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Arithmetic(s)--calling Jake S and kcurie!

by rg Sun Nov 11th, 2007 at 06:24:43 PM EST


A while back Jake S wrote the following in the School Leaving Age diary.

Teaching arithmetic requires a great deal of drilling, because to be truly useful it has to be almost a conditioned reflex. Similar to reading, in a sense. Reading should come so naturally to you - be such an integral part of how your brain works, if you will - that when you see a signpost, you should start reading what it says even before your conscious self catches up to what's going on. Similarly with arithmetics. When you see a problem, you should start solving it automatically, with the same instinctive part of your brain that makes you snatch your hand away from a heated cooking pot.

In the same thread, kcurie wrote:

So a more proper solution will be a 13-15 (3 years) compulsory of pure language and math for everyone (langue means understanding gramming and memorizingg vocabulary and comprehensive reading.. yes.. memorize is GOOOOOOD.. no matter what a stupid pshycologist will tell you... the good ones already chnged their opinoon at least on this one) and basic logic and math for any mature life...

A question for yez

What are the fundamentals of mathematics no fifteen year old should be without?

Now, maybe that is a list too far.  (I really don't know.)

Let me try....a quote from Migeru.

if people want to teach themselves you are just a facilitator

It's in the edge--the wide valley bottom--where learning occurs.

The slope yonder is called, "Can't, no, boring, too hard, I'm not good enough, it's not worth it, I don't like you" etc....

And that other slope, the one that people climb up and sometimes never come back, is called "Perseverance, weak!, do it again, wrong, you shouldn't have bothered, I'm afraid not, I don't think Max will make it" etc.

But also...ach..."Almost there!" and "Not right now, thanks", and "There are other slopes", "Sit down and look around",  and "Who's going your way?"

What I'm thinking is: I understand the "reading words" part, more or less.  It's a case of words being readily present (in my life), and the way words relate to situations around themselves, such as "EXIT" when you leave a place, and "ENTRANCE" when you go in.  Or "Yes" and "No", or "Where's" in the various Spot the Dog adventures.  

("Where's Spot?  Is he in....the hedge?"

No!  It's Mr. Squirrel!  

(That's not written in the book....so the repeated words that appear are recognised...heh!)

The hard thing is to teach how to read critically--what does that mean?

So I'm thinking there must be a maths evquivalent.  I can do the "How much is X?" maths (Spot the Mathematical Dog--"Is it more than one hundred?  No!  Is it more than ten?  Yes!"); I see that basic logic must be applied...

I'm thinking of a not-overly-tiring list of things that, if a person can do them, they can call themselves "numerate"....hmmmm....now I wonder if I could write a similar list to demonstrate that a person is literate...

Heh!  Maybe it's just me; 'twould be good though, I think, to list out the basics in all materials--we have a lot of knowledgable people here at ET, and they are knowledgable because they have embraced their subject, and so here I am, maybe I'd like to embrace it a bit too, but I can't be a professor of everything, so some subjects...I'll be happy with the basics, but with an element of rigour, where I can say, "I understand the basics of econmics" and therefore I can follow economic conversations...for example.

There are maybe divisions among folks deeply immersed in their areas of interest; but okay....if I had six months free time to learn "the basics" of, say, arithmetic; is that long enough?  Or maybe I don't need six months; maybe I can learn the basics in a month?

The very basics, I mean.

A silly example.  I declare (regularly) that it is possible to learn to touch type in three weeks, as long as you practice for half an hour every day, and you won't be able to do the numbers along the top, and you may be a bit sloppy with the shift key...but you'll know the basics.

Hmmm...

Pythagoras - Wikipedia, the free encyclopedia

Pythagoras' religious and scientific views were, in his opinion, inseparably interconnected. However, they are looked at separately in the 21st century. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were influenced by Greek Mythology. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog.

One of Pythagoras' beliefs was that the essence of being is number. Thus, being relies on stability of all things that create the universe. Things like health relied on a stable proportion of elements; too much or too little of one thing causes an imbalance that makes a being unhealthy. Pythagoras viewed thinking as the calculating with the idea numbers. When combined with the Folk theories, the philosophy evolves into a belief that Knowledge of the essence of being can be found in the form of numbers. If this is taken a step further, one can say that because mathematics is an unseen essence, the essence of being is an unseen characteristic that can be encountered by the study of mathematics.


Display:
What are the fundamentals of mathematics no fifteen year old should be without?

A good essay and a great question --- that I'll leave to others more numerate than myself to answer.  

But one non-arithmetic skill that I think is useful in arithmetic (an other areas) is estimation.  It seems to me it is a numerate sub-set of commonsense.  I wish I knew how to teach practical estimation if not commonsense.

by cbatjesmond on Sun Nov 11th, 2007 at 06:54:57 PM EST
In the U.S., at least, the biggest shortcoming with most students is that they have not memorized the multiplication tables. This is essential, because if you are in the middle of a problem and have to stop to figure out what 6x8 is, then you get distracted from the problem itself. Elementary school teachers don't like to confront this because it requires hours of drill, which is boring, and because they don't know the tables either.
by asdf on Sun Nov 11th, 2007 at 07:53:20 PM EST
I think it's one of those things that are a result of allowing calculators. The building blocks at the lower levels of thought don't get constructed and so the ways of thinking to build the higher levels dont get constructed.

Any idiot can face a crisis - it's day to day living that wears you out.
by ceebs (ceebs (at) eurotrib (dot) com) on Sun Nov 11th, 2007 at 08:38:53 PM EST
[ Parent ]
I spent hours trying to drill the times tables as a youth, and they never stuck.  For some people, those things come more easily than others.

I never learned to think of anything mathematical as a problem that could be coherently solved.  I never "got" numerical thinking, and I was never doing anything more than applying formulae or tricks to the questions.  I never really understood why anything was right beyond the most basic arithmatic questions.

When I took college algebra, the pinncle of my mathematical accomplishments, there was a particular type of problem that was just beyond my ability to comprehend.  I spent hours and hours fighting with those problems, and I remember on three separate occasions "getting it," and finally understanding how to do the thing.  I then forgot the next day, and had to go through the same process again.  I needed to pass the course, and didn't want to take it again, so I pushed through, but when it came to the test, I STILL didn't understand the problem type.  I had some time on the final, so I did my best, and without any understanding at all did a bunch of stuff to the problem that I didn't really understand, but seemed reasonable at the time, and eventually stopped, because I couldn't see anything more to do.  I'd actually gotten it right, to my shock and amazement, but I couldn't understand why.

Later, I took three quarters of statistics.  That was easy, really.  I never came anywhere close to understanding what on earth the formulas that produced correlations or standard deviations and whatnot were, but I could see quite clearly how to plug data sets into the equations, and could understand how to use the information they produced.  But the maths behind all that?  Pure and incomprehensible gibberish.  I thought about the people who had developed those formulae with a bit of awe, because the relations between the data going in and the information coming out was completely and totally mysterious to me.

On the other hand, I had a great intuitive sense for probabilities.

I've since gotten better at basic arithmetic, and can sometimes remember or intuit basic multiplication.  When pressed, I think i remember how to do long division.

However, I find intepreting statistical data to be quite easy, and think in percentages and fractions all the time.  However, that doesn't have any connection to math in my mind.

by Zwackus on Sun Nov 11th, 2007 at 11:29:33 PM EST
[ Parent ]
I can completely sympathize -- I can do it if forced, but math literally makes my brain ache.  

Oddly, I was quite good at word problems.  Perhaps math essay questions are exactly what I need...

Maybe we can eventually make language a complete impediment to understanding. -Hobbes

by Izzy (izzy at eurotrib dot com) on Sun Nov 11th, 2007 at 11:46:46 PM EST
[ Parent ]
There's nothing that I found more frustrating that my college undergraduate students being unable to deal with "word problems". That is, they could do all the mechanical stuff but could not turn a verbal statement into algebra or conversely. And so, they were utterly unable to apply math to other disciplines like physics, statistics, engineering...

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 07:39:56 AM EST
[ Parent ]
It is true that some excitement about math things helps a lot. (You add two apples to three apples, and get five apples. You add two thousand euro and three thousand euro, and get five thousand euro. Isn't here something remarkable?!)

Without a bit of excitement, math is pain. But wouldn't it be most fair to recognize that pain (to a certain extent) as something necessary in the learning process? Modern education focuses very much on making learning as "enjoyable" as possible, as if trying to make the impression that you can learn anything without any pain. This is akin to the ideology that all economics can be based on self-interest, with no need to worry about common externalities.

I would say: kids, learning math will probably be a greater or lesser pain to most of you. But be not afraid - the more you learn, the greater chance you will like something there. Go as far as you can - you may not need much of this math, but you may need to learn to go forward despite resistance. Your grandpas did not do too badly after forced educations after all.

And speaking of statistics: much can be learned by examining the numbers around in the media, even in the classroom. (The meaning of correlations or standard deviations is not that deep: those are just the simplest and most handy mathematical measures of something useful. Just as you can consider arithmetic, geometric, quadratic and many other means, so you can think of other measures of variation - if only that would make your life more exciting.)

by das monde on Tue Nov 13th, 2007 at 01:51:17 AM EST
[ Parent ]
I just can't remember whether we were actually drilled to learn multiplication tables at school, or only had to learn it as home assignment which the teachers only testing us later. At any rate, we knew it.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 06:05:16 AM EST
[ Parent ]
I learned the multiplication tables by drill and I almost always revert to Spanish when doing math in my head.  That may mean something to kcurie, or maybe trivial.

Our knowledge has surpassed our wisdom. -Charu Saxena.
by metavision on Wed Nov 14th, 2007 at 02:27:05 PM EST
[ Parent ]
Basic arithmetics : knowing how to perform the 4 operations without a calculator.

Multiplication of small numbers ought to be drilled at an early age, i.e. before 8. Long division ought to be learned early on, as it is the first big algorithm one learns and applies, and as such probably has an important pedagogical role.

Deduction and abstraction :

Someone commented on the School Leaving Age diary about how useless triangle names were. Maybe so. But an important aspect of mathematics is reasoning about abstract things ; geometry is a very useful tool for doing it. It allows reasoning with visual help, making it possible to discover and undertake demonstrations ; yet also makes an important point of abstraction (the figure as a representation rather than reality ; concepts such as the widthlessness of dots and lines).

Numeracy :

The ability to transfer from word problems into mathematics problems ; and using intuition to check one's mathematical results against likely real world results.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 07:32:17 AM EST
For arithmetic I would add Euclid's algorithm, simple divisibility rules (by 3, by 9, by 11...) and simple multiplication tricks (like how to multiply two two-digit numbers ending in 5).

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 07:37:56 AM EST
[ Parent ]
I kind of forgot about non-natural-numbers numeracy, i.e. getting a basic understanding of fractions, real numbers, and negative numbers, and how to use them ; fractions include the euclidean algorithm. Also, this part allows to push through an understanding about the difference between exact, symbolic computation and approximate computation.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 07:46:18 AM EST
[ Parent ]
An exercise in powers of ten. See the wikipedia series on orders of magnitude.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 07:56:52 AM EST
[ Parent ]
That's physics !

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:01:13 AM EST
[ Parent ]
You say that like it's a bad thing. ;-)

Any idiot can face a crisis - it's day to day living that wears you out.
by ceebs (ceebs (at) eurotrib (dot) com) on Mon Nov 12th, 2007 at 08:11:03 AM EST
[ Parent ]
Mathematics are pure. Physics aren't. Plus, one of the things to understand is that orders of magnitude don't really exist in maths ; there are as many numbers between 0 and 10 as there are between 0 and 0,000000000000000000000001.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:21:30 AM EST
[ Parent ]
The number 0 and the number 1 set a natural scale for pure (dimensionless) numbers. One could look at dimensionless numbers coming out of physics and put them on a logarithmic scale.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 08:24:51 AM EST
[ Parent ]
You don't teach arithmetic to children under 15 because it's "pure math" but because it is practically useful.

Plus, human mathematics is not pure.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 08:26:59 AM EST
[ Parent ]
Firstly, I was snarking.

Secondly, I think education should be valuer for its own sake rather than for its practical purposes. Realising one can do maths for maths's sake ought to be part of the syllabus ; one of the pitfalls of maths, and especially of maths for practical purposes, is that of mistaking them for a set of problem-solving techniques, which make understanding maths (and actual further problem solving) harder.

If up to 15 you only teach the maths that are practically useful, not looking into tome abstract details, it actually becomes very hard to do further maths, and since those one has learned are only a disjointed set of algorithms and quick answers, they are fast forgotten, too.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:56:01 AM EST
[ Parent ]
Could you give an example?  And....is it a visual experience, or is there some non-visual "place" where one revels in the pleasure of....maths....

(I mean, I an almost see it, I think, but I'm one of those who sees the application, I'm seeing it backwards maybe, from the machine to the parts to the materials to geology, into the atoms, and...out there in the land of the abstract....mathematics!

I almost called this diary "The Joy of Maths"...I keep thinking of applications...."What the numbers meant to Ka Ne Suss was that the thing was about to blow."

I'm intrigued!

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Mon Nov 12th, 2007 at 07:52:23 PM EST
[ Parent ]
When one advances enough, the pleasures of maths become non-visual ; Maths after all is the art of abstract symbol manipulation. Finding a pathway between statement A - hypotheses, to statement B - consequences, logically true step by logically true step. Yet at the same time the steps are not trivial ; there is the joy of the treasure hunt. Maths isn't about numbers, indeed very often it is numbers, and annoying computations, that may make maths boring...

As an example, that first of all demonstrations, that of Pythagorean theorem : how does the figure prove that in a triangle with a 90° angle, a²+b²=n² ?


Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:26:32 PM EST
[ Parent ]
is this homework?

