Teaching Physics to the Mathless
Some say that Galileo was to Italian writing as Shakespeare was to English. Not that I would know. I love his physics, but I have read him only in translation, as my Italian begins and ends with words that my grandfather said when angry. Grampa loved me, but the people whom you care about the most also frustrate you the most. Thus I often want to use those Italian words with my students, a quarter of whom grasp the mathematical language used in physics about as well as I understand Italian. More maddening still, many colleagues shy from acknowledging the intractable scope of this problem.
These students, physics majors all of them, do know math on some level. With effort they can manipulate symbols and formulas, but when they err they do not understand that they erred, because they never developed comfort with algebra and its rules. They do not understand that some uses of mathematical symbols are as wrong as writing “I uses the symbols” in English. Having never achieved fluency in algebra, they in turn lack fluency in calculus — passing grades only mean so much — and are hence lost when I use that mathematical language. Their incomprehension would be akin to me hearing a subtle Italian figure of speech.
There is nothing shameful about not grasping algebra, just as there is nothing shameful about not knowing Italian. However, not knowing Italian, I stick to translations of Galileo, and would not enroll in a class that analyzed the original works. I would likewise advise individuals who lack fluency in math to stick to popular physics books until they catch up in math. If catching up is too tall a task, there is no shame in majoring in something else and reading physics for fun. Nobody looks askance at me for studying physics professionally and reading other subjects for pleasure.
Sadly, many students neither catch up nor change majors, and we downplay the placement tests that predicted this. To hide their lack of high school math foundations, the class designated to remedy their deficits is named “College Algebra.” However, it attempts to fix in months a deficit that festered for years. Even when developmental math classes stretch over multiple semesters, the results are not much better. Two semesters of coursework for adults with freedom to mismanage their own time is no substitute for sustained practice in formative years. Yes, a celebrated few do overcome deficits, but you may be unsurprised to learn that many eighteen year-olds do not put forth the required Herculean effort. Nonetheless, polite academics regard this observation as blaming the victims of a world that never prepared them.
They would not regard it as blaming the blameless to say that athletic excellence requires years of intense practice. Everyone recognizes the difference between recreational running and intercollegiate competition. Nobody suggests accepting all aspirants onto track teams as compensation for inadequate high school gym classes. Joining a team is competitive, and requires adhering to rigorous practice schedules. Science and math professors, sadly, have no authority to mandate hours of intense problem-solving practice. We can assign homework, but heaven help professors who assign weekly homework loads comparable to an NCAA team’s ostensibly 20-hour practice schedule for a 3-unit class, or assign grades as stringently as coaches select team members. A mere professor dares not claim a coach’s authority.
Military instructors do enjoy boundless authority over students, but even they do not teach quantitative subjects to people with weak math preparation. Officers have the power to impose study schedules that college professors only dream of, the can-do attitude of fitness buffs, and a track record of training young people from every socioeconomic bracket to perform difficult tasks. Nonetheless, authority and effort only go so far, thus they use standardized tests to determine who can train as a nuclear reactor operator or cryptographer. If someone is not prepared for cryptography or nuclear propulsion, the military has honorable alternatives.
Yet when I suggest that students who lack mathematical foundations should reconsider their plans and pursue the other fine majors offered by our university, progress-minded colleagues regard me as a drill instructor would regard a flag-burning hippie. My colleagues’ motives are not fiscal; with our enrollment numbers we could afford to offer compassionate alternatives to the unprepared. However, if unprepared students are victims of bad systems then sending them to other majors is exclusionary. I could note that many floundering students are white suburban males, i.e. quite privileged, but that would just undermine a comforting narrative that equates inaction with inclusion. Galileo argued that the Bible tells how to go to heaven, not how the heavens go, but we seek both, going to heaven by teaching the education system’s lost sheep how the heavens go.
Such notions receive less traction at more selective schools. Professors and administrators may now say warmer and fuzzier things, but friends report that students still change majors early and often, as my classmates did. The sooner you find a major in which you will flourish, the sooner you can join affiliated extracurricular organizations and seek internships. Seeing college as a place to find a networking scene in which you will shine requires some social capital, yet those who profess the greatest concern for the least advantaged eschew such counsel, letting students flounder in order to appease a misguided conscience.
They have a fallback excuse when the social justice argument strains credulity: Besides the spiritual good of keeping diverse students in science, there is earthly concern for an alleged shortage of scientists. Such talk goes back at least to Sputnik, yet economic reality has never matched rhetoric. The next fallback is a concession that we have enough scientists, but “too many” are foreign-born, a narrative that sits uneasily alongside celebrations of diversity.
