Reform Now, Before It's Too Late! William Mueller
This is the sobering assessment of the state of American mathematics instruction offered by the Department of Education, reporting on a national study of contemporary educational practices. The indictment runs from the top to the bottom: Universities, colleges, high schools, and elementary schools all get failing grades. Summarizing the situation, the president of Cornell University has remarked simply that: "...our inferiority seems to me to be very marked." The report's conclusions are hardly a revelation, however, to the many mathematicians and educators whose impassioned critiques of popular mathematical textbooks and instructional methods, especially those used in calculus, have appeared in publications such as this one. Given the uproar, and the grim prognoses for the future, it would seem that every conscientious teacher of mathematics in this country should be duty-bound to reevaluate the misguided attitudes and ineffective educational methods that have taken hold in the classroom. At least, it seemed that way in 1890. That is the date on the report from the Department of Education (when it was called the Bureau of Education) [ This article offers a small sampling of voices from that era. Through a collage of quotations and citations, I've attempted to stir up a little of the dust that was being kicked around 100 years ago. The dust has never really settled. Indeed, I suspect that these voices will elicit an acute sense of retrospective déjà vu among modern readers. This being the case, I propose a simple question: To what extent can the
Here, for example, is Prof. James Harkness, who presented two papers at the session with titles that would blend in very nicely at any modern-day session on reform: "Courses in differential calculus and differential equations" and "The importance of some preliminary training in applied mathematics":
It is tempting to characterize Prof. Harkness' presentation in the parlance of our own times. He is "pro-reform", in the sense of rejecting outdated traditional methods in favor of "newer developments". He is, however, simultaneously opposed to "proofs that are no proofs", which a modern-day reformer might regard as a criticism of the simplified, suggestive arguments that guide students through so many of today's "reform" texts. It becomes apparent that one must be extremely careful when applying familiar categories to the past, since they may fail to capture many of the essential issues being raised. I suggest, in this look backward over a century, that simple categories are equally insufficient in the present. Witness the manner in which Prof. Harkness continues, mixing together what would be considered opposing sides of the modern debate:
Present-day reformers might cheer on the call for a de-emphasis of mindless symbolic manipulation; at the same time, they might object to the implied criticism of overly simplistic "application-oriented" curricula. Today, Prof. Harkness would be a man without a camp. It is equally difficult to apply modern categories to Prof. W. F. Osgood, who presented the papers "Shall we try to place the calculus as early as possible?", "How early in the course may the lecture method be used with profit?", and the provocative "Are the best results in graduate work secured from students who have devoted most of their undergraduate time to mathematics or from those who have combined a fair amount of mathematics with a more general culture?" He says:
The tone of these remarks might easily be mistaken today as "pro-reform"—there seem to be all of the requisite nods to the "notion" of limit, the central role of applications, and the inadequacy of traditional, formal textbooks. Yet Prof. Osgood is actually arguing against teaching the mere "notion" of limit. The "central ideas", he says, will be missed. This may not ring true to modern-day reformers, who are willing to base a great deal of the calculus curriculum on the belief that central ideas like continuity, differentiation, integration, etc. What becomes apparent as Prof. Osgood proceeds, however, is that the "central ideas" he has in mind are something entirely different. He would like calculus students to have a thorough preliminary grounding in the theory of arithmetic and real numbers—something that most modern-day reformers would prefer to leave to subsequent courses in analysis.
