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*To*: ablunden@mira.net, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Mike Cole <lchcmike@gmail.com>*Date*: Wed, 8 Jul 2009 20:57:05 -0700*Cc*:*Delivered-to*: xmca@weber.ucsd.edu*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:reply-to:in-reply-to :references:date:message-id:subject:from:to:content-type; bh=KIG0cV4jCcrUesRgs6AGt5t9UwtBS2WiWnqXszSoiGE=; b=nGM7vIn4jsCjP5DwpDYJa+QgIXIg9yYEyvYUfstfO3NYhokv1FXQFBXIyQv8s7R+wX jhkYyQdPpiDek5lfPUSadKZPFb04Ox+7GsWuOX2hffGkzpeeWYCEsDOv1JjHqw0tYxk7 gVVfmLUmtRr30c8hloHvPulf4br7o+MkZPoGg=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:reply-to:in-reply-to:references:date:message-id :subject:from:to:content-type; b=TGG1BSHgdkMPfKbtRlKd6BoszUbFd53mcgLogwrI2ZWWFm0wblwt8KkPsorcO70MXt UfffeQvG8/ireSiJHL5hiEuem+KdfHEYbL/fmDuYizLGfet5bGpp62kFvojDytPKGvvg XSxnfu5Hzn5Fh/h9aQGntOuCc7rY2Q1TdzI0U=*In-reply-to*: <4A553223.8020809@mira.net>*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <00dd01c9fefa$6dc53690$8b55d383@w2k3.fi.uu.nl> <2C7F9BCA-5807-42F9-8AB7-CC1F7EE9F9FE@umich.edu> <4A553223.8020809@mira.net>*Reply-to*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

Andy-- i am just back from Alaska and want to talk about several matters, but am falling down tired. Re the problem under discussion. You have now encountered Luria and context-variable learning AND DEVELOPMENT from another angle. I am waaaaay behind the 8 ball but will try to be on skype with you asap. mike On Wed, Jul 8, 2009 at 4:56 PM, Andy Blunden <ablunden@mira.net> wrote: > Thanks for that Jay. You make your points very strongly. But if it is true > that for the learning of mathematics: > > "almost all practices and procedures are highly context and content > specific, with what appears otherwise resulting mostly from certain sorts of > cultural habitus acquired by individuals. Classes of tasks and practices > have historically developed with certain kinds of similarities within > particular cultures. Cultures have preferred strategies or ways-of-doing > that are implicit, and members who have the appropriate trajectories of > cultural experiences tend to develop dispositions fitted to these." > > then this does not negate the value of lerning mathematics, does it? It > merely suggests that there may be other ways of achieving the same end. I am > guessing that participating in formal debate teaches similar dispositions. > Does this come up against the same obstacles? I rather suspect it does, but > of course I have no evidence. > > Isn't acquiring a fitting disposition effectively the same as > context-variable learning? > > Andy > > > Jay Lemke wrote: > >> I'm pleased there has been uptake from Ivan's and my concerns about school >> mathematics, and science. >> >> A few miscellaneous responses -- >> >> I don't doubt that there are many, even semi-technical, occupations where >> mathematical procedures are employed, and even where (though I suspect much >> less often) some judgment is required about how to apply them (thus >> requiring at least some theoretical understanding). I will be interested to >> see what Bakker's research shows about the most prevalent of these -- how >> advanced a level is used so widely that it is efficient to teach it to ALL >> students in schools? I really don't imagine very many people factor >> polynomials or solve quadratic equations, outside of higher level >> specializations. But these are empirical questions, whereas the content of >> the curriculum is NOT based on empirical findings of this sort, but rather >> on traditions of dubious validity. >> >> How to teach the mathematics that is widely used is then a separate >> question, and I think there is growing agreement that more realistic >> contexts are better for gaining wider success. There is still the very >> fundamental issue of whether translating such contexts into school >> activities can work well and generally (which I tend to doubt), or whether >> the learning needs to be taken outside of classrooms, or at least into mixed >> settings that combine experience and experience-based intuitions from >> non-school settings with some reflection and analysis work in classrooms, >> etc. Obviously SOME math