Historical Roots of Confusion Among Beginning Algebra Students: A Newly Discovered Manuscript

Abstract

and the related freedom of the algebraist. The history of algebra can facilitate students' adjustment to the spirit and direction of not only abstract algebra but contemporary mathematics in general. The perplexed student might find comfort, for example, in the realization that arbitrariness and freedom are youthful additions to mathematics, born of the early nineteenth-century research on abstract algebra and non-Euclidean geometries, and widely accepted only later in the century. Around 1800 there was but one algebra. In this algebra, frequently called universal arithmetic, letters stood for numbers or quantities, and the laws of arithmetic, such as commutativity of addition and multiplication, prevailed. By the middle of the nineteenth century, however, mathematicians had created many different algebras, including the quaternions, whose multiplication is noncommutative. From this perspective, the confused modem algebra student may be a victim of the circumstance of having been born in the twentieth century. Additional solace might be derived from the revelation that bewilderment and occasionally even rejection greeted the original formulation of the symbolical approach to algebra. Thus the perplexed student has historical roots! A study of early objections to symbolical algebra can also shed light on and stimulate discussion of the oft-unvoiced concerns of present-day modern algebra students. This paper offers a newly discovered nineteenth-century manuscript which documents confu- sion as an almost immediate response to the emergence of symbolical algebra in its limited form. (Like modern algebra, early nineteenth-century symbolical algebra admitted undefined entities; unlike modern algebra, symbolical algebra, in its original form, was governed by the laws of arithmetic.) The manuscript is a spoof-play on Augustus De Morgan's Elements of Algebra (5), one of the first undergraduate algebra textbooks to incorporate the symbolical (or limited modem) approach to algebra.