Why is the diagonal of a square not "2"

Bill Spight · **#41**

quantumf wrote:

DrStraw wrote:

The quality of the students I try to teach calculus to now is abysmal compared to the first time I taught it 37 years ago. The problem is that now everyone has to go to college and they all think they deserve a grade A because they paid their fees. The truth is that 75% of them should not be there but there is nothing else for them to do. All the manufacturing jobs have gone overseas and anything except minimal wage requires a "degree".

In my opinion it is an organized effort to dumb down the populace so that they don't have the intelligence to understand what is happening to them. I once had a student, about 5 years ago, pull out a calculator out of habit and punch in 0x0.

And yet, it's a widely documented fact that on average we're collectively getting smarter. This is across all countries, developed and developing nations. See http://en.wikipedia.org/wiki/Flynn_effect. So your conspiracy theory of a deliberate dumbing down of the population doesn't square the evidence. If you are genuinely observing a weakening trend among your students, and you are not just applying a selective memory, what else can be the cause?

You have given one possible explanation, namely a lowering of standards required to enter the university, but that only makes sense if the university is generally accepting many more students than it used to, or there are many more universities than before, meaning the average student across them will be of a lower standard. Perhaps these are both true.

Another cause may be the metrics you apply are less relevant now. For instance, high levels of arithmetical numeracy is perhaps less important now (given computers/calculators) and instead we expect students to be better at abstract or scientific reasoning.

When I was in high school I ran across a few of my grandmother's high school textbooks. They were more difficult than my textbooks, college level, even. It seems like high school was harder in her day. But, according to the Flynn effect, people my age were "smarter" in academic terms than those of her day. How can we reconcile this apparent contradiction? Maybe it is a question of selection. In her day, most people did not go to high school. That is why in theory an IQ of 100 meant a "mental age" of 13. The apparent dumbing down can be attributed to a greater number of students going to high school in my day. Perhaps the same kind of selection effect is going on today. In relative terms the cohort of people who graduated from high school in my day but did not go to college would be capable today of getting an associate degree. And many of them are doing so. Add to that the economic pressure to get some kind of certification beyond the high school level and you can get students going on to college who can't hack it, even if they are better academically than the average high school graduate of their parents' time.

Thinking about this in terms of go, suppose that the Flynn effect means that the slice of the population who would have been 7 kyu 50 years ago would now play at the level of 6 kyus from that time. However, because of rank inflation, they would now be 5 kyus.

Edit: In my day the average college graduate had an IQ around 125. Translated into mental age, that gives a mental age of 16.25, a bit below the mental age of the average high school graduate of my grandmother's day. They were actually smarter than that, because IQ norms are adjusted to the current population. But I doubt if they were as smart as the average college graduate of my grandmother's day.

DrStraw · **#42**

quantumf wrote:

And yet, it's a widely documented fact that on average we're collectively getting smarter. This is across all countries, developed and developing nations. See http://en.wikipedia.org/wiki/Flynn_effect. So your conspiracy theory of a deliberate dumbing down of the population doesn't square the evidence.

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed. But beyond that, it is not really relevant because I was not referring to the average level of the entire populace, but to the average level of those attending university. This also is consistent with my claim of dumbing down: people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them - it merely means that they have been trained to pass standardized tests instead of think for themselves.

Quote:

If you are genuinely observing a weakening trend among your students, and you are not just applying a selective memory, what else can be the cause?

You have given one possible explanation, namely a lowering of standards required to enter the university, but that only makes sense if the university is generally accepting many more students than it used to, or there are many more universities than before, meaning the average student across them will be of a lower standard. Perhaps these are both true.

Most definitely that is true. A few years ago the governor of Virginia mandated at all state colleges and universities increase enrollment by 14% in three years. Obviously, then, the top schools (UVa and VPI, etc.) get the best students and the second tier have to start at a lower level. The bottom tier school have to take what is left over and average standard must drop considerably. I have seen a drastic decline in just the last five years.

Quote:

Another cause may be the metrics you apply are less relevant now. For instance, high levels of arithmetical numeracy is perhaps less important now (given computers/calculators) and instead we expect students to be better at abstract or scientific reasoning.

The ability to do abstract thinking is what I was referring to. The overall ability of the general populace may have increased, although from my anecdotal evidence I would question that. However, when only 2% high school kids go on to college then their average IQ will be well in excess of 100. When well over 50% are going then the average IQ is going to be only a little above 100, even allowing for IQ inflation, and many of them are going to be below average IQ.

