Learning Mathematics in North America

Photo Jul 09, 3 15 13 PMThe Arnold and Deanne Kaplan collection of Americana is now available for research. Besides their fine work building the Arnold and Deanne Kaplan collection of Early American Judaica, available to researchers at Penn’s Herbert D. Katz Center for Advanced Judaic Studies, the Kaplans built a small collection of American non-Jewish material as well. It includes a collection of printed books and an eclectic assortment of manuscripts and photographic material. Among the manuscripts are account books, ciphering books, diaries, letter books, penmanship notebooks, and recipe books. Each of these categories is quite interesting and can be explored further on the collection’s finding aid.

The largest grouping in the manuscript series is a collection of 22 ciphering notebooks dating from 1764 to 1870. These exemplars provide a superb view into the ways mathematics was taught in North America through the mid-nineteenth century. As outlined by Nerida Ellerton and M.A. (Ken) Clements in their book Rewriting the History of School Mathematics in North America, 1607-1861: The Central Role of Cyphering Books, the “cyphering tradition” allowed students of various ages and abilities to prepare their own ciphering books by employing formulaic presentations of mathematical rules followed by the computation of particular practical exercises.

Among the characteristics of the American cyphering tradition identified by Ellerton and Clements, are the following:

  • The tradition remained relatively unchanged from the establishment of the colonies through the mid-19th century
  • Teachers often did not have formal qualifications in mathematics; in fact, mathematics was quite often taught by writing instructors, and the emphasis on penmanship can often be seen in the ciphering books
  • Teachers would prepare problems for students, students would solve these problems on scraps of paper, and once approved, the problems and their solutions would be transferred into the students’ ciphering books
  • Some ciphering books were sewn and bound by the students themselves, while others were produced commercially as blank books ready to be used
  • Entries were completed in ink with decorative calligraphic headings and followed a formulaic pattern of topics or rules followed by individual problems
  • Lessons and problems could encompass branches of mathematics including arithmetic, algebra, geometry, and trigonometry, and often emphasized useful applications in the fields of navigation and surveying
  • Few school-aged children actually attended school, instead learning at home or by individual instruction
  • Those who did attend school (especially in New England and Mid-Atlantic colonies and states) most often did so only during the winter months
  • Most of the ciphering books were created by boys; girls who did learn mathematics usually did not progress past numeration and the four operations on whole numbers
  • The use of printed mathematical books was not prevalent until after the mid-19th century

Most of these characteristics hold true for the ciphering books in the Kaplan collection. Only one book clearly comes from the copying of printed texts: the notebook of Jacob Samuel Hillegass indicates it was copied from An introduction to mensuration, and practical geometry by John Bonnycastle and A treatise on surveying by John Gummere. All of the notebooks were created by boys with the exception of two volumes created by one Amanda Maires. In a few cases the ages of the children can be extrapolated from inscriptions of their birth dates.

The level of attainment among the ciphering books is wide-ranging. Beginning arithmetical principles include exercises in notation and numeration, the four operations, compound operations, currency exchange, reduction, the various rules of three, vulgar and decimal fractions, and percentage. More advanced topics include exercises on loss and gain, barter, brokage, tare and tret, annuities, fellowship, equation of payments, and mensuration. Practical problems deal with cloth, land, liquid, long, and dry measures; avoirdupois and apothecary weights; and the work of masons, carpenters, joiners, slater, tilers, painters, and glazers.

Despite a certain formulaity to the notebooks, they are each distinguished by a unique script and decoration (or lack thereof), their layout and design, and their paper and binding structure, and many contain marginal notes, drawings, and inscriptions.

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