He seems willing to admit, however, that there is only so much room in the syllabus:
Prof. F. Morley, one of the editors of the
Professor E.H. Moore, the vice-president of the Society, was also concerned with "fundamental ideas" in calculus, delivering an address titled "Certain fundamental ideas which should be emphasized throughout the undergraduate course". He advocated an intuitive development of the fundamental ideas that would culminate in a more rigorous presentation:
It is interesting to consider, then as now, how the pronouncements of "reformers" actually translate into classroom practice. In his day, Moore was a much sought-after speaker on the issues of reform. For many of Moore's students, however, the implementation often fell short of the lofty goals. Moore's classroom is given a vivid evocation in [
The "fundamental ideas which should be emphasized throughout the undergraduate course" that Moore had lectured about did not, apparently, include "social conventions". Accordingly, one wonders if the "proud moment" described above, and described by Moore himself in his addresses, might have felt more like a stay of execution to many of his students. We know that the aforementioned Mary Winston, who would survive Moore's class and then leave the department, wrote home to her mother about the psychological trials of Moore's "reforms". [
Mathematicians and mathematics educators interested in the New Education found a forum for their ideas in the newly emerging mathematical publications such as
This was a basic principle of the movement: students should have direct experience, even tactile experience, with concrete representations of the objects of study. After presenting his ideas on how to apply this basic principle to the teaching of differentiation and integration, using the so-called "heuristic method", Prof. Durell provides a more expansive summary of The New Education:
In an appeal to the skeptical working mathematician, Prof. Durell quotes from an article in the
And, Prof. Durell adds:
In the classroom, the concrete representation of abstract ideas promoted by The New Education often took the form of mathematical models. "Mathematical models", at the time, did not mean anything like the synthetic cycles of application and re-evaluation so central to today's reforms—it meant actual, physical objects made of wood, string, and plaster. Geometric models were carefully crafted to demonstrate the interplay between the analytic and the geometric, the intricacies of which were very much on the nineteenth century's cutting-edge. Today, this same interplay is more likely to be illustrated through carefully crafted programs on computers and graphing calculators. (Physical "models", in the modern sense of "models of knowledge construction", continue their presence in the form of mathematical "manipulatives".) Even in the 1890s, however, there was nothing essentially "new" about providing students with multiple representations of abstract ideas. Mathematical models had been a part of mathematics instruction since at least the 17th century. For example, in a volume titled In America, the use of mathematical models goes back to at least the beginning of the nineteenth century, when the first common schools attempted to emulate the "objective" practices of their Swiss and German counterparts. Sets of geometric solids were sold to the new schools with claims that they would help teachers present a common curriculum. The market for these models was maintained well into the early parts of the 20th century with fervent calls for, successively, "Object-Oriented Instruction", "Technical Training", "Art Education", and "Exact Thinking". School boards and government commissions formalized the arrangement, making geometric models a required component of mathematics instruction in many states. For example, an inspector's report for the Connecticut State Board of Commissioners for Common Schools from the 1830s includes on its checklist the questions "Does the school have a set of geometric solids?" and "Does the school have a cube that could be divided to illustrate the process of taking a cube root?" [ American universities of the nineteenth century were, for the most part, teaching institutions that carried forward the lessons of the common schools, preparing (mostly) young men for their careers. In the middle of the century, America began to establish its first technical colleges, looking to the French military academies and the École Polytechnic for guidance. The French schools had been heavily influenced by the example of Gaspard Monge, who had introduced mathematical models into technical training with his "descriptive geometry" for representing three-dimensional objects in two-dimensions. The models of Monge and his students were, accordingly, elaborately stringed representations of various projective paths and their motions. Collections of these models were bought up by many of the new American technical colleges. The idea of a "research university" did not emerge in the United States until the end of the nineteenth century. Looking to Europe again, mathematics departments found inspiration in the German system, which was then promoting the construction and use of mathematical models in graduate education. Felix Klein had established a laboratory for the construction of mathematical models in Munich, employing the labors of many of his graduate students, and when he brought a large collection of these models to the German Universities Exhibit at the World's Colombian Exhibition in Chicago in 1893, sales to American universities were brisk. This was exactly the time when The New Education began to be debated in the American mathematical journals. Far from new, the idea of multiple representations of mathematical concepts already had a long history in both elementary and college education. Today, it has simply been repackaged, with calls for increased use of computer visualizations, and names like "The Rule of Four".