Any idiot can face a crisis - it's day to day living that wears you out.
by ceebs (ceebs (at) eurotrib (dot) com) on Mon Nov 12th, 2007 at 08:36:48 PM EST
[ Parent ]
Yes.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:58:11 PM EST
[ Parent ]
(a+b)*(a+b)-n2=4(a*b)/2 (from the diagram)

a2+b2+2ab=2ab+n2

so a2+b2=n2

Any idiot can face a crisis - it's day to day living that wears you out.

by ceebs (ceebs (at) eurotrib (dot) com) on Tue Nov 13th, 2007 at 07:48:34 AM EST
[ Parent ]
Yeah, this is a nice proof. But somehow it makes me feel like cheating, because it is only simple if you use a level of symbolic algebra that didn't exist in the old Greek days.
A slight variant on it, which is much nicer in my opinion, can be found here: proof #9 on
http://www.cut-the-knot.org/pythagoras/index.shtml
by GreatZamfir on Tue Nov 13th, 2007 at 08:01:19 AM EST
[ Parent ]
It did exist in the old greek days because they stated Pythagoras' theorem in terms of areas of squares built on sides, and addition of areas was a common technique.

At no point in the proof there is a nonhomogenoeous polynomial adding a length to an area, for instance. So Ceebs' argument can be written out in words involving areas.

I think that diagrammatic proof of Pythagoras' theorem may have originated in India?

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:10:50 AM EST
[ Parent ]
C'mon, it's not hard at all. It consists of producing a second graph from the above.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 06:38:59 AM EST
[ Parent ]
Woah!  Nothing is hard if you know how to do it.

I'm still pondering the idea of an "imageless" maths that invokes images (the diagram above), or a graph--something visual at any rate that stands for...the invisible maths behind the image...

So it may be easy, but easy is good (for me) if it helps me concentrate on the underlying aspect, in this case:

When one advances enough, the pleasures of maths become non-visual ; Maths after all is the art of abstract symbol manipulation.

(Thing is, I have pondered this and I am wondering whether maths' claim to be somehow bigger than the universe (maths gives us "the universe" + 1)--I mean the idea that maths "encompasses" the universe as compared to the universe being "bigger than maths"--Heh, I'll have to try and explain this again later, but I mean something like: "What does it mean that we can't beyond a certain exponential--I was thinking about mathematical models of the universe--there is the "empty box" model, we are in it and the sides are an endless distance away.  Then there is the "closed form" model, balls, saddles, but always (inevitably) seen from "outside"...heh...I'll post this just to remind myself that I had a thought in there somewhere.

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 07:40:16 AM EST
[ Parent ]
See the book proofs without words.

Notice that before the development of symbolic algebra in the middle ages, elgebra had always a geometric interpretation. Squares were the areas of squares. Linear quantities were the lengths of segments. Cubes were the volumes of actual cubes. Inhomogeneous polynomials (mixing quantities of different degree) didn't often occur.

Mathematics has always been visual, touchy-feely, intuitive, until the formalization in the 19th century.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:16:03 AM EST
[ Parent ]
Woah!  Nothing is hard if you know how to do it.

But this one should be really easy. We had to find this out on ourselves, I don't know, maybe as sixth graders.

I mean the idea that maths "encompasses" the universe as compared to the universe being "bigger than maths"

Do you know that Set Theory proves that there is no Universe?

Then there is the "closed form" model, balls, saddles

Saddles are a representation of open ever-expanding hyperbolic universes.

*Lunatic*, n.
One whose delusions are out of fashion.

by DoDo on Tue Nov 13th, 2007 at 08:40:52 AM EST
[ Parent ]
Mathematical education in the communist countries was notably more advanced than anywhere else. Stuff was learned about two to three years earlier in Russia or Yougoslavia than in France, for example.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:49:22 AM EST
[ Parent ]
I know it's about 26 years since i had to do anything like that. It's good to see that I still have the tools in my mental toolbox, (albeit a little dusty, plus there are probably newer shinyer mental tools out there somewhere which I havent aquired)

Any idiot can face a crisis - it's day to day living that wears you out.
by ceebs (ceebs (at) eurotrib (dot) com) on Tue Nov 13th, 2007 at 07:53:07 AM EST
[ Parent ]
When one advances enough, the pleasures of maths become non-visual ; Maths after all is the art of abstract symbol manipulation.

What? Nonsense: that's logic you're thinking of. Abstract symbol manipulation is a tool, most of the time, not an end.
by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 07:43:53 AM EST
[ Parent ]
Which parts of maths deal with stuff that are not abstract symbols ?

Logic is how you are allowed to manipulate abstract symbols ; the rest of maths is deciding about some abstract symbol, and then playing with them a lot...

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:13:44 AM EST
[ Parent ]
Almost all of it. Which parts of mathematics deal solely with abstract symbols? Which value of "abstract" are you using?
by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 08:21:28 AM EST
[ Parent ]
Which part of mathematics don't deal with abstract symbols, i.e. with arbitrarily chosen words or signs that point not to a "real" entity from the concrete world, but to a thought entity that behaves according to some abstract hypotheses ?

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:28:40 AM EST
[ Parent ]
I do a fair bit of category theory: I think that the set of natural numbers is a concrete example. To my way of thinking, in the context of mathematics, an abstract symbol is one that doesn't have any meaning behind it.
by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 08:31:54 AM EST
[ Parent ]
What do you mean? The free monoidal category on one object is indexed by the natural numbers.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:41:06 AM EST
[ Parent ]
Yes, and that's a concrete example as well.
by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 09:12:29 AM EST
[ Parent ]
I don't really see how N is concrete, i.e. how it has an existence in the real world.

In my view N only has the meaning we give it through axioms ; axioms which are rules on how to write proofs.

What do you mean by 'meaning' ? :)

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:45:32 AM EST
[ Parent ]
The natural numbers predate the Peano axioms bu how many millennia exactly?

Axiomatization is not the prerequisite for mathematics, it's the endpoint.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:08:19 AM EST
[ Parent ]
Euclid did have an axiomatisation of natural numbers... We switched axioms more recently, but axiomatisation is as old as mathematics.

Also, is R more concrete than the set of p-adic numbers? is Euclidean geometry less abstract than other geometries ?

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 09:22:13 AM EST
[ Parent ]
Actually, spherical geometry is more concrete than Euclidean geometry because it is the geometry of the visual field.

R is more concrete than the set of p-adic numbers. That is why it was invented centuries earlier.

And while Euclid and his contemporaries had axioms, mathematics had existed before them. The greeks may have invented the axiomatic-deductive method, but they did not invent mathematics.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:40:25 AM EST
[ Parent ]
Then why was Euclidean geometry invented a long time before spherical geometry ?

Which geometry is concrete to a blind person ?*

It seems you define concrete as intuitively accessible to the human brain... It makes god a very concrete concept nowadays.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 09:45:31 AM EST
[ Parent ]
It seems you define concrete as intuitively accessible to the human brain...
Where else does human mathematics come from?

As for spherical geometry being invented after euclidean geometry, I don't know what came first, but spherical geometry was highly developed by babylonian astronomers while the Babylonian value for pi was still the integer 3.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:55:12 AM EST
[ Parent ]
Is mathematics dependent on society ? We were wondering elsewhere on the relative intuitiveness of fractions and decimals. If concreteness is linked to intuitive understandability and conceivability, depending on whether a society insists on decimals or fractions one concept or the other becomes more concrete. Concreteness  isn't constant across human brains, according to your definition...

Also, is there mathematical truth independent of thought processes : is logic only a cognitive process ? Colman was contrasting logics with the rest of mathematics. Is logic "true" because it agrees with our thought processes - but many people think without adhering to the laws of logic. Why would logic be different from the rest of maths ?

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 10:16:45 AM EST
[ Parent ]
What do you mean by 'real world'?

Concepts of abstract and concrete can depend where you're looking at them from: N can be relatively concrete. In a moment we can consider what is concrete, precisely.

by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 09:11:53 AM EST
[ Parent ]
Real maths - as opposed to mechanical arithmetic - is a complicated little beastie and varies from field to field.

You have a couple of things going on - your intuition about the structures you're dealing with, your visualisation of them (which I don't mean in a way that's easily mappable to visualising real things, but you're using the same part of the mind), the symbolic representations and the available facts about the symbolic representations. So you're working on several levels, and different people enjoy different parts. Generally I think people are guided by intuition to propose things which they then need the symbolic machinery to prove - it's too easy to let your imagination run away with you. Sometimes your intuition is mistaken and the symbolic machinery will help you understand why.

by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 07:56:03 AM EST
[ Parent ]
Great link, thanks.

Where Mathematics Comes From - Wikipedia, the free encyclopedia

Human cognition and mathematics

Lakoff and Núñez's avowed purpose is to begin laying the foundations for a truly scientific understanding of mathematics, one grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path.



Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Mon Nov 12th, 2007 at 08:08:20 PM EST
[ Parent ]
Re : your link, I think the goal of primary mathematics education should be to get the pupil to the point where the cognitive, intuitionist mathematics, i.e. as an extension of basic cognitive instinct as described it the book, begins to fade, and conceptual, non-visual and abstract mathematics begin to be visible in the distance.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:35:56 PM EST
[ Parent ]
For that reason, it is my pet-peeve that general mathematics education should reach complex numbers.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 06:37:55 AM EST
[ Parent ]
Should our hypothetical fifteen year old know about complex numbers?

Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 07:42:16 AM EST
[ Parent ]
It depends on whether they should know enough geometry and algebra to motivate them, which isn't too much.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:19:15 AM EST
[ Parent ]
If general education ends with 15, then yes.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 08:44:05 AM EST
[ Parent ]
Why shouldn't it? You need complex numbers to solve the quadratic equation, and around 1800 the connection with planar geometry and the operation of rotation was discovered.

Now, whether the quadratic equation and planar geometry should be part of general mathematics education is a different story.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:18:39 AM EST
[ Parent ]
You can also solve the quadratic equation by re-organising it into a simple square equals constant equation:

Y^2 = b^2/(4ac^2) - c/a,
where
Y = x + b/2a.

That's how it was taught to me in highschool first grade, which was then ninth grade overall, e.g. we were aged 14-15. But planar geometry, if I guess right what that is, came much later, maybe only college, around the same time complex numbers. I'd pull complex numbers ahead.

*Lunatic*, n.
One whose delusions are out of fashion.

by DoDo on Tue Nov 13th, 2007 at 09:43:09 AM EST
[ Parent ]
Well, you're right, you can solve the quadratic equation by completing the square [here is another thing that needs to be taught to young students: completing the square - my American undergraduates had no idea what that was] and you can solve the cubic equation by using trigonometry (or hyperbolic functions) without using complex numbers.

Regarding planar geometry, historically
Complex number - Wikipedia, the free encyclopedia

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.
IMHO one of the nastiest things about 19th century mathematics was that arithmetic/algebra was considered rigorous but that geometry was considered too empirical. Therefore, work on the foundations of mathematics concentrated on basing all mathematics on algebra. For calculus this research program was called "arithmetization of analysis" and was completed by Weierstrass. Over the years this has had the horrible effect of emphasizing algebra in early education and analysis as the gateway to advanced mathematics, to the detriment of geometry which is then learnt with little intuitive backing.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:53:03 AM EST
[ Parent ]
Interesting. But I had 'intuitive' geometry early on, and arithmetised geometry (if I get this right, I am thinking of parametrised surfaces and such) only in college, actually almost in parallel with algebraised foundation of analysis.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 10:15:03 AM EST
[ Parent ]
you can solve the cubic equation by using trigonometry (or hyperbolic functions) without using complex numbers
Here's something else that people should know about. It may be a borderline case of whether it is too hard for under 15's, but logarithms and exponentials are not really all that much harder than trigonometry, and they are definitely very useful. Moreover, an intuitive understanding of rates of growth seems to be more useful than trigonometry to "the modern man".

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 10:20:16 AM EST
[ Parent ]
 

Exponentials and logarithms are even easier than the trigonometric functions. For one thing, they are almost arithmetic operations: you just solve for x in e^b=x, or e^x=b, with some peculiar number e, and continuously variable b. Secondly, properties like addition formula exp(a+b)=exp(a)*exp(b) are much simpler. And if you know differential equations, y'=y is simpler than y"+y=0, and y'=1/x is important as well.

Exponential/logarithmic functions might look less exciting than trigonometric functions, but dull looks are deceiving. Everyone who has to pay interest rates must know the exponential function.

by das monde on Wed Nov 14th, 2007 at 01:17:11 AM EST
[ Parent ]
But for some reason incomprehensible to me, hyperbolic functions like 2 cosh(x) = e^x + e^{-x} are considered harder than trigonometric functions even though they behave in the same way in terms of derivatives and algebraic identities.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Nov 14th, 2007 at 02:31:39 AM EST
[ Parent ]
However, hyperbolic functions don't have such a nice graphic representation.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Wed Nov 14th, 2007 at 02:50:53 AM EST
[ Parent ]
The graphs indeed look less exciting. But their steepness growth has to be appreciated.

You probably know the famous fable of exponential growth: Rice grains on a chessboard:

A courtier presented the Persian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement [demanded] over a million grains on the 21st square, more than a million million on the 41st and there simply was not enough rice in the whole world for the final squares. (From Meadows et al. 1972, p.29 via Porritt 2005)

This tale could be supplemented with the following:

Put a 1g granule of gold on the first square. On the second square, put 1.011 g granule of gold - a 1.1% increase. On the second square, put 1.022 g of gold - still 1.1% more. On the third square, put 1.033 g of gold - another 1.1% increase. If you go on like this, on the last square you put 1.011^63 *1g=1.992 g of gold - almost the double of what is on the first square. In total, there would be about 92.188 g of gold on the chessboard. That is worth about 1629 euro in today's markets (with 17.67 euro per gram). Not too terribly bad so far.

