calculator assistance (standard 5.5, p. 11), and at grade 6 and presumably thereafter, adult status having been reached, any divisor of more than one significant figure will demand calculators (standard 6.6, p. 13).
At the high school level, in "Algebra II," linear systems are only to be solved by matrix inversion using calculators, and the entire mathematical meaning of this part of algebra is thereby bypassed, including such practical applications as the recognition of linear dependency and the analysis of systems whose matrices are not square (p. 21). These are not particularly sophisticated matters, but they are fundamental. If linear algebra is to be taught at all, in even so simple a matter as discovering the intersection of three planes in space, the logic of the elimination of variables is of greater importance than the production of a list of numbers, which can be found by looking at the answer book, after all. Thus, Virginia is of divided mind in its advocacy of Reason, for on page 20 it also says, "The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion . . ." and goes on in some detail. When, as in the case of the set of linear equations just mentioned, a golden opportunity arises for using a deductive argument, the state misses the opportunity.
An "Advanced Placement" calculus course is described, but the rest of the high school curriculum as outlined is not a sufficient preparation for this level of work. In particular, the lack of needed exercise in mathematical reasoning in most of the curriculum, as the example of linear algebra indicates, is also visible in the treatment of geometry.
One unusually good feature of these standards is the repeated instruction concerning speech and vocabulary, that the student should use technical words fluently and correctly, and explain his work. A good exercise, and a model for other good exercises, is found on page 13, item 6.3, "The student will explain orally and in writing the concepts of prime and composite numbers." Again, under "Patterns, Functions, and Algebra," 7th grade level, "The student will use the following algebraic terms appropriately in written and/or oral expression: equation, inequality, variable, expression, term, coefficient, domain, and range" (p. 16). However, in speaking mathematically here the authors should have followed their own advice, and used the mathematical "or," which means the same thing, we believe, as their "and/or."
Standard 1 ("The student understands and applies the concepts and procedures of mathematics") has five components, of which 1.2 ("Understand and apply concepts and procedures from measurement") occupies page 57. Now component 1.2 has three subcomponents, one of which, named "Approximation and Precision" has two entries at each benchmark (i.e., grade level). The second entry in each benchmark is
At grade 4: estimate to predict and determine when measurements are reasonable, for example, estimating the length of the playground by pacing it off.
At grade 7: use estimation to obtain reasonable approximations, for example, estimating the length and width of the playground to estimate its area.
At grade 10: use estimation to obtain reasonable approximations, for example, estimating how much paint is needed to paint the walls of a classroom.
This sort of repetition is common throughout this nearly vacuous document, and is probably intended, by its slight variations, to suggest a progression of skills as the student grows older, but in few cases does it say anything instructive. In the case just quoted the three exercises are at least welldefined; more often they are so general as to defy interpretation. An example, at Standard 3 ("The student uses mathematical reasoning"), where component 3.3, headed "Draw Conclusions and Verify Results," prescribes at the 4th grade level, "reflect on and evaluate procedures and results in familiar situations"; at the 7th grade level, "reflect and evaluate on [sic] procedures and results in new problem situations"; and at the 10th grade level, "reflect on and evaluate procedures and results and make necessary revisions" (p. 63).
This document contains standards for other subjects besides mathematics, which takes up only 12 pages. It is marked "Washington State Commission on Student Learning, APPROVED--February 26, 1997," but the interior pages are marked "Work in progress." Since the present edition does contain some content, as for example in "Geometry" (p. 59), it is possible that the impulse behind the present adherence to vague generalities is not final state policy, and that future editions will contain even more that a teacher or textbook committee can actually use.
This document is intended, among other things, as a guide to statewide assessment, and places in boldface, for that purpose, certain crucial items. The language is everywhere direct and usually quite definite, e.g., for 1st graders, "Given two whole numbers whose sum is 99 or less, estimate the difference, and find the difference using various methods of calculation (mental computation, concrete materials, and paper/pencil)" (p. 46). Such clarity is maintained throughout, though it sometimes makes the document sound like the table of contents, or sometimes exercises, of a series of textbooks. At the high school level, in "Algebra II," standard A2.9 asks the student to "perform basic matrix operations and solve a system of linear equations using the inverse matrix method. Graphing calculators will be used to perform the calculations." This is definite and clear, but not advisable, except as part of a more comprehensive treatment of systems of linear equations, as has been commented above (see the note for Virginia). Furthermore, clarity and definiteness are not always present; one of the boldface instructions at the high school level is "solve problems using non-routine strategies" (p. 179). Even though this instruction occurs within the course description "Algebra II," it is not possible to tell what is meant; yet the state intends to examine students on this matter.
There is a thread called "Computer/ Technology" that appears at every grade level, K-8, and is apparently designed to make sure children learn to use calculators or computers. It is unfortunate that these topics appear within the "mathematics" portion of the Instructional Goals and Objectives, where they might give the unwary teacher the impression, sometimes mistaken, that these sometimes sociological exercises are designed to augment mathematics lessons. "Identify work produced by using technology as intellectual property and thus protected [by] copyright laws" (grade 7, p. 121) is plainly not mathematics. This entire thread should be thought through again as it is related to the mathematics curriculum. A distinct course, perhaps called "Computer Science," teaching children how to use computers and how to behave in their presence, would be another matter, of course.
