mm-126
I *am* sympathetic to the neophyte teacher; as I
said to Matt, allteachers must have a first year of teaching.
I also predict that shewill relax a bit as the course wears
on. I do know that many teacherscling to the lecturing style
because they are concerned about losingcontrol of the class,
not owing to discipline problems (this is acollege course),
but because students might take the discussion intounforeseen
directions. The course does not have a set curriculum.
Thewhole point of the course is to teach students how to
write proofs. PROMYS reads like a summer idyll of leisurely
investigationby intellectually compatible people who can
afford to spend severaldays at a time working out a few
problems. It was very pleasant, nodoubt.Smile: PROMYS is
idyllic only to those who like doing math for 15hours a day.I
was primarily reacting to the math with intimidation recipe
vs. thecharismatic sage on the stage dichotomy proposed by
the OP. There areall sorts of variations in between. Matts
present lecturer is a verypleasant, unthreatening person.
PROMYS combined discovery withcooperation. Its directors are
alumni of the Ross program at Ohio. Sothis model is used in
many other summer math programs.mattsmom-- newsgroup website:
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A possibly
related comment I recently heard a math professor at UCSCmake
(!st year Calculus):Sometimes I think its the smart students
who are getting confused.- In other words, the smart students
are smart enough to perceive thatthere are gaps in what is
being taught to them. For example, they arepresented with the
idea that an irrational number is one whose decimalexpansion
is innite. (Well, how do you know its innite if youcant go
all the way to the end to look?). Or they are are told that
anegative times a negative is a positive. (Excuse me?) The
not-so-smartstudents are quite willing to parrot this stuff
back, and not worryabout it. The smart students see that
there is some kind of problemhere, but they are not often
quite smart enough or knowledgeableenough to be able to
resolve or even completely express this. All toooften they
end up interpreting this feeling of theres -something -not -
quite - right - here as Im - not - getting - this . Andif
they really are in some ways stupid, they may then conclude
Imstupid or I cant do math. (My point here is that these
twopernicous patterns may be exactly what makes some people
stupid -likely enough theres nothing wrong with the
hardware)Of course a couple other things that make people
stupid are:(1) Trying to learn stuff that doesnt interest
them.(2) Trying to learn when theyre scared out of their
wits (Whathappens if I fail? Do I lose my self-respect? Do I
lose my status inthe community? Do I lose my Mothers
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=Hey, not
bad; this comment from MarkV95061:>A possibly related comment
I recently heard a math professor at UCSC>make (!st year
Calculus):>Sometimes I think its the smart students who are
getting confused.>- In other words, the smart students are
smart enough to perceive that>there are gaps in what is being
taught to them. For example, they are>presented with the idea
that an irrational number is one whose decimal>expansion is
innite. (Well, how do you know its innite if you>cant go
all the way to the end to look?). Or they are are told that
a>negative times a negative is a positive. (Excuse me?) The
not-so-smart>students are quite willing to parrot this stuff
back, and not worry>about it. The smart students see that
there is some kind of problem>here, but they are not often
quite smart enough or knowledgeable>enough to be able to
resolve or even completely express this. All too>often they
end up interpreting this feeling of theres -something ->not
- quite - right - here as Im - not - getting - this . And>if
they really are in some ways stupid, they may then conclude
Im>stupid or I cant do math. (My point here is that these
two>pernicous patterns may be exactly what makes some people
stupid ->likely enough theres nothing wrong with the
hardware)>>Of course a couple other things that make people
stupid are:>(1) Trying to learn stuff that doesnt interest
them.>(2) Trying to learn when theyre scared out of their
wits (What>happens if I fail? Do I lose my self-respect? Do I
lose my status in>the community? Do I lose my Mothers
love?)>A tutor must be patient with the student; and patient
enough to explain theimportance of a concept or skill to the
confused student. Putting highemphasis on a relatively short
time limit in months or years to mastersomething can work
poorly. Some students may be behind in their
Mathematicsdevelopment and should receive good instruction
EARLY and for a few years inorder to then be able to study
Algebra in high school (which now is consideredto be late).