Do we have another chessboard? On the first square of the 2nd board, we put 1.1% more than the 1.992 g on the last square of the 1st board. That is 2.014 g of gold. On the second square of the 2nd board, we put 1.1% more, which is 2.036 g gold, or 2.014 times more than on the 2nd square. If we go on like this, we always put 2.014 times more gold on the Nth square of the 2nd board than on the Nth square of the 1st board. In particular, we would have 4.012 g of gold on the last square of the 2nd board, and the second board would weight 185.67 g of gold.

How many chessborards do we have? Is it 10 in total? After increasing the ammount by 1.1% per next square, the 10th  board would contain 2.014^9*92.188g, which is ~50.27 kg (kilograms!) of gold. That is worth 888 thousand euro! All 10 boards would contain ~99.76 kg of gold.

Can we borrow 10 more chessboards? The numbers will be very similar to the original Persian story when we come to the 20th square there, since the numbers 2.014 and 2.0 are about equal.

The moral is that the "interest rate" of exponential growth tells you how fast you double the amount. With 1.1% of interest rate, you double in 63-64 steps. With 1% growth, you double in 69-70 steps. With 4% growth, you double in 17-18 steps. (Compute log(2)/log(1.04).) With 5% growth, you double in 14-15 steps. To compare two interest rates, you should compare the time scales of growth.

Is the exponential function more exciting now?

by das monde on Wed Nov 14th, 2007 at 04:10:36 AM EST
[ Parent ]
In France this movement to teach mathematics along algebraist lines was called "Maths Modernes" and was a failure. It had been reverted for a long time when I got in school. I do wonder how much it is a failure of Maths teachers, who were not able to adapt to new methods that themselves weren't necessarily very competent about. I am not so sure Euclidean geometry, with its triangles cut in pieces, is so much more intuitive than basic algebra.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 10:24:36 AM EST
[ Parent ]
They need to learn that arithmetic is not learnt for its own sake only.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 08:14:22 AM EST
[ Parent ]
Oh certainly, the ammount of people who don't want to learn anything beyond very basic maths, because theres no point to it and they'll never use it.

teaching the uses would improve maths teaching in the UK no end.

Any idiot can face a crisis - it's day to day living that wears you out.

by ceebs (ceebs (at) eurotrib (dot) com) on Mon Nov 12th, 2007 at 08:59:56 AM EST
[ Parent ]
Actually arithmetic usually is learned for its own sake only. Most people need to know basic arithmetic with a calculator, estimation without one, budgeting, maths-for-DIY, maths-for-travel (time, distance, speed, time zones, currency conversions), some basic statistics - enough to know when to yell 'Bullshit!' at what's in the media - and that's probably it.

After that you're usually on a specific vocational and/or academic career path with specialised requirements.

What is a non-mathematician going to do with abstraction? It means nothing to them, they almost certainly don't understand it, and it has no relevance to their lives.

The problem with suggesting that maths should be used to teach abstraction is that there are other ways to teach abstraction. Music theory can get very abstract by the time you're trying to write an orchestral score. Art can be abstract. Other languages can be abstract. (Personally I've always found my Latin A Level more useful than the other three.)

So what exactly is an understanding of mathematical abstraction going to give teens if they're not on a science/engineering track?

by ThatBritGuy (thatbritguy (at) googlemail.com) on Tue Nov 13th, 2007 at 11:14:35 AM EST
[ Parent ]
Well, first, how the hell would teens know whether they wanted to be on a science/engineering track if they've never seen any of either?

Second, your argument applies to all sorts of subjects. Why bother having an education system at all?

by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 11:57:33 AM EST
[ Parent ]
Third, people don't retain all they are exposed to, not even all they work on. So, if you have an idea of what you want 15-year olds to retain into adulthood, you need to expose them to a lot more as part of the curriculum.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 12:03:16 PM EST
[ Parent ]
Colman:
Well, first, how the hell would teens know whether they wanted to be on a science/engineering track if they've never seen any of either?

Oddly enough, a lot of them seem to know. They're the ones - e.g. - modding their PCs and writing machine code demos for them.

They don't need to have seen calculus ahead of time. What they need is intense curiosity. If they don't have that, putting them on a science and engineering track is a waste of time.

If they do, they'll be asking for new things to learn ahead of the official schedule.

Colman:

Second, your argument applies to all sorts of subjects. Why bother having an education system at all?

Yes, it does - which is exactly the point.

And yes, that's exactly the question to be asking.

What is education for, exactly? If you don't know start by agreeing an answer, deciding that this or that subject is 'good for people' just because it is (and because it's academic, and based on notions of education whose prototype is medieval) isn't a very imaginative response.

Teens could be taught all kinds of things in school - creativity, social skills, politics and activism, psychology, environmental awareness, business skills, media management, meditation and self-awareness - and a very long list of other skills that could turn them into active voters.

Instead we teach them quadratic equations - which 50% of them don't understand and 80% will never use again - and then wonder why they're such idiots when they vote. If they bother to vote at all.

Isn't that 'What is education for?' should be about?

Or shall we just continue with the rather irrational belief that being able to do algebra makes people intelligent, educated and informed, just because it does, see?

by ThatBritGuy (thatbritguy (at) googlemail.com) on Tue Nov 13th, 2007 at 12:37:50 PM EST
[ Parent ]
Not all of it is physics...

Orders of magnitude (numbers) - Wikipedia, the free encyclopedia

This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities.


We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 08:18:22 AM EST
[ Parent ]
Another great link!

Orders of magnitude (numbers) - Wikipedia, the free encyclopedia

Smaller than 10-36
  • Computing: The number 5×10-324 is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value.
  • Computing: The number 1.4×10-45 is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.

Orders of magnitude (numbers) - Wikipedia, the free encyclopedia

10-24

(0.000 000 000 000 000 000 000 001), short scale: One septillionth long scale: One quadrillionth)

ISO: yocto- (y)

[edit] 10-21

(0.000 000 000 000 000 000 001, short scale: One sextillionth, long scale: One trilliardth)

ISO: zepto- (z)

Orders of magnitude (numbers) - Wikipedia, the free encyclopedia

10-9

(0.000 000 001; short scale: one billionth; long scale: one milliardth)

ISO: nano- (n)

  • Mathematics - Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the US Powerball Multistate Lottery, with a single ticket, under the rules as of 2006, are 146,107,962 to 1 against, for a probability of 7×10-9.
  • Mathematics - Lottery: The odds of winning the Jackpot (matching the 6 main numbers) in the UK National Lottery, with a single ticket, under the rules as of 2003, are 13,983,816 to 1 against, for a probability of 7×10-8.

100

(1; one)

    * Mathematics: φ ≈ 1.6180339887, the golden ratio
    * Mathematics: e ≈ 2.718281828459045, the base of the natural logarithm
    * Mathematics: π ≈ 3.14159265358979, the ratio of a circle's circumference to its diameter
    * BioMed: 7 ± 2, in cognitive science, George A. Miller's estimate of the number of objects that can be simultaneously held in working memory
    * Astronomy: 8 planets in the solar system



Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Mon Nov 12th, 2007 at 08:17:20 PM EST
[ Parent ]
great diary, rg!

fun to read...

my left brain got squished in school, and math was my least favourite subject, because the teacher never changed his shirts and had terrible dandruff.

my laments of 'i'm stuck, sir' brought showers of little flakes, and they weren't manna of knowledge.

geometry i liked, cuz there was a satisfaction in sending in clean, accurate drawings.

science swam by in a fog of incomprehension, mostly. i remember a very aesthetic experiment involving growing beautiful crystals and the gorgeous colour of potassium permanganate, and a feeling of brain-seizure looking at the periodic table.

flash forward many decades, and now i see maths in a whole new way, which would have inspired me to try harder had i seen it then.

and with time and love to regenerate the braincells (and forget the shirt collars), now i notice my increased speed at visualising multiplications and divisions of 2 and 3 digit numbers, which veers on idiot-savant, to me at least...lol...

it's a beautiful thing, because it coexists with a sense of wonder at how much i must have changed in those years, without ever trying to develop maths skills per se.

it depends on a certain concentration and i can easily lose the thread and revert to my old 'duh' state, but when it flows it feels miraculous and very pleasurable indeed.

so i see why some mathematicians fall in love with numbers as a type of spell, like music.

...which it has deep connections with, but that's another diary...

thanks for this lovely diary, rg!

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty

by melo (melometa4(at)gmail.com) on Mon Nov 12th, 2007 at 09:30:34 AM EST
I've also noticed that I'm much better at a variety of basic things that stumped me horribly in the past.

I wonder how much of that is differential brain development - some people get that portion much earlier than others, and some people get a lot more of it than others.

While I don't deny that poor pedagogy can sour people on math, I also don't think its deniable that some people are just better at math and numerate things than others.  

Yet, interestingly, those traits don't seem fixed, as observed above.

by Zwackus on Mon Nov 12th, 2007 at 06:04:40 PM EST
[ Parent ]
I'm rubbish at maths. Although algebra and mechanics I am ok with.  I was taught well in my first school but when I moved, the second school undid everything to the point that at 8 years old I couldn't count anymore.  5 primary schools and significant periods out of school, either through 'displacement' for want of a better word, truanting or illness meant that the fundamentals that I should have learnt in primary school, were non existent.

My Father was fairly absent even when he was there and I don't remember him helping me with maths.  My mother left school at 14 and couldn't help me with homework. My schools let me down.

I was actually practising my long division today because a colleague was talking about her son's homework.  I can do it but it is hard. It shouldn't be.

by In Wales (inwales aaat eurotrib.com) on Mon Nov 12th, 2007 at 04:56:08 PM EST
I'm rubbish at maths. Although algebra ... I am ok with.
In my definition algebra is part of maths. Do you mean "arithmetic" when you say "maths" and if so do you have a word for arithmetic, algebra and geometry (and possibly calculus) all together?

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Mon Nov 12th, 2007 at 04:59:32 PM EST
[ Parent ]
I guess I do mean arithmatic. But 'maths' classes encompassed all of maths. So we didn't have calculus or algebra classes to make any distinction. Everything was 'maths'.

I can (could) do calculus within the context of mechanics and physics (but I just had to look it up on wiki to find out what calculus is).

There is no clear thread in my brain for me to follow to aid my understanding of maths.  I didn't go to enough lessons to keep up, basically.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 05:32:10 AM EST
[ Parent ]
Yet all that said, I am good at data analysis, with trends and with understanding tables and graphs and stats type stuff. Although I have never studied statistics either...
by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 05:34:42 AM EST
[ Parent ]
You know in modern mathematics arithmetic (number theory) is a branch of algebra? It's interesting that you can do algebra but you cannot do arithmetic.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:20:38 AM EST
[ Parent ]
Is it?  I see them as entirely separate things.

Afew can probably back me up on my being poor at arithmatic.  I was with him when he was helping the little ones with maths homework last year and they were better than me.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 08:24:02 AM EST
[ Parent ]
Can you do polynomials?

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:18:08 AM EST
[ Parent ]
I don't know what polynomials are.
* goes to check wiki *

Ah, yes I can do them.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 09:25:15 AM EST
[ Parent ]
So you can do algebra and polynomials but not arithmetic.

Can you do multiplication and division of polynomials?

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:37:33 AM EST
[ Parent ]
Fairly simple ones.  Are you talking about factorisation/expansion?

I remember struggling with quadratics. That's where I remember my maths education ending.  

I work under the assumption that were I to be taught properly and make the effort to learn and practise, I could grasp these things but currently it is all a mess in my head that I can't make sense of.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 09:48:18 AM EST
[ Parent ]
I remember struggling with quadratics. That's where I remember my maths education ending.
So how can you say you "can do polynomials"?

And I'm talking about factorization of polynomials, and (long) division of polynomials.

Can you multiply and divide polynomials but you cannot multiply and divide numbers?

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:57:37 AM EST
[ Parent ]
I've not heard of polynomials before. I looked it up and I recognise the form of the equations. I know the process I have to go through to work them out. But I know I couldn't deal with quadratics.

I can only multiply and divide basic numbers (eg 2x5 or 4x3) and it requires lots of thinking to get it right.  Most of the examples of polynomials I remember doing were no more complex with the multiplication than that.

Blame a shit education system, rubbish syllabus and the 'mainstreaming' of deaf kids that involves sitting them in classes with no support and then condemning them when they can't follow what is being said.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 10:24:36 AM EST
[ Parent ]
Aritmetic is exactly that: adding, subtracting, multiplying and division of numbers.
On the set of positive numbers you can add and multiply and always get a positive number. If you subtract, you have to extend your set with the negative numbers. If you also do division, you will need fractions. In this sense, subtration and division 'define' the negative numbers and
fractions (called rational number, with 'ratio' in it)

Aritmetic is often used to describe the 'algorithms' you use when doing those calculations, and that you learn in prmary school, such as adding and multiplying using carries ( 'remember one') and long division.

You mention you could do the same operations on polynomials. Thats's actually pretty sharp: polynomials do behave very much like numbers, so you can also so operations on them. 'Algebra' is the word mathematicians use for the study of the operations you can perform on mathematical objects (so, addition on numbers, or addition on polynomial, or even addition on    surfaces).

However, high schools use the word 'algebra' for a small subset of this, namely the rules to solve equations( if ax=cd, then x =cd/b etc.). Sometimes those equations have powers of x in them, but no other operations with x. Then they are called 'polynomial equations', and you could study them as objects in themselves. quadratic equations are polynomial equations with only powers of 2 or lower.