Nonetheless, diversity remains central to many professors’ reluctance to guide students to other majors. They worry about an alleged exodus of diverse students from science. Never mind that attrition into science from other fields exceeds attrition from science, or that undergraduate women do not change major at disproportionate rates. Not everybody was meant to study the subject that they marked on a form at age 17. Some individuals should switch from science to other fields, and others should and do switch into science. Indeed, some of my most successful students started in very different fields before choosing physics, and these new recruits to physics were quite diverse.
It’s understandable to lament sending people away from a desired path, but does changing major truly mean leaving an endeavor? Many people spend their lives honing skills in music, sports, art, and other fulfilling pursuits. Most earn neither degrees nor livings in such passions, but still find satisfaction, and nobody except the drummer’s neighbor objects. Those with a passion for science can work a “day job” and spend free time gazing through telescopes, writing code, observing wildlife, or doing self-study. Studying the natural world is a human right, but technical careers should be for those who can master technical skills. Just as one who cannot sing in tune after years of lessons should consider karaoke in lieu of Broadway, one who is weak in algebra after years of math classes should choose popular science magazines over advanced physics courses.
Still, I am stuck with unprepared kids whom my colleagues would not counsel away. A few semesters of “College Algebra” did not remedy a decade of poor math preparation, amazingly enough. We try to help along the way, but our time and authority are finite. Maybe they could succeed if we covered half as much material in twice as much time, but nobody will pay for us to spend the time, nor for students to stick around. Besides, the majority of the students should not endure such a languid pace.
Some problems of time can be solved via financial aid, enabling students to prioritize classwork over paid work, and there are good reasons to provide aid, regardless of major or level of preparation. But money alone can’t fix everything, so many colleges go farther via well-photographed “pipeline” programs (“pipeline” being a metaphor for educational systems providing future scientists) for disadvantaged but promising students. Chosen students get mentoring and internship placement to distinguish themselves when applying to graduate programs. Some of these students are excellent, but funders want to see that their dollars are saving the world, not just skimming the cream of the crop. Program directors thus have to take hard cases and try to turn them around.
So the pipeline programs pick and groom some weak students, and of course some beat the odds, and of course we celebrate their successes. Alas, such stories are not the full story. Too often a pipeline program proclaims the greatness of students who exhibit merely middling performance in any class where the professor is not funded by said pipeline program. They are middling in project-based classes, middling in traditional theoretical classes, and worse than middling on the Graduate Record Exam (GRE).
The GRE scores are damning, as the test reflects external standards, not my idiosyncratic notions. Yes, there are smart, accomplished people who test poorly. And, yes, some people test well but lack common sense. (My grad school classmates may have stories…) And of course we should not select solely on test scores. Nonetheless, while tests do not tell everything, it does not follow that they tell nothing. The GRE Quantitative Reasoning section, taken by students in all fields, tests high school math foundations, not advanced calculus. In a better world, physics majors would ace this test, but in this world many physics majors get mediocre scores on a math test taken by physics and French majors alike, strong evidence that their mathematical deficits are not mere figments of a cranky old man’s limited imagination.
The excuse that we are supposed to accept is that students have test anxiety, but too many flounder with basic math in settings besides the GRE. They entered with weaknesses that they never remedied, we passed them anyway, and when a test exposes this problem we claim that the test is unfair.
Disdain for standardized tests is ironic given liberal enthusiasm for European social policies: Europe, like most of the world, bases college admission primarily on tests. American academics, however, only know Europe as a place of tuition-free universities in nice cities with rail transit. Who wouldn’t want to grade papers in a café with fine espresso, fresh croissants, and historic architecture? Alas, experienced colleagues report that Europe is not academic utopia, neither for traditionalists nor for progressives. Some test-vetted students do use their freedom from tuition burdens to study, but others — being human — slack off, or continue ancient traditions of eroding town-gown relations via loutishness.
Besides the ironic Europhilia of test-averse professors, favored alternatives to tests sit uneasily with commonplace attitudes towards success and its determinants. Many professors proclaim that instead of criteria measuring smarts or preparation, we should reward “grit” and “growth mindset” (pop psychology terms for work ethic and determination). “Anyone can succeed if they try hard enough, so favor those who try hard over those who test well” goes the argument. Work ethic is of course crucial, but this is a strange attitude to embrace while portraying students as victims whose failures arise from systems rather than personal failings.
If anyone believes that all can succeed with sufficient effort, it is Asian American students and parents. I do not claim that Asian Americans are an undifferentiated mass of obsessive students; as humans, they have a wide range of mindsets, successes, and shortcomings. Nonetheless, ubiquitous private tutoring centers in Asian neighborhoods testify to a critical mass of parents who believe that time and effort are key prerequisites of success. Rather than praising their grit and growth mindset, some professors complain behind closed doors that Asians study “too much.” (An ironic complaint from professional scholars.) One factor is surely resentment of children who compete against their own. Tellingly, though, complaints about Asians are loudest from those who profess the greatest concern for social justice. If the “model minority” poses problems for preferred narratives then the model is a problem, not a paragon.