Math, especially Algebra, often works like a language; and
students can learnto READ this in the study of Algebra. The
graphical aspects common at thislevel help. Science
coursework also helps since it is so reliant often-timeson
quantitative expressiveness. Study of science serves to make
the Math morereal-but all my commentary here misses the
fact of possibly some students justnaturally having more
neurological difculty with written symbolism than
otherstudents. The quality of the instruction and the time
that it is given, and the studentsmotivation are all
important in learning.G C-- newsgroup website:
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=Paul,I made
a reply, but unfortunately titled it What is mathematics?
andit got attached to another thread with that name; you
might want tolook for it.Also, I posted a recent reply to
Math Tutor: how tofind a good onethat may interest you; its
along the lines of this topic.(Sorry but Im too lazy to say
this all over again here, and I dontknow how to grab them as
attachments yet.)You must be writing a book on this subject -
surely no one else wouldtake the trouble to express
themselves so fully, so considerately, andso well.Mark--
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=Im
trying to help a visiting 7th grader (remedial, I believe,
though)and her worksheet is using the term plain number, in
conjunctionwith Primer Number. Is it simply any number thats
not prime? Cantnd it anywhere on the web, and she doesnt
have her book with her.TIA!-- newsgroup website:
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=a@b.com asks
about:>her worksheet is using the term plain number, in
conjunction>with Primer Number. Is it simply any number
thats not prime?You will have to be more specic about plain
and Primer. You will haveto see her textbook andfind the
actual vocabulary it uses. What kind of topicdoes this
worksheet treat? Most readers at this newsgroup probably know
whata plain number is, but they do not often use that
terminology, so its meaningon the worksheet is not clear.
Whole, natural, rational, irrational, integer;the readers at
this newsgroup can make sense. Is Primer in some way related
to Secundar and Tercer? Or is it a slightcorruption of the
word prime? G C-- newsgroup website:
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=I am having
troublefinding the equation of a rational function where y^x
=x^y and goes through (2,4) (4,2) (2.48832, 2.985984)
(2.985984,2.48832)(2.25,3.375) (3.375,2.25)
(2.44140625,3.0517578) (3.0517578,2.44140625). Thenumbers are
rounded off ofcourse, but they all are true for y^x = x^y.
Any-- newsgroup website:
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=Correction:
I was mistaken where I said, So an innitely smallhyperreal
number cam be both equal to 0 but not exactly the same asjust
0. They are not equal, just as 0 is not equal to a
complexnumber with real part 0. That would lead to
contradicting the relevanteld theorems, since the sets of
hyperreal and surreal numbers eachare elds. (Example: 1/inf
= 0 implies 1 = 0*inf, contradicting thetheorem 0x = 0 for
all eld elements x. The multiplicative inverse ofa nonzero
innitesimal between 0 and all of the real numbers is
aninnite number greater than all the reals.)Paul > Stacia,
Im assuming that 1/innity is represented as 1/oo, where I>
use oo as the love-knot symbol, rather than 1/omega for lower
case> omega (more on this below). Here is a rather roundabout
way of> answering your question about 1/oo. There are two
books to read that I> recommend as absolute general interest
classics, even though they were> rst published in the early
eighties. They are about abstract> mathematics written by
mathematics Ph.Ds for the general public. I> recommend them
for all interested adults, especially parents who might> wish
to share the wonders of the universe of the Mindscape with
their> kids who might be ready for some pieces of it even at
age 10. (This is> especially if the child is advanced enough
to take Algebra at age 10.)> They are:> > INFINITY AND THE
MIND by Rudy Rucker and THE MATHEMATICAL EXPERIENCE> by
Phillip J. Davis and Reuben Hersh.> > In Ruckers book,
youllfind much about the tension between the> historical
potential innity vs. actual innity debate. Many> before and
some after Cantor, including some noted mathematicians,>
rejected the actually innite and accepted instead only the>
potentially innite. Rejecting innitesimals, the actually>
innitely small, led to the theory of limits, 200 years after
Newton> and Leibniz.> > Note: The love-knot denotation of
