The following article was initially drafted with a guest author, Kirill Peretoltchine, at the end of July 2012.
A giant statue in the opening ceremony of the Athens Summer Olympics in 2004, onto which laser images of geometrical shapes and scientific concepts were projected, was a powerful reminder of a bygone era. Ancient Greece was a birthplace of logical thought, education, mathematics, science… and democracy.
The Renaissance was marked by an explosion in the diffusion of ideas, and the naissance of the scientific method that has allowed us to explore this world. This was the time of Copernicus, Galileo, Michelangelo, and da Vinci — the last of which, far from being just a scientist and artist, was also an engineer and writer: the stunning definition of a Renaissance man.
And one of the founding fathers of the United States of America, Benjamin Franklin — also the founder of our alma mater — was a polymath himself. Politician, scientist, writer…
There is a reason we honour and respect figures like da Vinci and Franklin, even if we, enlightened with 21st century practicality, do not expect to educate the entire populace in their image.
We agree that there are serious deficits in the North American educational system that are in need of redress. We also concur that it is impractical to teach higher math effectively to every high school and college or university student. But we are firm in our belief that lowering the bar isn’t the answer. Andrew Hacker has a limited view of mathematics that fails to appreciate its value, and his solution of removing math from standards is flawed.
Math isn’t just for STEM
It’s true that the most visible applications for higher level math are in STEM (Science, Technology, Engineering, and Mathematics). A seemingly simple statement like , which even a typical 8th grader should be able to grasp as an algebraic equation with four variables, has bountiful implications in biology and chemistry.
But math has its uses outside of STEM, too. On the simplest levels, arithmetic serves practical purposes in life: personal budgeting, figuring out how much your bank is gouging you on your mortgage, calculating your smartphone bill, and so on. Basic algebra might be useful in, say, preparing for a road trip — how far will you get before your tank of gas runs out? Calculus finds applications in economics, financial modelling, etc. Take out calculus, and there go derivative pricing models. Probability/statistics seem to have even broader implications: they form the bases for all kinds of “analyses”, from presidential elections to life insurance.
None of these examples are STEM applications. We don’t need to rely on standard arguments, like the contention that theoretical calc and linear algebra, detached from physical application, develop certain modes of thinking.
Indeed, the following recording illustrates what happens when people fail at basic numeracy. Is this irrelevant to daily life? We really don’t think so.
Frederick: better pedagogy
The importance is that, when courses are taught at the general level (i.e. K-12), instructors should make the effort to connect seemingly useless proofs like with applications. Some teachers call this “motivating the lesson”, and the best ones always find something for the students to grasp onto, whether it’s a municipal water tower leaking onto the streets or optimizing the speed at which you drive a 10-year-old truck for fuel efficiency.
Of course, this applies not only in math—if the justification that math builds logical thinkers isn’t sufficient for mandating this education (as it generally isn’t), then an English teacher, too, should justify how analyzing Shakespearean literary themes or postmodern novels will help a student who hopes to be a biology researcher or a future nurse. There is room for pedagogy in all fields.
The lowest common denominator?
It is an uncontroversial position that statistics find broader uses in non-STEM fields than, say, calculus or linear algebra (here at Penn, statistics are taught in the business school in their own department independent of math). Yet, as some commentators point out in the NYT comments section, one cannot comprehend how standard deviation works without algebra skills. (How does make sense without understanding algebra?) Take out an algebra foundation, and all one becomes is an automaton applying an equation that someone else supplies.
Take out the algebra foundation, and quite possibly the student will never have what it takes to decide to become a chemist, an engineer, a financier, or a Nate Silver. We should retain such foundations because they enable decisions that would not otherwise be possible.
Yet the op-ed chooses this example to support its argument:
“… a definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or above.”
The system shouldn’t be designed to push kids into “entry-level worker” positions. That would be defeatist. It would mean responding to challenge by backing down—an approach that a recent NPR piece attributes to the North American perspective that “intellectual struggle in schoolchildren is … an indicator of weakness”.
Even if we accept that some people struggle with math and won’t use it, we should look for other ways to address this perceived problem.
Alternative Solutions for Reform
It is an easy matter to disagree. It is a far worthier challenge to find an alternative — a superior solution.
Kirill: “A German System”
The importance of math is difficult to gauge on an absolute scale; there are some professions for which it is directly applicable, some for which it helps greatly, and finally some for which, we are forced to admit, math is not particularly useful.
