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Paper
State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
Whether this or that rubric is represented at each grade level will be of little interest to the present report, in which we shall rather pay our attention to the topics that actually appear, and to what depth of knowledge or understanding is demanded in each appearance. Whether those topics we count appropriate appear under one or another psychological (or philosophical) rubric will not matter. This is fortunate, as the rubrics are not always strictly comparable among states. The age of the child is. Teachers and school districts looking for practical guidance tend to ask what to teach in the 7th grade, for example, and what book to choose for that grade level, rather than to ask what sequence of topics in probability, say, should obtain in the K-12 progression as a whole, or whether their state succeeds in classifying what is accomplished in each grade into a list of independent rubrics covering mathematics comprehensively. It is, in fact, curious that the NCTM, and hence the states, have spent so much effort in this direction, whose principal consequence appears to have been the fragmentation of the mathematics curriculum during each year of schooling into a multitude of themes, or "threads," with consequent loss of depth in each, not many of Bruner's "higher cognitive levels" having been in fact reached by the spiraling. This phenomenon has been characterized by one knowledgeable commentator, concerning the United States' poor performance on the 1996 Third International Mathematics and Science Survey (TIMSS), who remarked that the typical American curriculum is "a mile wide and an inch deep." The same can be observed from our children's performance on the NAEP mathematics tests. In contemplating each state's standards, we shall ignore most theoretical debates and simply ask the text to have this function: We imagine a newly arrived teacher in a remote district, of average knowledge and experience in teaching the mathematics traditionally taught at the grade level he works in. He has access to textbooks and other materials, but does not have previous experience in the schools of that particular state. He wishes to construct a syllabus and choose appropriate books (and perhaps technical tools, such as graph paper, compasses, computers, and calculators) in order to set up lesson plans and examinations. This (imagined) teacher is not for this purpose asking the state for instruction in pedagogy, or even in mathematics; he only wants to know what his state demands the student should know of mathematics by grades 4, 8, and 12; or perhaps by end of kindergarten, grade 3, grade 6, grade 9 and grade 12; or even grade-by-grade. The standards published by the state should answer this question. Choosing textbooks and deciding the mixture of lecture, homework, group work, and classroom discussion are not usually, under the American tradition of local control, the decision of the state at all; but it has increasingly become the custom that the framework for results and even a list of the desired results themselves should be of statewide uniformity, at least approximately. Our judgments of how the state has complied with this request, our ratings of the documents we call "standards," will be based on four major investigations:
(I) Clarity We shall explain these criteria in two stages: first, by giving them definitions (Section V); and secondly, by illustrative examples (Section VI). Following this exposition we shall present the ratings of the states in the form of a matrix of numerical grades (Section VII). The general explanation of why a state receives a grade it does, under any of the criteria, will be implicit in our descriptions of the criteria. Section VII, the numerical scores, will nonetheless be introduced by a further explanation of our ratings. The table of the grades themselves (i.e., the Ratings), brief as it is, represents the goal, and is the condensation of the judgment of the authors after study of the documents in light of the criteria.