can be taught successfully in classrooms, like some >> literacy skills, and some translation strategies are of value. >> >> But I would agree that the abstract approach to math and science, and the >> overstuffed topic curricula in these fields, is there more because (a) we >> know how to segment it and test it, and (b) it's a good way to keep a lot of >> people out of universities and professional jobs, while seeming to be >> completely objective and fair about a rigged system. >> >> Does it benefit the mind in more general ways? I am a splitter ... I don't >> believe much in transfer, generalization, general intelligence, >> multi-purpose higher mental functions, etc. I tend to think that almost all >> practices and procedures are highly context and content -specific, with what >> appears otherwise resulting mostly from certain sorts of cultural habitus >> acquired by individuals. Classes of tasks and practices have historically >> developed with certain kinds of similarities within particular cultures. >> Cultures have preferred strategies or ways-of-doing that are implicit, and >> members who have the appropriate trajectories of cultural experiences tend >> to develop dispositions fitted to these. >> >> In these terms, experiences with mathematics CAN support developing >> dispositions that make mastering other kinds of abstract reasoning practices >> come more easily. Symmetrically, mastering mathematics is easier if you've >> already had success with other implicitly similar kinds of tasks and >> strategies. Learning abstract decontextualized mathematics, however, seems >> to me one of the hardest ways into such a cultural complex of similar >> practices. And any benefit from working at mathematics seems to me to accrue >> only if (a) the work is enjoyable or at least has a supportive relationship >> to a desired identity, and (b) you are successful at it, preferably early >> on. >> >> All this applies to conceptual understanding of sciences equally as well. >> >> JAY. >> >> Jay Lemke >> Professor >> Educational Studies >> University of Michigan >> Ann Arbor, MI 48109 >> www.umich.edu/~jaylemke <http://www.umich.edu/%7Ejaylemke> >> >> >> >> >> On Jul 7, 2009, at 1:59 PM, A.Bakker wrote: >> >> Interesting discussion! Let me dwell on two projects in response to Jay >>> and >>> Andy. >>> >>> 1. what kind of math do we need at work? >>> >>> We have analyzed the mathematical knowledge required in 239 >>> intermediate-level professions (think of service engineering, florist, >>> baker, low level analyst in science labs, builders, car mechanics, salary >>> administration, secretarial work, hairdresser etc). Some of these do not >>> have to do any calculations at all (butcher in a factory just selecting >>> good >>> and bad parts of meat), but the vast majority of professions face simple >>> arithmetic, geometry (area, volume), data handling and risk, and >>> sometimes >>> formulas. Even at the lowest level of education, lab analysts face some >>> high-level statistics (F-test, t-test, correlation etc) in method >>> validation, precision of instruments etc. >>> >>> Although there is some truth in Andy's comments, this analysis gives a >>> more >>> nuanced image. Moreover, there is more than math at work and in daily >>> life: >>> math required for higher-level education. Vocational students without >>> enough >>> mathematical and scientific baggage have trouble getting through their >>> higher vocational education (nursing, teaching, management etc). however, >>> I >>> should note that our Dutch school system differs drastically from the >>> American one because our vocational education is big (60% of the >>> students) >>> and starts early (pre-vocational education at age 12). >>> >>> 2. basing science units on authentic practices >>> >>> Indeed, many math and science problems at school are not very realistic. >>> It >>> is in fact quite hard to design good ones. Over the past years we have >>> tried >>> to base educational units on authentic practices in which science or math >>> is >>> used (with activity theory in mind as well). We have 'translated' >>> authentic >>> goals to learner goals, adapted ways of working and knowledge required to >>> be >>> manageable to students (grade 10-12). The idea was to use meaningful >>> relationships between goals, tools, knowledge etc in outside-school >>> practice >>> as sources of inspiration for school units. Although we have had some >>> success, there are still many challenges in designing good units - even >>> if >>> we allow the learning goals to be drastically different from the >>> Standards >>> (say: insight into health and nutrition rather than say DNA, evolution, >>> cell >>> biology...). >>> >>> So I agree with Jay that content is a major problem, but even then we >>> have a >>> lot of work to do in terms of designing good alternatives. >>> >>> Chevallard has written interesting papers on didactic transposition, >>> adapting knowledge as used in the 'real world' to school situations. He >>> describes a contradiction that cannot be resolved completely: education >>> promises to prepare kids for their future and for society. At the same >>> time, >>> education cannot really fulfil its promise. What students learn is often >>> something that teachers can easily test. Chevallard argues that the main >>> reason that we still teach math and science is NOT that they are so >>> useful, >>> but they can be rolled out nicely in stages over the school grades and >>> can >>> be tested in objective ways. A lot of things that are very useful to >>> learn >>> do not make it to the curricula, simply because they are so hard to teach >>> and test (medicine, psychology, sociology, social skills etc.) >>> >>> Arthur Bakker >>> >>> -----Original Message----- >>>> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] >>>> On >>>> Behalf Of Andy Blunden >>>> Sent: dinsdag 7 juli 2009 13:30 >>>> To: eXtended Mind, Culture, Activity >>>> Subject: Re: [xmca] a minus times a plus >>>> >>>> Your key claims are beyond challenge Jay; you can get by >>>> perfectly well in all aspects of life without mathematics, >>>> apart from a basic understanding of the notion of quantity >>>> and some elementary arithmetic, except for a very small >>>> group of professions. It has annoyed me, this need to invent >>>> pseudo-problems that seem to demand mathematics, to >>>> "justify" the need to learn maths. It seems to me that it is >>>> requirement to pass maths exams to gain entry to a very wide >>>> range of jobs etc., which is the only real motivation for most. >>>> >>>> But can you tell me, is there no evidence that going through >>>> the process of learning maths in some way benefits the mind? >>>> in the same way that (as I understand it) learning to read >>>> and write has a permanent and effect on how people think? >>>> that mathematics is a kind of mental gymnastics. >>>> >>>> Andy >>>> >>>> Jay Lemke wrote: >>>> >>>>> Thank, Ivan, for responding in part to some of my concerns re teaching >>>>> math-as-math in schools. >>>>> >>>>> It's a big, old debate in education whether we should teach ideas, >>>>> concepts, and disciplines as abstract systems, in the hopes they can >>>>> then be used as tools to think with ... or whether that usually doesn't >>>>> work, puts kids off from the subject, and it's better to let concepts >>>>> appear more naturally in the context of real-world problems, issues, >>>>> activities which are not about math or science, but in which math-using >>>>> and science-using activities and practices can play a helpful part. >>>>> >>>>> The academic, and intellectual answer, as part of a cultural and >>>>> institutional tradition, is that we cheat students out of the power of >>>>> math and science if we don't give them the systems of abstract >>>>> concepts, >>>>> and that other approaches tend to degenerate into second-rate >>>>> practicalism that avoids theory, critique, alternatives, creativity, >>>>> etc. My own view, after a long time participating in, observing, and >>>>> trying to analyze the teaching of science, and to a lesser degree >>>>> mathematics, is that the powerful systematic conceptual tools are a >>>>> very >>>>> advanced stage of membership in one or more very specialized >>>>> communities, and are simply not of much use to most people. >>>>> >>>>> Maybe my view is a bit extreme. But I think it remains true that it is >>>>> not just a failure to find the magic method of teaching