Re: What Is Mathematics For?

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Re: What Is Mathematics For?

Jonathan Groves-3
On 8/2/2010 at 1:40 pm, Dom Rosa wrote:

> > The truly superb article, "What Is Mathematics
> For?,"
> > by Underwood Dudley has been published in the May
> > 2010 issue of the AMS Notices.
> >
> >
> > http://www.ams.org/notices/201005/rtx100500608p.pdf
>
> The August 2010 issue of Notices contains four
> letters about Dudley's article:
>
> http://www.ams.org/notices/201007/rtx100700822p.pdf


Dom,

I thank you for mentioning this issue of the Notices from
the AMS that contain these four replies to Dudley's article.

David A. Edwards from the University of Georgia says that relatively
few positions even in science and engineering rarely use much
mathematics beyond eighth-grade mathematics (whatever eighth
grade mathematics is exactly, I'm not sure, and that seems
to vary some from state to state anyway) and that they require
technical degrees as merely filters. He also mentions
Vivek Wadhwa's statement that America is producing more scientists
and engineers than there are job openings. I take that he is referring
to his article that I had found at
http://www.businessweek.com/smallbiz/content/oct2007/sb20071025_827398.htm.

I'm no scientist or engineer, so I cannot say whether such claims are
true or not. But I can say that our schools and even colleges are
in such a mess these days that high school diplomas and even many
college degrees are not worth more than the paper they are printed on.
How else do you explain the countless floods of college students
I see whose reading, writing, math, study, and even common sense
skills are still stuck in second or third grade? It disturbs me
to see all the discussions and assignments in class that appear
as if they were written by the students' children rather than
by the students themselves! I would conjecture that massive grade
inflation troubles employers enough that many cannot trust that
ones with only a high school diploma or college diploma truly
have meaningful diplomas because of many who do manage to graduate
without learning much of anything. I suppose that they can test
the skills of those they might be interested in hiring, but I imagine
that such testing is time consuming and expensive. I thought about
that when I have thought about a job as a statisician, but I believe
I would need a stronger background in statistics to qualify or at
least to give myself a strong chance of getting such a job. But I
then realized this dilemma if I choose to study some statistics
on my own: How can I show that I have learned more statistics than
what my degrees and transcripts show? I would not blame employers
in the least bit if they did not believe me because anyone can
make such claims. And it would take a lot of their and my time to
show that I indeed did study on my own. So this thinking has told
me that I am sure that employers want students with credentials
but that there is tangible proof of such credentials and that the
proof of such credentials is actually meaningful proof and not
simply a fancy version of some scribbled note by someone saying
that John Q. Smith really has these credentials and that I witnessed
this myself.

I doubt these claims myself, but I don't work in science or
engineering to know how to test this claim or to refute it.
The best I can do for now is to ask some colleagues I know who work
in science.

However, let us suppose Edwards' claim is true. Does this mean
that students who stop with arithmetic are really competent enough
to understand the mathematical side of science? Perhaps some are,
but few would be. I myself would doubt this seriously because
of the meaningless way that most elementary mathematics is taught
and, on top of that, with the massive grade inflation these days
so that many students can finish arithmetic with good grades but
understand very little of it. And I would venture that most who
work in education know that virtually all students finish arithmetic
with good grades but don't understand much of it. Those who know
what subject knowledge it takes to teach mathematics effectively
generally realize and agree that teachers should know mathematics
at a higher level than the level they will teach because the extra
mathematics courses help them learn (at least they should, but that
is not always the case) the mathematics they will teach much better
than otherwise. A few other reasons are often given as well, but
this one reason is an important one and pertinent to the discussion
here. If any form of mathematics is a significant part of the
job--whatever level that math may be, then the students should learn
the mathematics and should learn it well. Furthermore, as Sherman Stein
mentions in one of these replies, it is better to overprepare in mathematics
than to underprepare in mathematics in case of changing career
goals and also because further preparation in mathematics can help
students understand better what mathematics they will use.

One of the most important goals of learning mathematics that is
sorely missing from elementary math courses is teaching students
how to study and learn mathematics for themselves and how to learn
to mature mathematically. A major difficulty I see with students
in mathematics and statistics courses is that students have little
mathematical maturity and little idea of what it means to think
mathematically.

