Reform Now, Before It's Too Late!
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) . The president of Cornell University was C. K. Adams, who delivered his remark to the annual meeting of the New England Association of Colleges and Preparatory Schools, in 1888 [17, p.24]. The publications "such as this one" included, indeed, the MONTHLY, beginning with its first volume in 1894, and the Bulletin of the American Mathematical Society, which began publication that same year. Both journals, from the very beginning, printed articles, editorials, and reports that expressed a tremendous concern about the way in which American students were learning mathematics. Reading through these today, one might suppose that at the turn of the last century the minds of America's youth were imperiled, American productivity was being threatened, and, indeed, the future of American mathematics was at stake.
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 current state of mathematics education be in such a state of "crisis", if the terms of the debate have changed so little in 100 years?
Reformers Unite. By 1900, the American Mathematical Society felt that it was desirable to provide a forum for the "discussions" of education, especially as they related to the undergraduate curriculum. A special session was organized for the Seventh Summer Meeting in New York City, and excerpts from the presentations that were made have been preserved in the Bulletin . Collectively, these excerpts suggest something of the tenor of the "discussions" that were taking place across the country. Read a century later, there is a familiar sound to the prescriptions being offered for what was ailing American mathematics education.
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. can be taught effectively with an intuitive "notion" of limit.
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 numberssomething 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 Bulletin, spoke on "The theory of equations; spherical trigonometry; geometric conics" and "On a mechanism for drawing trochoidal and allied curves". He seems to agree with Prof. Osgood's call for developing firm foundations, but, in a line that would please most current reformers, only after the student has developed a more intuitive sense of the subject in an elementary calculus course. His thoughts on developing that intuition through "interaction" sound decidedly modern:
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 setting is a graduate course, and although one might imagine Moore treading more lightly on his undergraduates, the atmosphere is apparent enough:
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".  (Ms. Winston would go on to become the first American woman to earn a doctorate in mathematics at a German university, in Göttingen, under the direction of the much more encouraging Felix Klein.)
The New Education. In the late 1800s, teachers who were interested in improving the quality and relevance of instruction in their classroomsand there were many of themorganized themselves into a "movement" that came to be called The New Education. The historian Florian Cajori, writing in 1890, traces the movement's evolution :
Mathematicians and mathematics educators interested in the New Education found a forum for their ideas in the newly emerging mathematical publications such as The American Mathematical Monthly. In the very first issue, in January of 1894, the Monthly featured an article entitled "Application of the New Education to the Differential and Integral Calculus", by Prof. Fletcher Durell of Dickinson College . The issues it discussed were considered important enough for the article to be continued into the second issue. Prof. Durell begins:
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 Educational Review written by Prof. Simon Newcomb . Prof. Newcomb had written:
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 reformsit 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 Stereometry Or, the Art of Gauging ..., written in 1689 by Thomas Everard, self-proclaimed "Philomath", there are any number of concocted "applications" that require some kind of three-dimensional visual acuity. One of them requires finding the volume of an ale tun in the shape of a pyramidal frustum with bases that range from a "trigon" to a dodecagon. To help with the visualization, there is an appendix of drawings showing the volumes involved. If these drawings alone are not enough to awaken the reader's spatial imagination, then an advertisement inside the title page informs the reader that geometric models are available at a particular shop in London (and "only" at this shop), the address of which is duly noted. There was money to be made in "new" mathematics, even then.
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?"  Business was so good during the nineteenth century that model makers were able to diversify into much more lucrative catalogues of "Mathematical Apparatus", which included everything from the latest in "noiseless" drawing slates and elegant "pointing rods" to hand-crafted "numerical frames", made of the finest woods. There was eventually a backlash, however, as teachers began to complain that the expensive, and increasingly complicated apparatus was driving the curriculum. The parallels with today's classroom computers probably do not have to be pointed out.
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".
Harvard Calculus. The struggles to develop a "Harvard calculus" more than a century ago also have many modern resonances. In a biography of Benjamin Pierce, the chair of the Harvard mathematics department from 1831 until the time of his death in 1880, Prof. F. P. Matz wrote in the MONTHLY :
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 he offered to students, in 1848, had never before been offered to American students by any other professor in any other American college."
