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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.

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Arithmetic(s)--calling Jake S and kcurie! | 180 comments (180 topical, 0 editorial, 0 hidden)

Re: Arithmetics (4.00 / 4)

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

Re: Arithmetics (4.00 / 9)

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

Re: Arithmetics (4.00 / 5)

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.

As we journey through life, we should keep an iron grip, to the very end, on the capacity for silliness. It preserves the soul from dessication.

by ceebs (bunchofwankers (at) gmail (dot) com) on Sun Nov 11th, 2007 at 08:38:53 PM EST
[ Parent ]

Re: Arithmetics (4.00 / 6)

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 ]

Re: Arithmetics (4.00 / 4)

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 ]

Re: Arithmetics (4.00 / 5)

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 ]

Re: Arithmetics (4.00 / 6)

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 ]

Re: Arithmetics (none / 1)

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.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 06:05:16 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (4.00 / 6)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 07:32:17 AM EST

Re: Arithmetics (4.00 / 3)

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 ]

Re: Arithmetics (4.00 / 3)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 07:46:18 AM EST
[ Parent ]

Re: Arithmetics (4.00 / 3)

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 ]

Re: Arithmetics (4.00 / 2)

That's physics !

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:01:13 AM EST
[ Parent ]

Re: Arithmetics (4.00 / 3)

You say that like it's a bad thing. ;-)

As we journey through life, we should keep an iron grip, to the very end, on the capacity for silliness. It preserves the soul from dessication.

by ceebs (bunchofwankers (at) gmail (dot) com) on Mon Nov 12th, 2007 at 08:11:03 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:21:30 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (4.00 / 3)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:56:01 AM EST
[ Parent ]

maths for maths's sake (4.00 / 3)

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 ]

Re: maths for maths's sake (4.00 / 3)

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 ?

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

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

Re: maths for maths's sake (4.00 / 2)

is this homework?

As we journey through life, we should keep an iron grip, to the very end, on the capacity for silliness. It preserves the soul from dessication.

by ceebs (bunchofwankers (at) gmail (dot) com) on Mon Nov 12th, 2007 at 08:36:48 PM EST
[ Parent ]

Re: maths for maths's sake (4.00 / 3)

Yes.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:58:11 PM EST
[ Parent ]

Re: maths for maths's sake (none / 1)

(a+b)*(a+b)-n²=4(a*b)/2 (from the diagram)

a²+b²+2ab=2ab+n²

so a²+b²=n²

As we journey through life, we should keep an iron grip, to the very end, on the capacity for silliness. It preserves the soul from dessication.

by ceebs (bunchofwankers (at) gmail (dot) com) on Tue Nov 13th, 2007 at 07:48:34 AM EST
[ Parent ]

Re: maths for maths's sake (none / 1)

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 ]

Re: maths for maths's sake (none / 1)

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 ]

Re: maths for maths's sake (none / 1)

C'mon, it's not hard at all. It consists of producing a second graph from the above.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 06:38:59 AM EST
[ Parent ]

Re: maths for maths's sake (4.00 / 2)

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 ]

Re: maths for maths's sake (4.00 / 2)

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 ]

Re: maths for maths's sake (4.00 / 3)

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.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 08:40:52 AM EST
[ Parent ]

Re: maths for maths's sake (4.00 / 2)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:49:22 AM EST
[ Parent ]

Re: maths for maths's sake (none / 1)

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)

As we journey through life, we should keep an iron grip, to the very end, on the capacity for silliness. It preserves the soul from dessication.

by ceebs (bunchofwankers (at) gmail (dot) com) on Tue Nov 13th, 2007 at 07:53:07 AM EST
[ Parent ]

Re: maths for maths's sake (4.00 / 2)

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 ]

Re: maths for maths's sake (none / 1)

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...

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:13:44 AM EST
[ Parent ]

Re: maths for maths's sake (none / 1)

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 ]

Re: maths for maths's sake (4.00 / 2)

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 ?

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:28:40 AM EST
[ Parent ]

Re: maths for maths's sake (none / 1)

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 ]

Re: maths for maths's sake (none / 0)

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 ]

Re: maths for maths's sake (none / 0)

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 ]

Re: maths for maths's sake (none / 1)

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' ? :)

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 08:45:32 AM EST
[ Parent ]

Re: maths for maths's sake (4.00 / 2)

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 ]

Re: maths for maths's sake (none / 1)

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 ]

Re: maths for maths's sake (none / 0)

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 ?