Bill Spight · **#43**

Pippen wrote:

Aidoneus wrote:

One can take the stance that it is a useful fiction--that the order is in our heads rather than in reality-

That'd be my take. It remembers me of the guy that once wanted to prove creationism with the fact that an apple is perfectly rounded and sized for our hands, because God made it for us^^.

A friend once told me this story about his uncle, who lived in the American Bible Belt. His uncle picked up a hitchhiker one afternoon and after a while, lit up a cigarette. The hitchhiker said, "If God had wanted you to smoke, He would have given you a smokestack." The uncle pulled off the road and said, "If God had wanted you to ride, He would have given you wheels. Get out!"

illluck · **#44**

Pippen wrote:

That'd be my take. It remembers me of the guy that once wanted to prove creationism with the fact that an apple is perfectly rounded and sized for our hands, because God made it for us^^.

That's a new one. Ray Comfort had the argument that bananas are perfectly formed for human consumption and therefore God.

Aidoneus · **#45**

quantumf wrote:

And yet, it's a widely documented fact that on average we're collectively getting smarter. This is across all countries, developed and developing nations. See http://en.wikipedia.org/wiki/Flynn_effect. So your conspiracy theory of a deliberate dumbing down of the population doesn't square the evidence. If you are genuinely observing a weakening trend among your students, and you are not just applying a selective memory, what else can be the cause?

If small changes in IQ scores indicate anything, it is changes in poverty. This can be seen, for example, as differences among ethnic groups in the U.S. substantially disappear when adjusted for income. It is similar to the rise in longevity due to decreases in infant mortality.

Bill Spight wrote:

When I was in high school I ran across a few of my grandmother's high school textbooks. They were more difficult than my textbooks, college level, even. It seems like high school was harder in her day.

Yes! Try having a modern student read the 6th grade McGuffey Reader. (Widely used in the U.S. in the 19th century.) Or maybe take this graduation test from 1912: http://www.huffingtonpost.com/2013/08/1 ... 44163.html

BTW, my father was born in 1895 and had only a sixth grade education. I have no doubt, however, that he could have joined me in Mensa.

quantumf wrote:

You have given one possible explanation, namely a lowering of standards required to enter the university, but that only makes sense if the university is generally accepting many more students than it used to, or there are many more universities than before, meaning the average student across them will be of a lower standard. Perhaps these are both true.

There is no maybe about the much higher percentage of the population going to college. As mechanization destroyed the viability of small farming--first in the U.S. and then around the world--the displaced populations relocated to urban areas and found new occupations in manufacturing. (Steinbeck may have romanticized his Okies, but not the problem of distribution of goods under mass production.) I pity young people as the mechanization of manufacturing has been accelerating their displacement since the 1970s; e.g., by the 1990s, U.S. Steel produced the same volume of steel with one-tenth of the old workforce. Forget outsourcing, the volume of U.S. manufacturing has not declined. Automation is behind the loss in manufacturing jobs, and few have considered the long-term implications. We are now getting many U.S. students into extreme debt for non-STEM degrees (science, technology, engineering, math)--degrees that have little marketable value, but which have done wonders for the bottom line of public and private institutions through government guaranteed loans and grants.

quantumf wrote:

Another cause may be the metrics you apply are less relevant now. For instance, high levels of arithmetical numeracy is perhaps less important now (given computers/calculators) and instead we expect students to be better at abstract or scientific reasoning.

I haven't seen any improvement in abstract reasoning among incoming students. But rather than share my anecdotes, I will share this link concerning the failure to develop critical thinking during college: http://www.mcclatchydc.com/2011/01/18/1 ... s-not.html.

Aidoneus · **#46**

DrStraw wrote:

quantumf wrote:

And yet, it's a widely documented fact that on average we're collectively getting smarter. This is across all countries, developed and developing nations. See http://en.wikipedia.org/wiki/Flynn_effect. So your conspiracy theory of a deliberate dumbing down of the population doesn't square the evidence.

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed. But beyond that, it is not really relevant because I was not referring to the average level of the entire populace, but to the average level of those attending university. This also is consistent with my claim of dumbing down: people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them - it merely means that they have been trained to pass standardized tests instead of think for themselves.

