On 4/28/2013 5:17 AM, Richard Fateman wrote: > On 4/27/2013 9:58 PM, Helen Read wrote: >> On 4/26/2013 4:24 AM, Richard Fateman wrote: >>> >>> >>> I note that the MIT regular calculus, 18.01 >>> http://math.mit.edu/classes/18.01/Spring2013/ >>> apparently uses a computer algebra system, but not Mathematica. >>> I do not see how it is used or how it could be used on the exams. >> > >> I teach calculus in a classroom (we have two such rooms) equipped with a >> computer for each student. We use Mathematica routinely throughout the >> semester, in and out of class, and most of the students like having it >> and using it. We have a site license that allows the students to install >> Mathematica on their own laptops so they can use it outside of class. > > I was speaking of how the MIT course could use computer algebra. If you > look at the review problems > http://math.mit.edu/classes/18.01/Spring2013/Supplementary%20notes/01rp1.pdf > you see that quite a few of them are trivial if you just type them in to > a computer algebra system, and presumably would not be much of a > learning experience if in fact they were just typed in. Others require > explain/prove/give examples. > > I have no doubt that a calculus course could be constructed using > computer algebra systems --- I would hope it would be quite different, > emphasizing (say) the calculational aspects of the subject and then > observing the symbolic, almost coincidental, solutions to the same > evaluations. It sounds like you are doing something along those lines. > > It does not surprise me that MIT is still doing the same old thing; when > I was covering recitation sections, the main lecturer was Arthur > Mattuck. In 1971. The notes used in 2013 are apparently by Arthur Mattuck. > > It is presumably possible to do things at U. Vermont without overcoming > such massive inertia. I have encountered substantial inertia at UC > Berkeley in mathematics and engineering, too. > > On the other hand, the question remains for any of these courses as to > whether one can objectively demonstrate that students learn calculus > more than those in a control group not using computers. I am not > doubting for a moment that instructors who like computers prefer > teaching using them. (Including me.) Yet there are still math > instructors who, for whatever reason, prefer not.
It's a difficult thing to measure. Many, many years ago I developed a "reform" version of the "baby" calculus at UVM. (This is our two-semester, easier, calculus sequence taken by students whose majors require them to take calculus but not at the level that is required by say math or engineering majors. It is taken by more students at UVM than any other course, including English 001, and many of the students have terrible deficits in pre-calculus skills such as algebra and trigonometry.) For the "new wave" version of the course we used a textbook written by some folks at Clemson University that emphasized concepts over algebra, and used data driven examples to motivate the concepts. The emphasis was on understanding and interpretation, using graphing calculators to handle the drudge work. Almost everything was a "word problem" and rote skill-and-drill problems were downplayed (though we still assigned some). Many of the faculty freaked out over this approach, and we ended up with two separate tracks taught by different faculty.
One semester we did make an attempt to compare outcomes by putting some common questions on the final exams, but the more conceptual questions that those of us teaching the "new wave" course proposed were rejected by those teaching the traditional course as "unfair" questions that their students should not be expected to answer. Which tells you something right there. In the end we found no statistical difference between the two groups on the skill questions (e.g., product rule), and on the (very few) mildly conceptual questions that we were permitted to ask the students in the "reform" group outperformed (in a statistically significant way) the students from the traditional group. Nonetheless, there was so much faculty resistance that the "reform" version ended up being given a separate course number, and was eventually killed off because almost all of the client departments continued to require the original traditional version.
I haven't taught the baby calculus in ages (I stopped when the reform version was discontinued) and don't really know what they are doing with it these days, but my sense is that it is somewhere in between the two versions, but closer to the old traditional way.
> <snip> .... >> I find that >> overall they seem to end up with a better understanding of series than >> my students did years ago when all we did was paper-and-pencil >> convergence (which the students found to be terribly abstract). > > Can you quantify this? (This is somewhat unfair -- you are stating your > own observations and I'm asking you to be an expert on human factors, > learning, etc. I've often seen and participated in "innovation" in > teaching and rarely tried to prove the innovation had positive results! > Nevertheless, it would be nice to have "evidence".)
Unfortunately I have nothing more than anecdotal evidence.
>> My students do use Mathematica on exams, but not for everything. I make >> up exams in two parts. Part 1 is paper and pencil only, and I keep the >> computers "locked" (using monitoring software installed on all the >> student computers). When a student finishes Part 1, s/he hands it in and >> I unlock that particular computer (which I can do remotely from the >> instructor's desk), and the student has full use of Mathematica for Part >> 2. I can monitor what the students are doing on their computers from the >> instructor's station (and of course I get up and walk around and answer >> questions if they get stuck on something like a missing comma). We have >> a printer in the room so that the students can print their work and >> staple it to their test paper when they hand it in. > > I have no doubt that there are interesting calculations that are vastly > easier to do with the help of a computer algebra system. > > I would be interested to see what kinds of questions you can ask on a > calculus exam that (a) test something that students are expected to know > from a calculus course and (b) require (or are substantially assisted > by) Mathematica. >> >> I've been teaching this way since the late 1990s, and wouldn't dream of >> going back to doing it without technology. > > Another question, based on my own observations ... If you are on > sabbatical and not available to teach this course, does someone else > pick it up and teach it the same way? What I've seen is that when the > computer enthusiast is not available, the course reverts to something > rather more traditional.
My department enacted a policy that requires some use of Mathematica throughout the three semester "grown up" calculus sequence, and drew up a document of minimum Mathematica competence that should be achieved by all students. From my point of view Mathematica competence in and of itself isn't really the main point, it's just a means to a greater end. Still, most of the faculty are on board with this, and many are very enthusiastic and integrate Mathematica in ways that we believe benefit the students (again, no hard evidence -- but I don't have hard evidence for lots of the choices I make in teaching; all I have is years of experience and observation). If I were not around, there are enough others to carry on.