And few of them understand symbolic reasoning, which makes it difficult
for them to learn the reasoning behind mathematics. And that also
makes it difficult for them to learn algebra. In fact, so many students
I have seen have such little understanding of any form of symbolic
reasoning that they have little idea of what it means to use a formula!
Rather than stressing algebra as we do, why not emphasize more about the
teaching of symbolic reasoning? Until students can learn to make sense
of symbolic reasoning and learn to make sense of mathematical statements
with letters in them, algebra and other symbolic mathematics will make
little sense to them, and their learning will be wasted.

And we should emphasize logic and reasoning and conceptual understanding
in arithmetic. Far too many students take arithmetic and "not get it."
And far too many students end up thinking that mathematics is mechanical--
nothing but following recipes and plugging and chugging into formulas.



Jonathan Groves
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Re: What Is Mathematics For?

Robert Hansen
Jonathan, I can follow the argument that mathematics is used as a surrogate for the complexity in reasoned thought that many tech careers require, and it works rather well in that role. But some of the statements in those articles are very incongruent with my 25 years of experience with DOD companies and a few years at NASA. These companies (and entities in NASA's case) are obviously heavy consumers of engineering talent. And the fact that we hire a lot of foreign talent, whether by H1 Visa or via out sourcing is very real. I am certain of the number of resumes I will get if I put out a posting for a job, unfortunately a lot of the replies will be unqualified. Maybe with all this complicated technology we have exceeded the capacity of qualified graduates in this country and it is only natural that we tap into the unused resources of other countries (which are starting to be used more there as well).

We have more graduates than there are jobs available.

That might very well be true, in fact knowing the college bubble we have I would be surprised if that were not true. But the graduates are not qualified. I can say this because of the number of resumes and phone screenings we must go through to find qualified applicants. Part of that is taste but part of it is also qualification.

Engineers don't use math.

Well, first, you would have to be more precise with the term "engineers" since that covers a lot of disciplines. Assuming we are talking of all the technical/scientific versions of engineering, I can say that as a foundation they heavily use mathematical like reasoning in their deconstruction of processes and problems. Of course that is to Dudley's point of "what is math for". But if we are talking specifically about math and how much of it shows up "on the job", all of the algebra, most of the trig and (analytic) geometry shows up. It might show up in a spreadsheet or portions of code or on a paper pad but it certainly shows up. Remember how I said that most of the world uses spreadsheets, not algebra? Well, all of the world uses spreadsheets, but the engineer's spreadsheets use algebra. And when I say we use it I mean in applied ways. Things like the fundemental theorem of algebra are left with old school notes in our attic. But solving equations and developing mathematica!
l models (especially this part) is very active and you would be very disabled without that last ability.

In any event, algebra is a very active skill, but what about calculus? This is more complicated. If you are working on a project that brings differential equations into your model then you have to at least understand how they work, in a calculus/algebraic sort of way. And the three phases of (engineering) calculus (single variable, multiple variable, diff eq) do a good job of that IF the student puts in the work. How often do they show up? It depends on the project. I am sure that the iPhone antenna team is knee-deep in some pretty messy ones right now. But they are not like algebra which is like a grammar, diff eqs are more like anchor points in the model, they give theoretical basis ti the model.

Teaching statistics would be a good alternative to teaching algebra.

How the hell do you do a fair treatment of statistics without algebra?

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Re: What Is Mathematics For?

Robert Hansen
In reply to this post by Jonathan Groves-3
I might also add that our engineering ecosystem is not in the best of shape because we have lost so much manufacturing and the applied engineering that goes with those production lines. First it was just the labor side of manufacturing but now it is also the engineering side as well. Many of those graduates might have gotten a job if that ecosystem was still in place. As an engineer I find it a bit sad because I think that middle layer is what fed the mid (last) century growth in technology and engineering. And I don't mean growth in dollars, I mean growth in people. Now it is in dollars.
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Re: What Is Mathematics For?

kirby urner-4
In reply to this post by Robert Hansen
On Sun, Aug 29, 2010 at 11:38 AM, Robert Hansen <[hidden email]> wrote:

<< snip >>
 
We have more graduates than there are jobs available.


