Have you ever wondered about the real reason why math education in our schools is *so* awful? Why despite the best efforts of large numbers of parents, the schools seem to be incapable of figuring out why they're so dreadfully bad at recognizing the difference between a halfway decent math curriculum and a trendy piece of garbage?

Read below the fold for a perfect example of why. The short version: the people who are involved in running education in America consider it perfectly acceptable to be idiots when it comes to math.

My kids go to school in the Arsdley school district in Westchester County, New York. It's an outstanding school system overall, and I'm very happy that my kids go there. (It's pure luck - we don't live in Ardsley, but in the neighboring village of Dobbs Ferry, and the realtor who sold us our house didn't mention that we were actually across the line into the better school district.)

Anyway - yesterday, we had our annual school budget vote. The superintendent of the school district sent out the following letter informing us of the election results:

May 16, 2007

Budget Vote ResultsThe 2007-2008 school district budget vote passed by a margin of 62.3%. A total of 1372 residents voted, sixty-one more than last year.

The results of the $53,413,610 budget:

Yes: 856

No: 516

The result of the election is that 62.3% of the voters voted to pass the budget, and

37.7% voted to reject it. As I was taught, the "margin" for a yes/no measure is the percentage by which the vote exceeded the threshold needed to pass the measure - meaning that the margin in this election was 12.3%. I'd also accept the slightly different definition of margin as "the difference between the yes votes and the no votes", giving a margin of 24.6% ; it's not the correct definition for a yes/no vote, but at least it's a margin of some kind. But the absolute percentage of "yes" votes is simply *not* a margin. That's not what the word means!

Our district superintendent doesn't even have a *clue* of what a "margin" means in an election. It's possible to become *the* main person in charge of the school district - a person whose job includes *judging the qualifications of math teachers, and judging the quality of math curriculums* - while being totally clueless about even the simplest of mathematical concepts.

"Margin" is such a basic thing - I would *expect* my kids to understand what the "margin of victory" in an election meant by the time they finished elementary school. But the superintendent of their school district *doesn't*.

I'm willing to bet that that letter was spell-checked and grammar checked before going out to the mailing list. I've never seen a letter from the superintendent containing grammatical errors like "We done good on the state tests last year". I've never seen an "'s" where an "s" was correct. He's careful about those things - because they're important for a person who is presenting himself as an educator. But math? Hey, who cares if he can't do math? Who cares if he doesn't even have a basic understanding of the mathematical concepts that can have a direct impact on his job?

He probably learned his definition of "margin" from sportscasters, who commonly say things like "The Yankees beat the Red Sox by a margin of 3 to 2."

This is clearly gibberish, as the Yankees are incapable of beating the Red Sox. (So saith a despondent Orioles fan...)

You know, as I was perusing this (I got there by following the advice in an earlier comment thread and googling "math wars"), it occurred to me that the most bizarre, anti-learning math standards, textbooks, and curricula wouldn't actually harm children in terms of what they learned if the teachers teaching them knew math.

My wife (who taught at a private school for a few years after graduating from college) has a horror story of overhearing some of her fellow teachers conclude "the answer book must be wrong" after checking the answer to a number sequence problem. (As in, "what's the next number here") The sequence in question was 1, 4, 9, 16, ___ ?

They couldn't believe that it would jump all the way to 25; that must have been a typo.

The book

Innumeracyis therapeutic reading when confronting too much of this, even if it is a bit dated at this point.Daniel Martin:

John Allen Paulos wrote a sequel,

Beyond Numeracy,which is also a little dated by now but only in spots (it predates Wiles's solution of Fermat's Last Theorem, for example).Beyond Numeracytalks more about good math than bad, but maybe straight-up good math is even better therapy.Your example depresses me even more than MarkCC's, by the way (you should feel proud).

I really hate to seem to take the other side here, but I really don't think this has anything to do with mathematics. It's all about vocabulary. It's in the same class as "less" vs. "fewer" for the express lane at the grocery store.

So yeah, he's not using the right term. It isn't innumerate, though.

Unfortunately, being "bad at math" is not a social stigma in our society. In fact, I sometimes think that certain people wear it is a badge of honor ("You do the math"), yet these very same people would

neversay "Oh, I'm barely literate! My reading and comprehension skills suck!"I would say that yes it is a problem of vocabulary, but it's still innumeracy. Most of numeracy is understanding, and being able and willing to apply mathematical concepts. "Margin" is such an important concept, especially in the sorts of math that have the most impact on our daily lives (voting, economics, finance, etc), that it's reasonable to expect educated people to use it correctly.

You think that's bad?

I know of an elementary school teacher (2nd grade, I think?) in the 1980s who carefully instructed her class how to divide by zero.

(In case you're wondering, x/0 = 0.)

I have to agree with Mark here. It sends a bad message when a man who would never make a grammatical, style or punctuation error sees nothing wrong with using incorrect terminology for a fairly basic mathematical concept. Unfortunately this happens so often that I just mentally fill in the correct term/concept from the context without noticing that I have done so.

Ambitwistor:

No, no, it's nullity!

Matthew L.:

Agreed. In some cases, "innumeracy" means not understanding the vocabulary of math. Confusing the different kinds of "average" — arithmetic mean, median, etc. — is a related example, where sometimes the confusion is intentional and sometimes not. And what about the "margin of error" quoted alongside poll results; is a failure to understand that vocabulary item not innumeracy?

John:

I'll agree that it has something to do with not understanding a term. But the reason that I still think it's innumeracy is because it would be unacceptable for him to

notunderstand or correctly use basic terms about language. When it's a mathematical term, even a ridiculously simple one like "margin", if he uses it wrong, people will react (if they notice) by saying "Oh, he just doesn't know that math stuff".But if he were to send out a letter about the school board elections saying that "The budget measure in the elecshun passed by 12.3%", he'd be out of a job. Getting the math right and the spelling wrong would be an absolutely unacceptable embarassment, and he'd almost certainly wind up losing his job over it. But get the spelling right, and make a math error that a 5th grader should be able to get right, and no one even blinks. (I'm pretty sure that I learned about what a margin was in fifth grade, talking about the presidential election of 1976, and my school district was *not* known for having a particularly good elementary math program.)

One thing you didn't catch, the guy doesn't know the proper rules for rounding either. The votes in favor of the budget are 62.4% to the nearest tenth of a percent. But then I guess that can be expected from a man who doesn't know the correct word to use.

Before assuming the cause is stupidity, consider that the government actively encourages flat-out lying. This may simply be a flat-out lie at work.

If caught in the lie, the source could claim to have 'mispoken', which means he's lying about having lied.

VHurtig:

You're right — I hadn't caught that. 856 / (856 + 516) = 0.6239 plus some other digits, or 62.4% to three significant figures.

VHurtig and Blake:

You're right. I just eyeballed it, and came out to less than 63 1/2, and assumed it was right from there. But he even blew the rounding. Innumeracy indeed.

Some folks who read the Washington Post may recall an incident several months ago when one of the editorial page morons, Richard Cohen, wrote a column stating that it was unimportant to understand algebra.

Ironic as it is, I think you're also incorrect.

A

marginis a "measure or degree of difference" [Merriam Webster's] or a "difference between one amount and another" [Encarta]. The margin is, quite clearly, 340 votes, or 24.8% of the vote. You're 12.3% (170 votes) is the number who would have to switch sides, not the difference.If you want to argue it's an issue of definition, it is a poor definition if the percentage margin wouldn't match the absolute margin. Not only does a margin of 24.8% provide greater consistency, it's also the popular usage.

This definition becomes especially important when there are more then two options or abstentions in a plurality system.

BTW, I apologize for the diction error: my edits and submit got away from me.

Math is all about vocabulary. Since mathematical ideas are extremely precise, expressing them without precision is tantramount to not expressing them at all. Some of my Calc I students last semester were surprised to get 1 out of 6 pts for stating the Mean Value Theorem as "f'(c)=[f(b)-f(a)]/[b-a]." Even if they still don't fully understand

whythe hypotheses are important, I doubt that there are many left that don't understand that they are important (or, more pessimistically, at least that I think that they're important).Mathematics is a universal language precisely because we give words and symbols precise meanings that are defined using only previously defined words and symbols all the way back to the foundations.

"margin of 67-33" is an acceptable colloquialism. Leaving out the "-33" is ignorance.

Your district superindendent no doubt pumps his gas at a station with a sign that says "Please prepay before pumping", and after you point this out to him, he'll just say he'll go there irregardless.

Xebecs, don't even get me started on sportscasters. Just once I'd like to get through an entire pregame show without one of the hosts saying something mathematically idiotic. Every time they say "law of averages", they use it incorrectly.

Struggling with this in my own school. My son is in first grade, reading well above grade level, but seems to have some of my own issues when it comes to math (primarily number-related rather than concept related; I can handle vector calculus but habitually check my own addition on my fingers for any sum that wraps around a 9).

Go looking for remedial math support. Our summer school for kids who are trailing the grade standards is 90% reading and writing instruction. Google "learning disabilities" and you'll be treated to page upon page of info about kids who can't read. It took me quite a bit of digging to find out that such a thing as Dyscalculia actually exists -- and no one will take me seriously when I ask if it is something we need to test for. Doesn't matter, there are no tests. Because math doesn't matter, apparently. That's what calculators are for.

And then there was the time in 12th grade when I was deriving the simple money multiplier formula (M = 1/r) on the board in AP calculus, and the teacher said "whoa - where on earth did that come from?" in response to my whipping out the formula for the sum of a geometric series.

And before you ask, no, it wasn't a pedagogical device where he wanted me to explain things more. I really don't think it's absurd to expect that people teaching what should be college-level economics to have a decent grasp of high-school-level algebra.

That probably still isn't as horrific as the can't-recognize-squares example though.

Jackie Clarke on Faux TV's Hannity and Colmes just utterred the following:

"They do 25 hours of radio a week. That's what a year, a million hours."

I know we shouldn't get too picky, but being off by a factor of almost 800?

/sigh

c'mon now, we all have our issues regarding what should be basic knowledge people.

what does it mean to be "schizophrenic" about something? (no googling or wikipedia-ing now!)

what is "negative reinforcement"?

...had one in our local paper describing a "10% increase in unmarried couples co-habitating" since the 60's in an article dissecting a declining divorce rate. this was on the front page sidebar and seemed to not square with basic knowledge of demographics nor be a sufficient contributor to be worth highlighting. sure enough, reading onto the continuing pages it was supposed to be 10-fold increase....

"what does it mean to be "schizophrenic" about something? (no googling or wikipedia-ing now!)

what is "negative reinforcement"?"

O.k., I'll bite. Without googling and just pulling this stuff from my general knowledge, schizophrenia is a psychotic disorder but in colloquial terms being "schizophrenic" about something often just means holding different mutually inconsistent viewpoints simultaneously. Negative reinforcement increases the likelihood of a future behavior by the withdrawal of a stimulus (which is likely perceived as aversive by the organism in question).

How'd I do?

My point being that the average educated joe should know basic shit and particularly the average educated joe who is in charge of educating children should be able to use the term "margin" without looking like a total retard.

#20 - "Irregardless"?! "IRREGARDLESS"?!??!11

... sorry about that. You want either "regardless" or "irrespective". I hope you're not a school superintendent.

The really sad this about the failure to recognize squares is that you don't even need to know multiplication to get this one, just subtraction: since

n2 is the sum of the firstnodd integers, a careful look at the terms of the sequence would have revealed4 - 1 = 3

9 - 4 = 5

16 - 9 = 7

and 16 + 9 = 25.

#20 - "Irregardless"?! "IRREGARDLESS"?!??!11

... sorry about that. You want either "regardless" or "irrespective". I hope you're not a school superintendent.

The really sad this about the failure to recognize squares is that you don't even need to know multiplication to get this one, just subtraction: since

n2 is the sum of the firstnodd integers, a careful look at the terms of the sequence would have revealed4 - 1 = 3

9 - 4 = 5

16 - 9 = 7

and so the next term in the sequence must be 16 + 9.

Now I feel like Jacob Two-Two.

I'm starting to wonder if they stopped teaching how to round numbers at some point. I regularly see students plug something into a calculator, get an answer of "2.3587892", and give the answer as 2.35. It's gotten to the point where I'm shocked when students *can* round properly.

@Sara

Calculators, computers, and any other devices really should be banned from mathematics classes. There's really zero reason to use them. Students need to be taught how to estimate sines and cosines, decimals, e, pi, and the rest.

Regarding calculators, I could not agree less. There are situations in which calculators should not be used, but there are others where doing something by hand is just pointless. If one has to find, for instance, the mean and standard deviation of a few hundred pieces of data then they should be using a calculator. Doing that by hand is a complete waste of time. Similarly, calculator functions for probabilistic distributions are valuable, as one of the skills that has to be developed is an understanding of how and when to use the varying distributions, adding menial algebra to that is pointless. That doesn't mean that students shouldn't know how to do this stuff by hand, simply that they shouldn't always do so.

I disagree on the non-utility of calculators. My experience has been that when I got my first (programmable!) calculator in 9th grade it actually improved my mathematical intuition and sped up my learning. I spent a lot of time playing with the calculator, trying out different things. There were trig functions and other exciting novelties that I could play around with, plot various combinations of functions and see how they behaved. This would not been as easily doable using interpolation in tables (yes, it was that long ago) and thus not something I could have done as much. Also, constantly seeing the values for e, pi, sqrt(2) and so on burned them into my brain much better than having them turn up every few chapters in a book.

I did have a fairly good understanding of "math" to begin with (and perhaps I am unusual in actually reading the manual), but it seems to me that calculator use reinforced that understanding.

"

Now I feel like Jacob Two-Two.Or Jacob Two-Squared . . .

On the other side of things, when I was temping for a construction management company, we used to get letters from the Philly school district requesting that all contractors working for the schools check their work, as estimates were being submitted that were basically riddled with errors. Iirc, they even gave some examples - it wasn't pretty. And no, they weren't all overcharges, either.

Of course, one could argue that the district had only itself to blame . . .

Re calculators, new tools changes the skill set for good and for bad. Personally I would encourage use of the latest tools.

That would probably entail accepting that rounding will come to mean dropping the last presented digits or applying a software command. But error estimation and error propagation would remain skills to master sooner or later.

I am more concerned with the ability to make sound estimates, even without a tool at hand, and especially to roughly check the result from the tool. The Colmes episode illustrates how it remains important in all walks of life.

SarahA:

What is the difference between the difference to switch (or pass the budget) and the difference to the amount, though? 🙂

But I agree with your remaining points.

I stopped using calculators, for the most part, during college. After a couple freshman classes (intro solid-state chemistry, for example) almost all the problems involved symbol manipulation rather than arithmetic. Of course, the fact that I never had a calculator in reach meant that I could still do long division when necessary.

(And it

doescome in handy at odd times, catching other people's innumeracy. Remind me someday to tell you about the time the CEO killed 250 million Americans every year. . .)Come to think of it, another way to state this is that I am more concerned about getting things roughly (or rather sufficiently) correct than about knowing exactly how wrong they are. (The later unfortunately in most cases still connects to the "sufficiently correct" part of the former, though.)

A "glass half-filled or half-empty" characterization of concerns, perhaps.

Re #25: Mel, I'm no mind-reader, but I think you effectively punted on Drugmonkey's point about schizophrenia. I'm guessing his point was that, contrary to popular belief, its technical usage refers to something substantially different from multiple personality disorder.

Re #26 (and #27): Again, I'm no mind-reader, but I'm guessing that Science Avenger intentionally used the nonstandard "irregardless" as something that a person who misuses "margin" would say.

"Re #25: Mel, I'm no mind-reader, but I think you effectively punted on Drugmonkey's point about schizophrenia. I'm guessing his point was that, contrary to popular belief, its technical usage refers to something substantially different from multiple personality disorder."

Sorry, I missed that point, probably because Drugmonkey also asked for the meaning of negative reinforcement, which I've never encountered as a colloquial term that differs in meaning from the psychological term of art.

And I'm pretty sure the diagnosis of schizophrenia has absolutely nothing to do with multiple personality disorder. As far as I know (again mining that general knowledge without researching the topic) MPD is a dissociative condition whose status as a legitimate disorder is debated. Schizophrenia is a psychotic disorder with a strong basis in abnormal neurological functioning. Schizophrenics don't have multiple personalities ala the popularized Sybil; they have broken brains that can cause them to attribute internal stimuli to external agents.

And I'm not a mathematician or versed in the proper use of "margin" as a mathematical term of art. I'm just an average person who read the use of "margin" in that letter by that school administrator and thought "That's not a margin. What a fucking idiot that guy is."

Re#36, bingo. "schizophrenia" is (most?) often used colloquially to indicate indecision in normal people as a misplaced analogy with multiple-personality disorder. neither has anything to do with the symptoms of schizophrenia as a defined disorder.

Re#37 and #25, Mel gets negative reinforcement right, nice job. what Mel is apparently unaware of is a very pervasive colloquial (and not so colloquial) use of the term in place of the non-behaviorist connotations of "punishment", i.e. as "pain" or some such. (the behaviorist connotation of "punishment" simply refers to a decrease in the probability or strength of a given behavior in the future due to whatever contingencies have been put in place, it does not imply "pain" or any other stimulus/sensation in particular)

I'll have to disagree here. It depends very much on the class -- there's definitely no reason to use calculators in, say, 5th grade math. However, they can be quite useful in some later math classes.

I taught a business math class last quarter, where the students were asked to deal with polynomials modeling certain real-world economic systems, which require computations *I* would balk at doing by hand (unless I was really bored). This quarter I'm teaching Calc 3, and graphing calculators make discussion of parametric and polar curves so much nicer -- while we graphed some of these by hand, as a general approach that's nearly impossible.

I definitely hadn't ever heard "negative reinforcement" used as a pop synonym for punishment. I'd be especially worried if I heard it used that way by an educator who is supposed to have been trained in behavior modification. "Schizophrenic" as a pop term for all sorts of things seems more common. So I guess the question is whether "margin" has become a pop term that just means "an election was held and here are the raw results" and whether that school administrator actually knows what a margin really is.

I love watching language change and I'm not a purist who thinks language needs to be preserved in any particular form. When a term of art breaks loose and starts being used colloquially for something different, it is an interesting question where in that process misuse from ignorance becomes a new colloquial usage. In my experience, "margin" hasn't departed so far yet that it doesn't look ignorant to use it as it was by this poor school administrator, but maybe I just run with the wrong crowd.

Mark@ 10

"it would be unacceptable for him to not understand or correctly use basic terms about language"

I would be surprised if he can understand and correctly use basic terms about language, such as "adverb", "passive" or "tense". He can produce impeccably grammatical English: that's a basic skill that native speakers learn most of quite casually and naturally when they are two years old, but to be able to correctly discuss grammar is as much an artificial, learned skill as anything in mathematics.

Jim Roberts

Graphing by hand is immensely boring drudgework that requires patience and a steady hand, I have little of either, so I appreciate graphing calculators.

Because of an error (or a loophole) in requirements, our class were allowed to use not only graphing calculators, but real symbol manipulating calculators in my university college. It doesn't seem that it hurt us very much, we still had to understand what we were doing - we were occasionally given integrals that the software couldn't handle, for instance.

For some reason, though, our calculators didn't have very much in the way of statistics functions. We used old-fashioned tables for various probability distribution. Now I'm ashamed to admit, although I did get very good marks in statistics, I've forgotten practically everything. Perhaps as a software engineer I would have explored and remembered more if I had some software to play with - looking at open source stuff like R and Tetrad, I sure regret not paying better attention.

>(In case you're wondering, x/0 = 0.)

She must have learned that from Chuck Norris. He can divide by zero.

Type "1, 4, 9, 16" into the search engine of the Online Encylcopedia of Integer Sequences.

It returns 213 results.

For example:

A003132 Sum of squares of digits of n.

n a(n)

0 0

1 1

2 4

3 9

4 16

5 25

6 36

7 49

8 64

9 81

10 1

11 2

12 5

13 10

14 17

15 26

...

A006508 a(1) = 1; a(n+1) = a(n)-th composite number.

n a(n)

1 1

2 4

3 9

4 16

5 26

6 39

7 56

8 78

9 106

10 141

11 184

12 236

13 299

14 374

15 465

...

A073141 Product of the largest and smallest number having in binary representation the same number of 0's and 1's as n.

n a(n)

0 0

1 1

2 4

3 9

4 16

5 30

6 30

7 49

8 64

9 108

10 108

11 154

12 108

13 154

14 154

15 225

...

A110979 Squares equal to the sum of the first n primes minus 1.

n a(n)

1 1

2 4

3 9

4 16

5 196

6 839056

...

and so on.

The Bad Math is to assume that there is only one correct answer to this sequence question!