This 1997 "Standards" is a draft; the final version will serve as a syllabus for statewide tests; page xi states, "If content does not appear in the academic standards, it will not be part of a WSAS test." Some things that do not appear are the quadratic formula, the binomial coefficients, a geometric series, a Euclidean proof, a conic section, an asymptote. The curriculum, in other words, is weak; even the Pythagorean theorem comes in for only casual mentions, and the trigonometric functions appear only in the mention of "trigonometric ratios" in right triangles, as an incident in geometry to be mastered by the 12th grade level.
On page 27, "Geometry and its study of shapes and relationships is an effort to understand the nature and beauty of the world. While the need to understand our environment is still with us, the rapid advance of technology has created another need: to understand ideas communicated visually through electronic media. For these reasons, educated people in the 21st Century need a well-developed spatial order to visualize and model real-world problem situations."
Such a definition reduces geometry to a sort of empirical science in the service of graphic art, and shows nothing of the nature of the deductive structure which launched modern science. On page 29, the ideas of geometry are confused with the artifacts of "analytic geometry," as if slope and intercepts were geometric objects. While the definition (called "Rationale") for geometry misses the point of mathematics as a model for structural relations, there is not in fact much Inflation or False Doctrine in this document. It is praiseworthy that, on page 22, it is warned that "the tools of technology are not a substitute for proficiency in basic computational skills." Yet the document is evasive on this point, as where it is advised that "selecting and applying algorithms for addition, subtraction, multiplication, and division" is given equal billing with "using a calculator" (p. 25). If one wants an answer to the question of whether children are taught "long multiplication," the answer will not be found here.
The document is similarly unclear on many other points, to the degree that one cannot really determine the suggested content at most levels. In particular, while logic is mentioned, there is no evidence that logical argument is ever associated with substantive mathematical information; all subject matter mentioned is presented without regard to the connections of one mathematical idea with another.
Japan has a delightful consistency throughout its document. In the primary grades, each grade starts with a list of objectives for that grade, followed by "Content" ("Numerical Calculations," "Quantities and Measurements," and "Geometrical Figures"), followed by a few paragraphs entitled "Remarks concerning Content." In grade 3, the students "learn how numbers are set on the abacus ("soroban"), and to use it in simple addition and subtraction"(p. 9). In grades 5 and 6, "Quantitative Relations" is added to the listings under "Content," followed by the usual "Remarks concerning Content." Here there is no misguided notion of having 9 or 10 "strands" running through all of the grades (some American standards documents consider that computing areas in the 1st grade should form part of a strand called "calculus.") In the middle school (grades 7-9) each grade has exactly the same format: "Objectives," "Content," "Remarks." All grade levels are refreshing to read; they are models of "clear, definite, testable," and no grade runs more than a few pages. In grades 10-12, the format is repeated, but by subject matter rather than by grade level. There is also an "honors" track for those who are academically inclined, as well as a third track, very rigorous, for those whose interests lie in science and engineering. At the end, there are very definite guides about the sequence of all the courses, taken separately or in parallel.
All this is achieved in a mere 47 pages, less than a tenth of the longest American framework, though "framework" is probably a good description of the Japanese document as well, for it does contain pedagogical hints of importance. In particular, consider this quotation from the "Second Grade Content":
To enable children to develop their abilities to use addition and subtraction through getting deeper understanding of them.
which is followed by, inter alia
To understand that addition and subtraction of 2- and 3- digits numbers are accomplished by using the basic facts of these operations for 1-digit numbers and to know and use them in column form.
It is typical of the Japanese standards to incorporate, as this example does, the essence of the reasoning into the implied lesson plan associated with the content to be conveyed. That is, "adding in columns" is not merely a practical necessity made obsolete by the electronic calculator, as many educators now believe, but is (when properly taught, of course) an elucidation of the nature of the decimal system, and an illustration of how mathematical reasoning proceeds from minimal information to what appears to be enormously more. In this case, the 9X9 addition table memorized by the end of the 2nd grade enables a child, provided with a bit of understanding of the nature of the system, to construct a table potentially infinite in scope. The writer of this item shows this understanding clearly, and many other examples can be quoted to show the same quality.
Items of content are clearly paced from grade to grade, with lists of vocabulary to be acquired by the end of the year given as further indication of what the year is supposed to accomplish. By the 3rd grade children are to understand decimal fractions, graphs, angles, "radius," the sphere. By the end of grade 4, they are to know rules for rounding off approximations, adding fractions with common denominator, multiplying and dividing two-digit numbers, and by the end of the 5th, congruence, the making of a statistical graph, least common divisors and multiples, and "to know that the result of division of whole numbers can always be represented as a single number using fractions." This last quotation also carries a lesson in concept of "number" which most American curricula either avoid or take for granted.
Calculators are not used before the 5th grade, to give children the chance to understand the decimal system first. One of the stated "objectives" at the grade 9 level includes, "[to] acquire the way of mathematical ly representing and coping with, and to enhance their abilities of mathematical ly considering things, as well as to help them appreciate the mathematical way of viewing and thinking." This much, taken alone, is vague (though poetic), but is followed by specifics enough: factoring, the quadratic equation, functional relations, the Pythagorean theorem, and so on, all presented in such a way as to follow "the mathematical way of viewing and thinking," much more than in such a way as to convince students of the "real-life" applications. Real life comes in its own place, a few years later.