innity, which we see in> calculus, etc. associated with
limits, is not the actual innity, the> simplest of which is
denoted by lower case omega, the ordinal number> of the
natural numbers, nor is it what Rucker calls the Absolute>
Innity, what he describes as the sort of imaginary ordinal
number of> the class of all sets. A rather nice book by John
the context of his surreal> numbers, relates these three types
of innity in what is possibly the> most philosophically
mind-blowing equation ever conceived. It sets oo,> (the
love-knot, sometimes called potential innity) equal to
omega> (lower case omega, the simplest actual innity) rooted
to V, the> class of all sets, which Rucker calls the Absolute
Innity, the> innity of the mystics, denoted by him as
Omega, upper case omega.> (The square root means rooted to 2,
and the Vth root means rooted to> V.)> > As I understand it,
oo (the love-knot) is that one and only one> surreal number
that is between all the nite numbers and all the> innite
numbers. I think I recall seeing in Conways book 1/oo,
quite> different than 1/omega, also being dened in an
equation. Conway> accessible to the general public, talking
about the need for> mathematicians lib, and then goes on to
formalize it all for his> mathematician colleagues. Conway
and others have contributed to a> eld called transnite
number theory, which is the study of (among> other things)
innitely large and innitely small numbers.> > Rucker has a
Ph.D in symbolic logic, the language of set theory, and> so
innite sets and transnite (ordinal and cardinal) numbers
are> familiar to him. He relates this stuff to the general
public in a> masterful way. He relates the theologian
Anselms description of God> (no mater how high I can
conceive, I attain not to God, but only to> what is beneath
God) to the modern set theoretic reection principle> and how
it is used to talk about Omega. I cant recommend his book>
enough for regular folk.> > In the book by Davis and Hersh,
there is a great write-up about> Abraham Robinsons
accomplishment in the 1960s in making> innitesimals, the
actually innitely small, logically> mathematically sound for
the rst time in history. He created> non-standard analysis,
an extension of the reals. He calls this set> the hyperreal
numbers, as opposed to Conways surreal numbers, which> go
much, much farther in extending the reals. Davis and Hersh
describe> 0 as being the standard part of a hyperreal number,
the nonstandard> part of which is an innitesimal. So an
innitely small hyperreal> number cam be both equal to 0 but
not exactly the same as just 0. So> Newton and Leibniz were
right after all, in that innitely small> numbers do exist
(as do innitely large ordinal and cardinal> numbers).> >
Paul > > > My daughter and I were discussing integers,
innity and number lines.> > She asked the question,Why do
number lines have arrows on the end? I> > told her that the
line reaches out both ways into innity and that> > this
portion of the line was what we were working on. She
replied,So,> > this is like a fraction of innity only you
dont know what the> > denominator is because there is no end
to innity. I thought her> > observation was correct only to
be told by someone that it was> > incorrect because 1/innity
is undened. Any thoughts? She is 10 so> > please answer in
such a way that it can be explained on her knowledge> >
level. She is currently taking Algebra.> > > > Stacia
Taylor-- newsgroup website:
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One has to be a little careful when dealing with innities,
since> > some of the usual algebra rules do not apply. For
example,> > innity * zero is undened, and no single
denition will give> > you a consistent system. This is why
some math teachers insist> > that innity is not a
numberif you include it in your number> > system you lose
some of the properties that people expect of> > numbers.> >
For that matter, if you include _zero_ in your number system
you> also lose some of the properties that people expect of
numbers.> In many respects, the behavior of innity is no
worse than that> of zero.I am puzzled by the last two
sentences. Could you explain what youmean?Dom Rosa--
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=>>
innities, since> > > some of the usual algebra rules do not
apply. For example,> > > innity * zero is undened, and no
single denition will give> > > you a consistent system. This
is why some math teachers insist> > > that innity is not a
numberif you include it in your number> > > system you
