rouben@math13.math.umbc.edu (12/04/90)
Here are a few thoughts and ideas on the role of calculators
and computers in teaching freshman calculus. I am interested
to find out if there are others who share these thought, or if
there are some who disagree with me. Comments from both
teachers and students of calculus are welcome.
--
I have thought mathematics at several universities for the past fifteen
years. It's no secret that there is a great deal of dissatisfaction among
the mathematics faculty of the American colleges and universities with the
traditional approach to teaching freshman calculus. "Calculus reform" has
become a very trendy topic among the educators. The National Science
Foundation has a special program with emphasis on developing new approaches
to teaching calculus. Innovative instructional ideas have begun to emerge.
It seems certain that the future calculus syllabus will be substantially more
applications-oriented and will rely less on rote learning and drill-type
problems. For the better or worse, the students may not be required to
remember the formula for the anti-derivative of 'sec x tan x' but they will
be expected to set up solve, and interpret the equations of motion of a
planet around a sun. (A full circle back to the roots of calculus!)
Certainly calculators and computers will play significantly more prominent
role in the teaching, learning, and use of calculus than what they have
today. At one end of the spectrum of the advocates of the use of
"technology" in the classroom are the more conservative types who are content
with sprinkling the traditional calculus textbooks and courses with a
moderate amount of "calculator problems," e.g., problems dealing with
computing roots with Newton's method or approximating integrals with Riemann
sums. At the other end of the spectrum are those who almost throw out the
baby with the bathwater and advocate doing away with the lecture format
altogether and teach calculus with the help of sophisticated software and
computer algebra systems (mathematica, derive, maple, etc.) in a computer
laboratory.
I tend to think that the student who gains proficiency with a computer
algebra systems begins to treat the computer as a vital link in performing
calculus-related functions, much in the same way that most of us treat a
calculator a vital link in performing mundane arithmetical tasks (When did
you last extract a non-trivial square root *without* using a calculator?)
Now this may or may not be such a great idea, but who knows, it may be the
wave of the future.
My personal preferences lie somewhere in between; I would not want to go as
far as to teach differentiation out of the mouth of mathematica, but I would
love to introduce a heavy dose of non-trivial calculator applications in the
courses I teach. I would emphasize, for instance, the numerical computation
of areas in 2D and volumes in 3D. I will include numerical solutions of
differential equations, both ordinary and partial. I would do away with the
epsilon-delta definitions of continuity -- not because I don't like epsilons
and deltas, I am an analyst of sorts, but because at this level epsilons and
deltas obscure rather than illuminate -- I will approach limits numerically.
Programmable calculators are ideal for repetitive computations and
investigation of the limiting values of functions. Computing the limit of a
difference quotient numerically does wonders in driving the point home that
not all functions are differentiable; it is trivial to program a calculator
to evaluate the difference quotient for the functions |x| or x.sin(1/x) at
zero. Among other things it becomes clear that the domain of a functions
should be carefully prescribed otherwise the program may flash an error
message and halt.
The latest advanced calculators, such as Hewlett-Packard's HP48sx, allow
computation of roots of functions, numerical integration, graphing, and even
some rudimentary symbolic algebra! These capabilities, and their
programmability, makes these calculators more like computers than
calculators. One great advantage over computers is that is their total
portability. The student may carry the calculator to the classroom or to the
library or to the cafeteria or to the dormitory, and he/she can use it for
other courses too. Another advantage is that the student, by the virtue of
being the owner of the calculator, has a vested interest in learning how to
use it efficiently and effectively. In contrast, a computer at the calculus
laboratory or wherever, does not belong to him/her, the manuals for the
hardware, software, and peripherals are not generally easily accessible, and
it is not clear that any investment in time and effort to deeply familiarize
with the facilities will pay off.
The down-side of the ownership of calculators is their cost. The basic HP48
retails for close to $300. This may be a non-trivial amount in a freshman's
budget. Do I, as the instructor of the calculus course, require everyone
in my class to buy the calculator? Do I structure my course so that it
becomes difficult, if not impossible, to get by without a sophisticated
calculator? What about those who absolutely cannot afford the expense?
What if some buy less expensive models with fewer features? Doesn't it
put them in a sad disadvantage when doing homeworks and taking tests?
I would like to hear your thoughts and comments on this. Specifically:
A - Is the traditional U.S. style of teaching freshman calculus
in need of reform?
B - Will the use of computer algebra systems (mathematica, maple,
derive, mascsyma, mu-math, etc.) enhance the learning of calculus?
C - Will the use of programmable calculators enhance the learning
of calculus?
