As my youngest daughter enters senior year in a reputed high school in Albuquerque, and college applications become our daily bread, I reflect on what she has learned in that institution.
She has been exposed to 20th century authors like Ray Bradbury, and even 21st century Pulitzer Prize winners like Anthony Doerr. In American history, they have discussed the presidencies of Carter, Reagan, the Bushes and Clinton. She has dealt with 19th century chemistry knowledge – the periodic table was introduced in 1869 by Mendeleev – and been told about elements that were discovered less than a hundred years ago. Similarly in biology, she has been introduced to Mendelian inheritance, a theory first exposed by Mendel in 1865 and later rediscovered in 1900.
But when it comes to mathematics, she has been reduced mostly to medieval knowledge, has scratched the surface of Renaissance calculations – finding the roots of a second-degree polynomial, or the inverse of a complex number – and spent a full semester digging in Euclid’s Elements, which collects known facts of geometry from, or before, the fourth century before Christ. Thanks to her past good grades and sensible choices, in her final high school year, unlike many of the students of her cohort, she will learn something about one-variable differential calculus, whose modern development is usually attributed to Newton and Leibniz at the end of the 17th century.
The facts are that the high school in APS with the best four-year graduation rate – allows almost half of its students to graduate without taking a calculus course. The percentage from schools with the worst rates of graduation hovers around 10 percent. Differential and integral calculus should be taught in every high school in America to everyone, not just to some lucky or special ones. It is the basic language not only of engineering, but of every discipline – which could be in the social sciences – that tries to describe rates of change. With this tool one can further study subjects like probability and statistics, which are fundamental to the understanding of uncertainty or randomness, paramount in modern life.
There is nothing wrong in still studying trigonometry or logarithms in high school, but with the right emphasis and time constraints. I remember my own high school almost half a century ago in impoverished Spain under Generalissimo Franco. We spent quite some time handling logarithmic tables to be able to multiply large numbers, but we also covered the fundamentals of limits, derivatives and integrals. Nowadays we would not want our students spending time with musty logarithmic tables because we have cheap hand-held calculators, though we still want them to understand the notion of the slow-growing logarithmic function used, for example, in analysis of algorithms. We do not want to throw away the knowledge acquired through the centuries, but we need to refocus and to overhaul the curriculum of high school mathematics.
I teach advanced engineering math at UNM, and I observe how in many cases American students have to catch up to their international classmates who have been exposed to more mathematics in their high school years. After some time, things level off and all students perform in a similar manner, but just imagine how far our college students could go if their first year in college were a breeze because all high school graduates knew well their 17th century mathematics.
José L. Palacios holds a PhD in Mathematics from the University of California at Berkeley. He has been teaching in the Electrical and Computer Engineering Department at UNM for the past five years.