lose some of the properties that people expect of> > >
numbers.> >> > For that matter, if you include _zero_ in your
number system you> > also lose some of the properties that
people expect of numbers.> > In many respects, the behavior
of innity is no worse than that> > of zero.>> I am puzzled
by the last two sentences. Could you explain what you>
mean?You might read my little Parody: Zero is Not a Number
at<http://mathforum.org/discuss/sci.math/a/m/374575/374575>.(
BTW, I suggest that you read the item being parodied rst.)If
you want more discussion on this topic, we could play a little
game:Someone mentions something about the behavior of some
specic innity(say, for example, the unsigned innity of the
one-point extension of thereals) which they think should
disqualify it from being considered as anumber. I will then
attempt to exhibit an analogous behavior of zero.I may not be
able to do so in all cases, but Ill try. David Cantrell--
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=You said it yourself: For nite numbers eg what fraction of
100 is 5?we> >always get a nonzero answer. Thus, isnt it
rather intuitive that to geta> >_zero_ answer, the number is
_not_ nite, for a nite number willalways> >give a nonzero
answer?>> This may not be worth further dead-horse-beating,
but I think only> mathematiciansfind the relationship of zero
and innity intuitive.Perhaps its a dead horse among _us_,
but maybe there will be an interestedparty or two (perhaps
the OP). Although I certainly agree there is muchregarding
infty that is counterintuitive, what we are discussing in
thisthread is not one of them, IMO. But like you said, thats
a matter of tasteand varies from person to person. To meet the
standard of intuitive whatyou discuss below really only
requires an intuitive understanding of theproduct of 0 with
any real number is 0.>> In particular, I think the what
fraction of question, although> formally equivalent to
division, feels intuituvely different. Maybe> its easier to
talk about if we think in terms of percent rather than>
fractions.A percent is a fraction of 100 written differently.
Delete the denominatorof 100 and tack on the %sign. IOW x% is
justx(1/100)But if it feels better, OK lets talk percents.>
Intuitively, 0% of anything is zero. So if we think about>
innity as a limit, we have> 0% of 1 = 0since (0/100)(1) = 0>
0% of 10 = 0since (0/100)(10) = 0> 0% of 10000 = 0since
(0/100)(10000) = 0> 0% of a googolplex = 0since
(0/100)(googolplex) = 0It is quite intuitive from this
argument that for any nite x, no matter_how_ large, we
have(0/100) *x = 0Thus, and quite intuitively, the limit is
clearly zero.> ...> So the limit is zero!Yes, clearly! So
what is so counterintuitive about it? I see you havealready
answered that below...> (I dont expect a kid to use the
limit> terminology.Why not. We use the word, and I dont
believe its just a random wordyanked from a dictionary.
Limit has a certain intuitive meaning.Mathemeticians were
taking limits all the time on just an intuitive notion,before
the concept was formally dened by such people as Cauchy. So
doCalI students. They take limits all the time (in fact we
all do, I guess,from time to time) using the intuitive notion
rather than the denition orconsequence thereof.I would at
least expect them to be clear on just exactly what it we
aretalking about that is getting larger and larger and
exactly what it is (theresulting number) we mean when we say
limit or whatever other word wechoose. In this case, its not
a denominator thats getting bigger as Iwould probably be
inclined to argue if using an informal limit approach.Its
the number (say, integer) being multiplied by 0 thats
getting bigger.Clearly, the limit here is 0. You drew the
same obvious clear conclusion,so I fail to see the
counterintuitiveness of this approach as well. butbelow I see
you seem to address this specic point.> My point is that
theres a kind of discontinuity> in saying that 0% of innity
is any nite value -- not to mention> the weird indeterminacy
of the result.)I believe I now understand your concern but
feel you may have a slightmisunderstanding. We are informally
referring to...lim x-->oo 0*xNote** this is very different
from something likelim x,y --> 0,oo x*yTheres nothing
indeterminate about the former. The limit is 0 and is
quiteintuitive to see. On the contrary, the value that is
approached for theproduct of two variables expressons, where
one is going towards 0 and theother is increasing, is
indeterminate. Actually, our intuition may fail uswith the
latter, since it may incorrectly seem clear that it is just