I. Clarity refers to the success the document has in achieving its own purpose, i.e., making clear to the (imagined) provincial teacher what the state desires. Clarity refers to more than the prose. (Furthermore, some of the other evils of bad prose will have a negative category of their own in the fourth group, Negative Qualities.) (A) Firstly, of course, the words and sentences themselves must be understandable, syntactically unambiguous, and without needless jargon. (B) Secondly, what the language says should be mathematically and pedagogically definite, leaving no doubt of what the inner and outer boundaries are, of what is being asked of the student or teacher. (C) And thirdly, the statement or demand, even if understandable and completely defined, might yet ask for results impossible to test in the school environment, whether by a teacher's personal assessment or a statewide formal paper-and-pencil examination. We assign a positive value to testability. Thus, the first group, Clarity, gives rise to three criteria: I-A Clarity of the language I-B Definiteness of the prescriptions given I-C Testability of the lessons as described II. Content, the second group, is plain enough in intent. Mainly, it is a matter of what might be called "coverage," i.e., whether the topics offered and the performance demanded at each level are sufficient and suitable. To the degree we can determine it from the standards documents, we ask, is the State asking K-12 instruction in mathematics to contain the right things, and in the right amount and pacing? Here we shall separate the curriculum into three parts (albeit with fuzzy edges): Primary, Middle, and Secondary. It is common for states to offer more than one 9-12 curriculum, but also to print standards describing only the "common" curriculum, or one intended for a "school-leaving" examination in grade 11 or so. In such cases we shall not fault the document for failing to describe what it has no intention of describing, though we may enter a note mentioning the omission. Other anomalies will give rise to notes in Section VIII, which follows the actual ratings. We cannot judge the division of content with year-by-year precision because very few states do so, and we wish our scores to be comparable across states. As for the fuzziness of the edges of the three divisions we do use, not even all those states with "elementary," "intermediate," and "high school" categories divide them by grades in the same way. One popular scheme is K-6, 7-9, and 10-12 for "elementary," "intermediate" and "high" schools, while another divides it K-5, 6-8, and 9-12. In cases where states divide their standards into sufficiently many levels (sometimes year-by-year), we shall use the first of these schemes. In other cases we will merely accept the state's divisions and grade accordingly. Therefore, Primary, Middle, and Secondary will not necessarily mean the same thing from one state to another. There is really no need for such precision in our grading, though of course in any given curriculum it does make a difference where topics are placed. Thus, the second group, Content, gives rise to three criteria: II-A Adequacy of Primary school content (K-6, approximately) II-B Adequacy of Middle school content (or 7-9, approximately) II-C Adequacy of Secondary school content (or 10-12, approximately) In many states, mathematics is mandatory through the 10th grade, while others might vary this by a year or so. Our judgment of the published standards will not take account of what is or is not mandatory in each state; thus, a rating will be given for II-C whether or not all students in fact are exposed to part or all of it. (Some standards documents, as in the case of the District of Columbia, only describe the curriculum through grade 11, and we adjust our expectations of content accordingly.) The difficult question here is to define "adequacy" of content. This can only be done by a listing of some sort. The authors of this report have in their minds a standard of what is possible and desirable to be taught at each level of school mathematics, and it is this standard of ours against which the adequacy of the states' standards is measured. To set this "standard" forth in abstract terms, by definition rather than enumeration, is, we believe, impossible. We shall therefore attempt to do it by two rather indirect means, as will be explained in Section VI, Examples for the Criteria. III. Reason, the third "group," will have only the one entry: Reason.
Civilized people have always recognized mathematics as an integral part of their cultural heritage. Mathematics is the oldest and most universal part of our culture, in fact, for we share it with all the world, and it has its roots in the most ancient of times and the most distant of lands. The beauty and efficacy of mathematics both derive from a common factor that distinguishes mathematics from the mere accretion of information, or application of practical skills and feats of memory. This distinguishing feature of mathematics might be called mathematical reasoning, reasoning that makes use of the structural organization by which the parts of mathematics are connected to each other, and not just to the real world objects of our experience, as when we employ mathematics to calculate some practical result. The essence of mathematics is its coherent quality, a quality found elsewhere, to be sure, but preeminently here. Knowledge of one part of a logical structure entails consequences which are inescapable, and can be found out by reason alone. It is the ability to deduce consequences which otherwise would require tedious observation and disconnected experiences to discover, that makes mathematics so valuable in practice; only a confident command of the method by which such deductions are made can bring one the benefit of more than its most trivial results. Should this coherence of mathematics be inculcated in the schools, at the level K-12, or should it be confined to professional study in the universities? A recent report (17 June 1997) of a task force formed by the Mathematical Association of America to advise the NCTM in its current revisions of the 1989 Standards argues for its early teaching:
