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What we know is not much. What we do not know is immense.
(Allegedly his last words.)

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Mathematics Education at West Point: The First Hundred Years

An 1869 article in The New Englander by Robert Keep describes the teaching at West Point during the nineteenth century so well that we will quote it at length.

Let us now look in upon the daily recitations, and select for our examination the fourth or incoming class, which numbers from seventy-five to one hundred. At the commencement of the year it is arranged in alphabetical order, from A to Z, and divided into six or seven divisions or sections, each containing twelve to fourteen men. Each section has two recitations per diem, except on Saturday, when the second recitation is omitted. These two recitations, during the first half of the year, are in Davies's Bourdon’s Algebra, and French. The Mathematical hours are from 8 to 11; French from 2 to 4; and this general division of time holds good through all the classes. Mathematics have the heart of the day—the three best hours—and six recitations per week; English studies, including Law, Ethics, Tactics, and Modern Languages, usually the two hours from 2 to 4 P.M., and but five recitations per week.

To teach the Mathematics to the fourth class, there are three of four instructors besides the Professor, each of these instructors hearing two recitations in the three hours—i.e. an hour  and a half is allotted to thirteen men…. Five or six men are at once sent to the black-board, and having taken their places there and written their names, are given each a proposition to demonstrate, while another is called up to recite on the reading matter of the lesson in answer to questions. He is catechised until one of those first sent to the board signifies his readiness, which he does by facing the instructor and assuming the attitude of attention…

He begins by describing in general terms his subject, then he enunciates the theorem or scientific statement, and  lastly follows out the work he has put upon the board, indicating each point of progress by the pointer…

His demonstration concluded, he, having so far proceeded without interruption, is keenly questioned by the instructor. What has been imperfectly understood is elucidated, misconceptions are corrected, and he is at last allowed to take his seat…. Nor do the questions asked cover only the point of demonstration, but run over the review and back-review, including everything in any way hinted at by the recitation, so that one man may often recite half the lesson. The work is now erased, another takes the place with perhaps the same topic, and the second of those first stationed  at the board is called upon for his demonstration, and so on through the section…. It is not meant, it must be understood, that the instructor at West Point does not question and communicate, only that the recitation and instruction are separated…

The instructor marks daily upon a scale as follow: 3, thorough, 2.5, good, 2, fair, 1.5, imperfect, 1, bad, 0, failure, and each Saturday transfers his marks to a printed blank which shows the daily and aggregate rank of each cadet of the section for the week. These blanks are exposed every Monday, at noon, to view… The marks are thus shown every week, in every department, for the four years. Dependent upon this is the system of transfers, which has been in operation some forty years. The sections do not remain the same from week to week. [1]

These weekly grades were also used to resection the cadets. This idea of grading the cadets every day, is at the core of Thayer’s philosophy.  And this method of instruction, which quickly became known as the ‘Thayer method’, was in use in all departments for the entire century. In the technical subjects such as mathematics, science and engineering, cadet boards were graded on a daily basis through the 1980s.

Given a very limited faculty, this class-intensive form of instruction would have taken its toll on the professors.  Thayer outlined his solution to this problem in a letter to President James Monroe dated October 10th, 1828.

It must be very desirable, for the reasons mentioned in your letters, to relieve the professors from these duties. This may be done  1t. [first] by employing a number of young graduates who would not only act as assistant professors but also under the Instruction of the President would perform the more active executive duties. This class of persons would as teachers be eminently useful even now but will be found indispensable whenever the number of students shall amount to several hundreds. A professor can deliver lectures to many more than he can thoroughly teach.  I will illustrate the idea I would convey by supposing a case. A class of 80 students is to be taught Mathematics or Natural Philosophy devoting three hours of each day to the study of the subject at their rooms and three other hours with the Professor. One hour is to be taken up in the Lecture but this alone is not sufficient. Each student should demonstrate a proposition or explain an investigation at the Black-board and also be interrogated to see that he thoroughly understands the principles. This will require, as experience proves, not less that 15 minutes on an average for each student. Now it is evident that only eight students can be examined in the remaining few hours so that each can be examined only about once a fortnight which in effect is merely equivalent to no examination at all. What is to be done? Let the class be divided into at least four parts or sections and let each section attend 3 hours daily with an assistant professor to be examined upon the subject of the Lecture or lessons given on the preceding day. The Professor besides lecturing may either have the recitations of one Section himself or what would be the better practice, he might without taking the immediate charge, be present at the recitations, visiting each section in turn and only occasionally putting questions and giving explanations. You know that this is the system of instruction which has been practiced at West Point during the last ten years with what success I leave it for others to say.[2]

As Thayer indicates, these assistant professor were recent graduates, and in some instances, cadets from the upper classes who had shown a strength in mathematics.  As we will mention latter, some of these cadet instructors went on to other academic posts and helped to spread the USMA curriculum across the country.