I'd say this bespeaks poor engineering i.e. if talented "human resources" are simply going to waste, that bespeaks a lack of imagination in the vicinity of EPCOT, which kind of sums it all up (not that Florida is the only state with no good ideas around keeping work / study programs going -- could be the curriculum is still kinda weak eh?).
 
Engineers don't use math.


Careful or someone will take that out of context and think you're off your rocker.  Remember how you hold to these minority views, like "philosophy and science parted ways hundreds of years ago".  I don't know where you got that, but it's hard to defend, plus you've been waxing philosophical yourself ever since.

In any event, algebra is a very active skill, but what about calculus? This is more complicated. If you are working on a project that brings differential equations into your model then you have to at least understand how they work, in a calculus/algebraic sort of way. And the three phases of (engineering) calculus (single variable, multiple variable, diff eq) do a good job of that IF the student puts in the work. How often do they show up? It depends on the project. I am sure that the iPhone antenna team is knee-deep in some pretty messy ones right now. But they are not like algebra which is like a grammar, diff eqs are more like anchor points in the model, they give theoretical basis ti the model.


What about spatial geometry?  I'm very suspicious how you say "calculus/algebraic" and then intimate that engineering is somehow not a spatial discipline.  What happened to CAD and architecture, all those pictures of homes...

E.g. (re spatial geometry in engineering):

 
Teaching statistics would be a good alternative to teaching algebra.


Still, nothing spatial (though visualization of data is of course spatio-geometry (pie and bar charts and such)) and getting more so by the day (like in the movie 'Avatar').
 
How the hell do you do a fair treatment of statistics without algebra?


Tell us about complex numbers and their fractals.  Include them before high school is over, or no?  On any track?  IB should include them, given Mandelbrot is French and the French tend to be proud of French contributions, unless they're so totally hooked on Bourbaki they won't give fractals the time of day -- but that's hard to imagine.

I think if you live within a 200 mile radius of Spaceship Earth at EPCOT, yet never mention this in geography class, nor project zooming in on Google Earth (or related KH-derived project) that you're probably squandering public funds.  

So I should get in line with my complaint yes?  

Better to simply include the above in an Oregon context.  I also write about "EPCOT west" in my blogs, suggest visiting with Immersive media (those kinds of cameras the Google fleet uses, to make those Google Street Views -- though they also use other models besides Immersive's).


Kirby

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Re: What Is Mathematics For?

kirby urner-4
In reply to this post by Robert Hansen
On Sun, Aug 29, 2010 at 11:49 AM, Robert Hansen <[hidden email]> wrote:
I might also add that our engineering ecosystem is not in the best of shape because we have lost so much manufacturing and the applied engineering that goes with those production lines. First it was just the labor side of manufacturing but now it is also the engineering side as well. Many of those graduates might have gotten a job if that ecosystem was still in place. As an engineer I find it a bit sad because I think that middle layer is what fed the mid (last) century growth in technology and engineering. And I don't mean growth in dollars, I mean growth in people. Now it is in dollars.

Engineering is always morphing and today includes such as Bioinformatics, a branch of computer science in some taxonomies.

Sharing the knowledge of how to design shoes, goes with the territory, after the shoes have been made, i.e. an apprentice engages in the assembly of product then begins to master design, possibly to inherit the entire process (the meaning of "to apprentice" in a classic guild context).  

In other words, it stands to reason that a company like Nike would have these layers, with the more central ones being design and marketing.  But then what kind of shoes are we talking about?  I bring up Nike because of its headquarters in Oregon, but the art of shoe making is widespread and takes on different guises (not everyone is in the market for high end somewhat spendy sports ware).  

We also have Intel here.

If you're working at Intel and fly to Ho Chi Minh City to check on something, then to Cavite before returning to Hillsboro, it's not like "jobs have been lost".  These global companies have been somewhat expat for over a generation already as a result of USG's post-WW2 agenda to encourage capitalism.

With the advent of telecommuting, we could have many more Americans working for companies with headquarters elsewhere, perhaps in the context of work / study opportunities mediated by universities and their student exchange programs.  Market some brand of Swiss chocolate while coding fund accounting software for an ice cream factory in Havana, all without leaving your home base campus dorm in Berkeley or Seattle, accruing credits towards your degree in social networking media (aka "communications").

