Numerical or Analytical

I hope this isn't too far off topic.

I was wondering recently about the changes in how math is taught,
with more numerical emphasis than there used to be.

For those who haven't taken an SAT recently, not only are
calculators now allowed, they are pretty much required.
(Even more, a graphing calculator as some problems expect
you to look at a graphed function.)

Not so long ago, math as taught in math class was pretty
much all analytical. When asked to do an integral, the
analytical (anti-derivative) form was expected, not just
the numerical answer.

But the easy availability of hand calculators that do
numerical integrals changes the emphasis. If one only needs
to understand the numerical answer then there is no need
to do the symbolic integral. (Though enough calculators do
symbolic integrals, maybe that won't be far behind.)

To get closer to on-topic for comp.dsp, it seems to me that
a more numerical approach should be good. It always seemed
to me before that numerical methods were last to be taught,
though usually more important in solving actual problems.

Still, they class time available isn't changing much, and so
it seems likely that less is being taught in terms of
analytical math. (How to actually do integrals and such.)

This is not meant to start a flame war, but to consider the
good and bad for actual problems such as those in DSP.

-- glen

On Fri, 15 Jun 2012 23:05:39 +0000, glen herrmannsfeldt wrote:

> I hope this isn't too far off topic.
> 
> I was wondering recently about the changes in how math is taught, with
> more numerical emphasis than there used to be.
> 
> For those who haven't taken an SAT recently, not only are calculators
> now allowed, they are pretty much required. (Even more, a graphing
> calculator as some problems expect you to look at a graphed function.)
> 
> Not so long ago, math as taught in math class was pretty much all
> analytical. When asked to do an integral, the analytical
> (anti-derivative) form was expected, not just the numerical answer.
> 
> But the easy availability of hand calculators that do numerical
> integrals changes the emphasis. If one only needs to understand the
> numerical answer then there is no need to do the symbolic integral.
> (Though enough calculators do symbolic integrals, maybe that won't be
> far behind.)
> 
> To get closer to on-topic for comp.dsp, it seems to me that a more
> numerical approach should be good. It always seemed to me before that
> numerical methods were last to be taught, though usually more important
> in solving actual problems.
> 
> Still, they class time available isn't changing much, and so it seems
> likely that less is being taught in terms of analytical math. (How to
> actually do integrals and such.)
> 
> This is not meant to start a flame war, but to consider the good and bad
> for actual problems such as those in DSP.

Strictly numerical solutions without understanding or double-checking 
with the analytical can lead to meaningless analysis, or vastly bloated 
analysis.

Strictly analytical solutions without understanding or double-checking 
with numerical solutions can lead to never getting your work done, or 
missing out on some important insight gained by intuiting the meaning of 
a numerical simulation's output, or it can lead to missing the fact that 
you got a sign flipped back at step three, and now you're at step 100 and 
wayyyyyy off in the weeds.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

Tim Wescott <tim@seemywebsite.com> wrote:

(snip, I wrote)
>> I was wondering recently about the changes in how math is taught, 
>> with more numerical emphasis than there used to be.
 
(snip) 
>> Not so long ago, math as taught in math class was pretty much all
>> analytical. When asked to do an integral, the analytical
>> (anti-derivative) form was expected, not just the numerical answer.
 
(snip)
> Strictly numerical solutions without understanding or 
> double-checking with the analytical can lead to meaningless 
> analysis, or vastly bloated analysis.

I agree. 

> Strictly analytical solutions without understanding or 
> double-checking with numerical solutions can lead to never 
> getting your work done, or missing out on some important 
> insight gained by intuiting the meaning of a numerical 
> simulation's output, or it can lead to missing the fact 
> that you got a sign flipped back at step three, and now 
> you're at step 100 and wayyyyyy off in the weeds.

I agree. The thing is, math was taught that way for so many
years mostly because that was the only way. When a numerical
integral was pages of multiply and add, it didn't make sense
to ask students to do it.

Now, we have had integral tables for many years, but still had
to learn integration by parts and such. Students still learn
integration by parts, but maybe not quite as well.

-- glen
 

On Fri, 15 Jun 2012 23:46:49 +0000, glen herrmannsfeldt wrote:

> Tim Wescott <tim@seemywebsite.com> wrote:
> 
> (snip, I wrote)
>>> I was wondering recently about the changes in how math is taught, with
>>> more numerical emphasis than there used to be.
>  
> (snip)
>>> Not so long ago, math as taught in math class was pretty much all
>>> analytical. When asked to do an integral, the analytical
>>> (anti-derivative) form was expected, not just the numerical answer.
>  
> (snip)
>> Strictly numerical solutions without understanding or double-checking
>> with the analytical can lead to meaningless analysis, or vastly bloated
>> analysis.
> 
> I agree.
> 
>> Strictly analytical solutions without understanding or double-checking
>> with numerical solutions can lead to never getting your work done, or
>> missing out on some important insight gained by intuiting the meaning
>> of a numerical simulation's output, or it can lead to missing the fact
>> that you got a sign flipped back at step three, and now you're at step
>> 100 and wayyyyyy off in the weeds.
> 
> I agree. The thing is, math was taught that way for so many years mostly
> because that was the only way. When a numerical integral was pages of
> multiply and add, it didn't make sense to ask students to do it.
> 
> Now, we have had integral tables for many years, but still had to learn
> integration by parts and such. Students still learn integration by
> parts, but maybe not quite as well.

Well, something has to give.  I'll bet that in the Conquerer's time in 
England, folks knew some really nifty ways to add large numbers expressed 
in Roman numerals that we were never taught.