The biography didn't appear until 1895, and Prof. Matz may be guilty of applying contemporary categories, such as New Education, to Peirce's ideas; but if one is willing to accept that The New Education was only a new name for a collection of ideas with a much longer history, then Pierce and the Harvard mathematics department were undoubtedly on the forefront of the movement. Prof. Matz notes that "Such advanced courses of mathematics as One of Prof. Peirce's innovations was to take the unprecedented step of writing his own textbooks, including the two-volume set
Elaborating on Peirce's style of writing, Prof. Matz notes:
The historian Florian Cajori also makes note of the Harvard mathematics department's influence on The New Education. He identifies John Farrar, Peirce's predecessor as chair, as "among the first to introduce important reforms in the mathematical teaching in American colleges." Farrar's chief contribution to "reform," it seems, was in translating European texts in mathematics, and then teaching from these translated sources in his courses, rather than from the many "interpretations" that were then available in English. Farrar's classes certainly
Regarding Benjamin Peirce, Cajori says that his textbooks "were adopted as soon as they came from the press." Apparently, they formed the basis of an "elective system", which tracked students according to their interests and abilities. Cajori writes:
Peirce's textbooks, however, were not popular. Cajori states: "The facts are that Prof. Peirce's textbooks were found very difficult." The freshmen, who were required to study out of Peirce's texts, were particularly unhappy:
By 1848, only five students "survived" Peirce's
The minority opinion gives some sense of the nature of Peirce's "reforms". The "most direct route to the higher regions of the calculus" is quite the opposite of the circling approach through applications and multiple representations advocated by many of today's reformers. Peirce's "compactness", "terseness", etc., prized by a minority of mathematicians on the committee, was exactly the source of the student difficulties. Outside of Harvard, Peirce's texts had a wide influence. One of Peirce's students, the Rev. Thomas Hill, who later became a president at Harvard, writes [
Here Peirce sounds like a more modern reformer, freeing essential ideas from their inessential accouterments, inventing novel presentations to accomplish his objectives. Not all of Peirce's "novelties", however, were as influential as Rev. Hill suggests. One example was his insistence that the symbols p
Unfortunately, Peirce was evidently another example of an educational reformer who was not always able to translate his lofty ideas into effective classroom practice. Certainly, many students seem to have been unable or unwilling to keep pace with his demands. In 1925, the MONTHLY published a monograph dedicated to Peirce [
Eventually, Peirce was able to reach a compromise with the administration, convincing them that the elective system should allow "lower-level" students to voluntarily discontinue their study of mathematics after the first year. The president of the university deemed the compromise "acceptable to both students and parents", and Peirce promptly discontinued his teaching of all lower-division courses. (The instruction in many of these courses was taken up by his son.) A student recalls this time:
Indeed, "genius" is the awe-struck term often applied by the students that were able to stick it out in Peirce's advanced courses. They generally admitted, however, that he "inspired rather than taught", with one student recalling that "I could not understand much that he said; but it was
We do not have books dedicated to testimonials from the many students who were lost to Peirce's "reforms", but one account, given by a student who tried and failed to meet Peirce's demands, survives [
Indeed, up until World War I, Germany provided the model for unflattering comparisons of America's "deficient" achievements. (A role to which Japan has been assigned in more recent years.) To find out what was happening in mathematics classrooms around the country, Cajori drew up a questionnaire that contained what he felt were the most important questions of the contemporary debate. They included: - Are students entering your institution thorough in the mathematics required for admission?
- Is the percentage of students electing higher mathematics increasing or decreasing?
- Does the interest in mathematics increase as students advance to higher subjects?
- What mathematical subjects are preferred by students?
- Are topics assigned to students for special investigation?
- Is any attention given to the history of mathematics? Does it make the subject more interesting?
- How does analytical mathematics compare in disciplinary value with synthetical?
- What method of teaching the calculus do you favor, that of limits, the infinitesimal, or some other? Does the infinitesimal seem rigorous, and to satisfy the mind of the student?
- Do scientific or classical students show greater aptitude for mathematics? Which sex?
- In what other subjects are good mathematical students most successful? In what least successful?
- Do you favor memorizing rules in algebra? What reforms are needed in teaching the same?
- To what extent are models used in geometry? ...What reforms are needed in teaching the same?
- State the time of your special preparation for teaching mathematics, number of hours you teach per week, and what other subjects you teach.
