Jonathan Kenigson, PhD, FRSA
Senior Fellow of Natural Philosophy
Athanasian Hall, Cambridge LTD (UK)
Few documents in the annals of mathematical pedagogy have managed to arouse such passion as the 1989 document Curriculum and Evaluation Standards for School Mathematics. On the one hand, proponents of the so-called “Reform Mathematics” have sought at every turn to denigrate memorization of basic mathematical facts as unfitting for the inquisitive mind. On the other hand, reactionary reformists have – mostly in grassroots campaigns and private schools and colleges – called for a regress to the status quo ante bellum of “drill-and-kill” arithmetic. The Janus-faced figure of Western pedagogical failure is more a death by cuts than any Mayberry-faced Eden could account for. To the reactionary, the modern student of mathematics cannot add. To the reformer, the same student cannot think. To the professor, teacher, or practitioner forced to adopt the ham-handed blunders of so-called educational experts, most students cannot do either. Reformers are correct that mathematics should be applicable to students’ unique contexts. Reactionaries are correct that without fundamental proficiency in arithmetic, nobody can do anything more.
Reality on the ground is much more deeply nuanced than political affiliations can convey. The modern student of mathematics can neither add nor think, if either PISA scores or plain experience are to be given any accord. Educators who deeply invest in student success perceive this vatic truth without any explicit mention, and coyly accept it with grotesque nausea. Such educators persist in the quasi-heroic enterprise of peddling expensively crafted merchandise to wary consumers, many of whom have already voted with their feet that the world of hard knocks is a better school than any state or church can provide. For students in the Eurozone and U.S., free and useless is hardly cheap enough. What is termed “mathematics” in K-16 curricula today is not worthy of any such designation. Paint-by-number varies as much from Florentine Realism as “mathematics” differs from the Quadrivium Arts that constitute Mathematics as mathematicians practice it. The former chimera is something else altogether more insidiously masquerading as the latter: a grab-bag of disconnected techniques made to manipulate contrived symbolic expressions and solve “applied” problems that can only be “applied” to coursework exams, time-wasting homework assignments, and stressful quizzes that seem to be due every several days at best. After exams are passed with threadbare comprehension, the relevant content is quickly forgotten, and the so-called educated citizen is free to return to ignorance of how to count change, ignorance of fundamental personal finance, an inability to find a tip on a purchase, and an obsessive addiction to grade-point-averages that either open or close various “mathematical” doors even more worthless than the ones already wearily trodden through.
Reality is harsh, and failed promises are yet harsher. It is long overdue for politicians and administrators to recognize that students must know fundamental arithmetic and logical thinking by heart before proceeding any further. Drills and repetition are excellent means of accomplishing this end, so long as these tools are coupled with actual applications in the ambit of students’ daily experiences. It should not take three pages of “brainstorming creatively” to know how to find 15% of 40. Mathematics is not, at its foundation, a Constructivist enterprise. Go forward and legislate waves not to break, suns not to set, or gravity not to operate on the fifth floor of every building. See how effective these laws and statutes are, and revel in our common human exiguity. It is better to master arithmetic and logic with rapt attention than to forget expensive and contrived problems quickly and at great cost to an academic career. Perhaps more realistically, politicians (I cite you, state and federal Departments of Education) should accept that without student mastery of fundamentals, the grand edifice of university lower-division mathematics is yet a half-sunk visage. It is time to look upon our works as the tumbleweed and tinsel that they are and despair, fully and without reserve. Your mathematicians despair silently beside you. If they spoke, they would likely not be paid any heed.