Two Mathematics: Ever the Twain Shall Meet?

In his classic essay on the aesthetics of mathematics, A Mathematician's Apology (1940), the prominent British mathematician G. H. Hardy identifies two mathematics: real mathematics and "trivial" (or useful) mathematics. Although writing from a rarefied point of view, Hardy aptly describes a disparity that has existed for centuries, one that persists in twenty-first-century America as a division between the rigorous mathematics that real mathematicians study, appreciate, and extend and the contextualized mathematics of everyday life. The historian Patricia Cohen confirms a similar division, in colonial times, between commercial arithmetic and more sophisticated mathematics. Fearing that bourgeois lads would find mathematics too difficult, colonial and precolonial textbook writers attempted to ship arithmetic to its essentials. But, as Cohen points out (1982, 27-28), "in fact they cut it into incoherent bits and made it an arcane subject, almost impossible to learn. " To make matters worse, arithmetic was suffused with commercial meanings, which caused those not destined to a life in commerce to lcam no arithmetic at all. The endemic lack of arithmetic skill among tradesmen in the seventeenth and eighteenth centuries is indicated by the popularity of "ready reckoners," books containing page after page of tables showing multiples of the unit costs of common commodities at a variety of prices. As American democracy has developed, the quantitative demands on its citizens have grown; indeed, largely driven by the power of computers to amass and analyze data, these demands have exploded over the past two decades. The ability to deal with the quantitative demands of everyday life is what we call quantitative literacy (QL for short). Robert Orrill, director of the National Council on Education and the Disciplines, has described QL as a cultural field where language and quantitative constructs merge and are no longer one or the other, reflecting the continued suffusion of arithmetic with meanings from societal contexts. The two mathematics embodied in Hardy's real and useful components and in Gohen's sophisticated mathematics and commercial arithmetic are represented today by formal school and college mathematics (formal, for short), on the one hand, and what I will call QL mathematics, on the other. The influence of both Hardy's real mathematicians and long traditions of content has led to a mathematics curriculum that is dominated in grades 8-14 by a hurried and linear sequence of geometry, algebra, trigonometry, and calculus (GATC, for short). Generally ineffective in educating for QL, this GATC sequence is underwritten by the perceived educational needs of future scientists, engineers, and mathematicians, who comprise approximately one-fourth of the college population. With QL mathematics becoming both more demanding and more in demand, the pressure on formal mathematics to respond with more elective QL education is increasing. However, in addition to the enormous inertia in K-16 mathematics, several worries generate arguments against significant systemic changes in curriculum or pedagogy in service to QL. Some problems in educating for QL result from our repeating mistakes made in colonial America. We have maintained the circumstance of two mathematics by keeping distance between QL mathematics and formal mathematics. We have stopped the GATC sequence to essentials, cutting the material into incoherent bits that are difficult to learn, much less to use. We even misuse the twenty-first century analog of ready reckoners-handheld calculators-as a total substitute for doing and understanding arithmetic and algebra. Today, formal mathematics is institutionalized in American society, and QL mathematics, which should be the darling of practical-minded legislators, struggles for a toehold. The GATC sequence is firmly embedded in local, state, and national legislation via assessment requirements for K-12 progress, school-to-college transitions, and college degree requirements. …