Algebra is totally useless...unless there's ever a time where you don't know something and want to figure out what it is...

The view from the high horse is often distorted. I sympathize with your distrust of conventional theories and your assumption that people in general are misinformed, but not everyone is, and your views are not the only valid ones.



Some problems are best solved with a jackhammer, but not everyone should need to know how to use one.

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Patience, grasshopper.

Hi Aidoneus,

I was completely unaware that you had some criticism of any education system (until I saw your first post in this thread, just now).
Therefore I have an exactly neutral (or zero) understanding of it.

Off topic:

I do take delight in contradicting posts like this. Why?

Your original post said that Federal Reserve notes were not money, and said it without any argument or explanation. This is a view I'm familiar with. Bill too, I expect. I've tried to read authors who think that gold is the one true measure of value, though it hasn't been very enlightening. I don't find this view very persuasive, and I'm not alone.

So we disagree. Not the end of the world. Perhaps we can discuss our opinions. Very well, what do you have to say for your view? Just that someone who disagrees "needs to take a course in basic economics". And you say this not even to me (I admit that I'm often combative), but Bill, who is well-informed, careful, and kinder than most. You couldn't have picked a worse target for your condescension.

One of the worst trolls we've had on the boards, SmoothOper, made more effort to justify his opinions than you have in this thread. That's why I snark.

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Occupy Babel!

I dislike talk of what "everyone should know". The world of things "everyone should know" is too big for any one person to know (mathematics, social psychology, linguistics, history, economics, biology, comparative religion...). So why use a phrase that implies judgment?

I prefer to think of which things almost anyone would benefit from knowing. And we can talk about which of those things are most likely to be useful.

And with regard to math (or most areas of knowledge): you often don't see where it's useful until you really understand it, at which point it shapes every aspect of how you see a problem. So I think most people who think they wouldn't get much out of knowing math probably don't realize what they are missing.

The other problem is that most mathematical thinking you do in everyday life revolves around having good intuitions and informally stated problems. School math tends to do a bad job of cultivating that sort of thought.

But I also don't know how much of the mathematical thinking I do is (middle-school) algebra as opposed to other subjects. I just don't think of what skills I'm using when I think through a problem. The exception is trigonometry: I slept through that class and I've never stopped regretting it.

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Occupy Babel!

My criticism of our educational system is inextricably welded to my criticism of our economic/political system, which seems to be off limits (viewtopic.php?f=8&t=10566). If I may offer one caveat to this "debate," defining money is not quite the same as arguing about economics--though, of course, one topic almost always leads by a smooth continuance to the other. For instance, glancing at some books in one of my bookcases, Jack Weatherford's The History of Money tells a tale from an anthropologist's perspective. As such, it makes for light and entertaining reading. (As do his various books on Native Americans.) Another "grainy" book is J.B. Baskin and P.J. Miranti, Jr., A History of Corporate Finance, and a cheap Dover reprint of K. Lancaster, Mathematical Economics includes nearly 200 pages of mathematical review for this graduate-level textbook on economic modeling. Then there are the numerous books that I would call philosophical macroeconomics. All interesting from a historical perspective, at least if one does not become trapped into one mold. (I cannot remember who, perhaps Robert Anton Wilson, said that "where belief begins, thought ends," but I hold the concept near and dear.)

If anyone is still reading my post--and I haven't been banned for injecting political/economic theory into our forum--I urge people to research the American System of economics (perhaps starting here: http://en.wikipedia.org/wiki/American_S ... ic_plan%29, or maybe http://www.dailykos.com/story/2012/12/0 ... can-System or https://www.youtube.com/watch?v=DcsRf7b6zes) to see a middle way between fascism (corporate-state wedded monopoly) and laissez faire (Austrian School). From Alexander Hamilton to Henry Clay, A. Lincoln, Henry C. Carey (https://archive.org/details/miscellaneouswor00care), FDR, and JFK, the American System has offered a blueprint for other nations to reach economic independence. (As opposed to the British Imperialist/IMF model of exporting raw materials and importing finished goods. Compare economic development in East Asia to Central America.)

Of course, IMHO, we are now firmly in the second epochal machine-led transformation of society. Just as farm machinery displaced farm labor from the 19th century to the Great Depression in America, moving workers into industrial factory jobs in the city, automation is transforming the world economy. Except for the U.N.'s Agenda 21 (drastically reduce human population for "ecological" reasons, without giving an explicit road map to achieve this "dream"), or America's pragmatic "answer" to lock up as much surplus labor as possible, few economists/politicians have addressed the fundamental issues of distribution of goods when (not if!) most jobs can be done by machines. I recall reading a story by K. Vonnegut in which people were compelled to wear out items at ever faster rates to keep up with the cornucopia of modern production. (A rather different nightmare than the currently popular ecological dystopias.) Foolish youth that I was in the early '60s, I argued with friends that automation would bring about a renaissance of personal liberty, creativity, and freedom from want. I'm afraid that reality has severely dampened my optimism. (Yeah, right, old age has made me a cynic!)