One of the most important goals of mathematics courses is to teach students logical reasoning. This is a fundamental skill, not just a mathematical one . . . It should be recognized that the foundation of mathematics is reasoning. While science verifies through observation, mathematics verifies through logical reasoning. Thus the essence of mathematics lies in proofs, and the distinction among illustrations, conjectures and proofs should be emphasized . . . If reasoning ability is not developed in the students, then mathematics simply becomes a matter of following a set of procedures and mimicking examples without thought as to why they make sense. (This task force, which continues its work in 1998, is headed by Kenneth Ross, former President of the Mathematical Association of America.) Even a small child should understand how the memorization of tables of addition and multiplication for the small numbers (one through 10) necessarily produces all other information on sums and products of numbers of any size whatever, once the structural features of the decimal system of notation are fathomed and applied. At a more advanced level, the knowledge of a handful of facts of Euclidean geometry--the famous Axioms and Postulates of Euclid, or an equivalent system--necessarily imply (for example) the useful Pythagorean Theorem, the trigonometric Law of Cosines, and a veritable tower of truths beyond. Any program of mathematics teaching that slights these interconnections doesn't just deprive the student of the beauty of the subject, or his appreciation of its philosophic import in the universal culture of humanity, but even at the practical level it burdens that child with the apparent need for memorizing large numbers of disconnected facts, where reason would have smoothed his path and lightened his burden. People untaught in mathematical reasoning are not being saved from something difficult; they are, rather, being deprived of something easy. Therefore, in judging standards documents for school mathematics, we look to the "topics" as listed in the "content" criteria not only for their sufficiency at this and that grade level, and not only for the clarity and relevance of their presentation, but also for whether their statement includes or implies that they are to be taught with the explicit inclusion of information on their standing within the overall structures of mathematical reason. A state's standards will not rank higher by the Reason criterion just by containing a thread named "reasoning," "interconnections," or the like, though what we seek might possibly be found there. It is, in fact, unfortunate that so many of the standards documents we examined contain a thread called "Problem-solving and Mathematical Reasoning," since that category often slights the reasoning in favor of the "problem-solving," or implies that they are essentially the same thing. Mathematical reasoning is not found in the connection between mathematics and the "real world," but in the logical interconnections within mathematics itself. Since children cannot be taught from the beginning "how to prove things" in general, they must begin with experience and facts until, with time, the interconnections of facts manifest themselves and become a subject of discussion, with a vocabulary appropriate to the level. Children then must learn how to prove certain particular things, memorable things, both as examples for reasoning and for the results obtained. The quadratic formula, the volume of a prism, and why the angles of a triangle add to a straight angle, for example. What does the distributive law have to do with "long multiplication"; why do independent events have probabilities that combine multiplicatively? Why is the product of two numbers equal to the product of their negatives? (At a more advanced level, the reasoning process they have become familiar with can itself become an object of contemplation; but except for the vocabulary and ideas suitable and necessary for daily mathematical use, the study of formal logic and set theory are not for K-12 classrooms.) We therefore shall be looking at the standards documents as a whole to determine how well the outlined subject matter is presented in an order, or a wording, or a context, that can only be satisfied by including due attention to this most essential feature of all mathematics. IV. Negative Qualities, the fourth group, looks for the presence of unfortunate features of the document that injure its intent or alienate the reader to no good purpose. Or, if taken seriously, will tend to cause that reader to deviate from what otherwise good, clear advice the document contains. We shall call one form of it False Doctrine, a phrase that almost explains itself, but will need some examples. The second form will be called Inflation because it offends the reader with fruitless verbiage, conveying no useful information. Under False Doctrine, which can be either curricular or pedagogical, is whatever text contained in the standards we judge to be injurious to the correct transmission of mathematical information. As with our criteria concerning Content, our judgments can only be our own, as there are disagreements among schools of experts on some of these matters. Indeed, our choice of the name "false doctrine" for this category of our study is a half-humorous reference to its theological origins, where it is a synonym for heresy. Mathematics education has no official heresies, of course; yet if one must make a judgment about whether a teaching ("doctrine") is to be honored or graded zero, as we are required to do in the present study, deciding whether an expressed doctrine is true or false is a mere necessity. The NCTM, for example, officially prescribes the early use of calculators with an enthusiasm the authors of this report deplore, and the NCTM discourages the memorization of certain elementary processes, such as "long division" of decimally expressed real numbers, and the paper-and-pencil arithmetic of all fractions, that we think essential, that should be second-nature before the calculator is invoked for practical uses. We must assure the reader that, while we differ with the NCTM on these and other matters, our own view is not merely idiosyncratic, but also has standing in the world of mathematics education, as can even be seen in some of the documents under review. And even were that not true, it is still our duty to make our own stand and to make it clear. We cannot simultaneously credit two opposing positions on what should be learned. While in general we expect standards to leave pedagogical decisions to the teachers (most standards documents do so, in fact), so that pedagogy is not ordinarily something we are rating in the present study, there are still in some cases standards containing pedagogical advice that we believe undermines what the document otherwise recommends. Advice against memorization of certain algorithms, or a pedagogical standard mandating the use of calculators to a degree we consider mistaken, might appear under a pedagogical rubric in a standards document under consideration. Here is one of the places where our general rule not to judge pedagogical advice fails, for if the pedagogical part of the document gives advice making it impossible for the curricular part, as expressed there, to be accomplished properly, we must take note of the contradiction under this rubric of False Doctrine.