So, in mathematics aritmetic is the study of numbers and the operations you can do on them, while algebra is the study of operations on all mathematical objects, including numbers.
But in school, aritmetic is 'learning to calculate' and  algebra is 'solving equations'

My guess is that you learned aritmetic in primary school, algebra in high school up to quadratic formulas
and a bit of calculus (functions and their derivatives) to finish it. That's basically the standard package. The sad thing is that these things are very much related, but high school maths doesn't do a good job of showing it.

by GreatZamfir on Tue Nov 13th, 2007 at 10:58:05 AM EST
[ Parent ]
My guess is that you learned aritmetic in primary school, algebra in high school up to quadratic formulas and a bit of calculus (functions and their derivatives) to finish it. That's basically the standard package. The sad thing is that these things are very much related, but high school maths doesn't do a good job of showing it.

That is more or less it.  The issue with arithmetics is that I went to 5 primary schools and spent long periods of time out of school, especially between the ages of 7-9.  I started off fine but with no consistent teaching methods and no continuity I didn't get those basics.  

I am capable of doing arithmetic but I have never done the rote learning in order to find answers quickly.  It even takes a few seconds to get 2x9 which in my book is appalling.  Give me an addition like 134+97 and my brain starts going round in circles if I try to work it out mentally without pen and paper. I should be able to work it out quickly without giving myself a headache in the process.

I suppose it could be argued that if I made the effort, I could rote learn my multiplication tables, practice my arithmetic and then it wouldn't be an issue any more.  But there isn't enough of a need or motivation to do so.  What I was trying to illustrate really is that not setting up the foundations properly leads to lack of numeracy or at least missing 'bits' making the more complex stuff harder to grasp and apply.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 11:17:33 AM EST
[ Parent ]
I suppose it could be argued that if I made the effort, I could rote learn my multiplication tables, practice my arithmetic and then it wouldn't be an issue any more. But there isn't enough of a need or motivation to do so.

So, what are you missing, then?

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 01:18:53 PM EST
[ Parent ]
I was only trying to answer your questions!
I don't especially feel as though I am missing too much but I feel that I ought to be able to do basic arithmetic far more easily than I can and I am annoyed at parents and an education system that let me down.  I wouldn't want any child of mine to struggle so much with maths.
by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 04:29:20 PM EST
[ Parent ]
The difference is that arithmetic is taught as something you 'should' be able to do in your head, at least in part. And relevant basics are learned largely by rote.

School level algebra is (almost) easier because it's a set of algorithms you apply to set problems. You either know the algorithms or you don't. Even if you don't know the algorithms, you can go a long way with some miminal basics.

You're not usually expected to do algebra in your head.

Psychologically they're actually different skills. Arithmetic relies on both sort term and long term memory very much more than algebra does.

In fact it's a problem for all maths teaching - it doesn't pay nearly enough attention to practical psychology. (I don't mean human potential or therapy or anything abstract - I mean the basic brain mechanics of symbol manipulation and memory.)

If you really want to teach maths, rather than simply using mathematical knowledge as a caste marker (which is largely how it's used now), the way to do it would be to pick apart the different psychological requirements at each level and develop a program to practice them.

Otherwise it looks like one big glop of largely unrelated and rather random skills held together with a rather tenuous and never-quite-explicit philosophical view of the world.

It's almost impossible to teach pre-college pre-adults from that basis without confusing them.

by ThatBritGuy (thatbritguy (at) googlemail.com) on Tue Nov 13th, 2007 at 11:30:28 AM EST
[ Parent ]
I like the caste marker idea. However, do you think it marks high or low caste? It definitely doesn't mark the highest caste, more the intelligent workers who are not bosses caste.
On your comment about different skills, my personal experience is that to really understand a mathematical subject, I have learn to do some parts of it more or less automatically, that's when I really 'feel' the subject. For people who work a lot with equations, high school algebra is probably just as ingrained as aritmetic, if not more (after all, actual calculations are done by calculator...). Reading In Wales's comments, he or she appears to be someone who has algebra ingrained, but not aritmetic.

Perhaps the difference you describe is more about the difference between often-used skills and rarely used skills? Or perhaps it works dfferent for different people

by GreatZamfir on Tue Nov 13th, 2007 at 11:57:34 AM EST
[ Parent ]
Reading In Wales's comments, he or she appears to be someone who has algebra ingrained, but not aritmetic.

I'd agree with that. Algebra became ingrained through doing physics and chemistry A-levels. Which I hope makes Migeru less baffled about why I struggle with arithmetic but I am fine with algebra.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 12:11:14 PM EST
[ Parent ]
I think maybe I should make explicit a project of mine:

I'd like to gather information on what professionals/experts (etc.) consider to be the basic building blocks of their various subjects.

I'm always intrigued by structure; and as I stated in the diary, I can't become an expert in many (or any) fields, but I think it is possible to learn (well) the basic building blocks of many.

kcurie suggested that all kids need to be taught between 12-15 is (very rigorous) maths and language--20h maths/30h language per week.  The language part I understand--and can imagine various curricula.  For maths--or arithmetic--I wouldn't know what to add beyond "plus, minus, times, divide".

Yet if, as suggested in Jake's comment, "numeracy" is as important as literacy, then...I'd like to see stated what elements are involved, then I can compare my knowledge, see if I have any gaps to fill--and then for any people I might know under fifteen, I'll have an idea of "Okay, I know the basics" at least in the field of numeracy.

So I'm asking in that specific kcurie sense, which I understand to be "You can be an adult at 15; what you will need from education is deep literacy and deep numeracy" (or maybe I misunderstood completely!), which is an approach that others may disagree with but which I would like to examine further.

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Mon Nov 12th, 2007 at 05:13:18 PM EST
In my personal opinion, the "basic blocks of math" is definitely not the same as the things everyone should know at 15. Especially geometry and calculus, both very basic math elements, have very little value for most people. They will never use calculus in life, and their innate geometric skills are up to any task. Many carpenters can build perfectly triangulated constructions without a whim of Pythagoras.

My personal list of what people should know. This is specifically aimed at people who do not use any advanced mathematics in their normal lives, but at the basic things even the most innumerate people should get a hang of.

1. Add, subtract, multiply, division.
Sure. Including tables. And lots of practice with everything, especially with simple calculations, to give 'the feel'. People should be completely clear that if they can do 4 * 8, they can do 40 * 800. Truth is, many people are not fully aware of this.

I guess people should learn the multiplication and long division algorithms, but it is important to realize that these algorithms were 'developed' for efficiency, not for robustness. The multiplication algorithm starts at the back of the numbers and works forward, meaning that the largest chance of mistakes lies at the front, where the most significant numbers are. In the same vein, standard long division gives you only the number of digits, and therefore the order of magnitude of the answer when you are completely finished. If you are disturbed in the calculation, you can easily make a factor 10 mistake

Second downside of these algorithms: hardly anyone ever uses them anymore for more than simple calculations, while they are designed for very long calculations, by people trained to work meticulously. In any situation where the answer matters, you will use a calculator.
This automatically brings us to 2:

2. Estimation.
If people can do quick calculations giving the correct order of magnitude, and the first digit give or take one, they can verify answers of their calculators.  If can they can do the first 2 digits correct, they will be within a few percent of the answer, and that is good enough for most things.

There are many estimation methods, and I think they should be taught stronger than the standard algorithms. In modern society, estimation is the more important skill.  First and for all, people have to be able to find orders of magnitude. You can't read newspapers if can't `feel `the order of magnitude of 3 million euros for 1500 people.
Also important is rounding. When people see 7847, they should not think `complicated number' but `a bit less than 8000'.  
On a side note, I sometimes wonder whether we shouldn't simply learn floating point notation to children. 1.3 e5 is definitely not more difficult to learn to understand than 130.000. In practice we usually round to powers of 1000. Think of how money is reported: 682 million ( 3 digits accuracy); 68 million (2 digits, but sometimes 68.2 million); 6.8 million, but rarely 6.82 million,  682.000 ( 3 digits, but different notation), 68.000 ( hardly ever 68.200).  Because most people do not know a decent notation for large numbers, they have to rely on words, leading to the trouble with American and European billions, and the messing around with prefixes. Do you know how much smaller a nanometer is than a millimeter?

Further estimation techniques: tables. Everyone should know both the multiplication table and the addition table by heart, and `know' the answer in the same way they `know' the meaning of a word when they see it, without thinking. We learn children not to spell words when reading, this is exactly the same.  
On top of the tables, we should teach some extra, useful sums. 2x25, 3x25,4x25 are standard calculations in our society. It is also good to let them practice the tables of 11, 12, 15 and 20. If only to calculate +10%, +20%, etc. Think what a paradise the world would be if people were good at calculating the effect of budget increases and price reductions!
If people know these, they can do `first digit' calculating: round all numbers to 1 digit, know the answer by heart, think about order of magnitude (number of zeros), round to 1 digit ( more is fake accuracy). I think this is the only level of arithmetic that people can really learn to do with the same ease as reading.

If you combine the above techniques, you get what I call the `good enough algorithm for multiplication' Many people use some version of it for quick calculations.

 I assume everyone knows the standard algorithm ( in 432x67: 7x2=14, write down 4, carry 1. 7x3= 21, add carry gives 22, write down 2, carry 2, etc. When finished with 7x432, write down a zero below this, and do 6x432 in front of it, you know the drill)

The `good enough algorithm' is this:
Round both numbers to 2 digits. I.e. 430 and 67
Calculate first digit times first digit, including order of magnitude.
So : 400*60 = 24000. Write down ( preferably, compare the order of magnitude with the `first-digit method, 400*70=28000)
Do second digit of number times first digit of number two.
30*60 = 1800. Write down
 Second digit of number2 times first digit of number1.
7*400=2400
Add all numbers, round to 2 digits. DO NOT calculate second digit times second digit, it is too small to be significant.
Result is 28000. The correct answer is 28944. Accidentally, the answer is the same now as the first digit method, but that is a bit of coincidence. This method is guaranteed to be good to within a few percent, the first-digit method is not guaranteed to be good.

Upsides of this algorithm: it is fast, especially on paper, doesn't require error-prone carries, and you start with the most important numbers.
Downsides: it is not exact. If you want exact, use a calculator ( or the standard algorithm). Also, you have to be aware of the order of magnitudes during your calculation. But that is in my opinion not so bad, spend the time gain on thinking about magnitudes, and the change of errors is small)

OK, enough about estimation. There are loads of other techniques, especially for division.

3. Types of numbers.
Already mentioned, I think. Start with integers, including zero. Let them learn the table of zero. Treat like all other numbers. Add negative numbers.

In my opinion, decimal numbers should come before fractions. People understand and `feel' decimal numbers in a way they don't `feel' fractions. Besides, decimal numbers are a special case of fractions. Once they are introduced to fractions, they will see infinitely repeating fractions like 1/3=0.333... . That might be enough in the direction of real numbers for all normal purposes. ( real numbers are `numbers with infinite decimals', like points on a line)

Focus on the meaning of numbers: you can't add apples to pears; 1000 apples is an integer, that is a counting number. But 1000 meters is a measured value, and will be 'imprecise' (once again, a hint at real numbers). Values usually need units. 1000 meters is 1 kilometer. When you multiply or divide, units change too. 50 kilometers in 2 hours is 25 `kilometers per hour'.  10 pizzas for 7 people is ten-seventh `pizzas per person' and is approximately 1.4 `pizzas per person'

In the case of fractions, my experience is that not-mathematically inclined people have enormous trouble with them. I suspect the far majority of people will never calculate 1/8 + 1/7 in a situation were transforming to (approximate) decimal numbers is not sufficient. Besides, people who have trouble with adding fractions will have no idea what to think of 15/56. Is it more than a quarter? Let them use decimals for complicated fractional calculations.

What is important in my view: Converting to decimals; Understanding that 2/4 is 1/2; 1/8 is smaller than 1/7; The equivalence between fractions and division. Especially this last part is often not fully understood by people, many people really have to think about `2 times 5 divided by 3 is the same as two-thirds times 5'.

4. Basic geometry. What are curves, areas, volumes. Area of rectangle, volume of rectangular beam. What is an angle. Sloped side of a straight-angled triangle is shorter than its straight sides combined. Pythagoras is nice, but doubt if people really -need- it.

I guess I am forgetting important things here, but guiding principle should be: do people really need it? If not, it is perhaps fun and interesting, but for people with little mathematical feeling, it is not fun, not interesting and only distracting from the stuff that is important. We should be careful not to throw the important stuff and the fun stuff in one basket, so that people call it all `complicated math' and forget about it.

5. Selected advanced topics. Outside of arithmetic, there are some areas that are important.

Basic, very basic algebra. Numbers represent values, and those values can also be represented by symbols. People understand this part easily. It's a miracle, but everyone understands that the letters GDP stands for a number that represents a monetary value ( a lot less people realize it is represent a flow, money per time, but that is another story).  The hard part is showing that you can manipulate the symbols, and use that to get  answers about the values. There is a lot of algebra people understand if you fill in the numbers, but not in pure symbols.
Still, people should understand `if a=bc, then b=a/c', and `if a= b+c, then b=a-c'. Preferably they understand that you divide both sides by c. But still, it would already be great progress if no university-level educated person got scared from simple formulas.

Next: logic. Venn diagrams should be taught to everyone and their relation to normal, language based arguments. For those who do not know them, imagine a circle representing all dogs, and one representing all animals. The dog circle with be entirely in the animal circle, showing that `all dogs are animals', but `not every animal is a dog'. Draw the circle of `long haired animals'. The overlap between dogs and long-haired animals are `all long haired dogs'. You can now easily talk about `long haired animals that are not dogs', which is the rest of the long-haired circle. Add a few extra overlapping circles, say `objects heavier than 10 kilos' , and the language to describe them would be horrible, while the drawing is still clear.)