I can't type fast enough to stay in this conversation. :lol:

But I am in absolute agreement. Schools now focus so much on metrics that students seem to learn more test-taking skills than critical-thinking skills.

oca · **#47**

Pippen wrote:

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

RBerenguel · **#48**

oca wrote:

Pippen wrote:

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

When starting in maths you eventually get some definition of the reals where this is one of the most stricking things... And then you just kind of "get it." I.e. for me the reals were introduced as completion by Cauchy sequences (well, I got them introduced as Dedekind cuts earlier, but on my degree they were as Cauchy sequences.) For the non-maths, a sequence {x_n} is an infinite, ordered set of numbers (or things, doesn't matter) like 1,2,3,4... A Cauchy sequence is one such that if we pick some arbitrary elements of it, say, x_n and x_m such that n and m are very large (say, n and m larger than N) then x_n and x_m are very close. When formalised, this has epsilons and conditions, but essentially "once you look very far, elements are incredibly close."

This would seem to imply Cauchy sequences have a limit (i.e. elements are closer and closer and closer... isn't there an "ending" element, then? Such an element X should satisfy that as n grows larger, X-x_n is smaller and smaller) and they do in complete spaces (i.e. a metric space where this happens is called *complete*.) But turns out that Cauchy sequences of rational numbers don't have as limit a rational number (:o! this is tricky to prove, but there are many counterexamples.) Since to get the reals you fill up the rationals by completing these sequences so all have "ending points," 0.9999... for me is the limit of {0.9, 0.99, 0.999, etc } which has clearly as limit 1 (since 1-any of these elements is smaller and smaller)

oca · **#49**

I also like to compare N* (positive integers,without 0) and Z* (integers, without 0)

So :

for each number n in N*, Z* contains that n and -n, so... Z* should be bigger than N* ...
but their will always be a number in N* to count each number of Z*

That's so weird when put into words...

leichtloeslich · **#50**

oca wrote:

I also like to compare N* (positive integers,without 0) and Z* (integers, without 0)

I think you're confusing notation here. If R is some ring, then R* usually denotes the multiplicative group of that ring.

For the integers Z that would be Z* = {1, -1}.

(You probably got confused by the fact that if F is a field, then F* is indeed just F without 0 (which is kind of the definition of a field, really))

lemmata · **#51**

Aidoneus wrote:

We are now getting many U.S. students into extreme debt for non-STEM degrees (science, technology, engineering, math)--degrees that have little marketable value, but which have done wonders for the bottom line of public and private institutions through government guaranteed loans and grants.

Government college loans: Putting more young people into debt with unrealistic and unprofitable career fantasies. Sad. Furthermore, when those people can't pay up, unborn babies are on the hook for the bill, too.

DrStraw wrote:

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed.

Indeed! Some IQ test questions actually require you to take invalid leaps of logic.

lemmata · **#52**

oca wrote:

Pippen wrote:

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

I believe that this sort of proof is often taught it schools. I learned to hate it as teaching tool. The proof focuses on symbolic manipulation and often leaves students with the impression that there is some mystical math voodoo.

The kind of "meaning" that RBerenguel demonstrates in his post is what we should aim to transfer to students when we teach that 0.999...=1 to high schoolers, but the current standards are terribly inadequate for that purpose.

John Fairbairn · **#53**

I think the biggest maths mystery in the world has not been mentioned here, and furthermore whatever explanation it has seems to fly in the face of the usual snobbery about IQs, Mensa and top universities.

It is this. A significant proportion of the populace who would normally not be regarded highly if talk was restricted to IQ and degrees, and who would look blank if you asked them whether they were interested in maths, have the ability to stand in front of a dart board and INSTANTLY, as soon as a dart lands, calculate all scores backwards, as well as calculating what further arcane combinations of scores (also backwards) are required to finish. They can further do this when addled by alcohol in a noisy pub while carrying on normal and robust conversations. Furthermore, they can do this while keeping in mind whose round it is and who is drinking what, and checking accurately that the barman has not diddled them with their change.

I have never been able to work out which I admire more: that or Ginger Rogers' ability to dance everything Fred Astaire (supposedly Hollywood's best dancer) did, but also backwards and in high heels.