Two other false doctrines are excessive emphases on "real-world problems" as the main legitimating motive of mathematics instruction, and the equally fashionable notion that a mathematical question may have a multitude of different valid answers. Excessive emphasis on the "real-world" leads to tedious exercises in measuring playgrounds and taking census data, under headings like "Geometry" and "Statistics," in place of teaching mathematics. The idea that a mathematical question may have various answers derives from confusing a practical problem (whether to spend tax dollars on a recycling plant instead of a highway) with a mathematical question whose solution might form part of such an investigation. As the first (January 21, 1997) Report of the MAA Task Force on the NCTM Standards has noted,
It should also be recognized that results in mathematics follow from hypotheses, which may be implicit or explicit. Although there may be many routes to a solution, based on the hypotheses, there is but one correct answer in mathematics. It may have many components, or it may be nonexistent if the assumptions are inconsistent, but the answer does not change unless the hypotheses change. Again, constructivism, a theoretical stance common today, has led many states to advise exercises in having children "discover" mathematical facts, or algorithms, or "strategies." Such a mode of teaching has its values, in causing students better to internalize what they have thereby learned; but wholesale application of this point of view can lead to such absurdities as classroom exercises in "discovering" what are really conventions and definitions, things that cannot be discovered by reason and discussion, but are arbitrary and must merely be learned. Students are also sometimes urged to discover truths that took humanity many centuries to elucidate, the Pythagorean theorem, for example. Such "discoveries" are impossible in school, of course. Teachers so instructed will necessarily waste time, and end by conveying a mistaken impression of the standing of the information they must surreptitiously feed their students if the lesson is to come to closure. And often it all remains open-ended, confusing the lesson itself. Any doctrine tending to say that telling things to students robs them of the delight of discovery must be carefully hedged about with pedagogical information if it is not to be false doctrine, and unfortunately such doctrine is so easily and so often given injudiciously and taken injuriously that we deplore even its mention. Finally, under False Doctrine must be listed the occurrence of plain mathematical error. Sad to say, several of the standards documents contain mathematical misstatements that are not mere misprints or the consequence of momentary inattention, but betray genuine ignorance. Under the other negative rubric, Inflation, we speak more of prose than content. Evidence of mathematical ignorance on the part of the authors is a negative feature, whether or not the document shows the effect of this ignorance in its actual prescriptions, or contains outright mathematical error. Repetitiousness, bureaucratic jargon, or other evils of prose style that might cause potential readers to stop reading or paying attention, can render the document less effective than it should be, even if its clarity is not literally affected. Irrelevancies, such as the smuggling in of political or trendy social doctrines, can injure the value of a standards document by distracting the reader, again even if they do not otherwise change what it essentially prescribes. The most common symptom of irrelevancy, or evidence of ignorance or inattention, is bloated prose, the making of pretentious though empty pronouncements, conventional pieties without content. Bad writing in this sense is a very notable defect, though not the greatest, in the collection of standards we have studied. Some examples will appear below. We thus distinguish two essentially different failures subsumed by this description of pitfalls, two Negative Qualities that might injure a standards document in ways not classifiable under the headings of Clarity and Content: Inflation (in the writing), which is impossible to make use of; and False Doctrine, which can be used but shouldn't. How numerical scores will be assigned under these headings will be explained in Section VII below; here we signal only their titles, as used in the Table of Ratings there: IV-A False Doctrine IV-B Inflation
Note: The examples given in this section are taken from state standards as named, but it must not be imagined that these are the only states, or the only quotations, that could have been used to the same purpose. Each example, for good or bad, can be matched by many others, from many other states. Criterion I: Clarity I-A. A standard should be clear:
Demonstrate understanding of the complex number system (Arkansas, "Number Sense" strand, grades 9-12); rather than unclear:
Connect conceptual and procedural understandings among different mathematical content areas (Washington, Standard 5.1, Benchmark 2 - grade 7, p.65). "Conceptual understandings" and "procedural understandings" might mean something, likewise "different . . . content areas," but "connecting" the two former "among" the latter is just too hard to understand, or make use of as a guide to classroom activity. I-B. A standard should be definite:
