"But there's a very famous theorem in topology called the Jordan curve theorem. You have a plane and on it a simple curve that doesn't intersect and closes—in other words, a loop. There's an inside and an outside to the loop." As Riehl draws this, it seems obvious enough, but here's the problem: No matter how much your intuition tells you that there must be an inside and an outside, it's very hard to prove mathematically that this holds true for any loop that can be drawn.

In the course of teaching hundredsof first-year law students, Monte Smith, a professor and dean at Ohio StateUniversity’s law school, grew increasingly puzzled by the seeming inability ofhis bright, hardworking students to absorb basic tenets of legal thinking and toapply them in writing. He came to believe that the manner of his instruction wasdemanding more from them than their mental bandwidth would allow. Studentswere being asked to employ a whole new vocabulary and a whole new suite ofconcepts, even as they were attempting to write in an unaccustomed style and anunaccustomed form. It was too much, and they had too few mental resources leftover to actually learn.

To signify that an angle is acute, Jeffreys taught them, “make Pac-Man withyour arms.” To signify that it is obtuse, “spread out your arms like you’re goingto hug someone.” And to signify a right angle, “flex an arm like you’re showingoff your muscle.” For addition, bring two hands together; for division, make akarate chop; to find the area of a shape, “motion as if you’re using your hand asa knife to butter bread.”

Washington State mathteacher Brendan Jeffreys turned to gesture as a way of easing the mental loadcarried by his students, many of whom come from low-income households,speak English as a second language, or both. “Academic language—vocabularyterms like ‘congruent’ and ‘equivalent’ and ‘quotient’—is not something mystudents hear in their homes, by and large,” says Jeffreys, who works for theAuburn School District in Auburn, a small city south of Seattle. “I could see thatmy kids were stumbling over those words even as they were trying to keep trackof the numbers and perform the mathematical operations.” So Jeffreys devised aset of simple hand gestures to accompany, or even temporarily replace, theunfamiliar terms that taxed his students’ ability to carry out mental math.

In the field of topology, homology is the formal way to count holes. Homology associates to each shape an algebraic object, which can be used to extract information like the number of holes in each dimension.