"...sportscasters, who commonly say things like "The Yankees beat the Red Sox by a margin of 3 to 2...."

The line that bugs me is:

"Here we are, bottom of the 4th inning, and there's no score."

Excuse me! The score iz 0-0. Zero is a number. Baseball game tied zero to zero is a score, and maybe a good pitcher's duel and defensive game.

Confusion of 0 with null set, perhaps...

I ended up writing about my encounter with a similar kind of innumeracy. . . enjoy!

There are an infinite number of well formed formulas generating sequences of natural numbers beginning with 1,4,9,16. In the real world, we apply Occam's Razor to all such situations and seek not the only, but the simplest, answer. Those who posed the question might have said this, but in life, people don't always give you complete specs and you need to fill in some yourself.

I suppose that implicit in such "find the next number" problems is the condition that the sequence have minimal Kolmogorov complexity.

This summer (between professorships) I am a High School Math teacher, in a school of needy, impovershed teenagers literally next to a freeway. I am in the trenches on the War with Innumeracy, and took a pay cut to do so.

My problems with School Administrators go beyond innumeracy.

Skipping over my own school days, there was the administrator who refused to suspend the insane student who literally tried to twist my son's head off, in elementary school. Said School Administrators claimed that they had no authority to do so, as the perpetrator, following some WWF maneuvers from TV, was not carrying a weapon or drugs at the time.

We transferred my son to another school.

Then there was the School Principal who tried to censor my son's Valedictorian speech -- my son was #1 in his class, and school Student Body President, and had been accepted to university at age 13 (thirteen). This is no "Bong Hits 4 Jesus" either.

It took me 10 months to get the bad-paying High School job from when I walked in the door of the school district HQ, and they refused to look at my CV, letters of recommendation, or other credentials, on the grounds that they were a "paperless office."

I was phoned 3 days before the summer school started to ask if I'd filled out the 50-page packet of forms that they neglected to give me.

It was this 3 days before school start that I was first informed of my need to pass a TB test, which takes 3 days.

I went to a private physician, and paid to have him available on the weekend. He railed about ignorant bureaucrats who try to make medical and legal decisions for which they are utterly unqualified.

He wrote: "Jonathan Post does NOT have active TB. He should continue teaching."

The School Administrators stamped and certified this writing.

Then, in one of my first day of classes, I was ordered out of the classroom, and off campus, on the grounds that my TB test was "equivocal."

I paid to have an immediate Chest X-ray, and for a Radiologist to come to the clinic and review it. Again, they wrote that I did not have TB.

Did I ask the School Administrators where they got their Law Degree and Medical Degree? I did not. I wanted to, but, instead, I thanked them for having the best interests of the children in mind.

I was tempted to let them do as they wanted, those School Administrators, and let them order all students whom I'd been close to to have their own TB tests, and let the parents start phoning in their anger. Poor parents might not have been able to get lawyers.

School Administrators innumerate? Among other problems, yes, usually.

I'll keep you posted, as the battle rages on, and I work with the walking wounded in the Good Math, Bad Math wars.

The Children's Crusade. Maybe an unfortunate phrase.

Also, we'll see if my first paycheck, on July 10, 2007, is correct. 10 months is a long wait.

Sucks, JVP. You have my sympathy and support.

I would have loved to be in a high-school class taught by you.

At age 62 and retired, I would like some good math education. A year ago I finally "knew" something, but I dont know where to follow up on it.

Consider the plight of the one-eyed, paranoid olympic swimmer.

As the date of the contest approached, our paranoid swimmer became more convinced that he would be cheated out of his great opportunity. He petitioned the committee to allow him to race in the far right lane, as his one good eye was his left eye, and he could observe when the other contestants began and finished, so as not to be cheated. His petition was granted, but his paranoia was not alleviated. The ease of accomodation fueled his suspicion that the committee planned to cheat him another way.

He developed a plan to outsmart the committee and based his plan Euclid and Pythagoras. He measured the length and width of each lane, and the diagonals. Assuring himself the lines separating the lanes were equal and parallel, and depending on Euclid's fifth postulate, that parallel lines meet at one point, at infinity, he laid himself by the pool and stayed awake to prevent anyone from making midnight changes.