One of Prof. Peirce's innovations was to take the unprecedented step of writing his own textbooks, including the two-volume set Elementary Treatise on Curves, Functions, and Forces. Prof. Matz writes:
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 sound different. Here is a reminiscence by one of his students :
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 Curves and Functions. A committee was impaneled once again, charged with digging to the bottom of the "so very decidedly unpopular" status of mathematics at the University. The majority on the committee reported that "the text-books were abstract and difficult", and that "there are other mathematical works of no small merit, which embraced the same subjects as the text-books now used, which were much less difficult of comprehension." However, a minority on the committee came to a different conclusion:
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 and e , generally accepted at the time, were "inconvenient". Peirce advocated the symbols and , respectively, to take their place. 
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 , and it is filled with the recollections of his students. Time and again, they recall his chaotic classroom presence and his angry impatience with detail. His blackboard technique often involved only a small corner of the board, which he would quickly erase at the sign of trouble, with the instruction that his students should make the necessary corrections when they went over their notes. One student recalls:
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 splendid." Another starry-eyed account of Peirce toward the end of his career goes like this:
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 :
The Government Report. In 1890, a decade after the end of Benjamin Peirce's tenure at Harvard, with his texts still widely used and The New Education beginning to receive a great deal of attention, the U.S. Bureau of Education commissioned a report on the state of mathematics instruction across the country. Its author was Florian Cajori, who would go on to become one of America's leading educators and a thoughtful writer on the history of mathematics, its teaching, and its role in liberal education. Part of the justification for the government study was given in the quotation from Cajori at the beginning of this article, which came from the report's introduction. It goes on:
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:
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:
and, in an out-lying opinion from Furman University,
Concerning the students' preferred mathematical subjects, most teachers will recognize the timelessness of the response:
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:
On the negative side:
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 not make the subject more interestinga clear case."
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:
There were, in general, many more schools that did not recommend the memorization of algebraic rules than those that did, and the reasons sound familiar:
and, in an interesting metaphor from the University of Missouri School of Mines,
There were also many observations about the kind of teaching that best complemented the non-mechanical approach:
One respondent noted:
Responses that did advocate the essential benefits of memorization and drill noted that:
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
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,
When student preferences were reported, however, the infinitesimal method was generally preferredit 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:
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:
Still, some respondents make it quite clear that they not only prefer analytic methods, but find synthetic methods a dangerous corruption of mathematical practice:
It is also of interest to look at the response to the educational "technology" of the timemathematical 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:
Other collections, such as the one at Dickinson college, seemed pitiful by comparison:
Outright hostility toward the use of models was rare, though not unheard of:
Typical among the responses from those that were not making use of models were questions of their value beyond the level of "beginners". A few respondents, however, simply stated that they were personally more comfortable "drawing figures on the board". One respondent said that models weren't used because
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:
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
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.
The "New" Reforms. As we turn into another millennium, one hundred years after the "reforms" of the New Education, the American mathematical community is filled again with commencement de siècle angst about its collective educational endeavors. Well-intentioned efforts to pass mathematics on to the next generation have "reformers" moving in many different directions. Collisions are inevitable, and our conferences and journals are filled with the noise. We hear that mathematics education is in "turmoil", facing a "crisis". We hear about the "math wars". With all parties convinced that nothing less than the good of mathematics is at stake, the noise grows ever louder.
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 centurymany centuries, in factand they will not be settled in this one. The debate endures because there is something inherent in mathematical practice that always leads to these disagreements. Mathematics, and the experience of mathematics, is chaotically, wonderfully, multitudinous. Nevertheless, the many motions within mathematics education have always managed to find a collective average motion that carries the subject fruitfully forward. Mathematics survives.
In 1912, Bertrand Russell, as even-headed a member of our profession as we are likely to find, wrote about the "problems" of philosophy [21, p.161]. He wondered about the value of a lifetime of struggle that could never reach a final, satisfying conclusion. Though he may have been talking about "philosophy" in general, we could just as well substitute "pedagogy", or whatever else it is that is keeping us up at night: