# Why geometry is important

#### WBahn

Joined Mar 31, 2012
26,398
For the second one, I like, "You're acute angle, yourself."

#### nsaspook

Joined Aug 27, 2009
7,869
For me geometry has always been the key to logic and intuition when solving problems. I've always had poor arithmetic instinct but could usually easily see the geometric form of the problem when refactored (by transforming arithmetic symbols into mental shapes) into a solution using space intuition. The problem with using geometric intuition is it's very easy to get fooled unless you have the arithmetic skill or experience to know when it's wrong.

#### cmartinez

Joined Jan 17, 2007
7,214
And for me, geometry helps me visualize complex problems, even arithmetical ones.

Maybe it's a little off topic, but I just want to pull my hair every time I ask a student what his definition of an integral is, and 99% of the time I get the stupid answer of "it's the area under a curve, of course". When I tell them that that's one of its uses, but it's not the definition itself, they just stare at me with that Homer Simpson look in their faces.

Why don't the teachers bother telling them the essential idea that is the basis of what they're teaching, instead of showing them only the possible applications?

EDIT: and I find your acute joke rather obtuse...

#### WBahn

Joined Mar 31, 2012
26,398
Maybe it's a little off topic, but I just want to pull my hair every time I ask a student what his definition of an integral is, and 99% of the time I get the stupid answer of "it's the area under a curve, of course". When I tell them that that's one of its uses, but it's not the definition itself, they just stare at me with that Homer Simpson look in their faces.

Why don't the teachers bother telling them the essential idea that is the basis of what they're teaching, instead of showing them only the possible applications?
There's two major sides to that coin.

If you do tell them all of that foundational stuff it goes in one ear and out the other -- they are only interested in the applications (or, more to the point, the applications that are on the homework and the exams). So what you are seeing is what they took away from the course as opposed to what was presented. And, to be fair, we are all that way to greater and lesser degrees -- when we are learning something new we seldom can absorb all of it, so we naturally absorb primarily the stuff that is important today and quickly forget that the other stuff was even mentioned.

The other side of the coin is that, increasingly, classes are filled with people that require extensive handholding and repetition just to gain any skill at all at using memorized formulas in the basic applications. They lack the fundamental math literacy to even start to actually understand the essential and foundational idea. Which of course means that that will leave that course even less prepared for the next course which has to be dumbed down even more. Schools, for a variety of reasons, are simply not interested in saying, "These are the standards. If only 10% of the class meets the standards, then 90% of the class will fail."

#### atferrari

Joined Jan 6, 2004
4,169
Maybe it's a little off topic, but I just want to pull my hair every time I ask a student what his definition of an integral is, and 99% of the time I get the stupid answer of "it's the area under a curve, of course". When I tell them that that's one of its uses, but it's not the definition itself, they just stare at me with that Homer Simpson look in their faces.
What other answer do you expect César?

#### boatsman

Joined Jan 17, 2008
186
How many students nowadays can calculate anything without an electronic calculator. Bring back the abacus!

#### WBahn

Joined Mar 31, 2012
26,398
How many students nowadays can calculate anything without an electronic calculator. Bring back the abacus!
Depressingly few. It is not uncommon at all to find students that struggle with adding single digit numbers in their head -- and it IS becoming increasingly uncommon to find students that CAN multiply single-digit numbers in their head. I had one student last semester -- admittedly at the extreme, but had plenty of company in his neighborhood -- that couldn't tell me what value needed to be subtracted from 53 to give 5. He finally guessed that it was 3 since that would cancel the 3 in 53 leaving 5.

They are being mathematically crippled by schools that believe that having them use calculators from first grade is somehow teaching them all about technology.

#### KL7AJ

Joined Nov 4, 2008
2,229
For me geometry has always been the key to logic and intuition when solving problems. I've always had poor arithmetic instinct but could usually easily see the geometric form of the problem when refactored (by transforming arithmetic symbols into mental shapes) into a solution using space intuition. The problem with using geometric intuition is it's very easy to get fooled unless you have the arithmetic skill or experience to know when it's wrong.
I'm the same way. I always grasped electronics intuitively, which I then later applied to math. I could never be a theoretical mathematician.

#### WBahn

Joined Mar 31, 2012
26,398
For me geometry has always been the key to logic and intuition when solving problems. I've always had poor arithmetic instinct but could usually easily see the geometric form of the problem when refactored (by transforming arithmetic symbols into mental shapes) into a solution using space intuition. The problem with using geometric intuition is it's very easy to get fooled unless you have the arithmetic skill or experience to know when it's wrong.
I'm a bit of everything -- jack of all trades, master of none. Sometimes a geometric interpretation works best, sometimes a physical (usually visual) analogy, sometimes the raw math. I find the most satisfaction when a purely mathematical model yields the most comprehension, but this usually only comes when I work out the math for myself so that I can see how the final equations evolved. I'm definitely not the kind of person that can just look at an equation and "see" what it means -- I've got a good friend that CAN think that way and is it almost scary how he can get insights into things that he has very little experience with just by seeing the math that models the system.

#### Glenn Holland

Joined Dec 26, 2014
705
I'm an old fashioned geek and I do most of my heavy duty studying using hard copy textbooks.

So called "Tactical learning" -IE- pushing a pencil over paper, underlining sentences and paragraphs, writing notes, and making drawings has been the basis of a solid education. It's my perception that most people are functionally illiterate and they cannot communicate in an orderly manner.

Here in California, I too frequently hear young people use the expression "Like you know" at the beginning of a statement. One example was a silly girl's comment to the news media about a brush fire near her home: "I looked out the window and it was like you know there was all this fire".

I'd like to literally scream at them "Why in the hell can't you learn basic English grammer?

#### wayneh

Joined Sep 9, 2010
16,511
Why don't the teachers bother telling them the essential idea that is the basis of what they're teaching, instead of showing them only the possible applications?
I'm not sure I would have grasped calculus anywhere nearly as quickly without the area-under-the-curve explanation. I'm a visual learner/thinker and to me, that IS calculus. Sure, I later learned triple integrals to get the weight of a snow drift or rotational integrals for round stuff, but in my head it's still the area under the curve.

I hated the pre-calculus stuff about limits and series. It all seemed utterly useless to me until I'd actually seen the calculus. The usefulness of calculus seemed immediately obvious to me. Only then did I wish we could go back and look at limits again, once I saw where we were going. But alas...

#### cmartinez

Joined Jan 17, 2007
7,214
I'm an old fashioned geek and I do most of my heavy duty studying using hard copy textbooks.

So called "Tactical learning" -IE- pushing a pencil over paper, underlining sentences and paragraphs, writing notes, and making drawings has been the basis of a solid education. It's my perception that most people are functionally illiterate and they cannot communicate in an orderly manner.

Here in California, I too frequently hear young people use the expression "Like you know" at the beginning of a statement. One example was a silly girl's comment to the news media about a brush fire near her home: "I looked out the window and it was like you know there was all this fire".

I'd like to literally scream at them "Why in the hell can't you learn basic English grammer?
This is getting complicated... but I consider knowledge a social issue, and not a personal one. The main reason being that we've all inherited almost every single thing that we know, and we have not actually learned or discovered anything ourselves... Even the language that we use to communicate between ourselves has been a gift.

We were taught how to add, subtract, divide and multiply. We were taught history, chemistry, physicists... you name it... and afterwards, we speak about those subjects as though we actually researched them ourselves! I think that we're failing to recognize the giants whose shoulders we're actually standing on!

Yes, I consider the most recent generations very much wanting in basic issues such as communication and arithmetic. But the big Q here is, whose fault is it? Aren't we the one's responsible for the education of the next generation of our peers?

I mean, the western world wouldn't be suffering this invasion of brats if it weren't because their parents (that's us) allowed them to become brats in the first place.

What has our generation done wrong?

#### Glenn Holland

Joined Dec 26, 2014
705
This is getting complicated... but I consider knowledge a social issue, and not a personal one. The main reason being that we've all inherited almost every single thing that we know, and we have not actually learned or discovered anything ourselves... Even the language that we use to communicate between ourselves has been a gift.

We were taught how to add, subtract, divide and multiply. We were taught history, chemistry, physicists... you name it... and afterwards, we speak about those subjects as though we actually researched them ourselves! I think that we're failing to recognize the giants whose shoulders we're actually standing on!

Yes, I consider the most recent generations very much wanting in basic issues such as communication and arithmetic. But the big Q here is, whose fault is it? Aren't we the one's responsible for the education of the next generation of our peers?

I mean, the western world wouldn't be suffering this invasion of brats if it weren't because their parents (that's us) allowed them to become brats in the first place.

What has our generation done wrong?
Part of what's wrong is teacher's unions that view their job as simply a tenured government job with a guaranteed salary and their position is virtually earthquake proof.

The California Faculty Association (CFA that represents instructors in the state university system) and another union that represents instructors in the community college districts are both set to strike in March of this year.

The CFA wants a 5% raise and the other wants a 3% raise. This happens over and over and over - some publically funded education union going on strike to get more \$. And they always claim it's for the student's benefit.

#### GopherT

Joined Nov 23, 2012
8,012
...99% of the time I get the stupid answer of "it's the area under a curve, of course". When I tell them that that's one of its uses, but it's not the definition itself, they just stare at me with that Homer Simpson look in their faces.
Are you sure it is not a Thomas Simpson look on their face? Cousins (even distant cousins) can look alike. Those kids may know more than you think.

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#### cmartinez

Joined Jan 17, 2007
7,214
Are you sure it is not a Thomas Simpson look on their face? Cousins (even distant cousins) can look alike. Those kids may know more than you think.
Oh.... I looked it up, and it was definitely not a Thomas Simpson look...

#### #12

Joined Nov 30, 2010
18,216
I have a favorite statement: "Your lame excuse tells me you have never known real hardship."
To locate its relevance here I must go back a long way. A hundred years ago, my grandmother said, "Root hog, or die." There was a time when you either worked for everything from survival to knowledge or you did without, and doing without was often fatal. Darwin was busy. I am the first generation of wusses. I didn't have to work to survive, but the lesson was fresh in my parents. I worked like my survival depended on it.

Today, the nanny state threatens to jail anyone who spanks their child and so, hardship is an evening without mind numbing video of one sort or another. People hand their children toys that cost a weeks pay and the children don't think it's worth anything because they didn't pay for it. There are lots of people that sit at a computer and become wealthy. Why work when life is that easy? "I don't have to learn. Somebody will hand it to me." "The world owes me because I didn't ask to be born." "My parents were rich and my husband will be too."

I tried teaching for about 20 minutes. When I turned around, there were two people that weren't asleep.
Well, guess what. Life is easy compared to smelling the mule that is pulling the plow, and there is no way a modern child is ever going to follow a mule around a corn field. Some of them think corn and fish and paper towels originate in the back room of grocery stores. There is no mental connection between what exists and how it got in their hand. It's eFn magic and the magic will work for me, too!

Right now, I'm happy to think I will die before all this perpetual, "growth" collapses. When it does, Darwin will work again.

#### cmartinez

Joined Jan 17, 2007
7,214
“It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a photograph, or a telephone or any other important thing—and the last man gets the credit and we forget the others. He added his little mite — that is all he did. These object lessons should teach us that ninety-nine parts of all things that proceed from the intellect are plagiarisms, pure and simple; and the lesson ought to make us modest. But nothing can do that.” – Mark Twain

#### Glenn Holland

Joined Dec 26, 2014
705
I have a favorite statement: "Your lame excuse tells me you have never known real hardship."
To locate its relevance here I must go back a long way. A hundred years ago, my grandmother said, "Root hog, or die." There was a time when you either worked for everything from survival to knowledge or you did without, and doing without was often fatal. Darwin was busy. I am the first generation of wusses. I didn't have to work to survive, but the lesson was fresh in my parents. I worked like my survival depended on it.

Today, the nanny state threatens to jail anyone who spanks their child and so, hardship is an evening without mind numbing video of one sort or another. People hand their children toys that cost a weeks pay and the children don't think it's worth anything because they didn't pay for it. There are lots of people that sit at a computer and become wealthy. Why work when life is that easy? "I don't have to learn. Somebody will hand it to me." "The world owes me because I didn't ask to be born." "My parents were rich and my husband will be too."

I tried teaching for about 20 minutes. When I turned around, there were two people that weren't asleep.
Well, guess what. Life is easy compared to smelling the mule that is pulling the plow, and there is no way a modern child is ever going to follow a mule around a corn field. Some of them think corn and fish and paper towels originate in the back room of grocery stores. There is no mental connection between what exists and how it got in their hand. It's eFn magic and the magic will work for me, too!

Right now, I'm happy to think I will die before all this perpetual, "growth" collapses. When it does, Darwin will work again.
Unfortunately, you're right about Darwinism. Here's my guess about how the human species will eventually get wiped out or at least severely "thinned". It is suspected that a huge meteorite extincted the dinosaurs and there have been several near misses in the past century.

If another large meteor impact occurs, most people cannot adapt to a prolonged effects of such a large scale disaster and they'd go belly up like roaches. By the way, I'm actually prepared for most natural disasters (for example a severe earthquake in San Francisco) and I could live for at least 6 months without modern conveniences if push came to shove.

#### WBahn

Joined Mar 31, 2012
26,398
Unfortunately, you're right about Darwinism. Here's my guess about how the human species will eventually get wiped out or at least severely "thinned". It is suspected that a huge meteorite extincted the dinosaurs and there have been several near misses in the past century.

If another large meteor impact occurs, most people cannot adapt to a prolonged effects of such a large scale disaster and they'd go belly up like roaches. By the way, I'm actually prepared for most natural disasters (for example a severe earthquake in San Francisco) and I could live for at least 6 months without modern conveniences if push came to shove.
My guess is that you are not nearly as well prepared as you think you are.