Re: Re: learning calculus through mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg108191] Re: [mg108046] Re: learning calculus through mathematica
- From: "Ingolf Dahl" <ingolf.dahl at telia.com>
- Date: Wed, 10 Mar 2010 01:45:01 -0500 (EST)
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- Reply-to: <ingolf.dahl at telia.com>
I must share another anecdotal experience, that I got today, but as experience it is not unique. I am working in the field of liquid crystals, and was investigating a paper, where they measured the optical retardation of liquid crystal cell, and they could vary both the wavelength of light and the inclination angle of the illuminating light. Using this data, they fitted three parameters, characterizing the optical properties of the cell. This really looked good to us, because we want to measure the same parameters. However, I used two days with Mathematica to solve the differential equation they presented, and from that I could (using their formulas) use NIntegrate to get the retardation as function of all involved parameters. Then I did some simple plots to check how the retardation varies when the fitting parameters are varied. Unfortunately I found that the retardation changed in (almost) the same way, independent of which parameter I varied... So the problem was as well specified as the equation x + y + z == 50. It is possible to get a very good fit of the x, y, and z to this equation, but the value and relevance of one particular solution is null. Thus it would be possible to specify only one, but not three, parameters from the experimental data. So why had not the authors of the paper seen the same? I am 100 % sure that it is not due to my superior intelligence. Maybe my experience from solving a similar problem on a Olivetti Programma machine 40 years back helps a bit, but the thing I think that really matters is that I have appropriate knowledge about the relevant CAS tool (Mathematica, of course), and I guess that the same was not true for these authors. It is not appropriate to reveal their names here. So the relevant test for the math students is maybe not to test them with small 2x2 equation systems that can be solved with paper and pen. Give them instead slightly more involved problems, and a computer with some CAS systems installed, and measure then the time and the quality of the solutions. The set of problems you can solve if you have a CAS system to your disposal is immensely larger than the set of paper-and-pen problems, especially if you are trained to use the abilities of the CAS. Ingolf Dahl > -----Original Message----- > From: Helen Read [mailto:hpr at together.net] > Sent: den 7 mars 2010 10:04 > To: mathgroup at smc.vnet.net > Subject: [mg108046] Re: learning calculus through mathematica > > On 3/5/2010 4:33 AM, David Park wrote: > > Certainly not every student should be learning Mathematica at the > earliest > > possible age. But maybe those who are seriously interested in a > technical > > career and are motivated should. Maybe it wouldn't be a part of > regular > > secondary school education, but be done on their own, or in math > clubs, or > > via mentoring over the Internet. > > > > Maybe it's true that CAS have not made a significant positive impact > in > > technical education. Does that mean people should give up? Maybe we > haven't > > properly learned how to use them yet. When new technologies come in > they are > > often used to just make the old approaches more efficient. Usually > what is > > needed is entirely new approaches. Instead of mass lectures and mass > exams, > > maybe there should be more self study, more mentoring and more > mathematical > > essay writing. As things stand now I have the sneaky suspicion that > students > > just don't know Mathematica well enough and it is another obstacle to > > getting through the course. > > I teach my calculus students Mathematica more or less by immersion, > while they are learning calculus. We use it routinely throughout the > course, and every one of them is competent at using it by the middle of > the semester; it really only takes a couple of weeks to get them up and > running. The very weakest students might think of Mathematica as an > obstacle, but even they are mostly OK with it, especially now that we > have the Classroom Assistant palette, which eases the learning curve a > great deal. > > I have the good fortune of teaching in a classroom (we have two such > rooms) equipped with a computer for each student, however, as well as . > This means we can use Mathematica any time we want. Often times I will > introduce a topic, do some chalk-and-talk, then give the class some > examples to work on with the aid of Mathematica, and let them discover > a > lot of things by doing the examples. Then we'll discuss the examples > afterward, and do some additional chalk-and-talk analysis. Or sometimes > I'll give them pencil-and-paper exercises to work on, and have them > check their work in Mathematica. I know this is all anecdotal, but I do > find a lot of value in this, and I think my students do too. > > -- > Helen Read > University of Vermont
- References:
- Re: learning calculus through mathematica
- From: Helen Read <hpr@together.net>
- Re: learning calculus through mathematica