Apparently, for example, the 3-4-5 method of creating right triangles was a common technique among farmers for dividing up plots of land for offspring in Ancient Egypt. Math in terms of chronotopes, so to speak.
It isn't a new idea to teach the history of math, of course, or mention history in math classes, but the social history of math usually isn't organically combined with learning math itself. (Have there been attempts to try this? Perhaps in the old USSR? Some experimental programs in the '70's? Some maverick teachers here and there?) In my case, this reading and learning project was mostly the history of just Western mathematics, being what was mainly accessible in bookstores in English.
Learning about the history of math is of course no substitute for learning math skills with paper and pencil, and other habits required for doing math competently - or learning how to use it in applications, like solving physics or business problems. Just like playing a musical instrument or getting good in a sport, math takes a lot of practice. I was good in math in high school and took some college calculus, so I already had a feel for it. But I found thinking about math and topics like calculus in this historical manner extremely helpful in a whole new way. It took a lot of the mystery, abstractness and strangeness out of it, and gave me a more concrete sense of human numerical and geometrical activity in history. It gave me a sense of where doing and talking about math as we teach and practice it today really comes from. And it gave me a new appreciation for the aids we have to help us do math today (I am still in awe of spreadsheet programs and what they can do). Of course, already being a historical materialist with a well-developed sense of history helped me enormously in this little project. Perhaps that is a prerequisite.
As for efficiency ... well, this historical approach would probably be even worse than the three you describe, Jay! I doubt this method of teaching math and calculus could help schools boost their test ratings, lower costs, etc. Done well, it could most certainly generate a lot of discussion, which would only add to its expense. (And undoubtedly some students would hate this approach, so there is always that, too.) And even worse, some might consider trying to use the historical method to teach algebra, geometry, trig, calculus downright subversive ...
- Steve On Aug 14, 2010, at 9:27 PM, Jay Lemke wrote:
I think that part of the argument that activity-based learning reduces college "attention spans" comes from the fact that after getting used to learning in meaningful activity contexts pre- college, students are then shoved back into the mother of all "direct instruction", i.e. the college lecture, and the boredom of it all is overwhelming and they are looking around for something else to do (surf the web, do email, chat, etc.). They have been "spoiled" for the utter idiocy of most lecture teaching. I imagine that many of them find their way around this eventually and make it into smaller seminar-style classes which are at least tolerable.When I was at the U of Chicago as an undergrad (long long ago), all undergrad classes were in small discussion seminar sections with faculty (not grad students) and there might have been an occasional once-a-week lecture on some "extra" topic by a notable speaker. This was of course a VERY expensive mode of education, which someone paid a lot of money for me to enjoy. Some exceptions were made for introductory science classes, as the science faculty claimed to be just TOO busy to staff small sections, and besides there was nothing "discussable" in introductory science. Or so they said. I was a physics major and found it really hard to keep paying attention to equations scrolling down a blackboard, animated by chalk and talk. I am sure that my "attention span" for this had been adversely affected by the classes in which we had animated discussions of all my other subjects, and no chalkboards.As to Calculus, there are at least three ways to learn it. By ingestion (and indigestion) of direct instruction, which enables you to perform the operations without understanding why you're doing it or why they work. By embedding it in abstract mathematical reasoning (cf. Anna Sfard), which enables you to think with and about calculus, but only as a matter of pure abstraction totally divorced from any application or use (or "transfer") to any "applied" (i.e. non-mathematics) domain. And by learning it as a tool to think with in the analysis of situations and problems in some concrete domain (like physics), which enables you to use it in that domain, imagine how it might be used in other such domains, but probably not be able to conceptualize it as "pure" mathematics or connect it to more abstract mathematical systems.I know because I learned it by the first two methods, and taught it by the third one. In my experience the first method produces the least results for the fewest students (but "passable" results for many), the second is totally impenetrable except for a vanishingly small minority who happen to have a very rare "habitus", and the third works pretty well for a high percentage of students from quite varied backgrounds, but (a) is rejected by math departments as not mathematical and (b) cannot compete in terms of "test results" per semester with the first method, largely because the tests are designed to show off masses of superficial "knowledge" and not knowledge-for-use. Method 3 can catch up with and surpass method #1, but it takes more