that is the >>>>> problem with math-as-math and science-as-science in the curriculum. It >>>>> is the content itself. Or, really, the lack of content, the lack of >>>>> engagement with real life activities that are meaningful and important >>>>> to the students, in the modern math and science curricula. And I do not >>>>> see the solution as inventing clever artificial problems and topics >>>>> that >>>>> seem to be relevant to real-life, but which are in fact just excuses to >>>>> do more math-as-math and science-as-science. >>>>> >>>>> A mathematician or a scientist can find, show you, highlight, apply >>>>> their conceptual tools to nearly anything. Some reasonable level of >>>>> abstract awareness of those tools can emerge from encountering, in some >>>>> detail and depth, several domains and examples or projects in which the >>>>> concepts have been highlighted for their usefulness (and that includes >>>>> usefulness for critical thinking, for imagining alternatives --- not >>>>> just for engineering practical constructions or solutions). But this >>>>> comes at the end of a long learning process, and almost as a kind of >>>>> side-effect, and not at the beginning or as the primary purpose or >>>>> goal-of-activity. >>>>> >>>>> There is math and science in jumping jacks and football, in mountain >>>>> climbing, in raising a pet or growing some food, in figuring the cost >>>>> of >>>>> better garbage collection in the neighborhood, in organizing a block >>>>> party, in understanding when to go to the hospital or what counts as >>>>> evidence in a court case. It might be better to say that there are >>>>> issues of quantity and degree, of probability and risk, of nutrition >>>>> and >>>>> cause and effect in all these domains and phenonena, and that the >>>>> workarounds and tricks and mnemonics and practical methods accumulated >>>>> across them all tend to implicate some more general strategies --- >>>>> which >>>>> we could just tell you, but then the odds are you wouldn't understand >>>>> or >>>>> remember or know how to use them for yourself. >>>>> >>>>> I am not talking here about advanced levels of education, but about >>>>> introductory ones ... up to about the age of 15 or 17, or up to the >>>>> point at which interest and possibility tend to focus students toward >>>>> some preferred specialization. Then the balance shifts, again not all >>>>> the way toward abstract disciplines (as, for example, medical education >>>>> has struggled to sort out for a long time now), but a bit more toward >>>>> the justification of more emphasis on theoretical learning, as part of >>>>> membership in a specialist community of knowers/doers. >>>>> >>>>> What are the practical situations in which you need to multiply a minus >>>>> times a plus? not textbook imaginaries, but for real? If you had some >>>>> broad and in-depth knowledge about such a situation, would it then be >>>>> so >>>>> hard to make sense of how signed numbers multiply there? And how far a >>>>> step is it, and how necessary a step for all to take, from an induction >>>>> from several such well-understood situations to the pure >>>>> mathematicians' >>>>> abstract arguments about how signed numbers multiply everywhere, or >>>>> really, nowhere?? >>>>> >>>>> JAY. >>>>> >>>>> >>>>> >>>>> Jay Lemke >>>>> Professor >>>>> Educational Studies >>>>> University of Michigan >>>>> Ann Arbor, MI 48109 >>>>> www.umich.edu/~jaylemke <http://www.umich.edu/%7Ejaylemke> >>>>> >>>>> >>>>> >>>>> >>>>> On Jun 30, 2009, at 6:50 PM, Ivan Rosero wrote: >>>>> >>>>> Here's a familiar exhortation: >>>>>> "We need as many engineers as possible. As there is a lack of them, >>>>>> invite >>>>>> to this study, persons of about 18 years, who have already studied the >>>>>> necessary sciences. Relieve the parents of taxes and grant the >>>>>> scholars >>>>>> sufficient means." >>>>>> >>>>>> According to my brief cyber-sphere search, these are the words of >>>>>> >>>>> Emperor >>>> >>>>> Constantine. >>>>>> >>>>>> So, anyway, we all know what road that empire took. I doubt it was >>>>>> lack of >>>>>> engineers though :) So, given the very similar verbiage spilling out >>>>>> of NSF >>>>>> these days, I agree with Jay, perhaps slowing down