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(90% chimp + 10% bee) [1]
The mental habits needed to learn mathematics are not qualitatively different from the learning strategies crucial to success in other fields. Aware of this fact, an instructor might be tempted to settle for cultivating domain-specific knowledge, leaving it up to other departments (or the students themselves) to make use of those parts of mathematics they find applicable. Rather than come down squarely on one side of the pure math/applied math divide, I borrow from the Dogme approach of English language teachers and let the students' interests dictate which scenarios (if any) our class should use to illustrate course content. In one of my first sections of finite math, for example, the day jobs of one student (in real estate/construction management) provided the setting for writing the standard form equation of a line and interpreting its coefficients.
Coming up with relevant examples based on experiences shared by only one or two students runs the risk of leaving behind the rest of the class. In anticipation of this outcome, I distribute more broadly the task of holding all students accountable, with first-day exchanging of contact information, opportunities to collaborate on note-taking, and group problem-solving sessions (both graded and not graded). These smaller collaborative units reinforce the message that mathematical knowledge is acquired by tapping into our "hive instinct"--a legacy of humanity's evolution that does not receive nearly enough attention in the mathematics classroom.
Our students are probably aware of the inheritance they've received from generations of thinkers before them, as Yuval Harari points out in a 2017 book review [2]:
Sloman and Fernbach take this argument further, positing that not just rationality but the very idea of individual thinking is a myth. ... No individual knows everything it takes to build a cathedral, an atom bomb or an aircraft. What gave Homo sapiens an edge ... was not our individual rationality, but our unparalleled ability to think together in large groups.
What gets lost in this attribution are the unrecognized instances of distributed thinking even in the day-to-day tasks that students traditionally associate with individual effort. WEIRD countries especially (Western, Educated, Industrialized, Rich, Democratic) suffer from a tendency to downplay humanity's instinct for tribalism, except in some sanctioned domains like professional sports and organized religion. Only those students with military service on their resumes are able to give first-hand testimony to the amazing accomplishments that emerge when a collective of average individuals dedicates its resources to a common purpose. [1]
Tragically lost when we moved all our classes fully online were the impromptu triggers of this "hive switch" we could employ when the students' dedication needed a boost. Having all the students engage in synchronized movement, or repeat a mnemonic device out loud, would forge bonds that keep them invested in the class and eager to come back the next week. Such emotional attachments might be possible to build over Zoom, but if so they are much more fragile than their in-person counterparts.
To quote the African proverb, "If you want to go fast, run alone. If you want to go far, run together." This wisdom applies no less to learning mathematics than to cultivating a field of rice, yet I constantly encounter students who dismiss study groups as a waste of their time. At least with in-person classes I could use kinesthetic learning activities and jigsaw readings to reduce the students' wariness of their classmates, and the resistance to study groups would melt away. On Zoom it's just too easy for a student to turn off the camera or to step away (citing household obligations).
Another downside of conducting classes online: "the Internet is so quick that we can confuse its memory for our own" [3]. To push back against students' unrealistic appraisal of their long-term memories, I emphasize the embodied nature of our intellect and ask that students slow down their digestion of new material with analog methods like handwritten notes, physical models, forest bathing, and other corporeal involvement. My statistics class this semester took a keen interest in the empirical basis for this recommendation [4], [5], [6], adjusting their study habits accordingly and with noticeable effect on their subsequent performance.
[1] Haidt, Jonathan. *The Righteous Mind*. New York: Vintage Books, 2012.
[2] Harari, Yuval. "People have limited knowledge. What's the remedy? Nobody knows." New York Times, 18 April 2017.
[3] McBain, Sophie, "Head in the cloud.", New Statesman, 23 February 2016.
[4] Jabr, Ferris, "Why the Brain Prefers Paper." Scientific American 2013, Vol. 309(5), 48--53.
[5] Mueller, Pam A. and Daniel M. Oppenheimer, "The Pen is Mightier than the Keyboard: Advantages of Longhand Over Laptop Note Taking." Psychological Science 2014, Vol.25(6), 1159--1168.
[6] Bui, Dung C. and Joel Myerson, "The role of working memory abilities in lecture note-taking." Learning and Individual Differences 2014, Vol. 33, 12--22.