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Mastering math and science through mechanistic thinking

Throughout high school, my teachers emphasized that a deep understanding of history demanded that students be able to explain how and why events transpired as they did. From a more pragmatic perspective, a C history essay detailed what happened. A B essay explored how events happened. And an A essay—obviously what I was shooting for—argued both how and why history unfolded as it did. The scheming cynic in me simply ensured that “because” appeared in every thesis statement I wrote for a history class in high school, for then I would always be addressing the why. However, this lesson extends far beyond the history classroom. Understanding both the how and the why aids learners in mastering multiple subjects, and it’s something that I strive to implement in my teaching style.

The human brain favors simple, unifying explanations for disparate facts to compensate for our inherently limited mental capacity and bandwidth. In many cases, this hardwired tendency is adaptive. For example, humans are exquisitely sensitive to visual stimuli that resemble snakes, likely because our distant ancestors used to be their prey. In fact, when blurred photos of different animals are slowly clarified, humans accurately identify snakes before any other animal (Kawai and He, PLoS One 2016). In other circumstances, this propensity for simplifying assumptions is harmful, as with stereotyping. Nevertheless, neuroscientists have repeatedly demonstrated that a coherent narrative enhances our ability to recall past events (Cohn-Sheehy et al., Current Biology 2021). Simply put, humans remember stories that make sense. My own experience as a student and tutor has convinced me that we can leverage this innate inclination to become better learners. By understanding the inner mechanisms of an event, concept, problem, or technique, we can more easily and more durably master it.

One of my favorite examples of this type of thinking comes from the realm of world history in Jared Diamond’s Guns, Germs, and Steel: The Fates of Human Societies. Here, Diamond argues that geography accounts for why technological advancement and human disease evolved more rapidly in Eurasia than in Africa or the Americas. He begins with the straightforward observation that Eurasia exists on an east-west axis, whereas both Africa and the Americas exist on a north-south axis. Because regions at the same latitude tend to have similar climates, crops and domesticable animals spread more easily along east-west axes than north-south axes. By sheer luck, Eurasia ended up with the animals and crops best suited to domestication—wheat, rice, cattle, pigs, chickens, horses, goats, and sheep. Compare this to North America’s native turkeys and South America’s potatoes, guinea pigs, alpacas, and llamas. Because of the impassable jungles of the north-south isthmus connecting the Americas in today's Panama, trade and the exchange of domesticable animals largely did not occur in the Americas in the pre-Columbian era. Only partly domesticated maize seems to have been able to penetrate this geographic obstacle, where its domestication was completed independently in North and South America (Kistler et al., Science 2018). In Africa, humans have attempted to domesticate zebras on multiple occasions yet failed each time. Despite their resemblance to Eurasian horses, zebras appear to be resistant to domestication.

Because of the ease and variety of domesticable foodstuffs in Eurasia, humans transitioned from a nomadic hunter-gatherer lifestyle to a sedentary agricultural one earlier in these regions of the world. The rise of agriculture led to a food surplus and enabled certain members of society to specialize into non-food-producing roles, leading to more complex social hierarchies. Intimate contact between humans and domesticable animals spurred epidemic disease due to interspecies transmission. For instance, measles is derived from bovine rinderpest virus, and the devastating smallpox virus is believed to have arisen from contact with camels (Babkin and Babkina, Viruses 2015). This trend continues in the present day, in which novel animal-derived viruses tend to arise among those in close contact with domesticated animals. As a result of Eurasia’s superior domesticable crops and animals and legacy of exposure and acquired immunity to lethal infectious disease, humans from this landmass won out when the continents collided. Diamond’s argument is both memorable and compelling because it invokes the simple mechanics of geography to craft a coherent narrative about the history of humanity.

When teaching math and science, I attempt to emulate this sort of first-principles reasoning. I challenge my students to understand not only how to solve problems and answer questions but also why their method works. For example, a few nights ago, when working with a student on probability, she and I encountered a problem that asked how many three-letter “words” were possible using the standard English alphabet if no letter could be repeated. She answered correctly: 26P3 = 26 × 25 × 24 = 15,600 words. "Great," I said, "but why did that work?" She couldn’t say, so I drew it out: “Imagine there are three slots for each of the letters in your hypothetical ‘word.’ For that first slot, you have 26 options, any of the letters from A to Z. But once you make the selection for your first letter, you have eliminated that letter from the running. For each of the 26 first-slot options, you have only 25 letters remaining for the second slot and then 24 for the third.” Her eyes lit up. And for the remainder of the session, she adopted this more intuitive approach and excelled. She earned an A on her test the next day.

This emphasis on intuitive reasoning over rote memorization also works well in the natural sciences. Organic chemistry intimidates the many students who view it as a laundry list of reactions that must be committed to memory. When I took the course as a Yale freshman, I instead framed it as a protracted exercise in understanding chemical behavior. Chemistry, particularly organic chemistry, boils down to the movement of electrons. Comprehending how different atoms and functional groups interact with and attract electrons allows the chemist to predict how molecules will behave. My organic chemistry courses at Yale reinforced this approach by firmly grounding the material in reaction mechanisms. Whenever we learned a new reaction in lecture, our professor would map out the step-by-step mechanism using curved-arrow notation. (Here, curved arrows indicate the movement of pairs of electrons among or within molecules.) Organic mechanisms are chemical narratives, allowing the chemist to peer into the inner workings of a reaction. Through a deep understanding of reaction mechanisms, I found that I could recall many reactions without spending hours memorizing them. I was also better prepared for exam questions that challenged me to apply my conceptual knowledge to unfamiliar reactions. Remarkably, much of this knowledge has stuck with me to the present, likely because I was able to develop an intuition for it. I aim to pass on this proven approach to organic chemistry to my students.

Many students struggle with the natural sciences and mathematics because the material is often presented as intangible, almost alien. History, literature, and the social sciences may come more naturally. Humans are social creatures. We are programmed to recognize emotions, relate to others, and understand motives. We excel at following plot lines and spotting holes or inconsistencies in a storyline. While we may never live in Ancient Rome or hear our kingly fate foretold by a trio of witches, we somehow can still relate to that experience. In my approach to tutoring math and science, I strive to capitalize on our inherent aptitude for narrative thinking, our hankering for why things are the way they are. Of course, numbers and atoms have neither emotions nor motivations. Yet, like human characters, they do behave in predictable ways. By unraveling these tendencies and understanding the principles that underlie them, we can optimize our study habits and mastery of the natural world around us.


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