By now, I guess I have lost any readers that I started with. In my old age, from one perspective I seem to be drifting into "blowhard" crank status, or, just maybe, a voice crying in the wilderness. (My hubris showing, again?)

When I saw the title of this thread, I thought, Me!

hyperpape, you make a good point about the value of mathematical thinking and how, unfortunately, school math does not do a good job of imparting it. One reason, OC, is that most secondary school teachers are not very good at it, either. If they were, they would have better paying job options.

A wonderful example of mathematical thinking is the famous story of young Gauss in the first grade of a one room schoolhouse. The teacher, probably needing a break, gave the students the task of adding the numbers from 1 to 100. In a couple of minutes Gauss handed in his paper. What he had done was to add 1 to 100 to get 101, then add 2 to 99 to get 101, etc. He had done that 50 times and the sum of 50 101s is 5050. Done!

That was not only very clever, it was an example of divergent thinking. Take that, Weather Woman!

Those of us with a little algebra can derive the expression, N(N+1)/2. Those of us with a little geometry can take a triangle like

*
**
***
****
*****

and fold it to get this rectangle

*****
*****
*****


How and why do the algebraists do that?

Why? Because it's fun. Take that, Weather Woman!

How? Let me address the more general question of algebraic thinking.

DrStraw has mentioned the importance of the idea of equality. Rearranging the *s in the triangle did not change how many there were. That number remained equal for both figures. When Gauss rearranged the addends that did not change the sum.

I think that one of the most valuable ideas in algebra is that of an unknown. Now, in real life there are lots of things that we do not know. The unknown is vast. It is a real cognitive leap, in the face of that vastness, to think about a specific unknown, when we may have no guarantee that it exists.

Closely related to the idea of an unknown is the idea of a variable. To find an unknown we may set up an equation with a variable and solve the equation for that variable.

Another important idea is that of a function. One variable may be the function of another variable.

There is a lot more to algebraic thinking, OC, but one thing that algebraists do is to look for functions. Given the problem of adding the numbers from 1 to 100, the algebraist replaces the 100 with a variable (N) and looks for a function of N which is equal to the sum of the numbers from 1 to N. It is N(N+1)/2. Voila! (Well, not quite voila, you have to prove it.)

This way of thinking, of taking the known and leaping into the unknown, looking for functions and relations, is powerful and often practical. It is not only problem solving behavior, it is problem seeking behavior. It is something that algebra courses could foster, and, IMO, should foster.

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The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

The young Gauss story is one of my favorites, but then so is this one about Von Neumann (source The Legend of John Von Neumann, P.R. Halmos, reposted at http://stepanov.lk.net/mnemo/legende.html):

Then there is the famous fly puzzle. Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time a fly that travels at a steady 15 m.p.h. [EDIT: this should be 10 mph, of course; error in the original, which I just noticed! EDIT2: Or maybe the original story was 5 mph fly and 15 mph bike, I don't recall.] starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover ? The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann; "all I did was sum the infinite series."

Actually, the fly has to go faster than the bicycles.

I am reminded of a time when, as a kid, I was camping with a friend. After lunch I cleaned our cups and left them out to dry. He asked me why I had left the cups right side up. I replied so that the remaining water inside them could evaporate. My friend, who later became a nuclear engineer, turned the cups upside down. So that the remaining water could run out, OC.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Word problem:

If a man and a half can dig a ditch and a half in a day and a half, how long does it take ten men to dig a ditch?

Answer:

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

It depends on how big the ditch is. If it is small then only one person can be in there digging at a time. If it is big they can all be in there digging together.

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

Yeah, you're right! Of course, the fly cannot go slower than the bike! Here is another, mathematically correct, though physiologically impossible, version of the anecdote (which is probably also apocryphal, http://www.math.ethz.ch/~efonn/stuff/anecdotes.html):

Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?

The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles. That's all there is to it.

When this problem was posed to John von Neumann, he immediately replied, "150 miles."

"It is very strange," said the poser, assuming the professor had immediately seen the simple solution, "but nearly everyone tries to sum the infinite series."

"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

I think that is usually stated something like dig a "post hole" to avoid DrStraw's objection.

Moe, Larry, and Curly are painting your house. (Beware low bids.) Moe could finish the job alone in 4 hours, Larry alone in 3 hours, and Curly alone in 2 hours (he has a special bucket technique). They all start painting on different sides. However, while Moe and Larry are using red paint, Curly is using white paint. When will your house be all red?

That was not an objection. It was thinking outside the box, something most algebra students have trouble doing.

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

But then, you are a Dr !

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I think I am so I think I am.

What on earth is half a man doing digging anyway?

Don't you know that this is Walking Dead week?!

Edit: I'm sorry, I forgot that you are in England, so my "joke" may not have translated across the pond. Walking Dead is an American tv show about zombies, though almost half the lead actors seem to be British, like so many recent American cable shows or at least the better shows.

The lead is definitely British, and his American accent is impeccable
(at least to my ears). But I've only seen parts of season 1.
Some day, I'll get a TV again.