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 09:22:13 AM EST
[ Parent ]

Re: maths for maths's sake (none / 1)

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 ]

Re: maths for maths's sake (none / 1)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 09:45:31 AM EST
[ Parent ]

Re: maths for maths's sake (none / 0)

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 ]

Re: maths for maths's sake (4.00 / 2)

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 ?

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 10:16:45 AM EST
[ Parent ]

Re: maths for maths's sake (4.00 / 3)

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 ]

Re: Arithmetics (4.00 / 2)

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: Arithmetics (4.00 / 3)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Mon Nov 12th, 2007 at 08:35:56 PM EST
[ Parent ]

Re: Arithmetics (none / 1)

For that reason, it is my pet-peeve that general mathematics education should reach complex numbers.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 06:37:55 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (4.00 / 3)

If general education ends with 15, then yes.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 08:44:05 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (none / 0)

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.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 09:43:09 AM EST
[ Parent ]

Re: Arithmetics (4.00 / 2)

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 ]

Re: Arithmetics (4.00 / 2)

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 ]

Re: Arithmetics (none / 0)

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 ]

Re: Arithmetics (none / 0)

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 ]

Re: Arithmetics (none / 0)

However, hyperbolic functions don't have such a nice graphic representation.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Wed Nov 14th, 2007 at 02:50:53 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (4.00 / 2)

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.

The concept that socialisation has to be linked to business relationships is a great victory for business relationships, not for socialisation...

by linca (antonin POINT lucas AROBASE gmail.com) on Tue Nov 13th, 2007 at 10:24:36 AM EST
[ Parent ]

Re: Arithmetics (none / 1)

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.

*Traitor*, n.
A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.

by DoDo on Tue Nov 13th, 2007 at 10:15:03 AM EST
[ Parent ]

Re: Arithmetics (4.00 / 5)

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 ]

Re: Arithmetics (4.00 / 3)

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 ]

Re: Arithmetics (4.00 / 3)

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 ]

Re: Arithmetics (4.00 / 2)

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 ]

Re: Arithmetics (none / 1)

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 ]

Re: Arithmetics (none / 1)

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.

As we journey through life, we should keep an iron grip, to the very end, on the capacity for silliness. It preserves the soul from dessication.

by ceebs (bunchofwankers (at) gmail (dot) com) on Mon Nov 12th, 2007 at 08:59:56 AM EST
[ Parent ]

Re: Arithmetics (4.00 / 2)

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 ]

Re: Arithmetics (none / 0)

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 ]

Re: Arithmetics (4.00 / 6)

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!

Lobbyists are people too...

by melo (melometa4(at)gmail.com) on Mon Nov 12th, 2007 at 09:30:34 AM EST

Re: Arithmetics (4.00 / 5)

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 ]

Re: Arithmetic(s)--calling Jake S and kcurie! (4.00 / 5)

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.

Ad astra per aspera

by In Wales (inwales aaat eurotrib.com) on Mon Nov 12th, 2007 at 04:56:08 PM EST

Re: Arithmetic(s)--calling Jake S and kcurie! (4.00 / 3)

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 ]

Re: Arithmetic(s)--calling Jake S and kcurie! (4.00 / 2)

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.

Ad astra per aspera

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 05:32:10 AM EST
[ Parent ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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...

Ad astra per aspera

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 05:34:42 AM EST
[ Parent ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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 ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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.

Ad astra per aspera

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

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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 ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

I don't know what polynomials are.
* goes to check wiki *

Ah, yes I can do them.

Ad astra per aspera

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 09:25:15 AM EST
[ Parent ]

Re: Arithmetic(s)--calling Jake S and kcurie! (4.00 / 2)

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 ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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.

Ad astra per aspera

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 09:48:18 AM EST
[ Parent ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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 ]

Re: Arithmetic(s)--calling Jake S and kcurie! (none / 1)

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.

Ad astra per aspera

by In Wales (inwales aaat eurotrib.com) on Tue Nov 13th, 2007 at 10:24:36 AM EST
[ Parent ]

Re: Arithmetic(s)--calling Jake S and kcurie! (4.00 / 2)

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