D - Should the cost of the calculator be a factor in deciding whether
to prescribe it as a required tool for enrollment in a course?
E - Have you thought, or have you been a student in a "non-traditional"
type calculus course? What was your experience?
E - Other thoughts and comments.
--
Rouben Rostamian Telephone: (301) 455-2458
Department of Mathematics and Statistics e-mail:
University of Maryland Baltimore County bitnet: rostamian@umbc
Baltimore, MD 21228, U.S.A. internet: rostamian@umbc3.umbc.eduts@cup.portal.com (Tim W Smith) (12/04/90)
Students don't need a $300 dollar HP to do these things. There are calculators from Sharp and Casio that do a lot of these functions for a lot less. As long as you make sure that you don't require all the features of the $300 HP, so that students with less money can still afford something that does what they need, you can probably get away with requiring a calculator. Heck, it won't cost anymore than if you required an extra textbook, considering the price of books nowadays :-(. Tim Smith
hb136@leah.albany.edu (Herb Brown) (12/04/90)
In article <4608@umbc3.UMBC.EDU> rouben@math13.math.umbc.edu () writes: >Here are a few thoughts and ideas on the role of calculators >and computers in teaching freshman calculus. I am interested >to find out if there are others who share these thought, or if >there are some who disagree with me. Comments from both >teachers and students of calculus are welcome. >-- >Rouben Rostamian Telephone: (301) 455-2458 >Department of Mathematics and Statistics e-mail: >University of Maryland Baltimore County bitnet: rostamian@umbc >Baltimore, MD 21228, U.S.A. internet: rostamian@umbc3.umbc.edu At The University at Albany, the Mathematics Dept created a Computer Classroom (not a lab, but a classroom; we also have several computer labs sprinkled throughout the campus) whereby we offer several different mathematics courses including Calculus. This was the first semester of operation. I taught two courses in the Computer Classroom: Calculus I and a course called Basic Analysis. (There were two other Calc I courses, a Classical Algebra course, a Lin Prog & Game Thy course, one in Numerical Methods, and a Stat course.) The classroom is designed so that no computer is physically between the student and the instructor. The students SIMULTANEOUSLY interact between their computer, the instructor's computer, the blackboard, and their fellow students. My experience thus far has been one of excitement and jubilation. (I have been in this business for nearly 20 years and remember these feelings when I first received my doctorate and began teaching.) Let me give you an example of what became possible in this Calculus course that I did not do (or even attempt to do) in previous ones. I like to discuss (briefly) the concept of an inverse function, although I am aware that it is a difficult concept. I've always attempted to convey the concept visually by drawing (or attempting to draw) both the function, it's inverse, and the identity function (y = x) on the same set of coordinates. In order to maintain a semblance of interest I would pick an 'easy' function to deal with, i.e., one that would permit me to compute its inverse by hand before ALL students nodded off. This semester, since we are using Maple software, I chose the function x^3 + x + 1 Now, solving a cubic in class is, well, how should I put it ..... However, we had MAPLE! So, after discussing the ideas of an inverse, I presented this example and asked for help in solving for the inverse. More than one student said "Let Maple do it!" That's exactly what I wanted to hear. We asked Maple do some dirty work. It did, giving us three answers. After analyzing each answer we discovered the inverse function. (P.S. They were ALL AWAKE!!) We then plotted the three functions I mentioned above and got to SEE what an inverse function looked like. I think most of those students now know something about an inverse function and its graph. One additional comment needs to be made here. When I say "We did this and we did that" I literally mean WE. Each student has access to his/her computer and does the calculation or plot on that machine and gets to SEE the results immediately. Herb w -- ---------------------------------------------------------------------------- Herb Brown Math Dept The Univ at Albany Albany, NY 12222 (518) 442-4640 hibrown@leah.albany.edu or hibrown@cs.albany.edu or hb136@ALBNYVMS.BITNET ----------------------------------------------------------------------------
pashdown@javelin.es.com (Pete Ashdown) (12/04/90)
rouben@math13.math.umbc.edu writes: >I would like to hear your thoughts and comments on this. Specifically: >A - Is the traditional U.S. style of teaching freshman calculus > in need of reform? Yes. The first time I took Calculus, there was a lot of fear and stigma involved. I didn't do so well. The teacher emphasized method over understanding. I took Physics after Calculus, which emphasized understanding over method, in addition, I bought a HP-28, then later a HP-48. I started to _understand_ what Calculus meant and how it was used. The second time I took Calculus, it was a breeze to go through. No fear, no stigma, and I understood what I was doing, rather than spewing crammed methods. >B - Will the use of computer algebra systems (mathematica, maple, > derive, mascsyma, mu-math, etc.) enhance the learning of calculus? Yes. Broderbund's introductory calculus software is a good example of this. You get to see a visual representation of tangents, areas, derivatives, etc. Although it is pretty limited, I would