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Re: More /.{I->-1} craziness

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  • Subject: [mg106680] Re: More /.{I->-1} craziness
  • From: Richard Fateman <fateman at cs.berkeley.edu>
  • Date: Thu, 21 Jan 2010 04:50:50 -0500 (EST)
  • References: <200912300915.EAA17299@smc.vnet.net> <hhhmn8$o9t$1@smc.vnet.net> <hhpl28$9lf$1@smc.vnet.net> <hip8gf$t4d$1@smc.vnet.net> <8304354.1263643340634.JavaMail.root@n11> <hiuur1$919$1@smc.vnet.net> <hj130e$bcn$1@smc.vnet.net> <201001191014.FAA29127@smc.vnet.net> <hj6qf0$8qo$1@smc.vnet.net>

Murray Eisenberg wrote:
> It's been some years since I've seen studies comparing performance of 
> students taking a traditional calculus course, on the one hand, and 
> those taking a Mathematica-using course, on the other hand.  As I 
> recall, the studies showed the latter group performing at least as well 
> as the former, and sometimes better, at traditional pen-and-paper 
> skills. 

Rather than looking at all the material you suggest, I will take your 
word for it, since it generally agrees with what I recall.  What I 
recall is that, essentially, the group of students using computers did, 
on the whole, substantially the same as the students not doing so.

  In addition, the latter could solve more complex or more
> realistic problems than the former.

This would be an interesting consequence (did they use the computer to 
solve more complex problems?)  I can see it either way -- a student who 
has more confidence and has seen problems solved where the answer did 
not magically drop out and become some remarkably simple expression -- 
may be more willing to attempt to attack a more ambitious realistic problem.

Often calculus students (indeed, I recall a graduate course in applied 
math where this was true) could guess that any time the solution of a 
homework problem required a large expression or a number with more than 
3 digits, that he/she had made a blunder somewhere. The right answer in 
the back of the book was always small.
> 
> You might want to take a look, e.g., at the link to "A guide to the 
> studies done on the Mathematica-based courses" at matheverywhere.com. 
> This concerns the "Calculus& Mathematica" project created by Jerry Uhl, 
> Horacio Porta, and Bill Davis, at University of Illinois and Ohio State 
> University.

The success of such courses obviously depend on the enthusiasm, energy, 
and charisma of the teachers. To what extent does it depend on the 
computing aspect?
In the case of this particular link, this leads to a business, where the 
professors are apparently selling courseware.  I'm not saying this is a 
bad thing. Just that I would not expect statements on that web site to 
present nuanced opinions on pro/con teaching math with computers :)
> 
.. snip..

> I've based several courses myself upon Mathematica use. Those of us who 
> do so often just grow tired of trying to justify what we're doing to 
> those who are dubious or skeptical or just plain ignorant of what's 
> possible and how. 

I haven't been in the business of teaching calculus for a long time. (I 
taught math at MIT before I taught computer science at Berkeley). My 
experience is that math teachers are, on the whole, quite conservative.
My own calculus lab (this was in 1971-2) was dropped when I was not 
available to teach it. I used a computer algebra system, obviously not 
Mathematica.

In the absence of a controlled experiment, it is hard to convince 
skeptics.  Even the experiments that might be tried would probably be 
flawed -- e.g. two sections of the same course -- may be defective if 
the better students self-select to come to the "experimental 
computer-based" course. What we would like, I think, is some mechanical, 
automatic, guaranteed-to-win, technique  (computer based or not!) that 
would take all students, including below average, and get them to have a 
superior understanding and appreciation of mathematics.  If someone 
could convincingly show that using Mathematica (or anything else) was a 
sure-fire mechanism, then people in charge of schools who are in a 
position to hire/fire teachers, might take notice.

There have been period efforts to improve education. e.g. I participated 
  in conferences on "Calculus for the New Century".  (uh, I think that 
was  the previous one!)..  Using graphing calculators -- still a big 
sell -- was one innovation.

I personally don't doubt the efficacy of using computers to actually DO 
calculus problems. I also suspect that by far the most lucrative sales 
of programs like Mathematica are to schools to make them available for 
courses like calculus, for those semesters in which someone uses them 
(perhaps for an additional calculus lab.).
But do students learn more?  Note that most calculus students just want 
to pass the course so they can graduate and never use calculus for 
anything -- so they do not really want to learn "more" than necessary :) .

Calculus is not the only course, though if you are selling educational 
software, it is probably the biggie (or maybe pre-calc, but that's not 
so much fun and often remedial).

I would like to believe that my colleagues who teach engineering of 
various sorts would see computer algebra systems as more or less 
essential tools in their fields, and would feel an obligation to make 
room in the curriculum for the teaching of such software. While I do not 
have the 50 years of experience of AES, I do have 35+ years of 
experience in a similar situation.  Maybe an occasional guest lecture is 
the best I could do. I suspect the engineering (undergraduate) 
curriculum is even less prone to experimentation than math.

One article I have read on using computers to teach math (modern 
algebra, not calc.) included a study on whether students learned more 
(with or without computers).  Observations:  students stopped learning 
the computer aspects when they realized the final exam was going to be 
held in a room without computers. On the final exam, the students 
with/without computers did about the same.

The authors conclude (with a straight face..) that more money should be 
spent on developing the computer programs so that they would be more 
successful.

Good luck.

RJF




> My own experience is that those who don't want to 
> "believe" simply will not believe.  And often they don't want to make 
> the effort to reconceptualize what it is they're teaching (and why) and how.
> 
> On 1/19/2010 5:14 AM, Richard Fateman wrote:
>> Do you have any evidence that, taken collectively, the students know
>> more calculus? Can you show that they do better on the final exam than
>> students who haven't used computer systems?
>>
>>     Typically the calc teachers I've encountered want to know "what to
>> leave out to make room" for computer stuff.  I tell them to leave out
>> Logarithmic Derivatives.
>>
>> Some students like computers because they are neat, and may be
>> enthusiastic about this aspect of the course (though not all...).
>>
>> Maybe it is unimportant that they learn calculus at all, and they should
>> just learn about computers. This would be an important but divisive
>> claim:  i.e. calculus is unimportant; we should require that students
>> learn computer skills (Mathematica??) instead. Maybe David Park's point
>> is really somewhere along that spectrum, and we should hold students who
>> learn Mathematica to a lower standard regarding the traditional
>> curriculum.
>>
>> To be clear, I don't object to teaching students about computer algebra
>> systems.  I do so when I get a chance (in computer science courses).
>> I just am unaware of evidence that it makes them better calculus
>> students.  I don't doubt that a teacher using a computer to do graphics
>> can enliven a calculus class. And even students doing graphics on their
>> own (e.g. TI graphing?) can have fun. But can you show they learn more
>> calculus if they have Mathematica at hand?
>>
>> RJF
>>
>>
>> Helen Read wrote:
>>
>>> My students come into university level Calculus I or II with no
>>> Mathematica experience, and learn to use it in my calculus class while
>>> learning calculus. ...
> 


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