RBerenguel · **#54**

John, this kind of ability is essentially lots and lots (loads and loads

?) of practice. If every night you play darts at the pub, eventually you'll "know" all the numbers, just out of practice and a lot of repetition. But of course, in many fields lots of this kind of practice can be misunderstood for "intelligence," and the lack of this practice by a lack of it.

Personally, I'm astonished when I see a basketball (Petrovic) or soccer player (Zidane) that seems to "navigate" among the pitch like time works differently for them.

oca · **#55**

lemmata wrote:

oca wrote:

Pippen wrote:

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

I believe that this sort of proof is often taught it schools. I learned to hate it as teaching tool. The proof focuses on symbolic manipulation and often leaves students with the impression that there is some mystical math voodoo.

The kind of "meaning" that RBerenguel demonstrates in his post is what we should aim to transfer to students when we teach that 0.999...=1 to high schoolers, but the current standards are terribly inadequate for that purpose.

I agree, I would say that helped me to accept, not to understand...

quantumf · **#56**

DrStraw wrote:

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed.

I realize you didn't necessarily focus on this in your reply, but I'd still like to comment on this. As far as I know, there is no other well-known, widely-agreed measurement of "smarter", apart from IQ scores. As you imply, IQ scores are pretty well correlated with success, at least in terms of what we usually measure success by, such as educational level, income, status and longer life, at least compared to any other single metric.

DrStraw wrote:

people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them

I'm not really sure what you mean by that. Presumably one analyzes the world around you in order to best decide how to navigate a successful path through it. If IQ is correlated with success, then I would suggest that it's also correlated with an ability to analyze the world around you. All other things being equal, the smarter (by IQ) person is more likely to make the better choice in most dilemmas.

DrStraw · **#57**

quantumf wrote:

DrStraw wrote:

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed.

I realize you didn't necessarily focus on this in your reply, but I'd still like to comment on this. As far as I know, there is no other well-known, widely-agreed measurement of "smarter", apart from IQ scores. As you imply, IQ scores are pretty well correlated with success, at least in terms of what we usually measure success by, such as educational level, income, status and longer life, at least compared to any other single metric.

You got this completely backwards. That is not what I meant at all. What I meant was that of all the things which are necessary to succeed only a small part of that set is related to IQ. EQ is probably more important than IQ.

Quote:

DrStraw wrote:

people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them

I'm not really sure what you mean by that. Presumably one analyzes the world around you in order to best decide how to navigate a successful path through it. If IQ is correlated with success, then I would suggest that it's also correlated with an ability to analyze the world around you. All other things being equal, the smarter (by IQ) person is more likely to make the better choice in most dilemmas.

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

Even though I am supposed to have a very high IQ as tested (+3 sd), I am not a big believer in it as a measure of much more than the ability to pass IQ tests. There are millions of people more successful than I who have much lower IQs. And, yes, I am quoting a sample of one there, but I obverse the same with others as I look around me. My problem is that I don't think my EQ is much, if any, above average.

Bill Spight · **#58**

DrStraw wrote:

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

That question came up in one of my undergraduate courses. Studies indicated that IQ was positively correlated with income (IIRC, to take a common measure of success), but only up to an IQ of 125 or so. Among people with higher IQs there was no correlation between IQ and income.

DrStraw · **#59**

Bill Spight wrote:

DrStraw wrote:

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

That question came up in one of my undergraduate courses. Studies indicated that IQ was positively correlated with income (IIRC, to take a common measure of success), but only up to an IQ of 125 or so. Among people with higher IQs there was no correlation between IQ and income.

I thought we were talking about success, not income. The two are not at all the same. Success is, in my opinion, a measure of happiness. Many people with high incomes are not happy, and therefore not successful, because they spend too much time acquiring that income.

Bill Spight · **#60**

DrStraw wrote:

Bill Spight wrote:

DrStraw wrote:

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

That question came up in one of my undergraduate courses. Studies indicated that IQ was positively correlated with income (IIRC, to take a common measure of success), but only up to an IQ of 125 or so. Among people with higher IQs there was no correlation between IQ and income.

I thought we were talking about success, not income. The two are not at all the same. Success is, in my opinion, a measure of happiness. Many people with high incomes are not happy, and therefore not successful, because they spend too much time acquiring that income.

Oh, I quite agree that income does not equal success.

As for any correlation between IQ and happiness, my guess is that there is a small negative correlation. Ignorance is bliss. :mrgreen:

Why is the diagonal of a square not "2"

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