At the time of the race, his frazzled neves betrayed him, and he jumped the gun. His one good eye scanned the pool to discover that he was alone in error. Moreover, at the precise moment that his eye crossed the plane of the water, the lines separating the lanes met, not at one point, but an infinity of points, along their entire length. Euclid's Fifth Postulate was bunk.

Officials fished our now catatonic contestant from the water.

Our swimmer became psychotic upon discovering the effect of the observer on Euclidean geometry, that Euclids fifth postulate is neither eternal nor universal. The relative position of the observer determines the outcome of the observation, Einstein's Special Theory of Relativity.

Having learned from our contestant, I asked myself. Does this apply to any other of Euclid's postulates? Is it possible that Euclid's First Postulate is flawed? Given the First Postulate that a line can be drawn between any two points, I arbitrarily desired to draw a line from a point on my computer to one drawn by Euclid. What an absurdity, because of time, a point identified by Euclid cannot be determined. Oh pooh, I just fell into Einstein's General Theory of Relativity. If the observer and time are included in the universe, Euclidean Geometry is no longer intuitive. Perhaps more importantly, including time and the observer in the defined universe, General Relativity and the Heisenberg Uncertainty may be the same thing.

That is as far as I have gotten. I am not the brightest bulb on the Christmas tree, so I know others have worked out the problem. Would someone be kind enough to give me some references?

Love Lou

I have no idea what you are talking about, Lou.

Of course it's not. By swapping it out for something else we get all the various non-Euclidean geometries.

In this particular case, though, the swimmer was simply mistaken. The lanes did nothing of the sort. He merely assumed they did because he could no longer see the other lanes hidden behind the two surrounding him.

None of this has anything to do with GR. You're abusing the notation to attempt to claim that you can draw a line between any two points. You must first have both of the points within a Euclidean plane. Unless you can define a plane containing both the point on your computer and a point drawn by Euclid, you can't use his postulates in any way.

If you look at the universe with time transmuted to a space-like dimension (so that you

candefine such a plane), then it's non-Euclidean. The first four postulates will still hold, but you'll have some different fifth postulate. Again, though, this has nothing to do with GR. It *definitely* has absolutely nothing to do with the Uncertainty Principle.Thanks for the response, Xanthir. I appreciate it. I really am looking for some education, and dont know where to look. I was taught that math was eternal and universal, the true science. But my education occurred when dinosaurs were reptiles, before plate techtonics, when Jefferson and Washington were good guys, and there was no such thing as Mendelbrot sets, fractals or cascades, no Schumacker-Levy. The Universe was mostly empty, except for random clusters of light emitting matter which followed the ordered existence of Cosmos. Results which did not fall within prediction were dismissed as "outlyers;" Total results would regress toward the mean given a few more measurements (and sometimes pretending you didnt get the "outlyer" results).

My occupation and avocation was working with delinquent youth. At the height of my game, I was recognized as being good at it. Unfortunately, the research and "science" of the subject was less valuable than what I learned as an eight year old in the "upper Misson" of San Francisco.

I think my question is: How do we decide what the parameters are? What are our hidden assumptions, and will we get different results as we change them? I chose geometry to question, because I wanted a more concrete jumping off place.

My hope that I can understand the answers is based on the ability of the Mayans, a stone age people, to culturally accept an accurate calendar (more accurately, a set of calendars), and to use a unit of measurement, 1.005 meters, for their reflecting pools which is closer to measuring the quarter meridian of the earth than our accepted meter (which had to be demoted from its position as a measurement based on earths circumference in 1895).

I know that someone has dealth with the limitations on our mathematics when we assume that time is a straight line. When we arbitrarily assign time to the fourth dimension, rather than assigning it to be first, and other issues.

If anyone out there could direct me where to look, I would appreciate it.

Love lou

Lou Ohls,

I just 2 days ago completed teaching Algebra 1 to problematic teenagers in an inner city high school. You have wrestled far longer than I with these matters in the battle between civilization and chaos.

"When we arbitrarily assign time to the fourth dimension, rather than assigning it to be first, and other issues."

Not really so, except the convention at (x_1, x_2, x_3) = (x,y,z) and x_4 = t.

The non-arbitrary part is the signature, i.e. that t has the opposite sign as x, y, and z, so as to describe Minkowski space.

The association time with "the 4th dimension" was wildly popular in the later decades of the 19th century, already a cliche before H. G. Wells' "The Time Machine."