Also: graphs and functions. A function is a relation, a graph shows the relation. How axes can be changed to make graphs look different( changing scales, `breaks' in the axes. I would almost add the `rule of difficult graphs' : if you are the intended target of a graph, and you do not understand the graph, the person who made the graph is probably a fraud who wants to lie to you. There is more useful stuff here. With practice and many examples ( graphs!), even the most innumerate person can understand important things about functions, such as inverted function ( swap the axes), one-on-one functions (if you swap the axes, you have a nice function) , one-on-many relations( if you swap the axes, some points have multiple values, many-on one(3-d graphs, people understand these surprisingly well). I would leave many-on-many relations out of the picture, (if only because they are, well, hard to picture)

My last, and tentative suggestion would be `introduction to statistics', a sort of `sex education for new paper readers'. This course would have no math in it, just an explanation what can and cannot be done with statistics, how uncertainty works, an explanation of correlation (without math), that it doesn't tell of the direction of causation, and might be an artifact of the data.  

These are my suggestions for the minimum level of numeracy at 15, to be taught to everyone. Of course, most of it is already in normal curricula. The important part in my view is to find the tings that are NOT critical for everyone. Its my view that a clearer distinction between important-for-everyone math and math-for-people-who-are good-at-math would be useful, to prevent the people who have trouble with it from forgetting the useful stuff together with the things they will never use again.

In short, I have 3 main points:
+Core numeracy should be aimed skills that people will use in normal life.
+Arithmetic should focus on quick grasping of approximate answers and order of magnitudes. This should be so ingrained that people don't think about using their calculator. Also, the meaning of numbers is important
+ higher skills should be aimed more at `passive consumers of math', who are not going to do complicated math themselves, but will read about it in the newspapers.

by GreatZamfir on Tue Nov 13th, 2007 at 07:16:10 AM EST
[ Parent ]
Excellent comment!

Also, you have to be aware of the order of magnitudes during your calculation.

I realise now that I have problems with this.  Are there any quick methods?

(I note that 4x7=28, 40x7=280, so it's add all the zeros and add to the right, no?  40000x7000=28 plus seven zeros?  Heh...anyone got a link to some quick rules about gauguing magnitude?)

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 08:01:26 AM EST
[ Parent ]
In this particular field I think you should push beyond tricks ( while I beieve that in many (maths) subjects, tricks are actually quite good to start with)

42 means 4x10 plus 2x1. That's the basic principle.

So 7x42 means 7x4x10 + 7x2x1 = 28x10 + 14x1 280 +14=294

The crucial step is that the 28 gets multiplied by 10, which is the same as adding a zero to its right. Your 'count the zeors' rule would fail, because the zero behind the 4 is 'hidden'

For the large numbers, best way is to switch to scientific notation. It will take a bit of practice, but then your lucky for life.

40000 is then written as 4e4, meaning 4x10x10x10x10, so 4 tens, which is the same as 4 zeros, is the same as '4 times tenthousand'.
42000 becomes 4.2e4, or 4.2 times tenthousand ( makes sense, right?), so the e4 is the amount of zeros behind the first number.
7000 becomes 7e3
4.2e4 x 7e3 = 4.2*7 x 1e7
29.4e7=2.94e8. Since a million is 1e6 and a milliard is 1e9, 2.94e8 is 294 million. So now you can just 'add the zeros', by adding the number behind the .e., and then changing the number back to format 'one digit in front of the dot'

Another way to think, that you should learn at the same time, is to group in powers of 1000 ( below they call this the rule of three, I think).
1000 is 10x10x10, or 1e3. In prefixes this is 'kilo-'.
The next one is 1000x1000= 1e6= one million, prefix 'mega';
next 1000x1000x1000=1e9= one milliard (or one billion in America), 'Giga-';
1000x1000x1000x1000=1e12= one billion ( one trillion in the USA), 'Tera-';

After you group like this, 42000 is 42 thousand, or 42000 Watt is 42 kiloWatt, 7000 is 7 thousand, and the answer is 42x7 x thousandxthousand= 294 million.

It also works the other way around: 0.042 is 42 thousandth ( 42/1000), or 'milli-', so 0.042x 0.007 is 42 millithingies x 7 millithingies = 294 microthingies, meaning 294 millionth

All strategies come down to same thing: separate the calculation of numbers from the calculation with the powers of 10, or powers of 1000. The thousand, million,milliard method has advatage that it works nice with our language system: '42 thousand' has more meaning to us then 4.2e4 ( pronounced '4.2 times ten to the power 4'). However, the downside is that you still have to take the parts smaller than a thousand into account ( 42 thousand times 70 thousand is 42x70 x million, meaning you still have to do the 42x70 part)

Last, and simplest check: when multiplying two integer numbers (integer is whole , without decimals behind the point), the answer has as many digits as the two numbers combined, or the same minus 1. Minus 1 happens when the multiplication is not 'pushed over the ten': compare 2x42=84 and 7x42=294, because 2x4.2 is smaller than 10 and 7x4.2 is larger than 10.
But do not use the 'count the digits' rule unless you see how it fits in with the other ideas.

by GreatZamfir on Tue Nov 13th, 2007 at 09:29:25 AM EST
[ Parent ]
I'm of the belief that fractions must be taught before decimals.

Firstly, basic fractions aren't that much less intuitive than decimal ; see the American habit of using quarter pounds, eigth ounces, etc...

Secondly, and especially given the average incompetency of the primary school teacher at maths, it leads to pupil unable to understand that 0,333 is not equal to 1/3. The true representation is the fraction, not the decimal. And it is necessary to understand that the approximations of decimals (an infinitely repeating decimal is a much harder concept than a fraction) go hand in hand with those of calculators. It becomes very hard to do once you're in the habit of using decimals.

Also, a lot of people put the emphasis on "interesting" maths for the kids. A big problem is that much of maths is taught too late, to kids aged 11-16 rather than younger kids, 8-11, who could learn a lot, and are much more motivated. Long divisions used to be taught to 8-9 years old in France ; now it is taught only to 11-12 years old, and the result is that it is too late for most to get into maths.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:03:45 AM EST
[ Parent ]
Continued fractions should be taught before decimals. Which they can be if people have been taught the Euclidean algorithm to deal with divisibility and prime factorization.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:21:40 AM EST
[ Parent ]
Methinks decimal points seem more intuitive because of... heavy use in adulthood. IOW, I'd guess fractions are more intuitive for young children who just learnt numerical signs.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 09:01:25 AM EST
[ Parent ]
Long divisions used to be taught to 8-9 years old in France ; now it is taught only to 11-12 years old

Huh!? I think I learnt it in second grade. What the hell do they teach now for five long years? If drilling multiplication tables is also off the plan?

*Lunatic*, n.
One whose delusions are out of fashion.

by DoDo on Tue Nov 13th, 2007 at 09:05:46 AM EST
[ Parent ]
They don't teach anything any longer. That's old-fashioned and authoritarian.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 09:06:35 AM EST
[ Parent ]
You are right about basic fractions being easy, but I think that might only apply to the very few fractions that have a 'name', a half, a quarter, a third, a tenth, perhaps a fifth and an eighth, depending on the language. These feel like 'units', and behave more or less like apples.

But to move beyond that is immediately complicated, you will have to explain the concept of 'three-eleventh' and 'two-sixth is the same as one-third'. I get the impression children find this doable for powers of a half, but not for all other fractions.

So, I think both decimals and fractions are not easy to explain. But in my opinion decimals have become much more important in society, where even most fractions are expressed as percentages.
The second reason I promote 'decimals first' is that I think, as you might have noticed, that people in general have a bad grip on the order of magnitude thing/powers of ten thing. I get completely sick of people who 'do the calculation correct, except for a factor ten'. The factor tens are more importnat than the de details, and people don't 'feel' this.

My suspicion is that children learn that 123 is larger than 45 'because it has more digits', not because the 1 alone is worth more than 45. This problem becomes clear once people start using decimals Problem number one for all children: 0.25 is smaller than 0.3, even though 25 is much larger than 3.

I think the principles of decimals are important for integer numbers too, and should be taught as one single system. That is: the first digit and its position are important, the rest is adding detail. Children HAVE to understand this to be able to work with decimal fractions, and I think the lessons of these should be applied to integer numbers.

At the moment, children learn the opposite: they apply  the rules for integers on decimal fractions. They learn '0.3 is really 0.30, and 30 is larger than 25, so 0.3 is larger than 0.25'.
So, I would like to introduce decimal fractions as early as possible, not as a 'special' system for small  numbers, but as extension of the normal decimal system, so that people will have to learn the real meaning of the decimal system.

by GreatZamfir on Tue Nov 13th, 2007 at 10:03:16 AM EST
[ Parent ]
one with which I agree ;)

Our child's 3rd grade teacher is an asshat, teaching from NCLB mandatory texts, modified by her colleagues into loose-leaf worksheets that introduce topical methods and vocabulary from algebra, statistics, estimation, geometry, unit measurement (vol, wght, linear, currency)-- excluding applications of multiplication and division. In fact, recently her teacher informed me that the class would be instructed in how to find the area of 2-3 plane figures. When I inquired how this might be achieved exclusively by diagram and to what end without application of arithmetic formula, she told me she didn't know because she hadn't reviewed the lesson plan. uhhhhhhhhh....

Needless to say, I'd anticipated such a response (due to previous conflicts over scale for a "bar chart", percent "estimate" of trees in a google map, etc) and had already instituted homeschooling each week to fulfill conceptual gaps in the standard maths 'curriculum' as well as computational drills with real (positive) numbers.

At this juncture, the greatest challenge for me is helping my child to transform her competencies manipulating a whole number into a comprehension of operations on that object. We've had quite a few discussions about sybolism, i.e. decimal and fractional notation, and the "accuracy" of value placement (modulo or "mixed number") denoted by each. Such conversations would not be possible had we not first demonstrated long division of whole numbers, e.g. 2/3 or 2÷3, and "divisions" along a ruler and an arc. I am optimistic. I do not want to die, knowing she's still counting on her fingers or a calculator.

Thanks for you support.

Diversity is the key to economic and political evolution.

by Cat on Wed Nov 14th, 2007 at 11:47:26 AM EST
[ Parent ]
To put it differently, you mention the importance of seeing the difference between 0.333... and 1/3, and you say that 1/3 is the true representaion.

Now, this concept of a 'perfect third' is a useful mathematical concept, and good for children who will do more with mathematiccs in the future. But 95% of people will never do anything mathematical, they just want divide real things by three.

ANd in reality, there is only two things you can divide by three: integer multiples of 3, like 90 people in 3 classes, where you do not need fractions, and 'real valued' numbers, such as dividing one meter in three equal pieces. Since such a division will never be exact, learning that '1/3' is the real value is just as pointless as learning that is is 0.333. But the concept that, no matter how precise you divide, it can always be accurately approximated by a decimal fraction is, I think, pretty fundamental.

In other words, decimal fractions are approximations of pure mathematical fractions, but in real life pure mathematical fractions ( just as almost all mathematical objects) are only approximations to an even more complicated reality. Decimal fractions may have less mathematical rigour, they are more versatile in describing reality.

God may be real, unless He is integer. But He is not a fraction

by GreatZamfir on Tue Nov 13th, 2007 at 10:16:42 AM EST
[ Parent ]
Vehemently disagreed.

I disagree with the view of education as teaching things people use "in normal life". (What is normal life anyway?) I see knowledge as value, even if seldom or never used.

More specifically, when one learns mathematics, one doesn't just learn the symbols and algorythms at hand, but practises abstract thinking, too. If we always rely only on 'known', 'every-day' things, pupils' mind will forever remain inside the 'every-day' box. (And that's a very small box.) It's useful at the beginning, to get the gist of it, but later I think people should get a sense that there is something other.

Even more specifically, doing long calculus might be useful to get people appreciate calculators, if nothing else :-)

I think the bulk of your core numeracy should be stuff for every 10-year-old to know, not every 15-year-old.

*Lunatic*, n.
One whose delusions are out of fashion.

by DoDo on Tue Nov 13th, 2007 at 08:59:28 AM EST
[ Parent ]
I disagree with the view of education as teaching things people use "in normal life".

I don't think it's exactly this, it seems to me more like: "To understand this abstract point, lets see how--if at all--it might create effects in normal life."

I think, also, that there should be a distinction between

"Unlikely to be used in your day to day business--but worth learning"

and

"Impossible to conceive of using a "normal life" frame, but worth learning nevertheless (a mind expansion tool?)"

Something like 10 to the minus five hundred and seventy two would for me fall between the two in that I can sort of get there from my normal frame (teaching magnitude by something getting smaller--starting with "1" being "my size", so 1.0 e-7 (heh...please go gently) would equate to...something I can't see but might have heard of, and smaller and smaller until we're getting into the realms of "impossible to conceive", which is where I wonder about the role of "visual models"--if a model isn't internally visualised somehow, can it be held in mind?  I think so, but I'm not sure.  I can't think of an immediate example.

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 10:30:47 AM EST
[ Parent ]
Can you conceive a regular polygon with a thousand sides ?

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 10:33:31 AM EST
[ Parent ]
I expect someone could produce a picture of a many many sided polygon--and as you kept adding sides to produce what would--to the naked eye--approximate maybe a circle--

I could invisage that if I were close enough I would see the edges, but that at a certain number of sides either the object would be too huge for me to see the whole (like the earth seen from this room) or if I am too view the object, the sides will be too small for me to see...

I think I'm maybe missing something.  I am thinking of abstraction as being "beyond where we can conceive", yet if I can't picture a model in some way, I'm not sure what I'm conceiving.