On Fri, 15 Jun 2012 23:05:39 +0000 (UTC), glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:

>I was wondering recently about the changes in how math is taught,
>with more numerical emphasis than there used to be.

I can see it both ways. On the one hand, as I indicated in my response to the
"DSP with Matlab" thread a few days ago, I believe that becoming too dependent
upon solution tools and numerical answers makes it possible to completely miss
the understanding of what is going on "under the hood". On the other hand, there
is the very valid point brought up here,
http://dangerousintersection.org/2012/06/02/the-danger-of-giving-homage-to-mathematical-incompetence/,
that people need a far better understanding of how numbers work than most of
them have, just to get along in the modern world.

And I also know from personal experience that having a quick way to solve the
analytical equations and see graphically how the results change can give me far
better insight into the process than merely working with the equations
themselves.

Greg

Like Tim, I think they go hand-in-hand.
I took a lot of math courses but never really *understood* (some may say 
that this has not changed!).
I also took a lot of engineering courses and learned to plug numbers 
into formulas.
When we would ask the math prof. "where did this come from?", he would 
say: "This isn't an engineering course."
When we would ask the engineering prof. "how do you solve this?", he 
would say: "This isn't a math course."
(Well this did happen once each in 4 years).
I hope their attitudes have changed over the years.....

Eventually I took a short series of grad courses in "computer 
applications" where the emphasis was on analog and digital computer 
solutions to a variety of, mostly real world, problems.  Guess what? 
*That's* where I really learned something about differential equations, 
partial differential equations, etc. because you had to know what you 
were doing before you could formulate the approach for a computer.

Tim may laugh at this but while I now believe I understand multivariable 
and state space control approaches the courses in the 60's left me cold. 
  Too much matrix algebra and such and not enough practical stuff.  I 
tried twice and left out of boredom both times.

Fred

Fred Marshall <fmarshallxremove_the_x@acm.org> wrote:

(snip)
> When we would ask the math prof. "where did this come from?", 
> he would say: "This isn't an engineering course."
> When we would ask the engineering prof. "how do you solve this?", he 
> would say: "This isn't a math course."
> (Well this did happen once each in 4 years).
> I hope their attitudes have changed over the years.....

There are also many stories about the separation between math
and physics. The physics department doesn't teach the math, and
the math department doesn't teach it in the way that is needed
for the physics.

(snip)

> Tim may laugh at this but while I now believe I understand 
> multivariable and state space control approaches the courses 
> in the 60's left me cold.  Too much matrix algebra and such 
> and not enough practical stuff.  

> I tried twice and left out of boredom both times.

The one I least liked in college math was called "Linear Spaces."
That is, linear algebra abstracted as much as possible.

Now, later when I needed the linear algebra in the physics
and engineering courses I mostly learned it, but not quite as
well as if I had understood the "Linear Spaces." (I got out
the book recently and yes, that is the title of the book section.)

With a little connection to practical engineering problems I
would have had a much easier time with it.

-- glen

On 6/16/2012 4:25 PM, glen herrmannsfeldt wrote:
> Fred Marshall<fmarshallxremove_the_x@acm.org>  wrote:
>
> (snip)
>> When we would ask the math prof. "where did this come from?",
>> he would say: "This isn't an engineering course."
>> When we would ask the engineering prof. "how do you solve this?", he
>> would say: "This isn't a math course."
>> (Well this did happen once each in 4 years).
>> I hope their attitudes have changed over the years.....
>
> There are also many stories about the separation between math
> and physics. The physics department doesn't teach the math, and
> the math department doesn't teach it in the way that is needed
> for the physics.
>
> (snip)
>
>> Tim may laugh at this but while I now believe I understand
>> multivariable and state space control approaches the courses
>> in the 60's left me cold.  Too much matrix algebra and such
>> and not enough practical stuff.
>
>> I tried twice and left out of boredom both times.
>
> The one I least liked in college math was called "Linear Spaces."
> That is, linear algebra abstracted as much as possible.
>
> Now, later when I needed the linear algebra in the physics
> and engineering courses I mostly learned it, but not quite as
> well as if I had understood the "Linear Spaces." (I got out
> the book recently and yes, that is the title of the book section.)
>
> With a little connection to practical engineering problems I
> would have had a much easier time with it.
>
> -- glen
>
Glen,

Hah!  That was the same course by a different name!!

Fred

couldn't do without analytical methods... how else would I get the Jacobian
matrix for the numerical solver :D

On Sat, 16 Jun 2012 08:14:45 -0500, Greg Berchin
<gjberchin@chatter.net.invalid> wrote:

>On Fri, 15 Jun 2012 23:05:39 +0000 (UTC), glen herrmannsfeldt
><gah@ugcs.caltech.edu> wrote:
>
>>I was wondering recently about the changes in how math is taught,
>>with more numerical emphasis than there used to be.
>
>I can see it both ways. On the one hand, as I indicated in my response to the
>"DSP with Matlab" thread a few days ago, I believe that becoming too dependent
>upon solution tools and numerical answers makes it possible to completely miss
>the understanding of what is going on "under the hood". On the other hand, there
>is the very valid point brought up here,
>http://dangerousintersection.org/2012/06/02/the-danger-of-giving-homage-to-mathematical-incompetence/,
>that people need a far better understanding of how numbers work than most of
>them have, just to get along in the modern world.

    [Snipped by Lyons]

Hi Greg,
   That was an interesting article.  Thanks.

However, I cannot figure out how 
mathematical ignorance can cause problems 
in choosing a spouse.

See Ya',
[-Rick-]