 Aug 2023

hub.jhu.edu hub.jhu.edu

"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.
How does one concretely define "inside" and "outside"? This definition is part of the missing space between the intuition and the mathematical proof.

 Apr 2022


In the course of teaching hundredsof firstyear 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.
This same analogy also works in advanced mathematics courses where students are often learning dense and technical vocabulary and then moments later applying it directly to even more technical ideas and proofs.
How might this sort of solution from law school be applied to abstract mathematics?
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 Mar 2022


To signify that an angle is acute, Jeffreys taught them, “make PacMan 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.”
Math teacher Brendan Jeffreys from the Auburn school district in Auburn, WA created simple hand gestures to accompany or replace mathematical terms. Examples included making a PacMan shape with one's arms to describe an acute angle, spreading one's arms wide as if to hug someone to indicate an obtuse angle, or flexing your arm to show your muscles to indicate a right triangle. Other examples included a karate chop to indicate division or a motion imitating using a knife to butter bread to indicate finding the area of a shape.

Washington State mathteacher Brendan Jeffreys turned to gesture as a way of easing the mental loadcarried by his students, many of whom come from lowincome 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.
Mathematics can often be more difficult compared to other subjects as students learning new concepts are forced not only to understand entirely new concepts, but simultaneously are required to know new vocabulary to describe those concepts. Utilizing gestures to help lighten the cognitive load of the new vocabulary to allow students to focus on the concepts and operations can be invaluable.

 Dec 2021

www.quantamagazine.org www.quantamagazine.org

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.
A relatively simple definition of homology and what it is.