time and more, expensive "discussion".JAY. Jay Lemke Research Scientist Laboratory for Comparative Human Cognition University of California - San Diego 9500 Gilman Drive La Jolla, California 92093-0506 Adjunct Professor School of Education University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke On Aug 14, 2010, at 4:01 PM, mike cole wrote:Thanks, David.If anyone is interested I can send draft of paper for APA. It is similar tomy AERA address (but less interesting-- damned print!).Sure, crappy instruction can come from "we pretend to teach they pretend to learn" regimes. The examples I gave all have pretty good evidence in their favor and in many cases detailed differentiation of what gets cut out as acoherent program enters the sausage grinder.While I am certainly willing to believe that people get into Universities having acquired levels of learning that are very low ( I deal with transfer students from California colleges, and direct admitees into UCSD who cannot handle, for example, a book as complicated as *1984*, I do not believe that it is a plausible account of the average Canadian university's enteringclasses.Apropos, however, of your point. Recent news reports concerning unemployment indicate that there are a couple of hundred pretty well paid jobs going begging in the US right now because there is a dearth of people who can handle the work tasks. Not a new story -- one which puts many industries inthe business of paying new employees to learn a lot before they start working. I'll read your paper with interest. mikeOn Sat, Aug 14, 2010 at 3:44 PM, David H Kirshner <email@example.com> wrote:Mike, Thanks for sharing that anecdote. Unfortunately there is a mirror image in reform teaching to thedysfunctional portrait you presented of direct instruction of procedures disconnected from meaning-making: engagement in activity with no visionon the teacher's part of what or how learning is to be supported. Ithink there is good evidence that the Math Wars in the US initiated notfrom ideological resistance (that came later), but from true horrorstories of kids in dysfunctional reform classrooms, some of them getting to college unprepared as learners (getting into college is not always a sign of a successful K-12 learning experience). As a community, I don'tthink we've done a good job of articulating what it is that makes activity-based learning environments effective. This was the topic of my AERA paper in May, "The Incoherence of Contemporary Pedagogical Reform," which I attach in case anyone isinterested. (The meat of the paper starts about half way through at thesection titled "Theoretical Analysis.") David -----Original Message-----From: firstname.lastname@example.org [mailto:email@example.com ]On Behalf Of mike cole Sent: Saturday, August 14, 2010 2:39 PM To: eXtended Mind, Culture,Activity Subject: [xmca] The Grip of "Direct Instruction"Yesterday I presented a longish paper at the American Psych Associationmeetings here in San Diego. A lot of it was about what here I would refer to as "activity-based"curriculum projects -- their virtues, problems, and apparent inabilitytogain traction against recitation scrips and direct instruction. A majorgeneral finding was that when implemented as designers intend, such programwork, but they tend quickly to be undermined by teachers who stronglybelieve that direct instruction on elements not under control of a meaningful whole is THE only way to be effective. A person from Canada posed a question after prefacing her remarks by sayingshe agreed with all I said, and thank you, etc. She began by saying thatin Canada such approaches had gained a lot of traction in k-12 education, but they were causing a problem at theuniversity level. She phrased the problem roughly as follows: "We get alotof students who are great at collaborative learning, but it appears tostrip them of their attention spans. And, doesn't a subject like calculus REQUIRE direct instruction?" These comments/questions knocked me over. I have long disliked the discourseof short attention span in school kids, which appears to masquerade fartoooften as a proxy for "the kids will not sit still and control themselves doing stuff they do not understand and do not understand why they shouldtry to understand."But I never expected that the the charge of "reduced attention spans"wouldbe attributed to college students (who have succeeded in getting in to college, after all) with the causal factor inducing this "deficit" beingthat their former (successful) modes of learning engendered byactivity-centered instruction). Moreover, I was surprised that anyonebelieves that calculus can be taught by "direct instruction" with no effortmade to subordinate procedural knowledge to knowledge of the potentialmotives for learning.I think I was experiencing exactly the challenges confronting the manyreally interesting and successful innovators in education (we might starthere with Dewey, but I have in mind modern scholars) who want to makeeducation a meaningful process to students but who find that their efforts are rapidly deconstructed once they leave the home ground. Anyone else have observations of this kind? mike Two things struck me _______________________________________________ xmca mailing list firstname.lastname@example.org http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list email@example.com http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list firstname.lastname@example.org http://dss.ucsd.edu/mailman/listinfo/xmca
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