Re: The state of education in the USA.

On Apr 7, 8:34 am, tgdenn...@xxxxxxxxxxxxx wrote:
On Apr 4, 10:59 pm, Tim Norfolk <timsn...@xxxxxxx> wrote:





On Apr 4, 3:54 pm, tgdenn...@xxxxxxxxxxxxx wrote:

On Apr 4, 1:21 pm, Tim Norfolk <timsn...@xxxxxxx> wrote:

On Apr 4, 11:21�am, tgdenn...@xxxxxxxxxxxxx wrote:

An interesting assertion. I have been part of studies involving
technology and its effects in education for about 20 years.

"You have been part of" meaning you were studied or you were a
researcher who published the study?

I have worked on both sides - as the one trying to implement new
ideas, and as the one functioning as a control. I have not bothered to
publish in the area myself.

In many
such experiments, the immediate results showed that with self-selected
students using new technology (say, graphing calculators), the results
in specific courses looked better. Many of those gains did not carry
through to higher-level material, including abstract mathematics
courses, where low-level algebra skills are used less. In addition, as
those same techniques were applied to the general population, the
results always were that the novelty effect wore off, and the
standards declined.

I would be interested in some more detail on these studies. What
courses are you talking about (abstract mathematics)?  I also don't
get what you mean "applying the same techniques to the general
population".

Courses like Discrete Mathematics, Abstract Algebra, Advanced
Calculus, Linear Algebra and the like. Much of what is done there
involves some basic Algebra, with ideas from Calculus and Linear
Algebra. Without an understanding of variables and functions,
initially gained in Algebra I, no student can understand what is going
on. The same is true in any reasonable programming course. The latter
have been poisoned, so my colleagues tell me, by the web, as students
search for code to steal, rather than learning to write it themselves.

Since I don't follow this area closely, you will have to check the
literature yourself. The effect I am talking about is the well-known
one of novelty. When the new graphing calculators appeared, only the
"nerds" wanted one, and signed up for the courses where we tried out
their capabilities. Once they became part of the standard schoolbox,
interest in them wore off, and our students can do less with them than
I can do with pen and paper.

...

It doesn't bother me at all, since I don't expect that a non-
representative sample (SAT takers in the past) is going to predict the
results for the current population. I would be truly surprised if the
results *didn't* have to be re-normed, and down.

That might be true in some cases. But virtually every student in Ohio
who intends to go to higher education takes the ACT, and those
averages have gone down every single time, once the re-norming is
taken into account.

Yes, exactly. And when I took the SAT, the population of students who
intended to go on to higher ed was highly selected, particularly by
test results. As the population of SAT-takers more closely approaches
the general population, it is inevitable that test scores will
diminish.

This despite 60+ years of 'improvements' from the
Colleges of Education. If the life expectancy dropped by 5% every few
years, we'd be hanging doctors from the trees.

Not if, every year, the population they were treating included sicker
and sicker people.







Once again, what you are saying is kind of vague. Can you be more
specific, or perhaps give an online reference to these studies you
keep talking about?

See the above. Without a lot of rote knowledge of basic arithmetic,
algebra becomes hard to follow, no matter what technology is used. In
that case, the underlying ideas of variables and functions become
meaningless, and inapplicable. There is some good evidence that, if
the abstract process engendered by learning algebra (see Piaget) is
not accomplished by a certain age, it will never be acquired. This is
similar to the tragic cases of humans who have never been exposed to
speech by age 8 or so, and do not have the brain connections that make
speech possible - ever.

The mathematical education community awaits your great expertise and
wisdom, but consider the following

1. Mathematics is the only subject to have been consistently taught
for about 2,500 years
2. In mathemaqtics, that which doesn't work is tossed onto the trash
heap, or given only a passing reference as a failed idea, or one
supplanted
3. Euclid's answer to King Ptolemy, who was bored by his lessons
"There is no royal road to geometry (mathematics)".

In short, I have yet to see one single innovation in mathematics
education that left students better prepared for further study. If you
have, let's see your evidence. From my experience, it takes properly-
prepared students with a willingness to work harder than in the non-
science disciplines. If you honestly have the magic bullet, you could
get a bunch of money from Bill Gates, and pretty much every major
corporation, since  most post-graduate management programs include a
screening test of 8th-grade mathematics.