Given a pattern of numbers, predict the next two numbers in the sequence (Alaska, Framework, "Reasoning," Level 1: ages 8-10); rather than indefinite:
Apply principles, concepts, and strategies from various strands of mathematics to solve problems that originate within the discipline of mathematics or in the real world (Alaska Framework, "Problem-solving," Level 3: ages 16-18). The first example is extremely definite, though as a lesson it might be mathematically questionable. The second ("apply principles . . . to solve problems") is exemplary in its indefiniteness. An indefinite standard describes something a teacher has no way of completing ("bringing to closure" is a common phrase for the process of completing a lesson), even if it is quite easy to know if that particular lesson is relevant to the standard. The wording in this example also illustrates a common failing in definiteness shared by many states at all levels: the use of the word "problem" without a hint of the nature of the problem. Mathematics is preeminently the science of problem-solving, but the "problem" of adding twelve and seven (grade 1) is of a different order from the "problem" of determining the dimensions of a rectangle of given perimeter and area. Every 7th grade teacher in a certain state might understand exactly what is meant by "problem" when the standard for grade 7, under "patterns," uses that word; but this understanding is parochial, an accident of the curriculum and culture traditional in that place and time. Our hypothetical teacher from out-of-state will not know this meaning. When the authors of the present report also cannot discern what is meant, such a standard must be counted inadequate. The quoted second example from Alaska is actually even worse than this, since it occurs under the rubric "Problem-solving," and ought rather to elucidate that term than assume its meaning is already clear. If the rubric had been "algebra" we would at least know that algebra problems were meant, though this knowledge would still be insufficient to define a standard of accomplishment. I-C. A standard should be testable:
Analyze spatial relationships using the Cartesian coordinate system in three dimensions (New York, "Four year sequence, Modeling," p.27); rather than ineffable or not testable:
Students [will] utilize mathematical reasoning skills in other disciplines and in their lives (New Jersey Standard 4.4, by the end of grade 8). It should be plain that the second standard, while desirable as a long-range goal for the teaching of mathematics, is not something one can do more than conjecture about; a teacher has little way of finding out whether it has been accomplished. On the other hand, there are many tests for whether the student can "analyze spatial relationships," whatever interpretation one wishes to put on the phrase. And there are many. The standard concerning "spatial relationships" is not very definite, or clear, but whatever it is, it is testable. Even though "clear," "definite," and "testable" are not the same thing, they are a family of qualities that tend to go together. Standards that fail any of these three tests announce themselves to the reader as somehow unclear, albeit in different ways. We therefore group them together, and can think of no better overall title for the grouping than Clarity. The idea here is that our imagined teacher in a state offering standards highly rated by us for each entry under Clarity should be in no doubt about what to explore, explain and examine, how to judge whether a book covers the ground described, or what to test the children upon at each stage in their progress. A poor score, on the other hand, signals a standards document from which he can glean little or no help at all on what the state expects of him in his daily tasks, and will have to rely on his own insight and experience to guess at what it might be, even though the intended content of the state curriculum might be adequate and clear in the mind of the state's educational advisors. Criterion II: Content Content will be rated separately for the categories Primary, Middle, and Secondary, in most cases meaning K-6, 7-9, and 10-12, with a year's leeway at either end of each segment, according to the way each state divides its content standards. Our grades will observe the state's own divisions, and therefore will be applicable to slightly different segments of the K-12 program for different states. To decide whether a state is asking its children to learn the right things at the right times, and enough such things, and not an unreasonable amount either, the judge must have a set of standards of his own. The authors of this report do indeed have them, though for purposes of judging others we must be rather flexible, there being more than one way to go about securing a given result by the end of the schooling process. Much as we would like to print here a complete exemplary curriculum that conforms to our own standards, the task would be prohibitively long (as every state well appreciates that has appointed a committee to do just that) and the resulting document would overwhelm the present report. Nor is it possible for a few sentences from a sample standards document to illustrate sufficiency, which must be judged by its entire extent. Hence, we shall not offer any quotations to serve as good and bad examples, as we do for criteria groups I, III, and IV. Just the same, since the reader deserves some notion of where we stand on some of the crucial topics that must form