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Re: learning calculus through mathematica

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  • Subject: [mg108162] Re: learning calculus through mathematica
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Tue, 9 Mar 2010 06:25:57 -0500 (EST)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
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  • Reply-to: murray at math.umass.edu

I do thing we are essentially in agreement.

I agree that students should learn to do some symbolic calculations by 
hand.  And they have to do some longer symbolic calculations by hand, 
not for the sake of getting an answer that way, but for the experience 
of concentrating on a problem that requires many steps.

Three issues here are: (i) which symbolic calculations?  (ii) 
calculations of what complexity? and (iii) for how long should they have 
to do them by hand?

Two examples.

Re (i): Should we be teaching the techniques of indefinite integration 
the way it is ordinarily done in calculus courses -- given that symbolic 
integrators often use different algorithms from what we do by hand?

Re (ii): Is it worthwhile to take student's time, e.g., to integrate a 
rational function whose denominator is the product of, say, the square 
of an irreducible quadratic and the cube of a linear factor?  Isn't it 
enough for students to do MUCH simpler examples with paper and pencil 
and then leave these more complicated versions to a machine?

Or, take a typical Calculus I minimization problem about a drowning man 
in the lake, with the lifeguard running along the (straight) shore at 
one rate and then swimming toward the man at a different rate.  If the 
rates are cooked up right, then this is quite reasonable to ask a 
student to do with paper and pencil.

However, change the problem to a more realistic one of, say, finding the 
path a light ray takes in traveling in air and then in water, where the 
two rates are 30 cm/ns and 22.5 cm/ns, respectively. The equation for 
the critical point of the total time of light travel reduces to a 4th 
degree polynomial with some unpleasant coefficients. In principle, 
students could apply the quartic formula to find those roots, but is it 
in any sense reasonable to ask them to do it?  To me it seems like a 
perfect opportunity to resort to a CAS to work through the entire thing, 
if not just to solve (symbolically or numerically) the quartic equation. 
  [I first learned this example, and using Mathematica for it, from the 
wonderful calculus course and book developed by Frank Wattenberg.]

Re (iii): Once a student has demonstrated basic paper-and-pencil 
proficiency in calculus and has graduated to a full-fledged course in 
differential equations, is there any wholly defensible point in 
expecting hand calculation of integrals such as may arise, e.g., in 
applying the method of variation of parameters?

The usual kind of answer I hear to this is something like, "But it 
provides reinforcement of what the student learned in calculus by giving 
him a chance to practice it again."  Which begs the question, of course, 
as to whether and why the student should still have to do such "baby" 
calculation by hand at this point.

I could rant on further about this, but I'll stop here.

On 3/8/2010 6:11 AM, Andrzej Kozlowski wrote:
> I don't see any real contradiction between this and what I wrote. "Long
> symbolic computations" can well qualify as something that isn't "fully
> accessible" to many students without a computer. Also, I don't think you
> would dispute that all students should be able to do "short symbolic
> computations" by hand. If they don't, then, in my opinion, they will
> never fully understand what it is that Mathematica is doing for them.
>
> Ultimately it is a question of finding the right balance. There are lots
> of things in mathematics that one needs to do just once completely by
> oneself in order to develop an intuitive understanding of what is
> involved. Once this understanding has been developed, there is no need
> to perform ever again these often tedious computations and manipulations
> by hand. It seems to me that showing students that all such things can
> be done by a CAS before they have understood the basic concepts can
> sometimes be seriously harmful (of course mostly to the "the lazier or
> intellectually weaker" ones - the others would probably not be satisfied
> with mere "button pushing"). On the other hand, there are enormously
> many fascinating things in calculus and other areas of mathematics that
> can't be done by an average student without a CAS either at all or
> within a reasonably time period. I think all computer aided calculus
> courses should include such examples (and even perhaps concentrate on
> them) because it is only such examples that can really convince both
> students and sceptical academics that CAS can be seriously useful in
> mathematics.
>
>
> Andrzej Kozlowski
>
> On 7 Mar 2010, at 16:15, Murray Eisenberg wrote:
>
>> One reason is very simple: by using a CAS to do many long symbolic
> calculations, students can focus on modeling and the resulting and
> relevant mathematical concepts and methods -- not the details of
> carrying out long chains of algorithmic, algebraic steps.
>>
>> My 45 years of teaching make perfectly clear that, for most students
> in calculus, e.g., they are so involved in trying to get the symbolic
> manipulations right, they have little or any idea of why they're doing
> them.  They totally miss the forest for the trees.
>>
>> The other side of this situation, I regret to say from my experience,
> is that the lazier or intellectually weaker students are often incapable
> of rising above merely carrying out mechanically the symbolic
> manipulations -- many of which they get wrong anyway -- to have much of
> an understanding of the higher-level concepts involved.
>>
>> On 3/7/2010 4:06 AM, Andrzej Kozlowski wrote:
>>> I have never seen  or heard any convincing reason why using a CAS
> should
>>> make it possible to understand and learn better those areas of
>>> mathematics which are fully accessible to a student with only a pen
> and
>>> paper. In fact I can see a few reasons why the opposite might be the
>>> case. In many situations I can see clear advantages in performing
>>> algebraic manipulations "by hand" or even "in the head", which is, in
> my
>>> opinion, the only way to develop intuition. The same applies to
>>> visualisation - while being able to look at complicated graphics can
>>> often be a big advantage, I always insist on students developing the
>>> ability to quickly sketch simple graphs by hand on the basis of
>>> qualitative analysis of analytic or algebraic data. This is again
>>> essential for developing intuition and I am not convinced that doing
> all
>>> this by means of a computer will provide equivalent benefits.
>>
>> --
>> Murray Eisenberg                     murray at math.umass.edu
>> Mathematics&  Statistics Dept.
>> Lederle Graduate Research Tower      phone 413 549-1020 (H)
>> University of Massachusetts                413 545-2859 (W)
>> 710 North Pleasant Street            fax   413 545-1801
>> Amherst, MA 01003-9305
>
>

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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