## 1488: "Flowcharts"

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Quercus
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### Re: 1488: "Flowcharts"

orthogon wrote:Whilst we could debate just how hard or easy certain concepts are, I think we could probably agree that mathematics, and to a lesser extent natural sciences in general, are viciously cumulative. Properly understanding one thing requires a pretty thorough grokking of the principles on which it builds. This for me is a very strong argument for that other unfashionable idea in education: ability-based streaming. Time spent teaching concept B to a student who doesn't get the prerequisite concept A is almost completely wasted. (This might not always be quite the whole story. Many concepts have some kind of circular dependence. Normally you'd think of matrix arithmetic as a prerequisite to geometrical transformations, but it's possible that somebody might suddenly "get" matrices as a result of learning about transformations. But even there there's a kind of cumulative effect resulting from iteration: by going back and forth between two topics you can make a small progress in your understanding of each in each iteration,

I'm in total agreement with this - as long as the streaming is done per subject, and it's made relatively easy to move between streams if appropriate, I think the upsides far outweigh the downsides, at least for maths and science.

like climbing up the inside of an alcove by moving your feet alternately on facing walls - I forget the climbing term for this).

Stemming is the word you are looking for.

Eebster the Great
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### Re: 1488: "Flowcharts"

At the very least, you're never going to get anywhere in Calculus if you don't understand algebra. It is almost pointless to try. Even moving back and forth between calculus and algebra is just going to lead to confusion.

Matrices are a little different. I don't think they are the oldest way to describe geometrical transformations anyway. It's probably possible in principle, if not practical, to get through analytic geometry without studying matrix arithmetic.

Kit.
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### Re: 1488: "Flowcharts"

orthogon wrote:Whilst we could debate just how hard or easy certain concepts are, I think we could probably agree that mathematics, and to a lesser extent natural sciences in general, are viciously cumulative.

Well, math is funny...

Spoiler:

Quercus
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### Re: 1488: "Flowcharts"

I've always thought that good research mathematicians shouldn't be allowed to teach mathematics, at least below advanced undergrad level - I've been taught by research mathematicians before and in my experience they're terrible at it. You see, by definition, good mathematicians are those that have an intuitive grasp of mathematics, therefore they have no idea how they do any of it.

An (almost) verbatim conversation with my former maths professor:

Me: Could you go through how you knew to approach the problem in this way?
Prof: I just looked at it and the shape of the solution appeared in my head.
Me: That doesn't happen for me.
Prof: Try looking at it some more.
Me: Grrr....

For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

rmsgrey
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### Re: 1488: "Flowcharts"

Quercus wrote:For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

That's not necessarily true - the main problem is that ability at maths and ability at teaching are pretty much independent, so a good maths teacher has to score on both aspects, and someone who's only a good teacher is probably better for teaching most maths than is someone who's only a good mathematician (though at the top end, you need a good mathematician to be able to teach it, and there are otherwise good teachers whose grasp of maths is bad enough to cancel out).

Kit.
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### Re: 1488: "Flowcharts"

Quercus wrote:For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

Even if you "work out explicitly all the tricks", after you manage to grasp this, fractions become trivial matter and the way they are taught to children starts to look weird and counter-intuitive.

orthogon
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### Re: 1488: "Flowcharts"

Kit. wrote:
Quercus wrote:For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

Even if you "work out explicitly all the tricks", after you manage to grasp this, fractions become trivial matter and the way they are taught to children starts to look weird and counter-intuitive.

I'm a newcomer to field theory, but isn't the point that it's a generalisation of things like rational arithmetic? Generalisations are fundamentally more abstract and therefore more difficult to grasp. All the sources on field theory that I've come across use things like arithmetic over the rationals as an example serving as a gentle introduction. Most people find it difficult to swallow a fully general topic straight away without more specific cases as examples. Having said that, I suspect that different people are comfortable at different levels of generality. For example, I feel at home at the level of engineering and applied science, where I have a whole load of general tools (conservation of energy, linear algebra, differential equations) that I can apply to many different situations, enabling me to do a lot of things without needing to memorize too much about each particular field. By contrast, I find subjects like history rather uninteresting, because of the lack of convincing generalisations: the only way to know that stuff is to memorize it all. It may well be that the sort of people who make good mathematicians are happier at a higher level of generalisation. They probably find it ridiculous that I consider matrices to be a different thing than rational numbers1, and waste all those brain-cells dealing with them separately.

1 I know, one is probably abelian and the other isn't. Or one's a group and the other's a field. Or a ring. Feel free to insert a better example.
xtifr wrote:... and orthogon merely sounds undecided.