Cajori tallied the responses from 213 colleges, universities, and normal schools. Among the colleges and universities responding were Brown, Columbia, and Howard (not Harvard); M.I.T., Rose Polytechnic Institute, and the Naval Academy; Ohio State, Kansas State, and the State University of Iowa; The University of North Carolina, The University of Dakota, and DeLand University of DeLand, Florida; Mt. Holyoke, Mt. St. Mary's, and The Kansas City Ladies College; Bowdoin, Williams, and Miami of Ohio; The German-English College, The National Deaf-Mute College, and The Seminary West of the Suwannee River. There was also a response from Pierce Christian College in College City, California, mailed in by a certain "D. Hughes". Overall, the report provides a broad portrait of the general practice of mathematics teaching in this country one hundred years ago. The responses, like the questions, have a very familiar ring to them. For example, regarding whether or not interest in mathematics increases as students advance, we hear: - "...so long as the student sees the bearing of his work upon practical scientific investigation or can be assured that it has such a bearing"
- "All who understand the principles show a growing interest"
- "With the best students only"
and, in an out-lying opinion from Furman University,
- "...the more evolved or abstruse the matter, the greater the interest to those who succeed"
Concerning the students' preferred mathematical subjects, most teachers will recognize the timelessness of the response: - "Their preferences are generally for the particular subject which they have had the best elementary training in"
Some of the questions forced the respondents to take a side. Regarding the use of student research projects implied in the question about assigning "special investigations," the responses came out about evenly. On the positive side, we hear: - "They ...form a very essential part of the work"
- "Independent problems given in all the classes for solution, reported on paper"
On the negative side: - "Not to undergraduates"
- "The man who pursues original investigation with the average student will make a failure"
The report finds that very little attention was given to the history of mathematics at the majority of schools, even though it notes that "No one answered that it did Most schools reported that "scientific" male students had a greater aptitude for mathematics than "classical" female students, though the responses from the women's colleges disagree, generally indicating that their "classical" students show the greatest aptitude. The respondent from The Cooper Normal College in Daleville, Mississippi says: - "Males in quantity, females in
*quality*."
There were, in general, many more schools that did - "...not to ...young boys, who contract the incurable habit of learning it by rote"
- "Rules and principles are to be deduced from examples"
- "Algebra should be
*seen*" - "...less toughing, disgusting gymnastics"
- "The teacher should assist the pupil to make his own rules"
- "Drill on ...
*raison d'être*" - "More of an inductive method; and the abolition of much that may be interesting theoretically, but of little practical use"
- "Students should be taught to think! think!! think!!!"
- "The pupils learn to do by doing"
- "Less formality and more 'realism'"
and, in an interesting metaphor from the University of Missouri School of Mines, - "Rattle the bones of the algebraic skeleton ...and show it in its living, breathing continuity"
There were also many observations about the kind of teaching that best complemented the non-mechanical approach: - "Teaching needs to be less mechanical"
- "Less memorizing and more thinking, both on the part of the teacher and student"
- "...less blackboard work"
- "...putting into words the ideas conveyed by its symbols, equations, and operations"
- "Let us have live, enthusiastic, and competent teachers—such as will teach the subject rather than the text-book"
One respondent noted: - "In the larger colleges algebra is mostly taught by tutors, who hold temporary appointments, and do not expect to make teaching their life work. Algebra as well as calculus should be taught by a permanent professor."
Responses that - "The chief cause of failure in many cases is not doing enough miscellaneous examples for practice"
- "I recommend the memorization of rules, unless the pupils furnish a good, working rule of their own (a rare case)"
- "It is only practice that makes perfect"
- "...
*the German system*" - "I prefer formulas"
- "We need no reforms"
Only one respondent, a Prof. H.C. Davis from the Agricultural and Mechanical College of Starkville, Mississippi, refused to take sides; he noted calmly that - "The method of teaching must, I think, vary under different circumstances"
Thoughts on calculus instruction were more evenly divided, with a slight majority of respondents preferring the method of limits to the method of infinitesimals. This was usually because of misgivings about foundational issues. As one respondent said, - "The philosophy at the base of the (infinitesimal) method seems to involve one in a maze of absurdities"
When student preferences were reported, however, the infinitesimal method was generally preferred—it was sufficient to "satisfy the mind" of most students, who found it simpler and more intuitive. As the respondent from the Rose Polytechnic Institute related: - "...it is my experience that the (infinitesimal method) more quickly removes the logical difficulties in the way of the
*beginner*"
This dichotomy between what teachers and students found "best" and "logical" was brought out by the question "How does analytical mathematics compare in disciplinary value with synthetical?" The fact that the question is phrased in terms of "disciplinary" value already shows a teacher's point of view, and the emphasis begs for the predictable flood of responses favoring the analytic. When the responses are read carefully, however, the decision about which method to emphasize while teaching seems less clear-cut: - "I think both are necessary for a full mental development"