part of every mathematical education, it is worth stating a few desiderata briefly: We believe the traditional arithmetic of fractions and decimals should be complete by grade 6, but with more understanding of logical connections than was customary 50 years ago and with many more applications than the traditional storekeeping and mensuration skills. We wish the middle grades to introduce geometry and algebra and the logic of equations and their application, and not be the "review of elementary math, plus ratios" that has been common in recent years. We wish the secondary curriculum to be mathematics as the mathematics profession understands it: not a collection of rules for algebra, trigonometry, graphing and the like, but an organized body of knowledge, albeit mainly of algebra, geometry and the elementary functions, with application to human affairs clearly distinguished from the inner logic of the mathematics itself--and both of them fully represented. To describe a curriculum in so few words is, of course, insufficient; but we might point to several of the state documents as models. None of them is a model in all respects, but the reader may deduce from the scores (in Part VII, or in the Notes on that State in the following section) which states describe the content we consider sufficient. They are all publicly available (see Appendix). While not the highest overall, Alabama, Arizona, and Tennessee (at the 8-12 level) are high-ranking for content, and California, North Carolina, and Ohio are good models in both content and the other categories. We have also included high scores for Japan, which presents a document much different from any of the American states, but which is exemplary in content when rightly read, and interesting in other regards as well. We shall have more to say about the Japanese standards in the Notes on the States (Part VIII). There is no single scale for content; what is left out of one listing might simply have been omitted in order that something else might be included, so that states with equal scores will not have identical intentions concerning content. This, too, is a reason for not trying to include a model curriculum description here. Criterion III: Reason There is no single place, grade level, or strand, where one can find whether a state standards document exhibits the guidance being graded under this criterion. The mere inclusion of a category of instruction labeled "Mathematical Reasoning" or the like is no guarantee that its contents serve the purpose. To the contrary, evidence of the demand for reasoning is more often found elsewhere, if at all, inextricably bound up with the mathematical content. Quoted examples can only indicate in part the tone of the document as a whole, and it is the whole which gives rise to our rating. At the risk of being unfair to the two documents as a whole we shall nonetheless give a pair of such examples, one exhibiting a poor incorporation of the ideal of fostering mathematical reasoning, and the other a good and clear indication of the kind of lesson that is designed to teach such reasoning and give the student an appreciation of the coherence of mathematics. Vermont, which has a thread named "Mathematical Problem Solving and Reasoning," lists 17 specifics, some under the grade 9-12 column, some under the grades 5-8 column, and some under the grades K-4 column. Here are six of them (p. 7.4): Students [shall]
Indeed, any of the six instructions can be construed to apply to any levels whatever, including the editors of a mathematical research journal such as the Proceedings of the American Mathematical Society. And the editor of even that journal might hesitate before grandly asking an author to "extend concepts and generalize results to other situations," as was here prescribed for children at the level K-4. Surely the Vermont authors had something more specific in mind, but it does not come across in the framework they actually wrote. Some demands for mathematical reasoning do occur in other parts of Vermont's Framework, to be sure, notably in the section headed "Mathematical Understanding," where certain required skills are mentioned; and all of this must be taken into account in giving an overall score for Reason. A high score would require that the intended lessons in reasoning, or making connections, be part of a sufficient number of the specific content demands throughout the document as to make plain what lessons in reasoning are intended; but Vermont's section, quoted (in part) here, that includes the word "reasoning" fails to do so. It contains exhortations, not standards. On the other hand, Virginia does show the proper quality clearly enough in some places to make it convenient to quote a selection by way of contrast:
(Grade 2) The student, given a simple addition or subtraction fact, will recognize and describe the related facts which represent and describe the inverse relationship between addition and subtraction (e.g., 3 +__ = 7, .... , 7 - 3 = __). These examples from Virginia's standards are, with one exception concerning pure logic, simple instructions as to content, but presented in a manner that carries the proper message of the unity of mathematics, and the reasoning by which its parts are held together. Five such examples do not, of course, amount to a curriculum in mathematical reasoning, and in fact Virginia is, like most states, somewhat lacking in these qualities overall; but these five should serve as illustration of the standard desired; whereas mere exhortation, to "make connections," "use . . . a variety of reasoning strategies," and the like, are not sufficient guidance, and simply are not usable in the absence of a referent. An instruction of the sort that should appear more often is also found in The District of Columbia's (Mathematics) Curriculum Framework, where on page M-5, under "What a student should know and be able to do by the end of Grade 11," appears the following:
5. Explain the logic of algebraic procedures. This is not completely stated, and so its intent must be inferred; but most of the documents under review don't even go this far in reminding the reader that the logic of a procedure is essential to its intelligent use. A more complete statement of the same desideratum may also be found in Pennsylvania's standards draft for the 6th grade level "Algebra and Functions" strand. Students here are to:
1. write verbal expressions and sentences as algebraic expressions and equations, and solve them, graph them and interpret the results in all three representations. Again, Pennsylvania's proposed "Standards" offers, at the grade 11 level,
Prove two triangles or two polygons are congruent or similar using algebraic and coordinate as well as deductive proofs. The Pennsylvania instruction is a little vague, especially for "polygons," in that it does not specify the sort of hypotheses that are likely to have been in place when the proof was to be done; but the idea is plain, the lesson valuable, and the word "prove" a delight to encounter. Criterion IV: Negative Qualities IV-A False Doctrine A standard must not offer advice which, if followed, will subvert instruction in the material otherwise demanded: First Example For 7th grade, under Standard: 5070-07 ("The students will develop number concepts underlying computation and estimation in various contexts by performing the operations on a pair or set of numbers."), Utah asks as #3 of "Skills and Strategies,"
Develop an algorithm for multiplication of common fractions and mixed numbers by using models or illustrations; explain your reasoning. This suggestion is misleading in that "develop" implies that the conventions concerning the multiplication of fractions are somehow to be discovered by the students, and indeed then justified by them. In fact the very definition of multiplication for fractions is conventional, not natural and "discoverable." It is a triumph of the development of mathematics that the rule for the multiplication of fractions is both a consistent extension of the multiplication and division in the ring of integers, and at the same time interpretable as an operation concerning measurements in geometry and partitioning in finite sets. That the definitions and results are independent of the "fraction" representation of the rational numbers involved, i.e., of the denominator chosen, is another complication. As with any other convention or fact, this cannot be discovered, though its invention and its rationale should be thoroughly taught. That 7th-grade students should participate in discussions of these matters is of course to be desired, but a teacher who feels obliged to set up a classroom environment in which students "develop an algorithm . . ." will be wasting time, and fooling either himself or his students; while possibly neglecting the truly interesting features of the system of rational numbers and its possible uses. Second Example Arizona, in "Patterns, Algebra, and Functions," grades 9-12, asks students to analyze the effects of parameter changes in formulas defining functions, "using calculators." This curriculum obviously intends the student to understand the effects of parameter changes, but believes the use of (graphing) calculators sufficient for the purpose. It is not sufficient for the purpose. Attention to the symbolic expressions, individual point calculations and point plotting, are a necessity. True, calculators are a valuable aid, even a necessity, in serious use with complicated data in science and in business; but that is life, not instruction, and the proper use of calculators must follow, not precede, the logical, conceptual understanding of the effects of parameter changes. A standard that instructs teachers to confine themselves to calculators in exercises, where they will not suffice for instructional purposes, is destructive of the lesson intended in that part of the curriculum. Omitting the phrase "using calculators" might have made this standard a good one, for the desideratum (to understand the effect of parameter changes) is clear and valuable; but the method of getting there should rather have been left to the teacher. There doubtless is a place for calculators somewhere in there, but this Arizona pedagogical instruction preempts the choice in an unfavorable direction. The widespread prescribing of calculators in the early grades is also destructive of learning, for the algorithms concerning fractions and decimal multiplication and division are not mere "19th Century skills" now rendered obsolete by technology, but when properly taught inculcate "number sense" and the ability to estimate; and serve as essential preparation for the more sophisticated operations of later years, such as the algebra of rational functions. Third Example Alabama, in its general introduction to the detailed list of standards that follows, has a section headed "Use of Manipulatives," which contains this directive:
