''I do not know everything; still many things I understand.'' Goethe
Observations by me and others of our tribe... mostly me and my better half--youngsters have their own blogs
Observations by me and others of our tribe
... mostly me and my better half--youngsters have their own blogs
Anybody want to try to replicate the experiment?
I suppose by symmetry I should be willing to try to become fluent in Arabic (I just didn't have time to spare) or figuring out which colors go with which.
I've been reading her book, A Mind for Numbers, mostly on the basis that I am a lot better at pure math than I am at anything applied.(The jury is still out. It's very self-helpy in tone, but I'm finding some helpful tips.)
It's the reason that evidence and proof are not the same thing. Her personal example is indeed evidence for the theory of neural plasticity. However, it is also powerful evidence for a purely genetic theory of math/spatial ability, which was preserved into adulthood despite neglect and poor training in early years. Because her story is a single thread we can't know many possible variables. Coming from a literary, wordplay family, she may have picked up a sense that it was useless, beneath her, not-her-tribe, or even contempt for math. Such things happen. She may also only be remembering herself as incompetent in comparison to a much-greater verbal competence. Compared to classmates she may have looked better, but scored poorly because of laziness or disinterest.I don't claim that this second theory is more likely. I don't have the slightest idea - only that it is possible. What her story does speak against strongly is the theory that there are early critical periods for learning math, that if missed, are forever lost. There actually is something like this for languages, which everyone has known from simple observation for years. Younger children learn languages more easily, and learn them best in experiential, not schoolbook settings. Which is why we wait until children are 14 to teach them languages from a textbook and expect them to become fluent.
Quite true. We've no evidence to distinguish low ability from poor training. (My wife execrates her high school algebra teacher.) A third option is that she was simply not ready for that type of rigor at the year when the school schedule called for it.
From the link:Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.Lectures of STEM subjects deal with material that is often complex and difficult- material that needs time and practice to sink into a student's mind. Very seldom does a student have a grasp of the lecture's material at the end of the lecture. For example, while a thermodynamic equation may be algebraically simple,the ideas behind a thermodynamic equation are anything but simple, and take time and practice to achieve fluency or understanding. I add "time" because often simply sleeping on something can aid understanding or fluency- IF one has already put in the effort.
Yes. I remember listening with interest and "perfect understanding" as my statistical mechanics teacher explained various concepts. I was busy, and didn't tackle the homework in time. The quiz was "The atomic number of copper is 29. Calculate its specific heat." I "knew how" on Monday, but without that practice it hadn't sunk in. (I've since forgotten how to do it.)
jamesWe've no evidence to distinguish low ability from poor training. (My wife execrates her high school algebra teacher.)I had interesting experiences with "bad" teachers in high school. Before I started high school, my chosen career would have been historian or political scientist, an opinion which didn't change in 9th grade. As a result of bad experiences with the teacher I had for history in 10th and 12th grades, I lost interest in history or political science as a career. My 9th grade math teacher was a family friend of sorts- she came to my mother's memorial service. Unfortunately, she was a poor teacher. This was not due to lack of intellect, as she was a Phi Beta Kappa graduate of a Big Ten school. I ignored her in class and taught myself from the textbook, a New Math effort. While I was a good student in Math before high school, it bored me. The proofs that New Math required kindled my interest in Math, which became both easy and also fun for me. I conclude from this that while I had bad teachers in math and in history in high school, I was better able to surmount a bad teacher in math because I had greater inherent ability in math. While I very much liked the New Math, liking New Math was confined to the upper echelons of the class. Not every one likes to write proofs.A family friend who was an engineer blamed that particular math teacher for killing his daughters' interests in math. His daughters had that math teacher in junior high, a younger age which is less able to surmount a bad teacher than someone of high school age. The engineering professor at the link didn't realize she had abilities in STEM until she looked at STEM in a different way: becoming familiar with the material, just as with a language. But yes, she had some inherent abilities- which had previously been untapped. Her story about picking up Russian again after 25 years reminds me of my experience with math proofs. I wrote math proofs in high school from 9th grade on. I saw little to nothing of math proofs in college until I took a Linear Algebra course a quarter century after graduating from high school. The course involved a lot of proof writing. I found out that like riding a bicycle, one can resume writing math proofs after a long hiatus.
FWIW, I read Asimov's books on Numbers and Algebra before high school. They were quite lucid, and I found Algebra quite straightforward. Geometry was proofs back then--that was also clear sailing.It hasn't all been clear sailing--e.g. I've tried several times to get a handle on category theory. Maybe if I spent a month at it without distractions :-( Or maybe not; I'm not as young as I used to be, and mathematicians usually do best when young.
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