Kirby

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Re: What Is Mathematics For?

Kirby Urner-5
In reply to this post by Jonathan Groves-3
Someone:
> > Engineers don't use math.
> >
> >

Me:
> Careful or someone will take that out of context and
> think you're off your rocker.  

You were quoting someone else, right?  Just not really
showing that in your text.  You then tried to provide a
serious response.

I met with some teachers informally the other evening,
lugged a bag of geometry toys and a piece of Flextegrity
(applied material) for conversation purposes.  One of
the teachers had been on a Fulbright teaching in Bulgaria.
Another was about to head off to Finland in January, on
a grant to study the education system there, in addition
to Finnish.  Lots of kids were there too, and enjoyed
playing with the toys, making shapes from magnetic
"mites" (space-filling tetrahedra that in turn make
"sytes" and "kites").**

Some of those present knew of my work on an alternative
high school math track, at least vaguely.  It's not
like these initiatives have made many headlines.  Like
most daily papers, The Oregonian is pretty techno-
illiterate (in terms of the stories it tries to track).
There've been two stories on me and my brand of futurism,
both in the Metro section as I recall, but they ran
quite awhile ago, along with my op ed piece on copyleft
versus copyright practices (early days of open source).

Like it or not, a lot of my work falls under the heading
of "esoterica" (but we knew that).

Dr. Bob Fuller was through again recently.  He hadn't
yet seen my blog post on the AAPT conference, even
though he features in it prominently -- more proof that
even some of my most important influencers have only
"precessional" working relationships (a term of art
from the Bucky stuff).  He wondered about my thoughts on
all the "pre med stuff" in physics education these days.
He'd been at least a decade ahead on that one, got in
touch with me based on the "First Person Physics" meme
I was circulating.

At the radical core, we'd like to bridge to the humanities
with science fiction and speculate about possible futures
in ways that don't shut people down (because too
terrifying) and encourage thoughts about planning, not
just for one's individual scenario, but more for the
world as a whole.  

"We" are these "radical math" teachers who cover verboten
topics still off limits to most math teachers,
tetrahedral mensuration in particular (a signature topic).

Are we early adopters with a bell curve to follow, or a
fork in the road that others won't take, a branch that
will wither and fizzle out?

When it comes to speculating about the future, many
students besides me have come up with the "Global U"
model to supplement the "Spaceship Earth" model.  
We're all engaged in work / study from cradle to grave.

Yes, many of us live in terrible circumstances,
scandalously impoverished, and this is a sign of
curriculum weakness more than anything, not a lack
of physical resources or energy (in principle).  We
realize that humanity and its engineering is still in its
infancy and our advancement may have been retarded quite
a bit by unfortunate trial-and-error forays into not-
working solutions (ideologies, belief systems....).

Over-specialization, lack of overview, is one of the
curriculum weaknesses we need to address, which is
where I think philosophy comes in (more below).

The growing student exchange component is about
fostering a sense of peer groups that transcends
ethnic boundaries, which is what you already find
in these public and private sector enterprises we've
been talking about (it's a sense worth developing
- -- cosmopolitan R us).

But when you bring cultures together to study time-
lines, one has to accept that much of recent history
is rather dark (to say the least) and trying to sweep
that under the rug has a corrupting influence across
the board.  Too much math education has been complicit
in expunging time-lines and going for some "sanitized
view" where human history simply falls by the wayside.
Loss of lore = loss of relevance and a narrowing view
that's also incapacitating.

An alternative design is to look for points of contact
between technical subjects (such as SQL) and recent
history, such as the Eugenics movement and its influence
in fascist Germany as well as the USA.  Hollerith
machines were not SQL engines obviously (that technology
came later), but we're talking about a lineage, the
history of punch cards etc.

http://mathforum.org/kb/message.jspa?messageID=7165242&tstart=0

A parallel trend in philosophy, which includes ethics,
is to embrace computer science as the successor to all
that flurry around symbolic logic at the turn of the
last century.  Philosophy was caught up in the same
switch to Boolean logic that occurred in engineering.  
The idea of "computer languages" goes back to Leibniz
and before (the I Ching an influence, also the Hermetic
tradition).