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 11:10:43 AM EST
[ Parent ]
You can write the words, "a regular polygon with a thousand sides". With a bit more computation, you can compute its perimeter, its area, the angle between two sides. You could try to program a computer to print one. You can know mathematically everything about it. You can conceive it, simply not visualize it.

Of course a polygon something of the Euclidean plane, i.e. a branch of maths where we can usually visualize what we conceive. In 5 dimensions geometry, visualisation is nigh impossible. But we can still conceive stuff, in the way we can conceive a 1000-sides polygon.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 11:28:46 AM EST
[ Parent ]
We can. But the unstated implication here is that this kind of mathematical conception maps - in some unspecified, nebulous way - to other kinds of conception. And that's a Good Thing.

But does it? And is it? I'd like to see some empirical evidence to back up this idea. (At least, I would if I knew for sure what's being claimed here - beyond 'We should teach this because it's obviously good' - which seems a little extravagant to me.)

So far it's being stated as if it's self-evident. I don't think it's self-evident at all. And - as I said in an earlier comment - if you're trying to teach abstraction and conception, there are other disciplines you can use.

Maths isn't the only option, and it's not necessarily the best tool for the job.

by ThatBritGuy (thatbritguy (at) googlemail.com) on Tue Nov 13th, 2007 at 11:38:11 AM EST
[ Parent ]
As for the advantages of teaching abstraction through mathematics rather than Latin, it is that Mathematics also have practical applications in physics, bioloy, social sciences... that Latin may not have.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 12:03:48 PM EST
[ Parent ]
You can't move as far from common-sense reality with music (or anything else) as you can with mathematics. The point is to cut the tie.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 03:11:02 PM EST
[ Parent ]
You can't move as far from common-sense reality with music (or anything else) as you can with mathematics. The point is to cut the tie.

wow, what a comment...

do you mean maths can create a virtual tower of thought that's taller than with any other tool?

how do you know what you say so absolutely is true?

as far as goals go, moving as far from commonsense reality sounds worthy to me, in a 19th C explorer kind of way..

off to the antipodes...back for lunch...

which is how all rg's diaries make me feel, lol...

i think science will ultimately discover much more benefit studying the fusion between mathematics and music, ( add healing and neuroscience too for the full shebang) than the nuclear variety, but that's just goofy me.

sure cheaper on hardware too.

how do you measure distance from commonsense reality?

a yardstick has to be a yard long!

if common sense really existed, perhaps we'd have a different world.

love your posts, dodo, really thought-provoking!

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty

by melo (melometa4(at)gmail.com) on Wed Nov 14th, 2007 at 07:54:16 AM EST
[ Parent ]
You just imagine a regular polygon with undetermined N sides, and then keep in mind that N=1000 :-)

If you want to represent a 1000-gon to someone, just draw a circle, and let the other look for sides and angles ;-)   If you bought a 2km by 2km piece of flat land, you can try to make a serious 1000-gon. Draw a circle with the radius of 1km; the width of the circle must be of order 1mm. If you have a circle perfect enough, start from a point on it, and then you should try to mark points on the circle each 6 m 283.175 mm in straight line (or each 6 m 283.185 mm in arc length). To make it in practice, try to draw each straight side so that it would deviate from the circle in the middle by 0.5 cm (more exact value is 0.4935 cm, if you care). Then call the Guiness Book of Records ;-)

by das monde on Wed Nov 14th, 2007 at 02:06:51 AM EST
[ Parent ]
I don't think it's exactly this, it seems to me more like: "To understand this abstract point, lets see how--if at all--it might create effects in normal life."

Well, here's the thing. When I was teaching assistant in an American university,  textbooks were fashionably peppered with "practical examples", which the students had difficulty with. I reached the conclusion that most of the difficulties had to do with the students' inability to understand the topic from which the example was drawn, not any difficulty with the new mathematical technique being taught to them. So I don't think teaching "from applications" works either.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 11:37:54 AM EST
[ Parent ]
I think there is an important distiction between things every 15 year old should know, and things every intelligent 15 year old should know.

Of course, you can't teach these things to literally everyone, but the original post was about numeracy as an equivalent to literacy.

I was thinking of someone who can read at the level he or she could read the Da Vinci Code, but would not think it worth the effort. That's more or less the level we call 'functionally literate', and this is a suggestion for 'functional numerate'.

Sure, I would love to see more math than this, and all kinds of beautiful ideas, and how they are related to each other, and what not. But I am afraid I know quite a lot of university students ( in the humanities sector) who are not quite up to the level I have described here. Just think how many people you know who ould skip
y=ax+b
in a text as being 'complicated maths', and would never, unless you told them, realize it was a straight line.

by GreatZamfir on Tue Nov 13th, 2007 at 11:11:20 AM EST
[ Parent ]
Your comments--much appreciated, GreatZamfir.  I should state that I am exactly the sort of person "who would skip
y=ax+b
in a text as being 'complicated maths', and would never, unless you told them, realize it was a straight line."

Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 11:26:33 AM EST
[ Parent ]
Well, in my view, that is more an indication of a bad education system than of people incapable of understanding maths. Calibrating curriculum to this creates a downward spiral.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 03:15:07 PM EST
[ Parent ]
Yes, probably education should be better. But increasing the goals of education is not going to magically raise the quality of it.

I bet rg did have linear equations in his curriculum. He probably passed tests where he had to find the coefficients, etcetera. But just like so many people, he forgot, and skips any formula in a text. And he is still someone who is interested enough in math to post this diary. Now imagine the non-interested!

Untill recently, I was all in favour of more math, more abstract math etc in curricula. Show people how interesting it is, how everything connects to everything else, how powerful a tool it can be for many applications.

Trouble is, such a program might raise the quality of exact sciences students later on, and perhaps attract a few extra interested people. But the rest would only be turned away from maths more then before.

I guess that if most people remembered and, more importantly, were at ease with the basics they have learned in present curricula, it would already be a great victory.

by GreatZamfir on Wed Nov 14th, 2007 at 02:53:07 AM EST
[ Parent ]
I would vote for the rule of three. I think it helps understand most things when you're able to bring them together.

Rien n'est gratuit en ce bas monde. Tout s'expie, le bien comme le mal, se paie tot ou tard. Le bien c'est beaucoup plus cher, forcement. Celine
by UnEstranAvecVueSurMer (holopherne ahem gmail) on Mon Nov 12th, 2007 at 10:22:18 PM EST
http://en.wikipedia.org/wiki/Rule_of_three_(mathematics)

Don't fight forces, use them R. Buckminster Fuller.
by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 04:04:46 AM EST
[ Parent ]
All, yeah for rule of three - my weekly maths exercise in the supermarket. grr, why do they have to use, pounds, kilos, grams, items just to confuse you.

Maths has to be taught in context (I think). As you know, I am watching the forth series of The Wire in the moment.
And one of the characters is trying to teach maths to the kids. A rather desperate affair, but he manages to teach them fraction, by tapping into their interest in rolling dice and gambling. It turns out to be very successful.  

That's how the basics of maths should be (and probably are - hopefully) being taught.

by PeWi on Tue Nov 13th, 2007 at 05:28:58 AM EST
[ Parent ]
I need to hash out a diary on that. I've been teaching math to kids in their early secondary schooling - more or less unwilling, because I was asked to teach them about science & geography, but the math teacher has dropped out for a while.

And I'm struggling. Math is clearly not my subject to teach. I caught their interest when we did percentages using candies, that seems to have stuck... But how to go on? I'm with the comments that a bare minimum of mental pain is required at some point - but I also want to explore how one can creatively "lure" learners into the deeper meaning of math using daily and visual examples.

I think I'll evilly convert the gist of the comments into a Teaching Math Workshop Diary...

by Nomad on Tue Nov 13th, 2007 at 07:14:53 AM EST
[ Parent ]
as they taught me in my far and few pedagogical lessons I had. Connect to existing knowledge and/or interest - transpose the kerygma(1) into the Lebenswelt(2). Use experience and knowledge to pick up the audience where they are at in life.

(1) kerygma here loosley translated around two corners as: object of lesson

(2) why, oh why did they stop using German as the universal language of scientific discourse?

Lebenswelt means so much more than Reality - world we inhabit, attitude towards the world, its reality - as distinct to the dogmatically impressed pre-emperical Un-Lebenswelt of the NeoConners.

wiki on Lebenswelt

Jürgen Habermas has also furthered developed the concept in his social theory. For Habermas, the lifeworld is more or less the "background" environment of competences, practices, and attitudes representable in terms of one's cognitive horizon. It is the lived realm of informal, culturally-grounded understandings and mutual accommodations. Rationalization and colonization of the lifeworld by the instrumental rationality of bureaucracies and market-forces is a primary concern of Habermas's two-volume Theory of Communicative Action.

Lebenswelt is of course also always Umwelt.

Der Begriff Umwelt wurde in der psychologische Wissenschaft von Jakob Johann von Uexküll eingeführt. Umwelt umfasst sowohl die "Innenwelt" als auch die "Außenwelt" und deren wechselseitige planmäßige Anpassung aneinander. Für Uexküll war Umwelt ein System, das sich durch die Beziehungen zwischen Subjekt und Umwelt ergibt.

http://de.wikipedia.org/wiki/Umweltpsychologie

The term "Umwelt"(environment) was introduced into the subject of psychological science by Johann Jakob von Uexküll. The Umwelt (environment) includes both the "inner world" and the "outside world" and their mutually planed adaptation of each other. For Uexküll Umwelt (environment) was a system that is characterized by the relationship between subject and its Umwelt (environment).
(this translation is mine - could a site gnom please insert the translation html into the Allowed HTML section, so it can copied from there, rather than having to open another window, hunt for the code and loose ones comment, due to foreclosure (: of the wrong window? (Thanks))

Now I might be accused of masching up terms here, but thats only due to time restrictions (-:

by PeWi on Tue Nov 13th, 2007 at 08:07:22 AM EST
[ Parent ]
why, oh why did they stop using German as the universal language of scientific discourse?

If it hadn't been for Hitler we would all be speaking German.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:23:10 AM EST
[ Parent ]
! Good one. I like counterintuitiveness... even if here some would argue that if Hitler hadn't been there we'd be speaking russian...

Rien n'est gratuit en ce bas monde. Tout s'expie, le bien comme le mal, se paie tot ou tard. Le bien c'est beaucoup plus cher, forcement. Celine
by UnEstranAvecVueSurMer (holopherne ahem gmail) on Tue Nov 13th, 2007 at 08:33:14 AM EST
[ Parent ]
A lot of the people that made Germany a world leader in most scientific and engineering fields either fled Europe for the US because of Hitler, or were recruited by the US after WWII.

History - Shame at Gottingen

In the spring of 1933, the University of Gottingen, the seat of brilliant achievement in years past, became the focal point of Hitler's anti-Jewish policies.  Student demonstrations proclaiming the coming of the "new order" became an every-day occurrence.  Respected scholars were brutally expelled.  Some of the world's foremost physicists such as Max Born, James Franck, Eugene Wigner, Leo Szilard, Edward Teller, and John von Neumann were forced to flee.

Attempts were made by "patriotic German" physicists to prevent the expulsion of so many "brilliant" men, but all to no avail.  Even well-known Germans such as Heisenberg and Nobel winners von Laue and Planck were unsuccessful in their attempts at mediation.

The clearest account of the state of the once-great Gottingen University was given by the mathematician David Hilbert, by that time well advanced in years.  About a year after the great purge of Gottingen he was seated at a banquet in the place of honor next to Hitler's new Minister of Education, Rust.  Rust was unwary enough to ask:  "Is it really true, Professor, that your institute suffered so much from the departure of the Jews and their friends?"  Hilbert snapped back, as coolly as ever:  "Suffered?  No, it didn't suffer, Herr Minister.  It just doesn't exist any more!"



We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 08:41:57 AM EST
[ Parent ]
You're talking about English as a scientific language. But english is also the ligua franca of business... Would'nt it be fun if the two were different though.

Thanks for the link.

Rien n'est gratuit en ce bas monde. Tout s'expie, le bien comme le mal, se paie tot ou tard. Le bien c'est beaucoup plus cher, forcement. Celine

by UnEstranAvecVueSurMer (holopherne ahem gmail) on Tue Nov 13th, 2007 at 10:40:04 AM EST
[ Parent ]
You wouldn't, trust me. Compulsory Russian education this side of the Iron Curtain was a joke.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 09:13:12 AM EST
[ Parent ]
Isn't this a factor of being hungarian ? I'd not be amazed if them Russians had gone under the assumption that Iron Curtain = Slavic people = Already close enough to Russian - and thus never put much emphasis on Russian learning.

I remember in 12th grade a Bosnian pupil who had gone away from the war, it was 1996. He was much better than us French pupils at mathematics, and used to browse Russian maths textbooks, despite not having formal Russian education.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 09:27:41 AM EST
[ Parent ]
No. Russian education was meant to counter the spread of English as lingua franca in the West (and from there East), and applied everywhere -- but didn't achieve high levels of Russian-speakers anywhere. I have read similar stories from Czechoslovakia. Russian education was compulsory from fourth grade I think, But people just learnt enough for the tests and then forgot it all, and the teachers weren't that pushing either. (Which I now regret.)

Now Russian math textbooks, since they were the top, no wonder everyone tried to browse them. The Bronshtein-Sememdyayev Handbook of Mathematics is still the one most hotly sought after by finishing highschool and college students.

*Lunatic*, n.
One whose delusions are out of fashion.

by DoDo on Tue Nov 13th, 2007 at 10:02:33 AM EST
[ Parent ]
Perhaps the tale of linca's friend has more to do with that fact that Russian is probably a lot easier to pick up if your native language is a Slavic one, as opposed to Hungarian, and less to do with just how mandatory it was.