imagine the packages you mentioned are much more capable. Being able to "play around" with ideas to see how they work is extremely valuable. >C - Will the use of programmable calculators enhance the learning > of calculus? Yes and no. In my higher math classes, I tend to bang out derivatives on the 48 rather than waste time doing them by hand. I suppose if I were stuck on a desert island without my 48 and a Nazi guard threatened me to find a difficult derivative, I would be dead. However, we all know the "square root" arguement as you mentioned it. Should these tasks be designated for computers/calculators? Is "hand-math" a dying breed? In my opinion, I certainly hope so. If I can do a problem quicker and more accurately on a calculator, I'll use the calculator. >D - Should the cost of the calculator be a factor in deciding whether > to prescribe it as a required tool for enrollment in a course? Yes, and in the case of the 48, there should be a suppliment from the University or a significant educational discount. However, the 28 comes very close to the performance of the 48 for about $100 less. In the case of Calculus, I have found the raw computing power of the 48 not much different than the 28. >E - Have you thought, or have you been a student in a "non-traditional" > type calculus course? What was your experience? See A. >E - Other thoughts and comments. Its nice to see a math professor pushing these ideas around. I doubt any of the math professors on my campus know the capabilities of the 48, although many of them do extoll the virtues of math software. >Rouben Rostamian Telephone: (301) 455-2458 >Department of Mathematics and Statistics e-mail: >University of Maryland Baltimore County bitnet: rostamian@umbc >Baltimore, MD 21228, U.S.A. internet: rostamian@umbc3.umbc.edu -- / (Rotate head 90 degrees for full effect) | BUNGEEEEEEEE! |---------------------------------------------------------------------->=<o \ Pete Ashdown pashdown%javelin@dsd.es.com ...dsd.es.com!javelin!pashdown
mroussel@alchemy.chem.utoronto.ca (Marc Roussel) (12/04/90)
In article <4608@umbc3.UMBC.EDU> rouben@math13.math.umbc.edu () writes: >Here are a few thoughts and ideas on the role of calculators >and computers in teaching freshman calculus. I am interested >to find out if there are others who share these thought, or if >there are some who disagree with me. Comments from both >teachers and students of calculus are welcome. Everytime I see a discussion along these lines, I get uncomfortable. It seems to me that lumping all "freshman calculus" into one box and asking this vague question is dangerous. There are different audiences for freshman calculus and their needs are different. The original poster said something about doing away with epsilon's and delta's. I certainly think that this is appropriate to a crowd of non-specialists, but I shudder at the thought of taking rigour out of your top-of-the-line course. There should always be a course available for students who want to understand rather than merely to become proficient with a certain set of skills. I am a chemical physicist who uses computer algebra a lot. These things have their place, perhaps even in senior classes. I believe however that one needs to learn to do things by hand before one begins to use machines to automate the tasks. I would be very concerned if the next crop of scientists coming along had no way of verifying the output of their calculations independently of the machine. I am looking forward to the discussion to which we will all be treated. Sincerely, Marc R. Roussel mroussel@alchemy.chem.utoronto.ca
gao@ucrmath.ucr.edu (weiqi gao) (12/05/90)
In article <4608@umbc3.UMBC.EDU> rouben@math13.math.umbc.edu (Rouben Rostamian) writes: >Here are a few thoughts and ideas on the role of calculators >and computers in teaching freshman calculus. I am interested >to find out if there are others who share these thought, or if >there are some who disagree with me. Comments from both >teachers and students of calculus are welcome. ... >I would like to hear your thoughts and comments on this. Specifically: ... >E - Other thoughts and comments. I don't have any formed opinion on this subject, for I am just starting to teach calculus. However I would like to share the following story with you. Yesterday is the final examination for my freshman calculus class. One of the problems is to find out whether the graph of the function 4 3 3t - 4t f(t)= ---------- 4 3 4t - 3t has any horizontal asymptotes. As time went by the students all finished up, and I was with the last student in the room. -- Which one has gotten you stuck? -- The asymptotes. -- That one is easy, isn't it? You just have to take the limit! -- Yes, and I am taking it. (He pushes the keys in his calculator.) -- Does your calculator take limits? -- No, I just, ... sort of ... put in the numbers ... (He does this), it worked for me in the homeworks. He got the wrong answer! Weiqi Gao
horne-scott@cs.yale.edu (Scott Horne) (12/05/90)