- "With the majority of my students, more satisfactory results are obtained through the synthetical"
- "...as well ask whether braces or tie-rods are of most service in a bridge-truss"
- "The synthetical is better for younger students; the analytical for the more mature"
- "Analyzing the whole into its elements is valuable, but building the whole from elements is
*very*valuable" - "That depends on the peculiar natural bent of the pupil's mind"
Still, some respondents make it quite clear that they not only prefer analytic methods, but find synthetic methods a dangerous corruption of mathematical practice: - "In my judgment the analytical is so far superior to the synthetical that there is left little room for comparison. Permit me to say that reason wants
*light*, not*darkness.*" - "I can not say fairly, for my teaching has been wholly in analytical mathematics"
It is also of interest to look at the response to the educational "technology" of the time—mathematical models. The use of mathematical models in the classroom caused the same kind of divisive debates that surround modern visualization technology. Then, as now, usage varied from place to place and from instructor to instructor. The most enthusiastic endorsements in the report come from teachers who used the models to help students visualize problems in three-dimensional calculus and the "higher surfaces" of descriptive geometry. Some instructors had their students construct models as a regular part of their courses, and one respondent, from the State Agricultural College in Fort Collins, Colorado, was such a free-thinker that he used models "out-doors". Some schools boasted about the specifications of their expensive, up-to-the-minute collections: - "Full set of Schröder's (Darmstadt) models"
- "We have twenty-six fine models of warped and single-curved surfaces"
- "We have about one hundred dollars worth of models for pure mathematics"
Other collections, such as the one at Dickinson college, seemed pitiful by comparison: - "None except sphere and cone"
Outright hostility toward the use of models was rare, though not unheard of: - "...find them hurtful rather than helpful. 'Normal School' methods are a failure."
Typical among the responses from those that were - "I prefer that students should learn as soon as possible to form mental pictures"
The method by which students would learn this skill was unspecified. Sometimes the reasons for not using models in the classroom were more prosaic, though equally recognizable: - "They are used to a very limited extent, because the college is not supplied with them"
- "Only slightly, because we are not able to afford them"
- "Hitherto but little; henceforth very great (if the appropriation asked of the State be granted)"
The report also gives us an opportunity to check on the extent to which the academic community has progressed with labor-related issues. Reporting on the number of hours per week that teachers spent in the classroom and in preparation, the majority of responses fell somewhere in the range of 10-30 hours/week, though it was noted that "...this does not include office hours." At the extremes were the respondent from Niagra University, who taught only two hours per week and prepared in a mere half hour, and the incredibly energetic "D. Hughes", who logged in at forty-one hours per week. The teaching load was considerably lower at institutions where "assistants" were available to take on some of the load, and at Adrian College in Adrian, Michigan, the respondent reports that - "My assistant teaches all up to and including analytic geometry, moral science, &c."
As noted earlier, these assistants were usually part-time employees who had little chance of turning their teaching into a permanent career. One presumes that the teaching of "moral science" was handed over from one part-time employee to the next. It is difficult to summarize the incredible number of fascinating particulars that make up the report, and Cajori doesn't really try. What he does offer, however, are individualized portraits of some of the respondents. For example, he describes the innovative teaching style of Prof. T. H. Safford at Williams College, who used, and advocated, the "heuristic method". Cajori describes the method as follows: "The heuristic method is, in general, the method in which the pupil's mind does the work." He then adds, perhaps unnecessarily, "It is a slow method." Among other things, Prof. Safford used the "rule of three":
Although the "rule of three" mentioned here, as a subject in arithmetic, is not the same Rule of Three (or Four or Five) advocated by many of today's reformers, the integrated approach of Prof. Safford's "ideal programme of study" surely captures some of the same spirit. Prof. Safford himself assures that the flexibility of the program is part of its enduring legacy:
Although Cajori is not able to summarize adequately the many nuances of his findings, he is able to capture the anxious uncertainty of so many unsettled "discussions":
At which point Cajori goes on to list a "rupture" of newly published textbooks, on subjects ranging from algebra to differential equations, each one promoting its own prescription for reform.
If we could hear history's lessons, we might be more willing to see the current "crisis" as just another episode in a long historical "discussion" about the nature of mathematics itself. The issues haven't been settled in a century—many centuries, in fact—and they will not be settled in this one. The debate endures because there is something In 1912, Bertrand Russell, as even-headed a member of our profession as we are likely to find, wrote about the "problems" of philosophy [
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