Then again, everyone I met in Estonia (talk about weird languages...) spoke Russian pretty well.  Though Very Very Begrudgingly.  

"Pretending that you already know the answer when you don't is not actually very helpful." ~Migeru.

by poemless on Tue Nov 13th, 2007 at 11:27:06 AM EST
[ Parent ]
Unless a Serbian, he still had to learn or decrypt Cyrillic (or skip over it and focus only on the mathematic formalism).

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 03:18:46 PM EST
[ Parent ]
Learning an alphabet is the easiest part of trying to understand a language. Especially if Slavic phonology is already known, learning that "ch" is written with a certain letter rather than with a c with something on the top. Also, Russian has a straight phonology-spelling relationship, compared to many other languages...

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 03:27:48 PM EST
[ Parent ]
A monkey can learn Cyrillic.  It's really not very different from our alphabet.  Not like Arabic or Chinese or anything.  It's noun declensions and verbs of motion that will kick your ass

"Pretending that you already know the answer when you don't is not actually very helpful." ~Migeru.
by poemless on Tue Nov 13th, 2007 at 03:36:03 PM EST
[ Parent ]
Arabic isn't that hard unless you try to decipher calligraphy. The only hard writing systems to learn are the ideographic ones, and they are hard for the native speakers too.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 04:10:03 PM EST
[ Parent ]
My point is that Cyrillic is very similar (in some cases identical) to the Latin alphabet.  I've never attempted to learn Arabic or Chinese, so I have no idea how difficult it is to learn those alphabets.

"Pretending that you already know the answer when you don't is not actually very helpful." ~Migeru.
by poemless on Tue Nov 13th, 2007 at 04:33:40 PM EST
[ Parent ]
The Arabic alphabet is very logical and easy to learn (I did it in a day), but the problem is that learning the alphabet alone doesn't really do much for you.  The vowels are implied, so if you don't know the word you're reading (or at least know the voweling patterns) then you can't even pronounce anything properly.  With the Cyrillic alphabet, at least you can sound things out right away, recognize place names, etc.  That takes longer in Arabic.
by the stormy present (stormypresent aaaaaaat gmail etc) on Tue Nov 13th, 2007 at 04:37:55 PM EST
[ Parent ]
Whereas Arabic has an alphabet not entirely dissimilar to the european one - after all, the first letters are Alif and Ba - Chinese doesn't have an alphabet. You pretty much have to learn a different sign for every single word, which makes it hard and tedious.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 04:38:16 PM EST
[ Parent ]
My east german friends and relatives all had russian as first language (some never learned any english at school), but they almost all refused to ever speak it - so I don;t think that it was truely successful, but then I had friends that travelled to Russia and their russian was at least as good as any average english pupil on the other side of the curtain who was traveling to the uk or the states (with the added complication that learning a gramatically complex language (russian) rather than a vocabulary rich (english) is always more complicated.)

I have to agree with the maths so. again my same age group relatives were far more advance in their maths and the maths books were also much better. I used some for my Maths exam preparation, came in very handy....

by PeWi on Tue Nov 13th, 2007 at 11:25:05 AM EST
[ Parent ]
My parents' generation too had some practical use of Russian, more than my generation: the Seventies-early Eighties were the time of barter trade and cross-border black markets. But that didn't require as high a level as I'd attest to the average German English pupil.

*Lunatic*, n.
One whose delusions are out of fashion.
by DoDo on Tue Nov 13th, 2007 at 03:25:29 PM EST
[ Parent ]
But without the War the Russian empire would have been different too. Anyway those counterfactuals can't lead us very far.

Rien n'est gratuit en ce bas monde. Tout s'expie, le bien comme le mal, se paie tot ou tard. Le bien c'est beaucoup plus cher, forcement. Celine
by UnEstranAvecVueSurMer (holopherne ahem gmail) on Tue Nov 13th, 2007 at 10:41:03 AM EST
[ Parent ]
the cartoon at the top is pure genius, lol!!!

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty
by melo (melometa4(at)gmail.com) on Tue Nov 13th, 2007 at 02:42:44 AM EST
Wow!!!

great diary..incredible comments... let me sketch why one would need 3 years of 20h per week of math and logic...

First regarding logic since it is the easy part..

-kids must learn the table of logics. Logic gates and all that stuff.. they must know how a computer works...and when soemeone is saying something inconsistent. if a kid does nto know how a computer works he willbe hopeless in the future... in the scientific area.. he/she will think it is magic and in the political area they will believe the pundits... argghhh

-Kids must learn how solve problems by templates and formulae.. there are comments of people here who could not understand deeply a porblem but they could solve it by template... a kid MUST be able to do that.. if they can NOT do that.. they will never be able to do it everywhere. If they DO, one day when they are older they may ask themsleves what they are really doing and why.

-And finally they can study logic structures whcich do not depend on a real object, conmutative algebra, nonconmutative algebra, vectorial spaces.. this just to study abstract ideas for the first time. i think, thsi should be their first abstract ideas..a nd not a what a philosopher said long time ago.. first these abstract ideas.. after that you can proceed with serious history or physics

And training logic and templates is a hard thing to do...

And now maths... 3 years to leanr

-Four basic operations... and at ease... and repeat everyday problems ... multiply substract and fast... practice, and do it fast. tricks to multiply by 10 or divide by 10. You need it working everywhere and everyday..a dn kids know it.. event he drop outs ask their teachers to explain that..e ven the ones going to prison. they REALLY WANT it.. so repeat adn repeat and repeat..a nd sicne the drop outs know about it they learn adn build confidence... and do it as they should do it fast..

-Percentages and the rule of three.. if one kid does not knwo how to make conversion among units and udnerstand deeply what a percentage means.. and specially waht a percentage over a percentage mean (10% of 45 % of something) he wil not be able to have a domestic economy... and a lot of kids just do not know.. and a lot of adults just cannot plan ahead the expenditures (really!!!) or understand the money they will get inthe bank for an account. Actually I ahve seen journalists reprots where they gave completely stupid information (and wrong) becasue they do not understand what a ratio really means!!!

-trigonometrics.. basic.. headed to have visual intuition... what is really an angle, when two angles are equal, how one relates different areas .. why an area is "square" meter?.. sinus, cosinus, tangetn...how are they going to buy a flat? I guess they will never work the land.. other wise how are they coing to measure area? why angles are related to vision  How the glasses work...? all the basic stuff

-Solve basic equationa and be able to verbalize them... They must be able to transform any everyday problem in polynomial algebra... and then use the template they know to solve it...Obtain reaults from partial information.. and ehre is when the idea of thinking by oneself comes.. if you are not able to gather raw information and see if its enough to get something you need... then you can not think by yourself. So one of the best think tot train  that is by writing system of equations.

-Real numbers functions. One has to knwo what a function means, how to plot.. and practice with thousands of fucntions out there... form the prices of cars, to eveolution of human populations.. and being abel toa nalyze it... knowing when it grows or slow down (derivative).. observing asymptotic behavior and such...

And that's it.. at least thsoe areas.. if they do nto control that.. they will be completely lsot in their lifes .. or lost on some important areas of their lifes.

This is the minimum topic they must address.... then there is the question about complex numbers (do they really need it in everyday life? ..and the same for real spatial vectors...this is about applying the vector space structure learnt in logic to compute lines , planes, intersection... I am not sure if this two topics should be there...so any opinion?.

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

by kcurie on Tue Nov 13th, 2007 at 06:00:15 AM EST
In eprcentages and rule of theree I include fractions by default, of course.

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

by kcurie on Tue Nov 13th, 2007 at 06:01:49 AM EST
[ Parent ]
This makes me feel a little better. Apart from logic and templates perhaps, I can do most of these things.

I can deal with my finances and taxes and I understand ratios and percentages and trig, how to estimate, and how to plot stuff and understand trends and to apply this to every day stuff.  I function fine in real life. I can cut through the crap of manipulated data in the media.

I function well enough to scrape through the maths bit of my physical chemistry PhD but I know it's not good enough.  

I dread being asked in my viva to show how such an such an equation on P and Q forms or scattering constants gives me the size and shape of my micelles.  Someone else has proved it, it is widely accepted and if something isn't working with applying the formulae, I'll ask a mathematician if we need to review it.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 07:38:00 AM EST
[ Parent ]
I think you can use templates to solve problem... (if you find in this situation..then do this and this.. and if you find this at this step then do this.. like a program code).

Regarding logic and logic gates..then basically you do not know how a computer works? If you do not know I think it will be very funny to look at it..

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

by kcurie on Tue Nov 13th, 2007 at 08:05:50 AM EST
[ Parent ]
I remember doing logic gates because I loved electronics and I know I understood it at the time. off the top of my head now I am not sure I know what a logic gate is - yes, no, if, and, or (and mixtures of that, right)?

With a lot of maths things, the words mean nothing, but if someone shows me this is calculus or trig then I'll remember.  For some reason I am not good at retaining certain details, like what things are called.

How much of what you absorb as a child but can't recall consciously as an adult is actually still there and in use as a fundamental piece of knowledge without even realising it?

eg if you ask me to find 8 words or phrases that mean the same as any word of your choice, I can probably not recall more than 2 or 3.  But if you gave me a list of 10 things that had the same meaning, most likely I would know and understand them all.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 08:19:03 AM EST
[ Parent ]
yes , it is exactly that..so you know what all kids would need to know for having a fruitful life.. but since you already have it, I doubted you did not know all of the stuff up there.... :) actually you know mch roe than that above.

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

by kcurie on Tue Nov 13th, 2007 at 01:29:58 PM EST
[ Parent ]
I totally agree with your points, probably including the 'first abstract concepts, before any philosopher' part.

However, at what level of student are you aiming? A very significant part the population will never study any philosopher, and I doubt they would study 'abstract logical object that do not depend on real objects'.
Trigonometric functions are another subject that will be hard to teach on any level but 'tan(x) is a button on your calculator'.
I am not sure your real function part would include differential calculus? If so, that is another subject I would not include in 'basic numeracy'. It is important, but you can definitely live without it.

I think your direction is very good, but you are going way beyond the level that would be comparable to 'functionally literate', and more towards the level comparable to 'appreciating high literature'.

by GreatZamfir on Tue Nov 13th, 2007 at 11:33:12 AM EST
[ Parent ]
Well, when kcurie and I were in high school 15 years ago, anyone who stayed in school past age 15 encountered philosophers, the trigonometric circle:

as well as derivatives, sequences and series, logarithms, e, and some analytic geometry. What? Too much? Yes, probably rather difficult for most. But they got past it.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 11:49:45 AM EST
[ Parent ]
Did you really study sec, exsec, versin, csc ? I had never heard about those...

(And I'm afraid of the amount of trigonometrical formulas you might get with those added to the mix)

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 12:02:17 PM EST
[ Parent ]
Yes, sec and csc. The other two I hadn't heard about. The Wikipedia article explains why they are obsolete.

If you know how to transform things you don't need to remember all that many trigonometric formulas either. But that's another issue: at that level it is true mostly they force you to learn a bunch of formulas as if they were all independent from each other.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 12:06:00 PM EST
[ Parent ]
Actually, I learned the formulas in Classes Prépas, where they are appropriately demonstrated and understood, just needed as shortcuts to fast computations - that is another problem with maths education : different people do maths at different speed, yet faster isn't necessarily better.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 12:15:11 PM EST
[ Parent ]
Yeah, here in Holland the system sends children to different levels of education pretty early on, after they are 12. I am not sure how good that is. What you are describing sounds more or less as the Dutch HAVO, which is the 'middle' direction (although half of all people are in the easiest direction, which has sublevels in itself).

Big question is, how much do they remember afterwards? I fully agree that people should be shown examples of things they might not understand, to stimulate them and in case they are smarter then they appear (I think the Dutch system is weak here).

But deciding what things should be taught to children is not the same as deciding what they really should know, even years later. My fear is that many people shun maths, because there are too many parts they don't even start to understand. Trig is stereotypically a subject that leaves people with an impression of 'it had to do with triangles, or circles, and something with your calculator, and pi, yes pi was also important'.

By the way, I learned a slightly different circle

by GreatZamfir on Tue Nov 13th, 2007 at 12:15:39 PM EST
[ Parent ]
I stole that circle from wikipedia, I also learned a simpler version.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Tue Nov 13th, 2007 at 12:58:29 PM EST
[ Parent ]
GreatZamfir:
Trigonometric functions are another subject that will be hard to teach on any level but 'tan(x) is a button on your calculator'.

Makes me wonder how any of us learned trigonometry before the 1980's ;-)

but seriously are we reaching apoint where most of the teachers never learned anything other than 'tan(x) is a button on your calculator' so don't have the conceptual tools to teach anything else to their students?

Any idiot can face a crisis - it's day to day living that wears you out.

by ceebs (ceebs (at) eurotrib (dot) com) on Tue Nov 13th, 2007 at 12:35:12 PM EST
[ Parent ]
Yes, good question! How did people learn trigonometry before the 80s? I mean, you can use the concept and play around with it, apply trig relations and stuff, but without a way to actually evaluate the terms, it must feel a bit pointless?

I know people used to have tables, but were they used in schools? Or did you just use the 'easy angles', like 30 degrees? Taylor series?

People are going to laugh at me now, but please, illuminate me!

I just realize this might explain the cot and csc functions that no one uses anymore. Without a 1/x buttton they actually have a use...

by GreatZamfir on Tue Nov 13th, 2007 at 12:45:43 PM EST
[ Parent ]
Tables. Though I think we just used easy angles before age 15 or so.
by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 12:47:07 PM EST
[ Parent ]
And that was late eighties. I don't think I was allowed use a calculator in state exams in 1988, though I wouldn't swear to it.
by Colman (colman at eurotrib.com) on Tue Nov 13th, 2007 at 12:49:22 PM EST
[ Parent ]
well it was mid 1970's when I was first introduced to it and it was just in at the deep end with tables and a lesson on how they worked

Any idiot can face a crisis - it's day to day living that wears you out.
by ceebs (ceebs (at) eurotrib (dot) com) on Tue Nov 13th, 2007 at 12:56:11 PM EST
[ Parent ]
We had little books of tables.

