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Marvin Minsky: What makes mathematics hard to learn? (2008)

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Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#151

Post by pdeuchler »

justinhj wrote: post #122
pdeuchler wrote: post #120

Earlier quoted context omitted.

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

If Richard Feynman taught our children physics that would be great too. Your bar is unrealistically high I feel. I'm pretty sure there's a large percentage of math teachers that want to teach math and enjoy teaching it. I'm not sure how much freedom they have to teach it their way though.

Obviously we can't all be taught by Feynman or Minksy, but given the most common answer to "What subject do you dislike the most" is "math" we probably need to do better. It's also a vicious cycle, we're raising our future math teachers to dislike math

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#153

Post by mindcrime »

marincounty wrote: post #119
kazinator wrote: post #62

Earlier quoted context omitted.

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 ave…

I now regret not taking calculus, but it's so much easier to learn something these days.

It's funny to hear you say that. I don't know how old you are, but lately I've been trying to teach myself some additional math that I never took before (like Linear Algebra) and refreshing on Calculus, and now - at the ripe old age of 42 - I swear I find it easier to learn this stuff than I did when I was in college 20 some odd years ago.

Maybe it's just a question of motivation, or maybe I have more context available now, or, hell, maybe I've gotten smarter... but whatever it is, I'm not going to complain, considering how you always hear about how our minds slow down as we age and things are supposed to get harder to learn.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#154

Post by samizdatum »

sedachv wrote: post #148
jrcii wrote: post #109

Earlier quoted context omitted.

>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.

> 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. The worst part is web development is a completely arbitrary made-up system without any abstractions you can learn and use in a different field of work. It does not even transfer to something as closely related as desktop/mobile GUI toolkits. I view "knowledge" like that a…

Really? Functional reactive programming, immutable data, unidirectional data flow, application state as a fold over a stream of actions... it seems like web development is seeing a really high uptake of valuable, transferable abstractions and ideas, far more than mobile GUI toolkits, for example.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#155

Post by theoh »

It has apparently been shown, years ago and on a limited sample, that avoiding arithmetic until 7th grade (in favour of general discursive education) can produce good results in terms of the ease and success of eventual learning of arithmetic: http://www.inference.phy.cam.ac.uk/sanjoy/benezet/

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#156

Post by sedachv »

samizdatum wrote: post #154
sedachv wrote: post #148

Earlier quoted context omitted.

> 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. The worst part is web development is a completely arbitrary made-up system without any abstractions you can learn and use in a different field of work. It does not even transfer to something as closely related as desktop/mobile GUI toolkits. I view "knowledge" like that a…

Really? Functional reactive programming, immutable data, unidirectional data flow, application state as a fold over a stream of actions... it seems like web development is seeing a really high uptake of valuable, transferable abstractions and ideas, far more than mobile GUI toolkits, for example.

JavaScript programming is, CSS and HTML not so much. I would also argue that the details of frameworks like Angular also fall in the "garbage knowledge" pile.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#157

Post by samizdatum »

sedachv wrote: post #156
samizdatum wrote: post #154

Earlier quoted context omitted.

Really? Functional reactive programming, immutable data, unidirectional data flow, application state as a fold over a stream of actions... it seems like web development is seeing a really high uptake of valuable, transferable abstractions and ideas, far more than mobile GUI toolkits, for example.

JavaScript programming is, CSS and HTML not so much. I would also argue that the details of frameworks like Angular also fall in the "garbage knowledge" pile.

Fair enough, you meant to refer to a subset of web development, rather than the entire field. It's a significantly weaker claim, though. I could pick out subsets of any field of work, and declare them non-transferable, and the details of most systems are going to be less transferable than the concepts undergirding them.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#158

Post by YeGoblynQueenne »

derefr wrote: post #115
YeGoblynQueenne wrote: post #107

Earlier quoted context omitted.

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 process…

Ah, nothing that complicated :) It's a data structure with named fields. Each field can be a different type and they are in specific positions in the data structure so that it forms a pattern with strongly typed components.

COBOL's type definitions are also patterns really - for instance you declare a field to be of type 9(7) which means a numeric field of length 7 and so on.

The whole setup gives you very fine control over the structure of your data. For example, you can describe lines of a document to be printed out with strongly typed fields to be filled in by your program, so a form basically.

It's also used as a template for documents like VSAM files. Those are flat-file databases and the COBOL program's working storage imposes structure on them.

Here's a small example:

01 FILE-RECORD.

__03 WS-ACCOUNT_____PIC 9(10).

__03 WS-SEPARATOR1__PIC X.

__03 WS-NAME________PIC S(25).

__03 WS-SEPARATOR2__PIC X.

[Underscores are formatting only, not COBOL syntax]

That declares a structure called FILE-RECORD with four named fields, one of numeric type with length 10, one of alphabetic type with length 25 and two of alphanumeric type with length 1. So that's one record in your file structure and you can fill it in with data from VSAM files and the like, then print it out, with the separator fields for formatting.

It's a strange thing, low-level and high-level at the same time, in a very nice way. It kind of grows on you. Well, it's grown on me anyway. Of course, YMMMV.

Btw, the numbers (01, 03) make the hierarchical relation between the root of the data structure and each of its fields explicit; I won't say I'm in love with that sort of syntax :)

Also, I'm a total no0b still, so I might have made some mistake above, apologies in advance if that is the case.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#159

Post by j2kun »

sn6uv wrote: post #106
j2kun wrote: post #56

Earlier quoted context omitted.

> Many students, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) High school geometry (at least in the US, and in my experience) is the worst of them all! Two-column proofs should be banned.

What's wrong with two column proofs? I think they're a great tool for getting into the absolutely rigorous mindset required for maths. Leslie Lamport even makes the case that professional mathematicians should use them to reduce publication errors [1]. My personal feeling (as a maths grad student) is that the utility of two column proofs depends on the field. For logic and algorithms sure but estimating integrals it…

I have never met a mathematician who feels that two-column proofs is the right way to teach high school geometry. Considering that little math taught in high school beyond basic algebra uses anything close to 'absolute rigor,' introducing rigorous proofs should be left for college when one actually has a reason to learn proofs for advanced mathematics (set theory, analysis, etc).

Geometry is taught correctly when proofs are used as a tool to convey a deep intuition about a mathematical pattern. Check out Lockhart's book [1] if you want to see what it's like when done right, though there are many more examples. Two-column proofs are simply a tool for the lazy/unknowledgeable teachers to fill a geometry class.

I am aware of Lamport's work, and it's a specific tool for a specific subfield in which there is a plethora of false results. Ignoring the fact that most of theoretical computer science research and most of math research more does not fall into the category that Lamport is critical of (distributed computing), these temporary issues about rigor in academic publishing should have no effect on high school pedagogy. Instead, we should listen to the world's finest math teachers, who pretty much all agree that two-column proofs are awful. A quote from Lockhart [2]:

> Geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.

[1]: http://www.amazon.com/Measurement-Paul-Lockhart/dp/067428438... [2]: https://www.maa.org/external_archive/devlin/LockhartsLament....

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#160

Post by j2kun »

anon4 wrote: post #71
j2kun wrote: post #56

Earlier quoted context omitted.

> Many students, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) High school geometry (at least in the US, and in my experience) is the worst of them all! Two-column proofs should be banned.

Two-column proofs I had never heard of that monstrosity before and having googled it, it looks worse than programming in COBOL.

Then be thankful you weren't subjected to a full year of two-column proofs as your daily routine.
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