Roger Penrose writes about how fractions are incorrectly taught in school in the preface of "Road to Reality" always the pie visual is used as an approximation which later confuses students.
Marvin Minsky: What makes mathematics hard to learn? (2008)
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Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#112This is interesting. I volunteer at a homework-help clinic for public housing kiddos. They've heard "the answer is right but you got there the wrong way" a lot. It's frustrating to work with the kids, because they say "if I do it that way, the teacher will say I'm wrong."
And, I basically don't have a clue about how they're being taught to do this stuff. Any sane person turns 2 x 15 into 2 x 16 - 2 or 10 + 10 + 5 + 5.
But this Common Core arithmetic is not quite sane. The intention is to teach reasoning skills, but the reality is that some curriculum-writer's idea of reasoning skills has become codified. Grumble.
You'll be missed, Dr. Minsky. So long, and thanks for all your teaching.
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#113It's hard to know exactly what would work for ALL students. Honestly, for me, the issue is "one size fits all" that the vast majority of our students get. I learned math just fine the way it was taught. Sure, it was boring at times, and I was mathematically inclined, but it worked for me. For others, it clearly doesn't work. We have to ask ourselves, what exactly do we want people to know who don't want to learn math…
Back when I was in high school, they used to break my class into to groups: those who study German and those who study English. I was in English class. I don't see why this won't work for mathematics. Break students into two groups: ones that never question anything and those who can't move forward until their why questions are answered in a satisfactory manner. The first group can study precalculus, computational analysis, matrix methods....basically anything that can be done by rote. The second group could be introduced to proof writing, beginnings of set/category theory, abstract algebra/topology, statistical inference...
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#114Roger Penrose writes about how fractions are incorrectly taught in school in the preface of "Road to Reality" always the pie visual is used as an approximation which later confuses students.
I remember asking my 4th grade teacher what the difference between fractions and division. She clearly didn't understand what I was asking. That was the moment I started seeking my own answers and stopped trusting my teachers blindly.
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#115Earlier quoted context omitted.
Two-column proofs I had never heard of that monstrosity before and having googled it, it looks worse than programming in COBOL.
Programming in COBOL is vastly underrated. If you're not speaking from experience, Google "working storage" (one of COBOL's nicest features) . Also: it runs on machines that look like Cylons. You can't beat that. :p
I can't quite figure WORKING-STORAGE out from the docs I found, maybe because I'm not sure what a "run unit" is—I'm assuming something like a Linux process group/session?
To contrast, LOCAL-STORAGE seems like a regular global variable (.DATA/.BSS) segment. So is WORKING-STORAGE basically like a set of seeming global variables, that actually point into (the equivalent of) a shared mmap'ed file, such that other processes in the same session/pgroup end up mmap'ing the same file and thus sharing memory that way, without having to clone around the memory handle itself via forking/threading/IPC?
In this case, it sort of reminds me of the OS-level version of an anonymous Erlang ETS table held by a supervisor and passed to all the worker children to manipulate.
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#116I had dinner with Prof Minksy, about 15 years ago. He asked what I was working on (evolutionary algorithms) we got talking about domains of knowledge. He said all knowledge has a half-life, the time it takes for half of what you know to be redundant or wrong. Math, he said, is the longest, measured in centuries or millennia. One should feel sorry for neuroscientists: they can go to the bathroom and half their knowled…
This is super interesting. I tried to look for any reference of this, but found nothing. Do you happen to know if this thought of his is available somewhere, in a book, a paper, etc? Or is it a concept borrowed from someone else?
There is a wikipedia entry on the concept:
https://en.wikipedia.org/wiki/Half-life_of_knowledge
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#117I had dinner with Prof Minksy, about 15 years ago. He asked what I was working on (evolutionary algorithms) we got talking about domains of knowledge. He said all knowledge has a half-life, the time it takes for half of what you know to be redundant or wrong. Math, he said, is the longest, measured in centuries or millennia. One should feel sorry for neuroscientists: they can go to the bathroom and half their knowled…
>neuroscientists: they can go to the bathroom and half their knowledge will be out of date Web development seems to be even worse than neuroscience in this regard. I feel like I have to perpetually study to be abreast of best practices.
Well, at least the languages you use stay pretty much the same? Javascript, HTML, and CSS are always going to be there. Although.. I suppose that could be considered both good and bad..
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#118My father made it memorization a fun quiz game. Maybe a couple times a week he'd ask whats "5 times 7" and so on. Or how to spell a word. There was no penalty for getting something wrong. You just wanted to learn the right answer to please your father and yourself.
> Maybe a couple times a week he'd ask whats "5 times 7" and so on.
Ah, the Henry Jones Sr. approach to parenting! I'm a fan.
Young Indy: I have to show you something!
Professor Henry Jones: It can wait. Count to twenty.
Young Indy: No, Dad, I-!
Professor Henry Jones: Junior!
Young Indy: One, two, three, f...
Professor Henry Jones: In Greek.
Young Indy: [rolls his eyes] Ana, theo, thea...
(from IMDB)
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#119Earlier quoted context omitted.
Disagree. Kids disengage from math for many different reasons. There is no single pattern to it. If it were that simple, it would not have remained an issue since schools were commonplace. Minsky solidly makes the point that when math is taught as nothing more than an endless series of tedious drills with no purpose in sight, of course it demotivates kids. No one likes pointless Sisyphean tasks. Many students, howeve…
Humanity as such absolutely loves pointless repetition, and abhors thinking about something hard, which involves focusing intensely on one subject in which where no repeatable progression of simple successes is taking place, leading to a mounting sense of discomfort. Just look at various cultural traditions, popular (and even serious) music, crafts, sports, ... sneeze ligion. Mindless repetition is everywhere. The sa…
For myself, I found math difficult. Never understood, even now, the need to spend so much time memorizing multiplication tables.
That said, I started over in a community college, and finished up to trig. with ease. I found trigonometry, surprisingly straight forward. I went on to finish a year of physics. I didn't need calculus for my major, and heard it was really hard. Plus, I didn't want to ruin my grade point average with a class I didn't need. I now regret not taking calculus, but it's so much easier to learn something these days.
My struggle with math was two fold. I didn't care in high school, and I never had a firm grasp on basic math. I look back , and once I truly inderstood basic math, especially fractions, and percentages; it all became so easy.
I see a lot of kids still struggling with math. They get pushed along, and avoid any class that has math in it. I do blame our U.S. teaching system in this case. Keep teaching basic math until the kid can teach it to to their classmates.
Then, and only then move on to algebra, and trig? As to calculus, I didn't take it so I won't comment, but when I was applying to health professional schools, I didn't find one that required Calculus. Probally for good reason. My physics classmates biggest, silent stumbling block was they were horrid in math, and looking back--they just didn't know the basics, so every step up the ladder became more mysterious. Most of my classmates seemed to the solving problems in physics by memorizing the homework; not truly understanding the problems.
Re: Marvin Minsky: What makes mathematics hard to learn? (2008)
#120While kids may find arithmetic to be boring and demotivating, they're able to do it. It's when math gets symbolic that you get the permanent attrition. For instance, most people never get to calculus. When you progress from arithmetic to abstract, it hardly gets easier . Moreover, memorizing. You go from memorizing multiplication tables to other kinds of tables, like tables of equations giving various identities, row…
Potentially unpopular opinion: Maybe our math teachers just suck? If we all had Marvin as our professor I doubt attrition would be as high, whereas most High School math teachers are simply analytically inclined people who realized they had better job security teaching math than english