Note before reading. The equations sometimes show on this blog as whitespace; one needs to click on them to see the associated jpeg file (It seems that this is a FireFox problem, i.e., IE shows equations fine!).
Introduction
In 1981 I wrote a paper (staying anonymous requires that I not cite it here) in which I proposed Computer Assisted Testing in Technical Subjects, with the idea of rigorizing the testing aspect of teaching/learning. More than 20 years later, it is finally coming into existence, with businesses (such as WebAssign) offering such tests, but without the proctoring required to make the system really work.
The idea was that students needed the ability to know when they were wrong during an examination, and needed the ability to come back when they'd studied more. They needed the ability to "give up" or say that "I don't know", so that they themselves would realize their deficiencies vis-a-vis the course requirements.
Many rejected proposals later, I'd given up on CAT but thought that publishing these thoughts might help get people to move more swiftly, if anyone ever reads them, in the direction I'd hoped to champion.
My idea is that schools should use discarded computers to create examination rooms, huge examination rooms, in which students could come to take tests when they were ready, and under proctored conditions, could prove that they knew what they were supposed to have learned.
Some threads of quantum chemistry understanding
It is hard to define what an education actually means, and in the current atmosphere of liberal versus conservative with respect to teaching "values" in the academy, it is amusing that virtually no one disentangles scientific from non-scientific learning. Political Science must be a priori controversial since it attempts a dispassionate examination of a passionate subject.
On the other hand, calculus is value free. No one, to my knowledge, thinks of it as controversial (except for biology which leads to genetics which leads to Darwin and evolution, which remains controversial).
So the discussion of education should bifurcate into two discussions, one concerning politically correct (or otherwise) topics, and the other addressing the engineering, physics, biology, and, of course, chemistry education which the nation needs.
We desperately need large scale, country wide, testing.
I recognize how controversial this is, but ascertaining what our students are learning in algebra, calculus, physics and chemistry (my areas of quasi-expertise) is essential to sustaining the effort our predecessors made possible through their efforts on our behalf. We need to know that what we expect to be learned has, in fact, been learned (and remembered).
That's different than knowing what was taught. All of us will concede that what's taught is rarely learned. What we need is a measure of what is learned.
Since the material we deal with is hierarchical in nature, there is no way that students can continue to level ``b'' if their mastery of precursor material taught in level ``a'' is insufficient to the task at hand. Learned foundational material which has been forgotten is useless, and
the ``learning'' was in vain, i.e., non-existent! Let me give you an example from one of the readings from a quantum chemical discussion of eigenvalues:
Consider the following:
The equations sometimes show here as whitespace; one needs to click on them to see the associated jpeg file (It seems that this is a FireFox problem, i.e., IE shows equations fine!).
Show that
This identity came up in a molecular orbital problem, and we expect any high school graduate to be able to show that the statement is true.Now, before continuing, I want to make sure you understand what I mean by "be able to show that the statement is true". I do not mean plucking something out of a multiple choice list. I mean actually writing lines of (what) algebra/mathematics/arithmetic/whatever and arriving at a proof that the statement is true. A multiple choice question of the type:
Which of the following statements is true:
- etc., etc., etc..
and then evaluate each of the choices until s/he finds the ``correct'' one. Multiple choice examination doesn't test what we think its testing, and its results are therefore useless!The entire motivation for Computer Assisted Testing was to make it possible to understand whether or not the examinee could actually do the work without the hints of a set of choices.
The argument was made that the so-called real world does not give multiple choices to scientific/mathematical queries. More important, even listing choices warps the intellectual environment of the measurement!
By the time students are studying "my subject'', quantum chemistry, it is too late to deal with elementary problems in mathematics and/or physics. We know very well that our students have a poor command of physics, and their calculus skills are atrocious.
Chemistry students have been forced to take calculus and physics without a rationale. They are skeptical that this material actually is applicable to chemistry, and Organic Chemistry, in the sophomore year, adds to the prejudice that numbers and mathematics are not part of "real'' chemistry.
Some thoughts on learning and teaching in the local environment
It has become abundantly clear to me that teaching and learning are orthogonal.