and taking a minute >>>>>> >>>>> or >>>> >>>>> two to rethink this wouldn't be bad at all. >>>>>> >>>>>> If I read you correctly Jay, one big worry you have is that we don't >>>>>> end up >>>>>> reifying mathematics (in the sense Constantine seems to be doing with >>>>>> engineering) in the frustration we experience with our almost complete >>>>>> failure in teaching it. >>>>>> >>>>>> It reminds me of mountain-climbing. For me at least, this is one hell >>>>>> of a >>>>>> difficult sport, and the few times I've ever participated, it has been >>>>>> >>>>> a >>>> >>>>> real big struggle to get to the top. And we're talking Mt. Washington, >>>>>> >>>>> a >>>> >>>>> measly ~6000 ft peek. Anyway, I struggle, sweat, almost pass-out, and >>>>>> finally I'm there. It is AWESOME, the joy is overwhelming. 20 >>>>>> minutes >>>>>> later, as my muscles cool down and my adrenaline levels-off, I stare >>>>>> >>>>> down >>>> >>>>> the thing and feel a creeping dread, even if the way down is many times >>>>>> easier than the way up. >>>>>> >>>>>> This story can go in many directions from here, as many as there are >>>>>> people >>>>>> who have made it (oh, God, this is cheesy) mountain-top. They are not >>>>>> universally happy stories however. >>>>>> >>>>>> I DO think it is useful to know some mathematics and have a host of >>>>>> scientific concepts to think with and through at our disposal. None >>>>>> of this >>>>>> is Bad (or Good for that matter) in and of itself. The Purpose, of >>>>>> course, >>>>>> is what is at issue. >>>>>> >>>>>> ZPDs are value agnostic. Mike and his team at LCHC are currently >>>>>> attempting >>>>>> to create ZPDs that can instill basic arithmetic in kids whose daily >>>>>> >>>>> (and >>>> >>>>> arguably far stronger) ZPDs pull them in many other (sometimes directly >>>>>> opposite) directions. Some of those ZPDs, however, are not in direct >>>>>> conflict with math. That is my hunch, or assumption. The task, then, >>>>>> >>>>> is >>>> >>>>> perhaps a bit simpler than creating new ones. >>>>>> >>>>>> Is it simpler to find and then piggy-back on, ZPDs that contain >>>>>> kernels of >>>>>> arithmetic in them? Susan Goldin-Meadow has pretty convincing >>>>>> evidence that >>>>>> specific motor activity can not only presage basic arithmetic, but can >>>>>> even >>>>>> aid in its acquisition. So, might not Jay's concern (if I read him >>>>>> right) >>>>>> that mathematics (and the whole lot of techno-science) becomes >>>>>> surreptitiously reified in our frustrated attempts to teach it be >>>>>> addressed >>>>>> from a different direction? >>>>>> >>>>>> Jumping-jacks anyone? >>>>>> >>>>>> Ivan >>>>>> >>>>>> On Sat, Jun 27, 2009 at 11:00 PM, Andy Blunden <ablunden@mira.net> >>>>>> >>>>> wrote: >>>> >>>>> >>>>>> I hope people won't mind if I continue to pick the brains of this >>>>>>> list on >>>>>>> the problem of my niece's progress in maths, or lack of it. >>>>>>> >>>>>>> It seems that the suggestion last time - that Marissa may have missed >>>>>>> important lessons while on holiday - may explain her poor performance >>>>>>> last >>>>>>> year in maths, even though maths has always been her weak subject. >>>>>>> She has >>>>>>> caught up a bit but she is still badly behind. >>>>>>> >>>>>>> It seems that the issue Mike has raised also applies: she is getting >>>>>>> homework that seem to presume she know things that in fact she >>>>>>> doesn't. The >>>>>>> only other negative in her school reports is that she doesn't >>>>>>> participate in >>>>>>> class discussion or ask questions when she doesn't understand >>>>>>> >>>>>> something. >>>> >>>>> >>>>>>> I presume the hesitancy about speaking up is probably the cause of >>>>>>> failure >>>>>>> to correct her maths problems and the teachers giving her homework >>>>>>> she >>>>>>> doesn't understand. >>>>>>> >>>>>>> She is now 15 and her maths homework is also beyond her father! :) >>>>>>> and the >>>>>>> crisis of the transition from childhood to adulthood around this age, >>>>>>> makes >>>>>>> it impossible for the father to get Marissa talk about it to him, or >>>>>>> engage >>>>>>> Marissa in games of 20 Questions or something to lead her to the joys >>>>>>> >>>>>> of >>>> >>>>> asking others. Discussion over