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These files are maintained strictly for historical reference, research, and recordkeeping purposes, and the content has not been altered or updated since archiving. Content on this site may not meet current digital accessibility standards. If you require an alternative, accessible format of any information found on this site, please contact Bruce Jones, bjones@ucsd.edu to request an accommodation.
I am not against what we might call this Scientific Theology altogether. Following it helps keep many bright and well-intentioned people striving to expand our narratives about many aspects of life when, absent the belief in the possibility of a master narrative, they might have given up long before making some useful discovery/ invention. But we ought not hold such an ideology uncritically. Niels Bohr, a founder of quantum physics, did not accept the idea that a single, coherent story could tell us all we need to know. He was no postmodernist, but believed that the nature of our interactions with the universe (or the universe-as-known/knowable-to-us) were such that we needed multiple viewpoints which, while complementary, could not be unified in a single logic. In fact their incommensurability with one another is just why having more than one gives us, in total but not through unity, a more comprehensive account. My experience of life leads me to agree.
Pedagogically, unified treatments generally require moving to much more abstract representations, as in the case of the Dirac synthesis of Schrodinger's wave view of quantum theory and Heisenberg's matrix view (both of which in turn were syntheses of the more intuitive earlier wave and particle models). In mathematics you can have Peano's axiomatic treatment of the integers, or the even more abstract formulations of the Bourbaki group. But I don't think I'd start primary school kids off with Peano or Bourbaki. They represent endpoints of long quests to achieve more coherent narratives covering more cases more "convincingly" (another cultural issue of values!). They build castles on top of castles on top of castles. Kids need to first build a castle they can live with, then discover its weaknesses, and then, if they wish, re-build. It is unwise to fall prey to the, equally ideological, belief that the most abstract account is always the best or truest just because it fits together the most pieces. We don't usually need something that comprehensive. In practical terms we need much less formal ways of proceeding, so long as they work. We almost never need the whole grand abstract theory to actually do anything, except satisfy an intellectual itch that is largely grounded in this cultural ideology.
It upsets many people to identify that ideology as patriarchal, or authoritarian, or bourgeois (headwork over handwork), or elitist, or even as a direct successor to christian theology (shaped by also having been a competitor to it). I personally happen to love abstract theorizing. But I hope I understand its limits, its drawbacks, and even its not always so morally defensible place in the historical culture and material order of things we inherit. It needs to be pursued judiciously and critically, and with the deeper recognition that it's dreams are impossible and sometimes dangerously misleading. Trust in limited, specific, local, concrete practice. And play with grand synthetic abstract formal theory as you would with fire.
JAY. Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke On May 2, 2009, at 12:10 AM, David H Kirshner wrote:
Foo Keong,I am fully supportive of efforts to create a master narrative that organizes all of the semantic representations of integers into a coherent scheme. This can be the basis for a curriculum that conveys something of the systematicity and intellectual rigor of mathematics. Unfortunately, I've not yet seen that done for integer operations. Of course, process/object reification--a la Sfard, Schwartz, Tall, Harel, etc.--is a wonderful resource toward that effort. But the argument needs to be framed in the particular, not the general. Although processes cohere into objects which later participate in higher level processes, these reifications follow specific trajectories. A classic example is the reification of an expression as a sequence of instructions (e.g., 3x + 2 meaning take a number, multiply it by 3, and add 2) into an expression as the result obtained through that process. As Sfard noted, only in mathematics does the recipe become the cake. So for the case at hand, it would be necessary to argue that the process of negation as take-away compresses into the object of negative as location (for example on a number line). Even then, the scope of the negation remains non-symmetric. The negative in 3 × -2 applies to the "2". The negative in -2 × 3 applies to the 2 x 3 (in the representation I introduced earlier that we're now discussing).But I think we're on the same team. David -----Original Message-----From: xmca-bounces@weber.ucsd.edu [mailto:xmca- bounces@weber.ucsd.edu] On Behalf Of Ng Foo KeongSent: Friday, May 01, 2009 2:46 PM To: eXtended Mind, Culture, Activity Subject: Re: [xmca] a minus times a plus So the negative sign in -2 × 3 is being interpreted as a /process/ whereas the negative sign in 3 × -2 is being interpreted as an /end-product/ (i.e. after taking away the blue chips from the zero-pairs, you get 2 red chips; -2 = 0 - 2). as an advanced learner i don't feel that these are different, because (using Anna Sfard's theory) i have /reified/ the process, compressed it as it were until i can treat it like an object without any problems. for a beginner, there is still a very wide gulf between the process and the end-product. is there another way out? is it the representation that is the problem, or should educators put more focus on the learner's learning experiences? F.K. 2009/5/1 David H Kirshner <dkirsh@lsu.edu>:Foo Keong,Yes, you can increase the semantic span of this approach by changing the media, as you suggest. But the basic semantic limitation still applies. The negative sign in -2 × 3 is being interpreted as a subtraction [-2 × 3 = 0 - (2 x 3)]--very different from the negative sign in 3 × (-2). Thus the lack of a commutative interpretation of multiplication in this representation is not completely solved by arraying markers in a rectangular configuration.David_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca
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