Don't ask me any more, I never knew what you were supposed to do with them.

by afew (afew(a in a circle)eurotrib_dot_com) on Tue Nov 13th, 2007 at 01:05:06 PM EST
[ Parent ]
How did people learn trigonometry before the 80s?

Surely you jest?

Firstly, we learned about measuring angles in radians. Then we learned the definitions of the various functions, and the associated geometrical drawings and the shapes of the functions. Then we memorized the trigonometric identities. Then we used our tables of logs of trig functions and our slide rules to solve problems.

No button pushing required, although slide rules* come in mighty handy.

*Also very useful in learning about decimal notation, logarithms, estimating, and significant figures.

by asdf on Tue Nov 13th, 2007 at 10:07:47 PM EST
[ Parent ]
No, I wasn't jesting. I realize there are many options, I just wondered which one you used. I always thought that slide rules were quite expensive ( probably because the expensive ones are the only ones that survived the big onslaught).

Then I thought that perhaps you might never have solved  problems but just learned the theory, which would explain   why old people tend to consider trigonometry as a synonym for 'abstract things'.

But please tell me, when the olden days used tables and slide rules, how exactly are calculators bad for education ( not saying you believe this, but many people claim this)? If anything, tables are even less instructive than calculators ( and definitely less than computers that can plot graphs!)

by GreatZamfir on Wed Nov 14th, 2007 at 03:01:53 AM EST
[ Parent ]
Well, definitely graphing calculators and computer software for geometry and calculus could be used to great effect in a classroom. But somehow all the courses I have been involved in that used computers entensively have seems kinda sluggish and clumsy.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Nov 14th, 2007 at 03:10:37 AM EST
[ Parent ]
I think this might be a fundamental economical problem. People who have any skills with these kinds of things (maths and computers and the ability to explain them to laymen) can earn good salaries elsewhere.
by GreatZamfir on Wed Nov 14th, 2007 at 04:22:56 AM EST
[ Parent ]
Yes, that is a major part of the problem. The other half is that teaching the uninterested is a thankless job.

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Nov 14th, 2007 at 04:42:34 AM EST
[ Parent ]
tell me, when the olden days used tables and slide rules, how exactly are calculators bad for education

There's a huge difference. To use a slide rule for multiplying and dividing, the first thing you have to do is convert your problem to one where all the numbers are between 1 and 10, with an exponent. Then you calculate the product with the slide rule and the exponent by adding the exponents. So even to do simple problems you have to understand--or at least use--exponential notation.

To use a slide rule to do trig problems, you have to understand how the functions change in the various quadrants (not taken into account by the slide rule), and you have to be fluent with at least three more scales on the rule. In addition, since all of the trig functions aren't on the rule, you have to know the relationships between them.

Since the slide rule itself works by logarithms, you get good experience working with them, and you can do things like converting between bases--if you know enough about how the slide rule works.

The calculations might be in error, so you have to do estimation to verify your work, and the rule itself is only accurate to about three figures, so you get lots of practice interpolating. Also there is the question of manual dexterity.

Finally, slide rules aren't expensive. There have been cheap plastic slide rules since at least the 1960s, and they're just as good as the fancy bamboo/ivory slide rules that the engineers used.

Tables are a bit of a cheat, but really all you get from a table is more accuracy. You still have to understand how logs work, because you don't have a table of trig functions, but a table of logs of trig functions (so you can do multiplication). Interpolation is important, and you still have to have a good understanding of the various trig identities because you usually don't have tables of all of the possible functions.

Nowadays, just press a couple of buttons and you get the wrong answer in an instant!

by asdf on Wed Nov 14th, 2007 at 09:12:11 PM EST
[ Parent ]
At the level of 15 trigonometrics is a must. i did at 15. precisely.. no problem. And I think it is really basic for your life.. and to see an example of what is behind those numbers int eh calculator.

Actually we also had the e number and the logarithm at this level (at the age of 14-15).. so trignometric functions ad the exponential fucntions are must see.. otherwise they do nto udnerstand what osclation int he market means.. or exonential growth means...

derivatives... mmmm...... this I really not know... I would say that the idea of limits is really encessary.. but use the limit to udnestand the derivative of a fucntion as another function.. no. you are right.. but to know if something is growing by comapring two adajcent clse number.. that for sure.

All in all we agree pretty much.

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

by kcurie on Tue Nov 13th, 2007 at 01:39:11 PM EST
[ Parent ]
I'm sorry (and I may be sorry that I asked this) but can you please explain how on Earth trigonometry could possibly be considered "basic for your life"?

Look, I got straight As in math my whole life, used to tutor other students who were having trouble, but I hated trig and struggled to get a C.  (In hindsight, I think this was more of a function of having a really awful teacher than any particular trig dificiency in the way my brain works.)  But I can promise you that in the 20+ years that have followed, I have not once needed to use trig in any way, shape or form.  Lots of other math, yes, I do use, but not that.

So I dunno, maybe it's basic for your life, but I do just fine without it.  Thank god. ;-)

by the stormy present (stormypresent aaaaaaat gmail etc) on Tue Nov 13th, 2007 at 04:49:41 PM EST
[ Parent ]
You really do not use trignometry in your everyday life???? wow.. well it can certainly be the case.. the case is tahat normally people delegate, if you deal with land proeprty is generally needed, optics, supermarkets, auditories, desfgm of any car, plane, highway, railway, computer.... all that stuff.. but it could certainly be that we have reached a point where all this work is sent to soembody who knows
It has already happened with probability and sttistics ( I can promise that most doctors do not have the foggiest idea about probabilty and they all work with cathadratics in the field...) so basically a lot of researchers just do not know what they are doing...

So the same thing can be happening with trigonometrics where you stay.... it si cerainy happening around here too..but not too much, engineers and architects and designers still do most of the jobs themselves and do nto send tit to special companies. I know that there are now like engineering buffet (like lawyers) who are inc ahrge of the strucutre calculations in buildings (where trigonometry is the basic tool)...

But still I find it weird...

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

by kcurie on Sat Nov 17th, 2007 at 02:42:32 PM EST
[ Parent ]
First, let me stress that I agree that all these subjects (trig, e, logarithms)are important and should be taught to as many people as reasonable.

But I do not believe they are so crucial to life that they should fall in any definition of 'numerate', just as reading and writing poetry is not what we understand by 'literate'. That is, I think most people can do without if they wish, and I am afraid many do wish so.

As for your examples of oscilations and exponential growth, I wonder how many people even know that there is a relation at all between oscilations and trigonometry.You can easily understand 'it goes up, then down' without sine waves ( and there are little pure sine wave in markets). Same for compound interest, 'every year I get interest, so I have more money, so the next year I get a little more interest'. That's the level people are OK with. If needed, they learn how to calculate the amounts, but many people can do without. Their algorithm is 'from the group of institutes I trust, I give it to the one that has high interest. If I want, I ask them how much it will become.'. It is probably a good algorithm too.

When I was in high school, my father built an elavated bed , for which he needed a complicated triangulation to stabilize it. I calculate the length and angles for him, he was happy. It's probably the closest he ever came to trigonometry in his life, and without me he would used the tried-and-true repeated fitting algorithm, each time sawing a bit from the beam until it would fit. It would have taken abit more time, but not worth learning trig for.

I think there is a second point here: all people in my class at that moment could have done the calculations. Most ( let's say the ones wishing to become doctors) have would not have done them, or even realized they could do them. The trouble would not have been the trigonometry, but the simple algebraic manipulations needed. People learn how to do those, but they never feel that it has meaning outside of math classes and tests. I think that this is the major problem in math education. Until people can AND WILL do simple algebra in real life, there is nothing they can do with trigonometry, or logarithms, or most higher functions.

by GreatZamfir on Wed Nov 14th, 2007 at 04:13:18 AM EST
[ Parent ]
It is known that with literacy there is such a thing as dyslexia.  My cousin is dyslexic, and as she explained it to me it has to do with the way the eye travels when reading--for a dyslexic reader the eye wobbles, or moves left-right too quickly.

At the biological level, the proposed aetiology of the visual dysfunction is based on the division of the visual system into two distinct pathways that have different roles and properties: the magnocellular and parvocellular pathways. The theory postulates that the magnocellular pathway is selectively disrupted in certain dyslexic individuals, leading to deficiencies in visual processing, and, via the posterior parietal cortex, to abnormal binocular control and visuospatial attention. Evidence for magnocellular dysfunction comes from anatomical studies showing abnormalities of the magnocellular layers of the lateral geniculate nucleus (Livingstone et al., 1991), psychophysical studies showing decreased sensitivity in the magnocellular range, i.e. low spatial frequencies and high temporal frequencies in dyslexics, and brain imaging studies.[40]

http://en.wikipedia.org/wiki/Dyslexia#Scientific_research

Has there been any study (or is there a word?) for (an) equivalent biolgical basi(e)s for problems with numeracy?

Don't fight forces, use them R. Buckminster Fuller.

by rg (leopold dot lepster at google mail dot com) on Tue Nov 13th, 2007 at 08:21:50 AM EST
Dyslexia is actually much more complicated than the biology aspect. When I read on very high contrast ie black on brilliant white, red on bright blue) I experience a similar effect with words and letters wobbling, but I am not dyslexic.

The way the brain processes information plays a large part in dyslexia. This is why some forms of dyslexia can be tackled by wearing tinted glasses or using coloured acetate overlays to reduce text contrast but severe dyslexia is linked with the way the brain processes visual information.  

I have a friend who is outstandingly clever and sharp but her dyslexia is very severe even though she is not lazy with trying - she tries to read as much as possible but the bottom line is that it doesn't go in and she relies on audio information.  Her spelling and ability to spots errors is appalling too.

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 08:31:30 AM EST
[ Parent ]
I'm mildly dyslexic, but have managed to avoid most of the symptoms through experience. but the words I have trouble with are the words I discovered before I gained my own coping strategy, so the main words I have trouble with are ones around four to six letters long. This is a result of the initial teaching alphabet, a 1960's system designed to help children read English more quickly. It's not used anymore because it did not succeed. however it's left me with the legacy that certain short words cause my brain to just lock up when I'm trying to work out how to write them. The one that causes me the most trouble is the word very. If i'm not relaxed, it can drop me mentaly into a paused state for 10 seconds while bits of my mind fight over how many copies of the letter R are in it.

as for the ability to spot spelling errors, I know with me it isn't just going to happen. I may have spelt the same word five different ways on the same page, but I just can't see it, my mind knows what word I intended, so just sees it when I look at whatever version is there.

Any idiot can face a crisis - it's day to day living that wears you out.

by ceebs (ceebs (at) eurotrib (dot) com) on Tue Nov 13th, 2007 at 12:53:15 PM EST
[ Parent ]
Oh and I forgot to answer your question - there is a maths equivalent called Dyscalculia
by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 08:33:58 AM EST
[ Parent ]


Diversity is the key to economic and political evolution.
by Cat on Tue Nov 13th, 2007 at 08:34:01 AM EST
Geometry (and physics, but I guess that's not math) is something that one actually ends up using IRL.  The ability to conceptualize space and what you can and can't do with it, and how.  I'm thinking about things like ... er, uhm...  home renovation...  :)

As far as wanting to teach abstract thinking - well, that's very important of course.  But it does seem to me that different people really do have different abilities.  I was exposed to math at an early age and ended up on some track that had me in classes with people older than I.  But I never ever got it.  And not for lack of trying, either.  There was just never a breakthrough moment.  Anything with numbers - precisely numbers - was pure torture and uninteresting.  Like those kids who fall through the cracks, I could figure out what was wanted of me and game it, but never with any understanding of what I was doing.  Likewise, I am sure there are people who find that math engages their mind in a way nothing else does (I've known these people), and if they were given some Dostoevsky to read, they could get the answers on the test right, but find it tedious and boring and never return to it again.  So what I am saying is that - I don't think there is some magical curricula we can apply like a blanket to all individuals and expect them all to develop some skill or appreciation as a result.  And I find that one's ability to absorb information is related to where they are in life.  If you suddenly get an urge to immerse yourself in a 6 month intensive maths course, you'll probably get a lot out of it.  If you're hormonally berserk, obsessed with the opposite sex, trying to figure out how to navigate social interactions, living with your parents and being told you have to grow up and figure out what you are going to do with your life now, and are put in a 6 month intensive maths course, your chances of gaining a lot from it are, I suspect, lessened.  I can't say I understand the fascination with teaching hypothetical people in a social vacuum.  


"Pretending that you already know the answer when you don't is not actually very helpful." ~Migeru.

by poemless on Tue Nov 13th, 2007 at 12:30:24 PM EST
But I never ever got it.  And not for lack of trying, either.  There was just never a breakthrough moment.  Anything with numbers - precisely numbers - was pure torture and uninteresting.  Like those kids who fall through the cracks, I could figure out what was wanted of me and game it, but never with any understanding of what I was doing.

well, you may be poemless, but that was fine prose, imo.

totally describes that feeling of being asked to empty the ocean with a leaky thimble feeling.

christ i'm glad that period of my life is over.

international politics and economics is just as puzzling, but at least i can come to ET and have a laugh while realising how dumb i am!

this diary and thread are vintage ET-

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty

by melo (melometa4(at)gmail.com) on Wed Nov 14th, 2007 at 07:58:28 AM EST
[ Parent ]


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