My mentor Richard Feynman became one of the great amateur experts on Mayan science. He gave new insights into how the Mayans combined Lunar and Venus periods in their calendar.

He also battled with California bureaucrats about elementary textbooks, which he conclusively showed to be wrong and silly.

Lou Ohls:I personally happen to think math isn't entirely without contingency with other knowledge, but it is customary to define is as outside science proper. Mostly because of its universal nature - only subsets are used in other sciences, without full formalization at that.

"Outliers" - and they are still there, as well as other statistical concepts. Perhaps you mean that new methods and models (e.g. bayesian probability) add to or replace old ones.

That is an interesting topic, at times useful to define boundaries for models or suggesting new ones, but at other times where we lack knowledge, a mere philosophical question.

IMHO it is more distracting than not if one wants to get "some education" in math and science as such. (But maybe the philosophical questions is your real area of interest.) There are so many results and theories to learn before getting around to systematically ask about changes in their parameters, even less in their assumptions, hidden or not.

Your swimmer should perhaps plumb the length and depth of the water before attempting to crawl. 😛 But of course the hardest trick is to balance your motivation and interest against the mind-numbing work of training and technique. In any case, good luck!

Thanks for the help. Now perhaps (perhaps not) I can be some help with the major topic, innumeracy of school administrators.

I have extensive experience with public agency bureaucrats, almost none with business, private sector. Innumeracy and illiteracy of ranking public bureaucrats is a major current cultural problem. Whether addressing the problem posited by MCCC or the one from Jonathan Vos Post, the pieces of the puzzle fall into place easier if one assumes that Darwin was mostly correct, that we descended along the monkey, primate line, but also that we carry much baggage from our inheritance.

Social dominance is a primary motive, with achievement of "alpha" status as the ultimate goal. If one watches a documentary on babboons or chimps on the tellie, the parallels are uncomfortably obvious with the behavior of your administrator(s). They are even more uncomfortable when assessed against the illiteracy of our present national "alpha male" and the sexual predations of our previous national "alpha male."

When the originally described administrator bragged of achieving his budget goals, that is all he was doing, bragging, just like the alpha babboon at the top of the tree, whooping, fangs bared, after a stressful event. Demonstrating his innumeracy was part of his boasting. He didnt even have to understand the vote to win it. Doing the math, understanding the vote, was for lesser monkeys farther down the tree.

In the nineteenth and early twentieth centuries, civilized men attempted to separate themselves from our genetic baggage, perhaps not successfully, through education and meritocracy. It was assumed that the boss worked harder, and longer hours, than the employees. The muck-rakers dispelled the myth, the Depression proved the muck-rakers had a point; nevertheless, for truly successful enterprises, the myth was reality. Quiet as it is kept, this dynamic for success has not changed; it is the backbone of the internationalization of commerce.

Our public bureaucracies are not going anywhere. Success in today's political environment may be failure after the next election. Since administrators dont know where they are going,and getting there is not a goal, maintaining alpha status is the only constant. Successfully overspending the budget is a sign of alpha status. While I was working with young bone-heads, on two occassions out team's budget was cut specifically because we were successful. The money had to be re-directed to bolster less successful enterprises. We also got billed for resources that went elsewhere. Other budget shenanigans I cant speak to. Later, in a union/management setting, I had the opportunity to argue with a "Director" of the California prison system. He was a top-notch "bean counter," but making a budget busting decision. He made the decision; he had the power to do so.

Administrators may not be as innumerate as they advertise, or they may be illiterate as well, leaving the details of mathematics and grammar to "bean counters" and "secretaries." We are clearly in a cultural era where merit is not an expressed value, except in sports, and incidentally elsewhere. Yet merit does have value to the individual, and we all benefit from it without celebrity.

The moral of this story is not to assume innumeracy, but to assume social dominance. If you need something from the bureaucracy, work with the status seeking of the administrator, do not challenge it. If you dont need from the bureaucracy, ignore their foibles.

Love Lou

It's surprising the number of adults who have accepted the fact that their children "aren't good" at math. It comes from their own opinions of math, and instead of encouraging the students and aiding in help, they place blame on teachers. In highschool, I can't tell you how many of my classmates have just accepted that they were "bad" and gave up trying settling for poor grades.