Side Topic: 'It's Not a Toom-ah'

[R.D. Silverman]

Originally Posted by R.D. Silverman

Responses?????

And some excellent side reading if people are interested:

http://www.de.ufpe.br/~toom/my-artic...c/ARUSSIAN.PDF

[wombatman]

Originally Posted by R.D. Silverman

And some excellent side reading if people are interested:

http://www.de.ufpe.br/~toom/my-artic...c/ARUSSIAN.PDF

I really enjoyed reading this. Another article along somewhat similar lines about the willingness to let students struggle:

http://ww2.kqed.org/mindshift/2012/1...tern-cultures/

[Chuck]

Originally Posted by R.D. Silverman

And some excellent side reading if people are interested:

http://www.de.ufpe.br/~toom/my-artic...c/ARUSSIAN.PDF

Thank you for pointing out an interesting article. That is what happened to me in college 45 years ago.

I could do the 0.9999 problem, but not the others. At this late date I will not frustrate myself by attempting to join the discussion group.

[wblipp]

Originally Posted by R.D. Silverman

And some excellent side reading if people are interested:

http://www.de.ufpe.br/~toom/my-artic...c/ARUSSIAN.PDF

This article includes four problems that Toom used as an indicator of mathematical preparedness for college math - and that most students in his business calculus classes could not solve. Already one person here has bravely admitted that he could not solve three of these. Is there interest in threads in this subforum dedicated to each of these problems? I believe these problem involve a handful of simple concepts that anyone here can master, but I also believe that the American educational system has failed to impart these simple concepts to many graduates.

Would PMs or email instead of threads be preferred? That avoids the problem of reading other people's answers instead of figuring it out for yourself.

If there is interest, then I also suggest that the professor (Bob) recruit some "teaching assistants" to handle these threads. I'll be mostly offline for the next two weeks, but volunteer to assist to the extent my limited access permits.

[only_human]

Originally Posted by Chuck

Thank you for pointing out an interesting article. That is what happened to me in college 45 years ago.

I could do the 0.9999 problem, but not the others. At this late date I will not frustrate myself by attempting to join the discussion group.

But to me, this seems to be one of the main reasons for this as a discussion group. I think it is for people who truly do work on it to help each other get past that tenacious tiny bit of trouble.

Originally Posted by wombatman

I really enjoyed reading this. Another article along somewhat similar lines about the willingness to let students struggle:

http://ww2.kqed.org/mindshift/2012/1...tern-cultures/

And another:
Richard Feynman on education in Brazil

From Surely You’re Joking, Mr. Feynman!:

After a lot of investigation, I finally figured out that the students had memorized everything, but they didn’t know what anything meant. When they heard “light that is reflected from a medium with an index,” they didn’t know that it meant a material such as water. They didn’t know that the “direction of the light” is the direction in which you see something when you’re looking at it, and so on. Everything was entirely memorized, yet nothing had been translated into meaningful words. So if I asked, “What is Brewster’s Angle?” I’m going into the computer with the right keywords. But if I say, “Look at the water,” nothing happens – they don’t have anything under “Look at the water”!

[Xyzzy]

Originally Posted by wblipp

Is there interest in threads in this subforum dedicated to each of these problems? I believe these problem involve a handful of simple concepts that anyone here can master, but I also believe that the American educational system has failed to impart these simple concepts to many graduates.

Yes, we are interested. We are unable to solve all (!) the questions in the PDF. (We "know" the 0.99… answer but we do not know why that is the answer.)

Originally Posted by wblipp

Would PMs or email instead of threads be preferred? That avoids the problem of reading other people's answers instead of figuring it out for yourself.

Obviously, one-on-one instruction is best, but it is inefficient. We can see threads working with a liberal application of spoiler tags.

[ewmayer]

Originally Posted by wblipp

This article includes four problems that Toom used as an indicator of mathematical preparedness for college math - and that most students in his business calculus classes could not solve. Already one person here has bravely admitted that he could not solve three of these. Is there interest in threads in this subforum dedicated to each of these problems? I believe these problem involve a handful of simple concepts that anyone here can master, but I also believe that the American educational system has failed to impart these simple concepts to many graduates.

I think Toom does engage in some rather 'selective prosecution' there - for the Tom/Dick/Harry linear algebra problem, yes, the student *should* have anal-retentively begun with 'let us denote the time needed by Tom, Dick and Harry by T, D and H, respectively,' but to pretend it is non-obvious what the student meant is being deliberately obtuse. (And Toom fails to mention whether the student actually solved the resulting system correctly.) At the same time Toom makes it sound trivial that 0.999... = 1, when in fact obtaining this result involves some highly nontrivial aspects of real-numerology and set theory - including 'faith-based' acceptance of at least one key axiom (probably several, but I've not the time to make a detailed list at the moment) pertaining to the reals. One must in effect resolve the seeming contradiction between the following two 'proofs' which yield opposing results:

Proof A: The distance d between the n-term (n finite) expansion and 1 is d = 10^(-n), which --> 0 as n --> oo, thus in the limit, 0.999... = 1.

Proof B: Using the same notation as Proof A, the number of distinct real points in the length-d interval separating the n-term expansion and 1 is infinite (in fact uncountably so). Incrementing n by 1 cuts the distance by a factor of 10, but the number of distinct reals in the new smaller interval is still uncountably infinite. Thus no matter how large we take n, there remains an uncountably infinite number of points separating the point corresponding to the resulting expansion from 1, hence 0.999... != 1.

Wikipedia describes the difficulties involved in such problems:

The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field (R ; + ; · ; <), up to an isomorphism,[1] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of classical mathematics.

The reals are uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted \mathfrak c and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted \aleph_0). The statement that there is no subset of the reals with cardinality strictly greater than \aleph_0 and strictly smaller than \mathfrak c is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory (ZFC), the standard foundation of modern mathematics, in the sense that some models of ZFC satisfy CH, while others violate it.

Thus, if Toom is going to fail a student for not explicitly declaring he meaning of the name-abbreviating variables, I fail him n times over (n large but finite) for not stating the many assumptions underlying his claim that 0.999... = 1. :P

[retina]

1/3 = 0.33333...
2/3 = 0.66666...
3/3 = 0.99999...

[ewmayer]

Originally Posted by retina

1/3 = 0.33333...
2/3 = 0.66666...
3/3 = 0.99999...

You also neglect to detail the many assumptions behind those '=' signs. Please provide a convincing refutation of my Proof B.

[Agree with wblipp that such 'sidetrack discussions' should probably be split off into separate threads - here we are veering out of the field on number theory as the subforum likely intends it to mean.]

[retina]

Originally Posted by ewmayer

You also neglect to detail the many assumptions behind those '=' signs.

Hmm. I have no idea what you mean here. I thought it was simply one thing is numerically the same as the other.

Originally Posted by ewmayer

Please provide a convincing refutation of my Proof B.

I go lost after the word "notation". I don't even understand Proof A.

[owftheevil]

Proof B seems to depend on the mistaken assumption that an uncountable set cannot be written as a countable union of uncountable sets.

As for the TDH problem, if you think about it for a minute, you will realize that if T,D, and H represent times to finish the job, then T + D =2, T + H =3, and D + H = 4 make no sense. In Toom's defense, i think this was what he wanted the student to realize.