What solutions do I propose? That depends on the problem we wish to solve. If we must somehow confer STEM degrees on students with inadequate math preparation, there are at least two possibilities. One would be more of everything, even things that I just dismissed. Time and effort are essential for building crucial skills, so let us pour, time, effort, and money into remediation, tutoring, study groups, and mentoring. Abandon all notions of 4 year/120 credit degree plans, financially prudent student/teacher ratios, and adult autonomy. Likewise forsake curricular egalitarianism, and put students into different tracks according to their preparation.
Students in slow tracks would get levels of tutoring and mentoring that no pipeline program has yet contemplated. Their incentive to work would be Scandinavian levels of financial support, healthcare, and childcare for students and their extended families, so their entire effort can and must go into math and science. The conditions of the support would be class attendance, study group participation, and homework completion. On some level it would surely “work”, in that degree completion rates and measures of learning would improve, at least on the margin. It would be expensive, and I cannot promise that participants would develop self-reliance, but some people would learn more than at present.
If such outcomes do not justify the cost in dollars and lost autonomy within your value calculus, there is another option, one of ancient pedigree and perpetual utility: Gentleman’s C’s, passing grades given to avoid trouble. Where once we made this compromise to appease rich men demanding credentials for thick-witted sons, now we appease legislators with more democratic concerns. We can certainly give even more C’s, provided that we suppress questions about long-term consequences.
Maybe, though, the problem is not the rate at which students pass physics but the rate at which they change majors. That is a cheaper problem to solve, and if done sensitively it would earn us gratitude from colleagues in enrollment-starved departments. Perhaps if more people studied humanity in its full complexity, instead of physics in its reductive simplicity, fewer colleagues would have faulted my drafts for lacking feasible solutions to hard human problems. Ancient and modern texts, the central concerns of humanities scholars, show that human nature is the timeless root of countless problems.
However, the obstacle to redirecting students is not solely simple-minded physicists, nor left-wing diversity concerns. The right flank of the body politic has written off the arts, humanities and social sciences, and now prefers to see students in natural science and engineering, or perhaps business. Conservatives were not always so earthly, but William F. Buckley is long dead. As for centrists, they have no ideological aversion to arts and letters, but science and engineering are practical, especially in a pandemic. Never mind that crafting health policy requires understanding economics and human behavior as well as viruses and immunity. And university administrators, ever practical, would rather not de-emphasize science fields with lucrative research grants, nor see students delay graduation by starting anew in other majors. With no natural constituency for urging students to reconsider their major, the only reason to do so is that students might actually benefit.
So here I am, with too many students attempting to learn physics while floundering in math, and no comfort from any narrative on offer. I might blame K-12 teachers who failed to prepare them. They might, in turn, blame school administrators, and, in unguarded moments, parents. Not all floundering students are poor, but some are, so we might also blame poverty and whichever politicians and social structures you blame for not solving it. Thus we blame all of society. “I shouted out ‘Who killed the Kennedys?’ when, after all, it was you and me.”
However, unless you are also a physics professor, it won’t be you who has to teach physics to students with weak foundations. It will just be me standing there, using symbols that too many only half-understand. Most will eventually pass with partial credit for scratching out half-correct answers, we will give them degrees, and everyone will pretend that this is a sensible situation. In reality, it is a bandage on a deeper problem that started long before college, a problem of weak foundations that everybody knows but none can discuss.
Our inability to discuss it does not mean that we have missed its early origins. Social service agencies run TV ads begging parents to read to their kids. Everybody knows that early exposure to rich vocabulary and other cognitive stimuli predicts academic success later in life. We also know that when individuals beat the odds they credit positive influences early in life. However, nobody knows how to craft programs that make exceptions standard. We settle for pipeline programs that intensively mentor small numbers of possibly exceptional kids, and proclaim that we could make exceptions of all If Only We Had More Funding.
Meanwhile, we pass people along, so I still have too many students who never mastered basic math. I have to teach them advanced physics alongside the students whom I did not mention here, and pretend that we have time to do all of this sensibly. Maybe I should not try to solve this problem. Maybe I should just teach what I can, and spend my remaining time on my own Special Program. I could form a reading group with a few of the students whom this essay is not about, and together we could study Galileo in whatever language we are able to use.
Anonymous Professor is tenured but doesn’t want his name on an essay critiquing a quarter of the students in his classroom.