OK, maybe not exactly, but the overlap between the two is small at best.
The social tendency not to harm students, the needs for institutions of ``higher'' education to fill their ever expanding seats, the faculty needs to have time to do the research which really pays the bills, etc., conspire to make teaching the least important
aspect of undergraduate schools. No matter what the publicists write (and say), these institutions regard their undergraduate charges as a burden.
Normal chemistry 127-8 is such a burden that faculty are asking immunity from having to teach it (several already have that de facto). The department has hired permanent sub-faculty to teach it; faculty with little or no research interest, who are dismissable at the stroke of a Dean's pen based on a single bad teacher evaluation (quoted from the Dean when meeting with the Department).
This permanent cadre of sub-faculty completely relieves the normal faculty of the burden of having to teach freshman. Several faculty have never taught freshmen anyway, but what the hell, let's institutionalize the immunity!
The fact that the State is only interested in the teaching of undergraduates, which presumably includes freshman, is irrelevant. What the devil does the State know about Universities, anyway? We know better!
What does it mean to be learned?
In the 12th century,
The equations sometimes show here as whitespace; one needs to click on them to see the associated jpeg file (It seems that this is a FireFox problem, i.e., IE shows equations fine!).
"Bhaskara demonstrated correctly thatWe don't want people writing this about Chemists! George Lang (not so) recently (on the WWW) wrote:an achievement, I might add, utterly beyond the collective intellectual power, say, of the English Department of Duke University. (It is pleasant to imagine members of the department sitting together in a long lecture hall, Marxists to one side, deconstructionists to the other, abusing one another roundly as they grapple with the problem.)''
D. Berlinski, A Tour of the Calculus , Pantheon Books, New York, 1995, page 38
"Most students I have talked to DO NOT link test performance with knowledge of subject.... The message that testing does not measure ability has come through to our students loud and clear. Thus our tests are not learning experiences or opportunities to receive valuable feedback about the students' depth of understanding. They are GAMES the students must play to break into the professional world."
CHEMED-L Internet Discussion Group on Chemical Education, Fall 1995.
It follows that it is better to use a third party as examiner, so teachers and pupils can be allies.
"The National Science Foundation agreed (that there was a problem in teaching/learning calculus) and spent $35-million from 1987 to 1995 on dozens of projects to update the teaching of calculus."
...
"Courses often consisted of bland lectures in which students learned how to calculate derivatives, and integrals. Students practiced the calculations at home, and on exams
professors asked similar problems with different numbers. Students, professors recall, were bored and disengaged."
...
"This approach (reform calculus instruction) really shies away from anything but superficial use of skills (Prof. R. L. Cohen, Stanford). For students who really
need to know math and use it, this wasn't nearly sophisticated or rigorous enough.''
R. Wilson, ``A Decade of Teaching 'Reform Calculus', Has been a Disaster, Critics Charge'', Chronicle of Higher Education, Feb 7, 1997, page A12.
I am convinced that examinations are the key to re-introducing rigor to American classrooms.
Here is a typical examination question (in Mathematics/Calculus):
The equations sometimes show here as whitespace; one needs to click on them to see the associated jpeg file (It seems that this is a FireFox problem, i.e., IE shows equations fine!).
Show that, given
one can obtain
by elementary methods.
Z. A. Melzak, "Companion to Concrete Mathematics, Mathematical Techniques and Various Applications", Wiley-Interscience, New York, 1973, page 177. (This is akin to the infamous "it can be shown by the serious student ... ".)
Offering help during examinations, CAT mode
If faced with the above question on an oral examination, the prepared student might like a hint. For instance, a hint might be "First integrate over x (on the right hand side) and see what you get."
If the student still needs help, perhaps one should suggest "Try integration by parts."
If the student wrote:
the examiner might query if the examinee has made a sign error, etc., etc., etc..
What we proposed here was that the computer could (and should) act as a surrogate for the oral examiner and probe with intelligent hints and corrections the extent of the student's ability to do the actual problem. Alternatively, we could offer the student a "do not know how to proceed" button which would make the student admit that s/he could not do the problem!