asobviously 0 as the former, but mathematically we know there
are exceptions.At any rate, its the former, not the latter,
that we are referring to here._That_ limit is clearly
zero.Using limit arguments, instead of percents I would
probably be inclined toargue fraction of innity in this case
probably means we have a_denominator_ that is growing without
bound. She even said as much in theOP (the denominator was
too large for her to identify due to no end onfty).lim
between arrows.This limit is intuitively 0. Indeterminate
forms never enter into thepicture. This also neatly addresses
that the particular length of thesegment matters not. Theyre
_all_ comparatively diddly squat (read...0)when stacked up
against innity. I again appluad the insightfullness ofthis
young lady to realize this, especially if the particular line
segmentshe was talking about was *not* given to be 1. Chances
are it wasnt. Ithink she was talking about the entire length
of the segment between thearrows. This is rarely 1 unit. But
one can always dene such a unit to be1 of that unit. Matters
not, though, since an answer of 0 makes the unitsirrelevent,
so WLOG she simply thought of the segment as being 1 long
whereon papert it may have spaned from -10 to 10, or -5 to 5,
etc. I supposemost, if bright enough to even wonder about such
things in the rst place,would instead ask the question in
terms of the actual length of the segmentinvolved.What is
very clear and intuitive from your approach is that for any
nitex, regardless of size, we have (0/100)*x=0, therefore if
such a fraction onnity even in terms of percents is to be
anything at all here, its 0.>> Now, the original question is
backward from this one. Its not> 0% of innity is what? but
rather what % of innity is this> nite length?Taking this
with a grain of salt, its sort of like solving for x
in(x/100) *oo = a, for some positive real a.x = 100a *
1/oo...which boils down, again, to what is 1/oo and from the
limit argumentthats 0.I guess what Im saying, not only here
but before, is it is conterintuitiveto argue that x can be
anything _other_ than zero. Any xed a whenmultiplied by a
fraction -->0, even when rst multiplied by 100,clearly
-->0.> And clearly any nonzero answer has to be wrong.> But
that doesnt make it intuitive that 0% is right! In
intuition> theres no excluded middle. The intuitive answer
is I have a> headache, because it cant be nonzero, and (see
previous paragraph)> it cant be zero either.IMO its pretty
intuitive that if we have ruled out the impossibile,whatever
remains (no matter how improbable) must be the truth. In
actually,what remains here I would far from call improbable.
What remains is clearlyand intuitively zero, since in your
informal limit argument we are neverletting anything =inty
not even formally for that matter. All we aredoing is letting
a certain real number increase, but it remains nite.Product
of 0 and any nite number is clearly 0, no matter how large
thatnumber is. The intuitive result is a/x -->0 as x-->oo--
Darrell-- newsgroup website:
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For the past twenty years, I have been teaching 7th grade
mathematics in apublic school system in the state of
Virginia.However, I have come to a decision. This is going to
be my last year teachingin this particular school system.I am
ready for a change.I have decided that I want to teach in a
school system which has adopted theSaxon Math textbook
series.Every time I look through one of the Saxon Math
textbooks, I feel a strongsense of identity. My reaction can
always be summarized in three words: This is me.I can be
described as an old-fashioned traditional style of math
teacher. Ibelieve in drill, rote learning, memorization, long
division, directinstruction, and all those other things that
most mathematics educators of theprogressive or reform school
despise.The Saxon Math textbooks speak to me, with a
compatibility with my personalphilosophy and teaching style,
louder than any other textbook series I haveever seen.Finding
a teaching job in a Saxon Math school system is now my new
mission inlife. My dream job.I would be willing to relocate
to anywhere in the United States for theopportunity to teach
middle school mathematics in a school which has adoptedthe
Saxon Math textbook series.It is my goal tofind a teaching
efforts must begin now.If any of you can provide any leads
which may result infinding my dream job,please feel free to
send them my way. I will be eternally grateful.Very
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