The fields that fall into the last category are those that, even today, do not explicitly require a college education, or even necessarily a secondary education. Workers like janitors, garbage handlers, and truck drivers comprise a necessary part of society — but whether they receive education, and to what degree, must be left to both individual choice and utilitarian economics. Indeed, an economic argument can be made that we waste resources teaching integrals and magnetic flux to students who neither appreciate that education nor require it for physical labour.
In this sense, we find ourselves accepting some of Hacker’s arguments. But these workers have no less need for math than they do Shakespeare or watercolour paint.
To be clear, we see enough benefit from teaching math to future businessmen, philosophers, historians, and artists that we reject optionalizing math in high school (or any earlier) based on intended field of study. We might, instead, think about changing the broader system and on a different scale, rather than attacking math itself.
A solution that we can propose, then — at least for secondary education — is to have a stratified education system, where students are separated by learning class into distinct groups. One such group (admission into which must occur by testing or another means of assessment) will continue to post-secondary university studies, while another group will instead pursue vocational training specific to their intended career, and bypass the (unnecessary for them) ordeal of learning advanced mathematics and sciences.
To summarize the suggested educational framework, here are a few key elements that we feel should be present:
- A strong standardized testing system to separate students into appropriate groups at entry. This must happen sufficiently early for the differences in curriculum to have an effect.
- Similar exit testing to differentiate members of society who have completed varying levels of secondary education.
- The ability to move between schooling levels.
If this so far sounds like a Brave New World-style dystopia, it really is not without real world foundation. The relevant example in secondary education today is the German school system, which comes closest to accomplishing these targets in an effective way.
After the completion of primary school (age 10), students can choose from the following options:
- Gymnasium/Gesamtschule: a type of school with a strong academic focus. Upon completion of an exit exam after grade 12 or 13, students graduate with an Abitur, which qualifies them for admission into university.
- Realschule: an “intermediate” school, with graduation after 10th grade. Graduates are awarded the Mittlere Reife, which is similar to the American high school diploma. After receiving this degree, many go on to vocational school, while others (who qualify) may attend additional schooling to receive an Abitur if they so choose.
- Hauptschule: the lowest level of secondary education. Students receive a diploma after 9th or 10th grade, after which they may enroll in a vocational school or begin part-time work/training. In some regions, different “levels” of the 10th grade are offered: completion of the higher level allows the student to receive the Mittlere Reife, the degree attained by Realschule graduates.
The beauty of the German system is that it allows for differentiation, but does not preclude the possibility for movement between educational levels. This differentiation has two clear goals:
- An optimized learning environment. Students who know from an early point in their life that they do not want to perform a highly specialized and academic job need not trouble themselves with unnecessarily “difficult” educational goals. This addresses the issue that the NYT op-ed brings up: failure rates for advanced classes will be lower if the population of students taking them is restricted to the academically capable and motivated. Students who wish to pursue this lower level of education will be grouped together with those of similar interests and goals, and will likely benefit from such an environment.
- Specialization of resources. It’s hard for a teacher to take into consideration such a wide spectrum of academic abilities as we see in the classroom today. This is inefficient for students on both ends of the bell curve: the more advanced students feel that the class is moving too slowly, as teachers are forced to move at the pace of the lowest common denominator, while those who are not quite up to academic standards may feel intimidated or discouraged by the successes of those at the top.
Detractors may question if this systematic differentiation would reduce social mobility long after the attainment of secondary education — in other words, if one would be limited by having “only” obtained a degree from the Hauptschule — but we see no reason that it would be any worse than what occurs in America when a teenager drops out of high school. Instead of setting just one standard that may be unattainable for many, this stratified educational system provides multiple paths to accommodate students’ inherent differences, and encourages them to exit secondary education with some indicator of achievement.
In our perspective, this is superior to the rigid inflexibility that North America affords at-risk high school students, and can only create more gradations to help those not currently served by a one-size-fits-all educational system.
The scope of this post has far exceeded a simple rebuttal to the question, “is algebra necessary?” Indeed, the answer to that problem cannot be restricted to a confined study of math — it must be in the context of other disciplines and the broader educational system.
We see value in mathematics across the board, from arithmetic to algebra, from calculus to statistics. Not everyone needs to understand moment-generating functions or higher-order differentials. Rather than cutting math because it seems to confound so many students, we instead offer actual solutions: first, on the smaller scope, a pedagogical emphasis on motivating learning, and second, more broadly, a stratified structure for secondary education; in both cases, we believe we can better provide everyone with the tools they need to thrive.
The authors, Frederick Ding and Kirill Peretoltchine, are undergraduates, respectively in the School of Engineering and Applied Science and the Wharton School, at the University of Pennsylvania. Both use math on a daily basis.