In article <4608@umbc3.UMBC.EDU> rouben@math13.math.umbc.edu () writes: > >I would like to hear your thoughts and comments on this. Specifically: > >A - Is the traditional U.S. style of teaching freshman calculus > in need of reform? Perhaps. But not just because some newfangled electronic equipment has come down the pike. >B - Will the use of computer algebra systems (mathematica, maple, > derive, mascsyma, mu-math, etc.) enhance the learning of calculus? No. They are not needed. Nor are any of the programs which supposedly teach calculus. The only advantage they offer is that some people will stay at a computer for longer than they'll stay at a desk with a textbook. Other than that, they're gimmicks. However, at the end of a course, you might like to show your students how wonderfully computer algebra systems can evaluate complicated integrals and other things. (Some might want to buy a copy, particularly if you can get an educational discount for them. I'd like a copy of Derive, but it's too expensive.) >C - Will the use of programmable calculators enhance the learning > of calculus? No, and perhaps it will enhance the non-learning of calculus. Numerical methods, though important and interesting, should not be the focus (or even *a* focus) of your course. >D - Should the cost of the calculator be a factor in deciding whether > to prescribe it as a required tool for enrollment in a course? Calculators are not necessary for learning calculus and hence should not be required. --Scott -- Scott Horne ...!{harvard,cmcl2,decvax}!yale!horne horne@cs.Yale.edu SnailMail: Box 7196 Yale Station, New Haven, CT 06520 203 436-1817 Residence: Rm 1817 Silliman College, Yale Univ Uneasy lies the head that wears the _gao1 mao4zi_.
mroussel@alchemy.chem.utoronto.ca (Marc Roussel) (12/05/90)
In article <1990Dec4.153552.29699@javelin.es.com> pashdown@javelin.es.com (Pete Ashdown) writes: >In my higher math classes, I tend to bang out derivatives on the >48 rather than waste time doing them by hand. >However, we all know the "square root" >arguement as you mentioned it. Should these tasks be designated for >computers/calculators? Is "hand-math" a dying breed? In my opinion, I >certainly hope so. If I can do a problem quicker and more accurately on a >calculator, I'll use the calculator. Let's take another example. Some (primary) educators have suggested that long division should be removed from the curriculum. However long division is the basis of polynomial division, and if you'd never done polynomial division, I don't think you'd ever be able to write a polynomial deflation routine (for instance). Perhaps the actual taking of derivatives is more like square roots (boring and arguably not terribly educational) than like long division (a technique with interesting generalizations), but the traditional introductory calculus course includes a substantial discussion of Newton quotients and of epsilon-delta proofs. These topics must not be lost in the shuffle to computer assisted education as they form the basis for understanding the generalization of calculus to higher dimensions. Marc R. Roussel mroussel@alchemy.chem.utoronto.ca
ags@seaman.cc.purdue.edu (Dave Seaman) (12/05/90)
In article <1990Dec5.030314.26463@alchemy.chem.utoronto.ca> mroussel@alchemy.chem.utoronto.ca (Marc Roussel) writes: > Let's take another example. Some (primary) educators have >suggested that long division should be removed from the curriculum. >However long division is the basis of polynomial division, and if you'd >never done polynomial division, I don't think you'd ever be able to >write a polynomial deflation routine (for instance). People manage to write eigenvalue/eigenvector routines and Gram-Schmidt orthogonalization routines and the like, even though these operations have no analog in elementary arithmetic. There is nothing particularly difficult about polynomial deflation, even if you have never seen long division in your life. In fact, polynomial deflation is easier than long division, because you never have to guess at a divisor and then go back and revise. -- Dave Seaman ags@seaman.cc.purdue.edu
horne-scott@cs.yale.edu (Scott Horne) (12/06/90)
In article <1990Dec4.153552.29699@javelin.es.com> pashdown@javelin.es.com (Pete Ashdown) writes: <rouben@math13.math.umbc.edu writes: < <<B - Will the use of computer algebra systems (mathematica, maple, << derive, mascsyma, mu-math, etc.) enhance the learning of calculus? < <Yes. Broderbund's introductory calculus software is a good example of this. <You get to see a visual representation of tangents, areas, derivatives, etc. I can do this with a pen. <Being able to "play around" with ideas to see how <they work is extremely valuable. So long as you're doing more than just playing. <<C - Will the use of programmable calculators enhance the learning << of calculus? < <Yes and no. In my higher math classes, I tend to bang out derivatives on the <48 rather than waste time doing them by hand. Does it do them symbolically or by numerical approximation? >I suppose if I were stuck on <a desert island without my 48 and a Nazi guard threatened me to find a <difficult derivative, I would be dead. I can't think of any "difficult derivative[s]"--that is, unless the person to whom they're difficult has a poor background in calculus. >However, we all know the "square root" <arguement as you mentioned it. Should these tasks be designated for <computers/calculators? Is "hand-math" a dying breed? In my opinion, I <certainly hope so. If I can do a problem quicker and more accurately on a <calculator, I'll use the calculator. Fine, but you should know how to do it without a calculator. Children can be taught to use calculators, but they won't learn anything about arithmetic that way. Likewise, it's fine to use a computer for integration, &c, when you know how to do it yourself, but you won't learn anything about calculus by typing integrals into a computer and copying out the answers. --Scott -- Scott Horne ...!{harvard,cmcl2,decvax}!yale!horne horne@cs.Yale.edu SnailMail: Box 7196 Yale Station, New Haven, CT 06520 203 436-1817 Residence: Rm 1817 Silliman College, Yale Univ Uneasy lies the head that wears the _gao1 mao4zi_.