In order to facilitate this style of instruction, the classrooms at the Academy were, and are to this day, fairly unique.  The 1896 Annual Report of the Superintendent describes them as follows:

Upon the walls in oak frames, their surfaces flush with the face of the frames, are twelve or fourteen slates, usually 4 feet by 3 feet 6 inches... They are all known by the generic name of blackboards.  From the lower part of each frame projects a shallow chalk tray, having at its bottom still shallower drawers, and above each drawer a galvanized wire grating. The chalk crayons and erasers, when not in use, are kept on the grating in tray, while the dust which there implements always generate falls into racks to support rulers and pointers.[3]

This means that there was a blackboard for each student (photo c. 1900) and several at the front for the instructor.  Several sources claim that Claudius Crozet introduced the blackboard to West Point in 1817, and that West Point was the first place the blackboard was used in the country.[4]  However, as noted earlier, the blackboard was in use at USMA in 1801.  Whether this was the first use of the blackboard in the United States is unclear. It was in use in various areas of the country by the second decade of the century: Philadelphia 1809, Boston 1812, Salem 1820, Bowdoin College 1824.  By the 1830s they were in common use in most schools.[5]

Blackboards (c. 1900)

Another of Thayer’s additions to the Academy was the institution of bi-annual exams, in the style of the École. Cadets were examined twice each year, in December and June. This took place in front of the Academic Board and, in June, the Board of Visitors also. One section of cadets was examined at a time. In advance, the instructor wrote the main topics of the course on slips of paper and then drew them randomly to decide which question was asked of each cadet. For example, there is an 1877 copy of Davies’s Algebra in the West Point Library that was owned by Cadet Acton. On the endpapers is written

Examined Jany 2nd 1879.

Subject – “Logarithms”   “Fessed cold”

The cadet slang “fessed cold” means that Cadet Acton failed the exam. This is confirmed by his Cullum[6] listing as x1882, a non-graduating member of the class of 1882. The question asked of him was to explain the topic of logarithms. Curiously, this section of the text is heavily annotated in a way that indicates the writer understood the nuances of the subject. But perhaps this is not Cadet Acton’s handwriting. The failure does not seem uppermost in his mind, for he continues the annotation:

This study was ordained in hell

To torment those who on earth dwell

And it suits its purpose well.

Glory hallelujah!!

Amen!

Amen!

Along with setting firm guidelines for academic instruction, Thayer set to work immediately to bring the curriculum to a higher level.  The topics taught during this period were: analytic geometry, plane and spherical trigonometry, mensuration, logarithms, conics, surveying, and fluxions.  In particular, the precursor to technical drawing, descriptive geometry became a mainstay of the curriculum for many years.  The Committee on Military Affairs of the Academy in 1834 describe descriptive geometry as: “a science peculiarly necessary in civil and military engineering, and which has been nowhere else cultivated with advantage or assiduity, save in France.”[7]  The first professor of descriptive geometry was Claudius Crozet.

[1]Keep, R., The System of Instruction at West Point: Can it be Employed in our Colleges, The New Englander, XXVIII (CVI), 1869, pp. 4-18. Available online at Cornell University Library “Making of America” website.

[2]The West Point Thayer Papers 1808-1872, ed. C. Adams, 1965.  Available online the USMA Library and at: http://www.dean.usma.edu/math/people/rickey/dms/doc/1828-10-10-Thayer-Monroe.htm

[3]Annual Report of the Superintendent of the United States Military Academy, 1896, p. 75.

[4]Edwin L. Dooley, Jr.,  Claudius Crozet: Disseminator of French Technical Education to the United States, Proceedings of the Consortium on Revolutionary Europe, 1750-1850, 1986, p. 452.

[5]Anderson, C., Technology in American Education: 1650-1900, 1962.