State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
The authors were commissioned by the Thomas B. Fordham Foundation to study and compare the "standards" or "frameworks" for school mathematics as published by the 50 States and the District of Columbia, and to grade them according to a set of uniform criteria representing the several purposes or qualities such documents should have. An initial judgment was needed even to solicit the correct publication, and we began by sending 51 letters, identical except as to the addressee. The text of the letter defines what sort of state publication we intended to study. Here is one of them:
About 30 states and the District of Columbia responded by sending printed or photocopied documents. Three or four states invited us to download such a document from a Web page of the state's education department, and a few replied explaining that the standards (or frameworks) documents we wanted were in the process of revision and not available. (The District of Columbia will hereafter be included as a "state" when we speak of the states generically.) Alaska wrote saying it printed such a document but does not send it out of state, so we secured a copy from a friend who lives there. We later discovered that Iowa, which did not respond at all, does not publish anything answering our description. In most other cases of non-response we nonetheless located a Web page containing the document freely available to the public, and we obtained it. During the writing of this report, other states completed and sent us their latest drafts, or published them on their Web pages, but Minnesota, Nevada, and Wyoming do not have their current drafts in quotable form. Other states having their standards in draft form are included in this study, though we have made clear that they are drafts, not yet adopted. In all, we have ended with documents from 46 states and the District of Columbia, and for comparative purposes one similar document from Japan. All states are named in the Appendix to this report, where enough bibliographical information is given to make clear just what we used for our commentary and where it was obtained, or by what state office published, and when. We hope we have not included any document that has been superseded by the time of publication. Of this, however, we cannot be certain.
It can be seen from the variety of titles in the Appendix that there is no uniform nomenclature, the words "Standards" and "Frameworks" often being used more or less interchangeably, along with "Curriculum Content Guide," etc. Certain States do define a difference. Alaska, California, Colorado, New Jersey, Pennsylvania and South Carolina are among those clear in this regard, publishing one document called Standards and another called Framework. In this usage, a "framework" will usually include the "Content Standards" or refer to it as a sub-sidiary document, but will also include guidelines for pedagogy, assessment, and other structural features of the mathematics program. In cases where a second document (usually a "framework") is in our possession but contributes nothing to what we are judging, we may omit its mention in the Appendix. Where the distinction or dual publica-tion makes a difference in the interpretation of our results, the second document will be listed. Also, some of the standards we received are printed in company with stan-dards for science and other subjects; we shall of course be reporting only on the mathematics sections, and such remarks introductory to the whole document as are relevant to mathematics.
Except where we are actually naming a document for reference we shall refer to all these publications generically as "standards." Whatever they are named, they are either the state's official answer to the description given in our letter of solicitation reprinted above, or what we construe as such an answer from the state's publications on its Internet web pages. As will appear in the description of the criteria we used to judge these documents, the presence of non-curricular information in documents labeled "framework" will make no difference in our ratings, since pedagogical and assessment advice and procedures are not part of what we consider in this study, except as their presence might cast additional light on the content (and sometimes "performance") standards that are the real subject of this report.
Standards are intended as a statement of what students should learn, or what they should have accomplished, at particular stages of their schooling. They are thus inferentially a guide to curricula, instruction and examinations, but are always something less than a curriculum outline by which detailed textbooks or examinations could be completely envisioned. They are also less than complete guides to instruction in another way, in that they do not intend, except incidentally, to prescribe or suggest actual pedagogical procedure. Detailed lesson guides, examinations, and prescriptions as to pedagogy are in most states left to local school districts, or published separately from the standards document.
A few states do give statewide tests at certain grade levels, and have designed their standards documents accordingly, and it appears that more states are planning to do this in the future; even so, a typical state standards document is intended to be compatible with any of a number of different textbook choices and methods of instruction. It is only intended to assure a uniformity of outcome, as might be measured by a statewide examination if there was one. Where such examinations are not mandated, a state's standards are still intended as serious advice to the local school districts.
While the standards documents under review usually omit prescriptions concerning pedagogy, but rather attempt to state simply what should be known (however arrived at) by students at each stage of their progress, there are documents that do importantly include pedagogical advice. In a few cases this advice is considered by us despite the above disclaimers, but only to the degree that this extra information casts essential light on the meaning of the content standards themselves.
For example, a state might prescribe the use of calculators in the teaching of "long division," and with such firmness that we are forced to conclude that the usual "long division" algorithm is not, or is hardly, to be taught at that point. In such a case, while the standards might also state that 6th grade students (say) should "divide and multiply numbers in decimal notation," we cannot credit the document with demanding that proficiency with the standard algorithm is intended unless it is so stated. Similarly with sample examination questions, classroom scenarios, etc., which are not part of our assessment except as they clarify the content demands. Our own interest is in the curriculum itself, or (more accurately) its intended result, as dictated or implied by the standards as published.
The best known document of the sort under review is not a publication of a state at all, nor of the U.S. government. It is the 1989 publication of the National Council of Teachers of Mathematics (NCTM) called Curriculum and Evaluation Standards for School Mathematics, and generally referred to as "The NCTM Standards." Since 1989, this document has been the single most influential guide to changes in the nation's K-12 mathematics teaching, and in the contents and attitudes of our best-selling textbooks.
The NCTM is a non-governmental professional association, founded in 1920, which has become one of the principal voices of its profession. Other voices are the two teachers' unions (NEA and AFT), the 51 state education departments themselves, all of which all have specialists in school mathematics, the U.S. Department of Education and, less officially, the faculties and deans of the major schools of education.
The NCTM publishes a monthly magazine, The Mathematics Teacher, and also other, more specialized journals, including a journal of mathematical education research. Its membership consists almost entirely of school teachers, professors in university schools of education, and education administrators and officials at all levels, and far exceeds the combined membership of the professional mathematicians' organizations. NCTM also publishes many other guides, yearbooks, and research reports, and it conducts national and regional meetings at which professional information is exchanged.
The most obviously missing voice in this listing of those influential in school mathematics today is that of the mathematics profession itself, as it might be represented by the three major professional organizations: The American Mathematical Society (AMS), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM). The MAA is devoted to the advancement of the teaching of mathematics at the college level, whereas the other two are principally interested in scientific research and publication. All three take an interest in the health of the profession itself; but none of them has traditionally considered school education at the K-12 level an interest of more than marginal priority. While a few members of NCTM are professors of mathematics in university mathematics departments, it is doubtful that one in a hundred of those persons who call themselves "mathematician" or belong to the American Mathematical Society, also belongs to NCTM or reads The Mathematics Teacher.
Yet a serious attempt to bridge the gulf of interests and influence between the world of K-12 mathematics education and the world of professional mathematics was made during the era of "The New Math," roughly the period 1955-1970, the effective bridge then having been the curriculum-writing projects and "teachers' institutes" financed (mainly) by the National Science Foundation at the time. This passed without lasting success, and it appears that, while a few mathematicians have always been present at councils concerned with school mathematics, or in grassroots projects of educational experimentation, or in textbook-writing teams, and while as professors they have been among the teachers of those who later become school teachers and professors of education, their influence on curricula and classroom practice has been negligible over most of the present century, including today. Professors of education, even mathematics education, are members of a "second culture" almost as distinct from the world of mathematics as C.P. Snow's literary culture was from that of his friends the physicists.
This separation is a cultural phenomenon more pronounced in the United States than in most European countries and Japan, and it is not really a necessary concomitant of the different professional responsibilities of the two groups. Indeed, another "bridging" effort on the part of the mathematical societies and NCTM is being made today, especially in connection with the movement towards standards for school mathematics, but it is only beginning. The results will be impossible to assess for some time to come.
The 1989 NCTM Standards, which was the work of NCTM and not the mathematics profession, is comparable to almost all of the state "standards" volumes under study in this report, and in many cases is their acknowledged ancestor; but it is not completely so. It is much longer than any of them (except for the New Jersey "framework"), and contains much advice on pedagogy, including "vignettes," i.e., imagined dialogues or classroom conversations, illustrating the way NCTM expects its recommendations might play out in practice. Its influence is manifest in most state standards, even where they confine themselves to curricular content. Its educational philosophy, usually briefly and inadequately referred to as "constructivism," and its categories of mathematics curriculum: e.g., "mathematics as 4 .5 problem-solving," "mathematics as communication," "mathematics as reasoning," "measurement," "number sense," etc., are all echoed, often strongly and by explicit cross-reference, in the state documents.
The NCTM Standards offers 13 such categories (each called a "standard") for Grades K-8, and 14 for Grades 9- 12. Mostly the list is the same for the two levels, but there is some variation: Some titles which are applicable to K-8 (e.g., "whole number computation") may become obsolete at the 9-12 level, while a few appear at 9-12 ("trigonometry") that could not reasonably have been part of the earlier work. The educational philosophy of NCTM follows the maxim of Jerome Bruner, an influential psychologist of education, who said that there is no subject that cannot be presented to a child of any age whatsoever in some intellectually respectable form. From Bruner's psychological theories arises the doctrine of the "spiral curriculum," according to which learning is best constructed by the individual in stages, each subject returned to again and again, but each time at a higher "cognitive level."
Thus, "Problem-solving," "Geometry," and "Reasoning" are titles applicable to lessons at all levels. "Statistics," for another example, is now a staple of kindergarten mathematics, though exemplified mainly by exercises of a data-gathering sort. ("Find out the favorite ice-cream flavors of your classmates.") And the rubric "Algebra" can also find its way into the earliest grades under the alternate title of "patterns" (often construed quite literally).
A second list of rubrics for mathematical accomplishment in the schools is provided by the periodic National Assessment of Educational Progress (NAEP), in which K-12 mathematical progress is measured by a Federal agency under the headings: 1. Number Sense, Properties and Operations; 2. Measurement; 3. Geometry; 4. Data Analysis, Statistics and Probability; and 5. Algebra and Functions. Some states have organized their standards with attention to these categories.