Most people who I have seen complain about mathematics education
weren't good at it, and extrapolate their ineptitude in the subject
(and current success) to 'mathematics is taught wrong', or
'mathematics is useless'. All I generally answer is that most of the
neat technology that they take for granted, including every aspect of
computers, imaging systems (MRI, CAT and PET scans), cell phones, and
modern pharmaceuticals, are only possible because of massive amounts
of mathematical modelling. To take imaging as one example, they seem
to think that the methods were invented by some MD puttering around,
when the truth is that the basic ideas incorporated deep mathematics
and physics, including Fourier transforms, and some incredible
software, the work of teams of mathematicians, physicists and
engineers. My own advisor worked on the first electronic computers,
helping to design the reactors for the Nautilus-class submarines that
kept America safe during the Cold War.

Because the average American takes such things for granted, and even
attacks the 'nerds' in the education system, we have fallen to at best
second place in most technology, and leads to at least some of our
economic woes.

Most people I've had this discussion with who take your position end
up using some version of the same logical fallacy as you do---although
you seem to have thrown in some strawmen along with the traditional
question-begging.

Everything you've said relies on *defining* success as doing the
little rote manipulations and performing the little rote algorithms
that you've been doing and teaching for  30 years. What you've ignored
is that people who *use* mathematics as a tool to develop all the
wonderful tech that you describe (scientists and engineers) use
software to solve their math problems.  There's nothing in the courses
you define as 'advanced' and 'abstract' that can't be done with
Mathematica or something similar.

So the proper design of an experiment (a study) would involve
comparing HS graduates who can pass your Algebra I test  taught
traditionally with those who have learned to get the same **results**
using software tools. And then we could follow them through  their
university careers,  perhaps distributing the groups between those
'advanced' courses taught traditionally and those taught using
software tools, and see who picks up on what faster or better.

And then, for those in-demand engineers you are talking about for
example, we could see who gets hired to rebuild our infrastructure and
solar energy plants and so on.

What would happen of course is that the software-users will do better
in all areas except when they are obliged to do something they haven't
learned to do, which is use pencil and paper. And they will be hired
preferentially since they have long experience with the industry-
standard tools of their specialization.

Now, since I've had this discussion before, I'm sure you will say
something along the lines of "but who will write the software?" and
"who will develop the new methods for solving problems?"

The answer is:  Not your students. It will be the same tiny fraction
of the  population that traditionally gets PhD's in mathematics and
physics and computing and goes on to do advanced research.  There is
no shortage of such people; in fact globalization has probably
increased that number.  But they probably don't go to your institution
since they all have scholarships to more selective schools.

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I believe that, at this point, I would rather summarize some of what
has been said here, and on other threads, rather than insert comments
everywhere. Before I do so, however,
I believe that it might be germane to discuss my career.

I have taught and tutored mathematics, statistics, physics,
engineering, computer science and martial arts for over 30 years. In
the course of my professional career, I have been involved in research
in computer graphics, numerical analysis software, educational
statistics, mathematics education, abstract analysis, bio-engineering,
civil engineering, polymer science, mechanical engineering, game
theory, data base analysis, approximation theory, medical imaging,
mathematical modelling of staffing, financial analysis, and several
other problems that I can't recall at this moment.

I believe that I have at least some knowledge of the application of
mathematics and the tools necessary to do so.

I have read many of your comments, indicating strong opinions in
education, mathematics, and technology, yet your argument style
indicates to me that you might be a lawyer.

Now let's get to some of your assertions, and some of the reasons why
I believe that you are incorrect:

1. Any deficiencies from school can be made up in college-level work
(this was in another thread some time ago, but I believe was your
contention).

Nation-wide, mathematics remediation has been a hot issue for 30 or
more years. Despite efforts using every approach that could be
imagined, no schools have managed to do much better than 50%. A
fortune awaits someone who has the 'magic bullet', but my experienced
opinion is that no such item exists. The problem has been avoided or
eliminated by many schools, either by deleting a mathematics
requirement, or re-defining remediation as college-level material.