A Spectroscopy problem
Here is a typical spectroscopy question:
Calculate the ratio of intensities expected for the (n=0,J=1 to n=1,J=2) line to the (n=0,J=2 to n=1,J=3) line at 25C. Assume that the rotational constant, B, is 10 recirocal centimeters.If the student answers:
(9 x e-B*(2*3)/kT)/(7 e-B*(3*4)/kT)
we can respond with a question about appropriate degeneracies of appropriate rotational levels.
If the student answers:
(9 e-B*(1*2)/kT)/(7 e-B*(2*3)/kT)
instead, then we can respond with a different question about appropriate degeneracies, etc., etc., etc..
A Quantum Chemistry problem
Another question:
What is the value of the 2px wave function at the point x=y=z=1 Angstrom?
Compare this to the question:
What is the value of the 2px wave function at the point r=1, theta= pi/2, phi = -3pi/4?
Should the help text inform the student about the relationship between the two points?
If the `perfect' examination is the oral examination with (decent) examiners, then an even-handed reproducible equivalent is Computer Assisted Testing, in which certain errors can be foreseen and "handled".
High Stakes Testing and the wrong debate
The following is lifted from a listserv
More than 30 years ago, the eminent social scientist Donald T. Campbell warned about the perils of measuring effectiveness via a
single, highly consequential indicator: ``The more any quantitative social indicator is used for social decision making,'' he said, ``the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor.'' High-stakes testing is exactly the kind of process Campbell worried about, since important judgments about student, teacher, and
school effectiveness often are based on a single test score. This exaggerated reliance on scores for making judgments creates conditions that promote corruption and distortion. In fact, the over valuation of this single indicator of school success often compromises the validity of the test scores themselves. Thus, the
scores we end up praising and condemning in the press and our legislatures are actually untrustworthy, perhaps even worthless.
Jerry Becker, "The Math Forum@Drexel", 2007
Eliminating the tests, rather than improving them, seems to be the answer "educators" espouse in this regard, much to my regret. The key is Computer Adapted Testing and Computer Assisted Testing, not abandoning testing for portfolios, video tapings, or whatever other silliness the Schools of Education invent.
Postscript to teaching Physical Chemistry after a haitus
I was allowed to teach P. Chem. P. Chem. in the summer recently, after a long hiatus (reason secret).
For me it was an eye opener. Consider that when I took the course, in 1956, 50+ years ago, the course was taught with closed book, closed notes examinations.
Admitedly, the quantum mechanics was less developed than now, in fact I actually don't think there was any in my course, although the newer Prutton and Maron had chapters on it.
Anyway, we learned to carry out derivations, call it memorize if you will, such that we could, from some starting assumptions, derive the equation that we needed to solve a problem. As a result, now, as I teach this material, I need no notes, no preparation, nothing but intellect and the path, i.e., what am I going to "teach".
I develop the equations, one after another, in sequence, one from the other, making a coherent statement of where we were and where we're going.
I point out to students the approximations made, so that one could extend the results to higher accuracy by undoing the approximations and making newer, less restrictive ones, and continuing on, maintaining tractability if possible.
My current students have to have open book, open notes examinations, and as I've done for years I have them do web problems, and base my exams on those questions. Usually, I clone n-1 questions from the web, and add one more they've never seen before, but I've learned my lesson, and now just clone questions!
Be that as it may, class time is taken up with doing every single problem out, so that exams become exercises in transcribing from their notes on these problems to the exam booklet. We've reached the ultimate in pandering!
And my students are incapable of carrying out derivations on their own! Worse, during my derivations, it is clear that my students know no calculus what so ever, i.e., they've retained nothing. We are bringing up a generation of incompetents (who will get A's and B's).
I've lost the desire to teach mainstream P. Chem. Of course every generation believes that their students aren't up to "snuff" whatever that expression means. But this is a serious deviation from past practice, where the hierarchical nature of the learning is becoming derailed. It implies that the future will be little more than extension of the present, with no "ah hah!!!!" moments, since our students are being trained rather than being educated. Heaven help us. (June 26, 2007)
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