Path: utzoo!attcan!uunet!lll-winken!ames!mailrus!tut.cis.ohio-state.edu!cs.utexas.edu!rutgers!ucsd!orion.cf.uci.edu!uci-ics!venera.isi.edu!smoliar From: smoliar@vaxa.isi.edu (Stephen Smoliar) Newsgroups: comp.ai Subject: Re: Question on Chinese Room Argument Keywords: Understanding, Comprehension, Learning Message-ID: <7653@venera.isi.edu> Date: 28 Feb 89 18:06:38 GMT References: <4298@pt.cs.cmu.edu> <45126@linus.UUCP> <125@arcturus.edsdrd.eds.com> Sender: news@venera.isi.edu Reply-To: smoliar@vaxa.isi.edu.UUCP (Stephen Smoliar) Organization: USC-Information Sciences Institute Lines: 85 In article <125@arcturus.edsdrd.eds.com> gss@edsdrd.eds.com (Gary Schiltz) writes: > >After I started college as an undergraduate in the mid 1970's, I >took my first calculus course. Coming from a small high school in >a small town, my math skills were minimal (a year or so of algebra), >so the whole course was very confusing. In all the time I was in >the course, I never did understand what calculus was all about. >However, I did know, for example, that a derivative was "the equation >you get when you manipulate another equation in such and such a way" >and an integral was "the equation you get when you manipulate the >equation in another way." > >I even had a fair amount of heuristic knowledge about how to solve >word problems. "Hmm, that problem [on the exam] looks like the one >we did in class. Let's see, first you take the derivative >of this and plug in these numbers and solve for this variable, and >then you circle the answer (and even if the answer is wrong, at least >I can get partial credit for showing my work, and if everyone else is >as confused as I am and they don't score well and the exam is graded >on a curve, maybe I can pass)." I seemed to be able to do fairly >good mapping of one problem to another based on its surface structure. > >Well, I did pass the course (now I'm ashamed that I didn't do what >was necessary to understand what was going on, but like a lot of 17 >year olds, I just took the easiest way). I later repeated the course >and understood what I was doing (and made a lot better grade). > >Anyway, from my gut level feeling (quite possibly useless, I admit) >about what understanding is all about, I really feel I had no >understranding of calculus during that semester. Just as the Brazilian >students didn't realize that symbols in physics equations actually >referred to things in the outside world, I didn't know that the >calculus was modelling anything. I truly had no idea that derivatives >had anything to do with rate of change, for example. But, from the >outside, it must have appeared that I had at least some understanding >of calculus; at least I was good enough at manipulating equations to >make the instructors think so. > I find this a very interesting anecdote because it may tell us some interesting things about both introspection and understanding. There is a school of thought which interests me very much and which Marvin Minsky discusses at some length in THE SOCIETY OF MIND which says that when we are trying to solve a problem, we look for a similar problem which we know how to solve and "complete the analogy," so to speak. This seems to be what Gary was doing in his calculus course, and I suspect he is not alone. Indeed, much of my freshman education seemed to be a matter of exposure to problems and their solutions, endowing me with a repertoire I could consult when I had to solve new problems. The first point I wish to make is that neither "looking for a similar problem" nor "completing the analogy" may be as easy to DO as they are to SAY. I think the source of Gary's embarrassment stems from the fact that his similarity metrics were based on what he called "surface structure;" and, indeed, I have encountered some anecdotes from tutoring scenarios which seem to be based on a student dealing with a surface structure "in the wrong way." Now what does that last phrase mean? I suspect what it means is that, to draw an analogy with language processing, we all have some ability to "parse" the "surface structure" of an example of a problem and its solution. However, some of us seem to have the ability to parse it better than others, at least to the extent that we can use the parse tree as a model for solving future problems. Perhaps this metaphor for parsing is what bridges the gap between what we might call "eidetic recall of a solved problem" and what we would call "understanding the solution to a problem." This brings me to my second point. At his "gut level" Gary felt, introspectively, that he really did not understand calculus. Now I know plenty of mathematicians who would claim that you cannot possibly understand calculus until you have been exposed to real analysis. (I had an analysis professor who liked to call his course "advanced calculus done right.") However, let me assume that Gary is an engineer, rather than a mathematician, so that his criterion of understanding has less to do with appreciating the "true" mathematics which underlies all that symbol manipulation and more to do know knowing how to manipulate the symbols in the circumstances of some pragmatic engineering problem. Having let the introspection cat out of the bag, I, for one, would like Gary to attempt to probe further as to just WHY, at that gut level, he felt understanding was eluding him. Did it have to do with problems he could not solve? Did his eyes glaze over whenever he saw integral signs in the pages of a book? Did he just feel that we was struggling more than his fellow students to solve problems? Perhaps if we probe these matters deeper, we may yet return to my initial point: that Gary's "gut level feeling" may leave something to be desired as a criterion for understanding. (One last question to Gary: Can you identify a moment at which you said, "NOW I understand calculus;" and can you recall the circumstances of that moment.)