Throughout their schooling, students should be involved in activities in which manipulatives are used to aid in conceptual and procedural understanding. . . . Use of manipulatives helps . . . clarify algorithms . . . Using manipulatives also richly illustrates the connection between concrete experiences and abstract mathematics. In the early childhood years, the use of manipulatives, such as blocks, Cuisinaire rods, and surely the abacus, can give a child the basis in experience that must underlie the abstractions of mathematics; and the world has used them since the dawn of mankind. But "throughout" their schooling is an error, and has moreover led to an unnecessarily large manipulatives industry allied to textbook publishing, whose products decorate the displays at teachers' conferences and professional journals. Mathematics is not raw experience; it is an analogue for experience. As it becomes more sophisticated, as with algebraic equations representing physical relationships, it must necessarily supplant that which can be manipulated; otherwise, the lesson of algebra is lost. A teacher who takes seriously the Alabama instruction as given will find himself looking for ways to employ such curiosities as "algebra tiles" in teaching the solution of equations, and literal balances for the understanding of just those methods that were designed to render literal balances unnecessary. Rather than look for ways to use manipulatives, teachers should strive for ways to wean students from their use, little by little, as a potential bicyclist must be weaned from training wheels. IV-B Inflation A standards document must not employ language whose purpose can only be to fill paper, nor must it suggest a profundity impossible for the level in question, especially if the indications are that the author doesn't understand the words being used. Such language weakens the respect the user of the document should have for valid parts of the document, even where it does not result in outright error or false doctrine. Bad writing in this sense differs from vagueness or other failures of clarity in that it is not simply mistaken, incomplete or obscure, but in that it is pretentious and cannot be taken seriously, or is empty of content altogether. If it does include a definite instruction, that instruction is generally impossible of execution, at least at the level of instruction in question. Example 1 In Michigan's Model Content Standards for Curriculum (Mathematics), Standard 8 states:
Students draw defensible inferences about unknown outcomes, make predictions, and identify the degree of confidence they have in their predictions. Then the first item under this standard, identically worded and repeated at each of the levels primary, middle school, and high school, is
Make and test hypotheses. As these four words constitute a short description of the entire science of statistics (and indeed of all science), this "standard" is impossibly broad, hence useless. Later items indicate certain particulars, e.g., "Make predictions and decisions based on data, including interpolations and extrapolations." This is broad enough, but does include some clues (e.g., "interpolations") limiting the intended lessons, though even here much the same wording is used at all three levels. This particular statement can be called, generously, "too vague," but "Make and test hypotheses" is worse, and is no guide to either curriculum or instruction. Example 2 Oklahoma's "Priority Academic Student Skills," at the grade 2 level, under the rubric "Mathematics as Connections" requires that the student will
Example 3. Hawaii's Essential Contents ("Geometry and Spatial Sense," Grades 9-12) claims to prescribe
Non-Euclidean geometries; and Hyperbolic and elliptical geometries. This shocking entry perhaps springs from its authors' having seen a popularization of these topics somewhere and deciding it to be a pleasant sort of thing to talk about in school. But, like capsule biographies of famous mathematicians and magazine articles about Fermat's Last Theorem, popularizations of arcane mathematical theories are not curriculum. A serious lesson in non-Euclidean geometries is impossible at this stage of schooling. Even Euclidean geometry is slighted in Hawaii's Essential Contents, and non-Euclidean geometry can no more be understood in the absence of Euclid's structure than night can be understood in the absence of day. (There is a possibility, evidenced in some advertising we have seen, that the [Euclidean] properties of the sphere are regarded as "non-Euclidean geometry" by some manufacturers of manipulatives to be used in the schools. If that is the origin of the mention of "non-Euclidean geometry" in several of the state standards, it is a poor use of the phrase, and as a lesson in geometry is inferior to the deductive Euclidean geometry it appears to replace.) Again, under Hawaii's rubric "Patterns and Relationships" we find that
Students develop algebraic thinking through . . . [among other things] . . . Topological concepts. In a curriculum guide that nowhere mentions an axiom, theorem, or deductive argument, this is also unrealistic, even apart from the problematic association of "algebraic thinking" at the high school level with such "topological concepts" as might possibly be found there. Are these "Essential Content" items for real? No. They are Inflation.
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