booker@network.ucsd.edu (Booker bense) (12/07/90)
In article <1990Dec4.153552.29699@javelin.es.com> pashdown@javelin.es.com (Pete Ashdown) writes: >rouben@math13.math.umbc.edu writes: > >>I would like to hear your thoughts and comments on this. Specifically: > >>A - Is the traditional U.S. style of teaching freshman calculus >> in need of reform? > Yes, having taken and taught calculus ( 3-years in G-school ), I think that most students don't learn any calculus in Freshman calc. They (the ones that pass) brush up on their algebra and learn to jump through symbolic hoops like trained monkey's, but they rarely get the intuition needed to use calculus effectively. They most useful thing we could teach a freshman student is a strong physical sense of what a derivative and a integral are and some initution about when an answer 'looks right'. I tried very hard to teach this in my classes, but it's not really possible in the 'barf back the formula' setting of most calculus courses. > >>B - Will the use of computer algebra systems (mathematica, maple, >> derive, mascsyma, mu-math, etc.) enhance the learning of calculus? > >Yes. Broderbund's introductory calculus software is a good example of this. >You get to see a visual representation of tangents, areas, derivatives, etc. >Although it is pretty limited, I would imagine the packages you mentioned >are much more capable. Being able to "play around" with ideas to see how >they work is extremely valuable. I think the key thing here is graphics. In the tv age, this is the only way to convey ideas that really gets to most students. One example of this that I think is quite good is the 'Mechanical Universe' tv course series that show up on PBS occasionally. They portray the physical ideas that motived the development of calculus in a very visually compelling manner . > >>C - Will the use of programmable calculators enhance the learning >> of calculus? > I would say no. In one course I taught, we did not allow students to use calculators. What's needed is pictures and the development of some physical notions that relate to the symbols. A calculator just provides numbers or answers faster , not better. > >>D - Should the cost of the calculator be a factor in deciding whether >> to prescribe it as a required tool for enrollment in a course? > I think a calculator is an unnecessary tool, a PC-Lab with appropriate software would be far more useful. > >>E - Have you thought, or have you been a student in a "non-traditional" >> type calculus course? What was your experience? > Yes, I was a student in a Independently Paced calculus class when I was a freshman at WPI those many years ago. The way it was set up was that you had a series of tests to pass and you could take them whenever you wanted. There was no class , but tutors were available in the lab were you took your test. I don't think that is a very good situation, I only learned enough to pass one test and go on to the next. It's too easy to get only a minimal understanding, particularly if you're good at taking tests. > >>E - Other thoughts and comments. > Another poster mentioned that epsilon and delta's should be part of 'any freshman course'. This is the most wasted week in freshman calc. None of the students understand it , they just don't have the background to understand an axiomatic proof. It would be far better just to leave it for the junior real analysis course, where maybe you'll understand it. In my case , I really didn't get a good feeling for eps-delta proofs until my graduate analysis class. I think this is another example of where math education has adopted the wrong approach. IMHO, elementary math courses should be taught from a historical basis, not the axiomatic one familar to most mathematicians. I.e. a freshman calculus course should start with the problem of determining the path of a projectile and see how that problem leads to the idea of a derivative. There is a place for formalism and precision, but it should come after the intution is in place not before. /* Booker C. Bense benseb@grumpy.sdsc.edu */