However, we're not talking about "machine intelligence"
(MI) taking over as the new boss.  We're talking about
humans amplifying their capacity to take in time-lines,
technical info, skills and lore, in order to respond
more effectively to the challenges they face.  The idea
of the MEMEX (Dr. Vannevar Bush, As We May Think, 1945)
is much more what came to be, versus HAL (Arthur C.
Clarke, 2001 Space Odyssey, 1968).

Philosophy is a source of "glue languages" i.e. ways of
connecting the dots across multiple disciplines,
especially in an age when "specialization" has been the
name of the game.  

Without overview, there's no coherent planning.  Without
planning, we lose a vision of what we're collaborating
to achieve.  I keep circling EPCOT because "Experimental
Prototype Community of Tomorrow" kind of sums up what we
need to be working on, pretty much by definition.  

Or call it campus housing (and food services).  

What mathematics will we need?  What models and
simulations?  These are the questions to be asking.
What big picture models might we share and internalize?
The "global energy grid" needs to be a part of that.
Just talking about "oil pipelines" hardly begins to tell
the whole story.

Kirby

** the four space-filling tetrahedra known to D.M.Y.
Sommerville in 1923 were the Mite, Bite, Rite and 1/4
Rite although that's not the terminology he used.
Michael Goldberg subsequently discovered these
accordian-like arrangements in triangular pillars
with a free variable i.e. a continuous "family".
His research connects to the geodesic dome story as
well, is cited by Coxeter in a 1967 paper.  Coxeter
wrote about Fuller's contributions in more detail
in a 1974 paper.

http://www.grunch.net/synergetics/virus.html
http://worldgame.blogspot.com/2009/11/kicking-can-down-road.html
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Re: What Is Mathematics For?

GS Chandy
In reply to this post by Jonathan Groves-3
Robert Hansen posted Aug 30, 2010 12:19 AM:

> I might also add that our engineering ecosystem is
> not in the best of shape because we have lost so much
> manufacturing and the applied engineering that goes
> with those production lines. First it was just the
> labor side of manufacturing but now it is also the
> engineering side as well. Many of those graduates
> might have gotten a job if that ecosystem was still
> in place. As an engineer I find it a bit sad because
> I think that middle layer is what fed the mid (last)
> century growth in technology and engineering. And I
> don't mean growth in dollars, I mean growth in
> people. Now it is in dollars.
>
"our engineering ecosystem is not in the best of shape because..."

Well, there are very few human-made systems that are in even 'adequate' shape (leave alone being in the "best of..."!)

But - even in all the surrounding gloom - there have been some pretty notable successes.  In regard to "engineering ecosystems", India is just about to have a huge and crashing failure at the CommonWealth Games (CWG) which we're hosting in Delshi during mid-late October, barely 45 days away.  As of now, major construction works are still going on, there have been MMMMMMMMillions of Rupees(maybe BBBBBBBBrillions or even TTTTTTTrillions!) stolen/squandered with no useful result at all, etc,etc, etc, etc...just Google "CommonWealth Games Delhi 2010" to understand the dimensions of the disaster now facing India.  In this context, I observe that:

- -- China hosted a hugely successful Olympics in 2008.

- -- S. Africa (a very underdeveloped nation compared to India) hosted a very successful Football (Soccer) World Cup just a few months ago;

- -- Great Britain is to host the 2012 Olympics in London and surrounding areas - and they are doing the final testing of all the facilities now, more than 2 years  ahead!  

(All three success noted above should, I believe, be regarded as essentially matters of developing effective 'engineering ecosystems' (for their specific purposes) - I believe that the phrase 'engineering ecosystem' should deal with something more than just the idea of 'manufacturing and applied engineering' - though surely 'manufacturing and applied engineering' are part of it).

Most of NASA's triumphs were, in fact, successes of 'engineering ecosystems' - and its few failures were failures of some part of the engineering ecosystems involved.

Just my 2 cents' worth - and for information, with a view to save Haim and others some small trouble of pointing to our coming disaster that India is due to be subjected (and to be able to say, "I said it first!)

GSC