It is also well-established that college success is best predicted by
high-school and college mathematics grades, and the analytical
portions of the SAT and ACT.

2. Improvement in mathematics education is being held back by the 'old
guard'.

There might be some truth to this, but a much larger problem is
unqualified teachers of mathematics and science. For example, one
colleague's daughter has a high school geometry teacher who spent 30
years as a kindergarten teacher, and doesn't know the material that
she is supposed to teach.

Notwithstanding the 'traditionalists', there are many researchers
trying to find better ways of achieving the mastery of abstract
reasoning made possible by studying mathematics. None have found
methods which work appreciably better than the current methods.

3. Success in mathematics is defined by rote manipulation.

In some courses, this is true. Unfortunately, there doesn't appear to
be a better way. Without a good grasp of arithmetic, the learning of
algebra becomes harder, since there are just so many symbols and
numbers that confusion is common. Without a good background in
algebra, trigonometry and geometry, the study of Calculus is
difficult. Without that expertise, one cannot understand Differential
Equations, the language of modern physics and engineering. In Calculus
and beyond, we often not only ask for 'an answer', but also for the
reasoning behind that result. This becomes much more important in
mathematical modelling, as all results are based on simplications, and
it is vital to provide analyses of the error estimates.

4. Since engineers and scientists use software to do much of their
work, this is all we should teach.

Some universities have tried this, with the result that some students
were very adept 'with the current software'. Many typical students can
push the approriate buttons, but their lack of pencil and paper skills
doesn't even give them the tools to determine when something isn't
typed correctly, let alone when the software fails. A simple example
is the difference between 1/2*x and 1/(2*x), both of which might give
reasonable-looking results, but with radically different meanings. The
basic skills in arithmetic, geometry and algebra form the alphabet and
grammar of the language that we call mathematics, like it or not.

5. Software such as Mathematica can do everything which is done in
advanced mathematics courses.

Not true, and blatantly apparent to anyone who has taken Discrete
Mathematics (where much of what is done is to formulate problems using
algebra, so they can be solved by software), and even more so in
Abstract Algebra (the language of Quantum Theory), or any course in
Real or Complex Analysis, including applications thereof, such as
Optimization Theory.

In fact, replacing lectures by software has been attempted at several
universities, such as VPI. There, with selected students, software
labs were extremely successful. Unfortunately, it appears that it was
as much the audience as the methods. Once such programs were adapted
to lower-level courses at other universities, they were no more
successful (and much more expensive) than traditional lecture methods.

6. We should test 'traditional' mathematics education against the use
of software.

A fine idea. Then, what should we do if the software approach fails,
and all of those students cannot go back and learn what they missed? I
am sure that you are willing to give financial support to students who
suffered as a consequence?

There is a deeper reason for education, rather than job-training in
specific software:

Over the years, there have been many studies in psychology, several
subjects in education, and even in the martial arts, contrasting
'abstract' problem-solving with problem-specific training. In the
studies, students were shown a single application of the material
learned, and then asked to solve a similar application. In every
study, the former group were more successful. This is why business
accreditation groups are encouraging business schools to move from the
case study approach.

7. Students using software 'will do better' (your unqualified
assertion), and will be hired preferentially.

Again, my experience with students using high-powered tools, including
the algebra-based graphing calculators, is that most are slower and
more error-prone with them than without.

8. My students will not be the ones writing the software.

My past students working at many software companies, including
Microsoft, and at places like the NSA, would, I suspect, disagree.

9. There is no shortage of people, who can develop software, in fact
globalization has probably increased that number. But they probably
don't go to my institution since they all have scholarships to more
selective schools.

For the latter point, except as a feeble attempt to be insulting, it's
just wrong. I have had many excellent students over the years, going
to the best graduate programs, Harvard medical school, and the like.
We even had one who founded his own very successful computer security
company, based on the skills that we helped him acquire.

You might be surprised at the answer to your first statement here. I
believe that it was a recent issue of Newsweek that discussed India.
In a piece on Bangalore, it was noted that India's education system
cannot turn out enough technical graduates, and so they are out-
sourcing to places like Romania. If, in a country like India, where
education may be the only ticket to success, they can't generate
enough trained people, what possible incentive could work here?

.

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