Many people believe that the ability to do mathematics is limited to a few people gifted with a special “math gene.” While it’s true that some people learn mathematics much more easily than others, mathematics up to freshman college level is within the grasp of almost all college students–provided they are prepared to work at it.
In 1974, a 43-year-old, feisty mathematics and physics teacher from Bolivia, named Jaime Escalante, came to teach at Garfield High School in a poor area of East Los Angeles. In the mid-1960’s, permissive school administrators had relinquished control and allowed predominantly Latino street gangs to stake out their territories with “placas” on school property. The student dropout rate had reached 55% and the poor academic performance of the students had put Garfield at risk of losing its accreditation. Teachers were told to prepare courses appropriate for poorly performing students. However, Jaime made an interesting assumption: these students, in spite of their poor background, might be capable of mastering the most challenging mathematics course in high school–AP Calculus (Advanced Placement). After 4 years of unrelenting effort, Escalante secured the support of his Principal and the Advanced Placement Coordinator to test his assumption. Garfield would offer the AP Calculus course and Jaime would teach it. In that year he recruited 5 students and 2 passed the course. By 1981, 14 of the 15 students in his AP Calculus class passed the challenging AP Calculus examination. The passionate coach would ask his students, “Where is the money?” Responding to their blank stares, he would say:
The money is in chemistry, physics, computers, electronics,’…I’ll teach you math and that’s your language. With that, you’re going to make it. You’re going to college and sit in the first row, not the back because you’re going to know more than anybody.
The next year, when 18 of his students passed the examination, the Educational Testing Service at Princeton, that creates and monitors the AP courses, became suspicious of the test results and had a dozen of Jaime’s students take another test under strict supervision. All the students passed, validating the previous results.
During the 1980’s, word of Escalante’s success spread; in 1987 there were 73 students enrolled in AP Calculus. AP courses in various subject areas proliferated at Garfield High, and with it, a new belief in student potential. In 1988 Jay Mathews published a book titled Escalante: The Best Teacher in America and this was followed by the iconic movie, Stand and Deliver–now a staple of Math Methods courses for teachers. In reflecting on his achievements, Escalante observed, “The key to my success with youngsters is a very simple and time-honored tradition: hard work for teacher and student alike”.
As mathematics courses become increasingly abstract in the senior undergraduate and post-graduate years, a smaller percentage of the population is able to master the discipline because the time required becomes prohibitive for those without a mathematical proclivity. Yet, even for those who are gifted in mathematics, the struggle remains, because the concepts underlying mathematical technique are deeply abstract, drawing upon special components of fluid intelligence, such as short term memory and visualization.
H. S. M. Coxeter, who has been called the “greatest classical geometer of the 20th century,” described how he discovered the snub cube, a 4-dimensional figure having 96 vertices, 432 edges, and 480 triangular faces: (Tetrahedra are pyramids with a triangular base.)
I had long been trying to extend to four dimensions the familiar construction for the snub cube…The snag was that, since the number of “nearest other white tetrahedra” was nine, the equality of their distances would impose eight conditions on the point to be selected: five more conditions than such a point could generally be expected to satisfy.
So, I went to bed and soon slept soundly. About 3 a.m. I awoke with the idea of using a symmetrical “isosceles” tetrahedron; … I could thus choose a point on the axis of symmetry and adjust its height so as to equate the distances of the two types of neighbouring point. I switched on the light and went into my living room to write it down, lest I might find the next morning that it had passed away like any ordinary dream. When morning came, there it was, ready for all the details to be filled in.
As Albert Einstein struggled to formulate a mathematical description of gravitation, he was about discover that his early disregard for mathematics would come to haunt him. Developing his General Theory of Relativity, would require a much deeper understanding of non-Euclidean geometry–a subject he had neglected during his studies at the Zurich Polytechnic. Fortunately, his friend and classmate, Marcel Grossmann, who had taken notes for Einstein when Albert skipped geometry classes, had become a leading international authority on non-Euclidean geometry. In 1912, five years after he began his work on the General Theory, Einstein hit an impasse and decided to visit his old friend. Desperate for mathematical help, Einstein pleaded, “Grossmann, you’ve got to help me or I’ll go crazy.”
Grossmann, ever willing and eager to help, suggested that Einstein embed his new theory in a 4-dimensional space that mathematicians call a Riemannian manifold. In this manifold, every event is represented by an ordered quadruple (x0, x1, x2, x3)–the first coordinate specifying its time and the last three coordinates specifying its spatial location. The challenge facing Einstein was to find a new non-Euclidean metric for calculating distances between points in this 4-dimensional curved space–a curvature that Einstein conjectured to exist in the presence of large masses. With the help of Grossmann and Einstein’s former teacher Hermann Minkowski, Albert was able to formulate his now famous General Theory of Relativity.
John von Neumann, one of the intellectual giants of the 20th century, had been a child prodigy who could divide two eight-digit numbers in his head at age 6 and was proficient in calculus at age 8. He served on the Manhattan Project that built America’s first atomic bomb and was the founder of the programmable computer. Ph.D. students would follow him down the hall, asking his advice on cracking their dissertation problem, as the revered professor walked between meetings. According to his biographer Norman Macrae, von Neumann was able, on many occasions, to set the student on the right track with a suggestion “off the top of his head.” Yet with all the mathematical intuition and fluid execution of mathematical methods, Johnny von Neumann once said to a student who confessed that he didn’t understand a particular mathematical technique, “Young man, in mathematics you don’t understand things. You just get used to them.” John von Neumann, an icon in mathematics and computer science understood that mastering mathematical ideas is a struggle for everyone.
Some people ask whether computers can do the math and relieve humans of the drudgery. Omni Magazine, in a famous 1979 interview with Richard Feynman asked:
Are physical theories going to keep getting more abstract and mathematical? Could there be today a theorist like [English scientist Michael] Faraday in the early nineteenth century, not mathematically sophisticated but with a very powerful intuition about physics?
Feynman responded:
I’d say the odds are strongly against it. For one thing you need the math just to understand what’s been done so far. Beyond that, the behavior of subnuclear systems is so strange compared to the ones the brain evolved to deal with that the analysis has to be very abstract … Faraday’s models were mechanical–springs and wires and tense bands in space–his images were from basic geometry. I think we’ve understood all we can from that point of view; what we’ve found in this century is different enough, obscure enough, that further progress will require a lot of math.
The importance of mathematics in the STEM subjects is incontestable. Furthermore, mathematics is within the grasp of most college level students, if they invest the time and effort. So, for anyone seeking a career in these areas, expect to encounter mathematical difficulties, but know that if you persevere through them, the rewards will be substantial.
