7 Matching Annotations
  1. Aug 2023
    1. "My teachers really gave me a glimpse of what math was like at the college level, the creative side of mathematics as opposed to the calculational side of mathematics.

      creative mathematics versus computational or calculational mathematics...

      we need more of the creative in early education

      partial quote from Emily Riehl

    2. "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.

    3. In an article she wrote recently for Scientific American, Riehl quoted John Horton Conway, an esteemed English mathematician: "What's the ontology of mathematical things? There's no doubt that they do exist, but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, despite having been a mathematician all my life. How can things be there without actually being there?"
    4. But then, so are numbers, for all their illusion of concrete specificity and precision.

      Too many non-mathematicians view numbers as solid, concrete things which are meant to make definite sense and quite often their only experience with it is just that. Add two numbers up and always get the same thing. Calculate something in physics with an equation and get an exact, "true" answer. But somehow to be an actual mathematician, one must not see it as a "solid area" (using these words in their non-mathematical senses), but a wholly abstract field of abstraction built upon abstraction. While each abstraction has a sense of "trueness", it will need to be abstracted over and over while still maintaining that sense of "trueness". For many, this is close to being impossible because of the sense of solidity and gravity given to early mathematics.

      How can we add more exploration for younger students?

  2. Jan 2023
    1. McCoy, Neal Henry. The Theory of Rings. 1964. Reprint, The Bronx, New York: Chelsea Publishing Company, 1973.

  3. Jul 2022
    1. Even physicists,when they leave equations behind and try to describetheir discoveries to the rest of us in plain English, findthemselves employing analogies, metaphors, and theother language tools we all use

      Within mathematical contexts one of the major factors often at play is the idea of abstraction: how can one use a basic idea and then abstract it to other situations to see what results.

      The idea of abstraction in mathematics is analogous to analogy and metaphor in literature.