Quercus
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### Re: 1488: "Flowcharts"

rmsgrey wrote:
Quercus wrote:For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

That's not necessarily true - the main problem is that ability at maths and ability at teaching are pretty much independent, so a good maths teacher has to score on both aspects, and someone who's only a good teacher is probably better for teaching most maths than is someone who's only a good mathematician (though at the top end, you need a good mathematician to be able to teach it, and there are otherwise good teachers whose grasp of maths is bad enough to cancel out).

That's all fair enough - I've never personally come across a really top notch mathematician who is also a good teacher, but maybe I've just been unlucky (the best maths teachers I had at university were physicists).

Kit. wrote:
Quercus wrote:For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

Even if you "work out explicitly all the tricks", after you manage to grasp this, fractions become trivial matter and the way they are taught to children starts to look weird and counter-intuitive.

Yes, but if you attempt to teach fractions that way you are a bad teacher. A big part of being a teacher is adapting yourself to what your students find easiest to grasp. That is probably going to make things weird and counter-intuitive to you, but as a teacher you need to be able to get inside a student's head and think "what would be the easiest way to approach this subject if I didn't have 90% of the background knowledge and experience (and potentially a fair portion of the natural ability) that I do have". It's not a particularly easy trick, which is one of the reasons why it's not easy to be a good teacher.

My contention was that it's potentially easier to do that if you already have direct experience of the learning process at the same level as most of your students, which is not going to be the case for gifted mathematicians unless they are teaching similarly gifted students.

Eebster the Great
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### Re: 1488: "Flowcharts"

I had a Physics and Calculus teacher who was very highly regarded and respected, and made sure 100% of his students scored a 5 on the AP. Yet he said he would probably make a terrible Algebra teacher. The idea is exactly what Quercus is talking about--that in school he had never had any problems with algebra, so he wouldn't know how to help other people work through it. When people came to him with questions about calculus problems, he could usually steer them in the right direction, explain what they were doing wrong, and generally sympathize and improve their understanding. But if people came to him with questions about algebra, his best reply would probably be "... just do it. It's easy."

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### Re: 1488: "Flowcharts"

Quercus wrote:I've always thought that good research mathematicians shouldn't be allowed to teach mathematics, at least below advanced undergrad level - I've been taught by research mathematicians before and in my experience they're terrible at it. You see, by definition, good mathematicians are those that have an intuitive grasp of mathematics, therefore they have no idea how they do any of it.

An (almost) verbatim conversation with my former maths professor:

Me: Could you go through how you knew to approach the problem in this way?
Prof: I just looked at it and the shape of the solution appeared in my head.
Me: That doesn't happen for me.
Prof: Try looking at it some more.
Me: Grrr....

For teaching you want someone who isn't naturally gifted, but got good by dint of much time and effort, so they had to work out explicitly all the tricks that the gifted mathematicians did implicitly.

The Zen of Mathematics.

This Comic is Bleeding into My Real Life; In what I think are Funny Ways.
I've started making Flow Charts. I'm Terrible at Flow Charts.

What funny Notes.
Arrows and Squares and lines drawn not under, but through important words.

I'm not submitting my Notes as a Navigation Tool.
viewtopic.php?f=7&t=111140

If you Keep Things Simple.

Other people want to Derive their Own Questions.
Hey! Look!
!Pyramid Power!

Spoiler:
EDIT: No.
Happiness stays Flat.
The line moves up.
No. ...

I don't know.
Life is, just, an exchange of electrons; It is up to us to give it meaning.

We are all in The Gutter.
Some of us see The Gutter.
Some of us see The Stars.
by mr. Oscar Wilde.

Those that want to Know; Know.
Those that do not Know; Don't tell them.
They do terrible things to people that Tell Them.

SawGani
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### Re: 1488: "Flowcharts"

Actually diodes are all connected anode to cathode and this ring shorts out all the connections to it.
The battery/watertank is connected by only one line, or it would explode.
A rectifier bridge would have all the diodes pointed diagonally upwards, not forming a shorting ring.
Besides, he has used up most of the title variations for "flow chart".

Neil_Boekend
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### Re: 1488: "Flowcharts"

SawGani wrote:Actually diodes are all connected anode to cathode and this ring shorts out all the connections to it.
The battery/watertank is connected by only one line, or it would explode.
A rectifier bridge would have all the diodes pointed diagonally upwards, not forming a shorting ring.
Brush up your basic electronics. It is a rectifier bridge and no short (unless you turn the battery around).
The battery only explodes if it is a Lithium based cell and not disconnected in time, but you should never charge a lithium cell with a dumb charger.
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