the dinner table is apparently also >>>>>>> unconducive to her participation. >>>>>>> >>>>>>> Does anyone have any ideas? I've run out of suggestions. I could >>>>>>> probably >>>>>>> help if I was there, but I'm 1000 km away. >>>>>>> >>>>>>> Andy >>>>>>> >>>>>>> Mike Cole wrote: >>>>>>> >>>>>>> SO glad you are interested in this, Jay. >>>>>>>> I have just made contact with Karen Fuson who has, lucky for us, >>>>>>>> "retired" >>>>>>>> from Northwestern and moved nearby. She is away for a week or so >>>>>>>> but then we are getting together. This is a problem that just may be >>>>>>>> tractable, theoretically interesting for sure, attractive of >>>>>>>> >>>>>>> experience >>>> >>>>> collaborators, >>>>>>>> and god knows, of practrical importance to lots of kids. >>>>>>>> >>>>>>>> mike >>>>>>>> >>>>>>>> On Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <jaylemke@umich.edu> >>>>>>>> wrote: >>>>>>>> >>>>>>>> Yes, Mike and F.K., these are very disturbing issues. Both that what >>>>>>>> >>>>>>> we >>>> >>>>> think we want to teach seems to depend on deeper (e.g. 4000-year >>>>>>>>> >>>>>>>> deep) >>>> >>>>> knowledge than it's realistic to expect most people to learn (or >>>>>>>>> want to >>>>>>>>> learn), and that how we teach even the most practical bits of >>>>>>>>> mathematics >>>>>>>>> (like 15 minus 8) seems to have gone so wrong that it's hard to >>>>>>>>> know >>>>>>>>> where >>>>>>>>> to start, especially for those we have most systematically failed. >>>>>>>>> We do indeed need to not give up. But we also need, I think, to >>>>>>>>> >>>>>>>> admit >>>> >>>>> that >>>>>>>>> it's time to seriously re-think the whole of the what, why, and how >>>>>>>>> >>>>>>>> of >>>> >>>>> education. Math is a nice place to focus because at least some of it >>>>>>>>> seems >>>>>>>>> universally agreed to be useful by almost everyone, because >>>>>>>>> professional >>>>>>>>> mathematicians and most people, including teachers and mathematics >>>>>>>>> educators, seem to hold radically different views about what the >>>>>>>>> subject >>>>>>>>> is, >>>>>>>>> and because success in teaching it, measured in almost any way, is >>>>>>>>> pretty >>>>>>>>> near the bottom of the heap. >>>>>>>>> >>>>>>>>> Yes, we can find somewhat better ways to teach the same stuff, but >>>>>>>>> maybe >>>>>>>>> it's the stuff itself (the content of the curriculum, viewed not >>>>>>>>> just as >>>>>>>>> information, but as activity) that needs to be rethought? along >>>>>>>>> with the >>>>>>>>> ethics and efficacy of who decides. >>>>>>>>> >>>>>>>>> No matter how many times you multiply a minus by any number of >>>>>>>>> >>>>>>>> pluses, >>>> >>>>> you >>>>>>>>> still get a minus. >>>>>>>>> >>>>>>>>> JAY. >>>>>>>>> >>>>>>>>> Jay Lemke >>>>>>>>> Professor >>>>>>>>> Educational Studies >>>>>>>>> University of Michigan >>>>>>>>> Ann Arbor, MI 48109 >>>>>>>>> www.umich.edu/~jaylemke <http://www.umich.edu/%7Ejaylemke> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Jun 6, 2009, at 6:12 PM, Mike Cole wrote: >>>>>>>>> >>>>>>>>> Hi Foo Keong-- It is so generous of you to even try to explain! And >>>>>>>>> your >>>>>>>>> question re math seems to me >>>>>>>>> relevant to other areas of knowledge as well when you ask, "Can we >>>>>>>>> condensefour thousand years of >>>>>>>>> human development into an easily digestible four minutes for >>>>>>>>> learners." >>>>>>>>> >>>>>>>>> Could we consider four years, just for whole numbers? Davydov >>>>>>>>> starts with >>>>>>>>> Algebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay and >>>>>>>>> others >>>>>>>>> believe his method of introducing youngesters to math has some >>>>>>>>> extra >>>>>>>>> power. >>>>>>>>> As I understand it, others on xmca are dubious and look to other >>>>>>>>> sources >>>>>>>>> of >>>>>>>>> difficulty. Karen Fuson, in her article on "developing mathematical >>>>>>>>> power >>>>>>>>> ins whole number operations" focuses on introducing number >>>>>>>>> >>>>>>>> operations >>>> >>>>> through very simple, familiar, imaginable, >>>>>>>>> events where exchange is involved. >>>>>>>>> >>>>>>>>> Its odd to me experiencing the cycle of time, the "coming back to >>>>>>>>> >>>>>>>> the >>>> >>>>> beginning and recognizing it >>>>>>>>> for the first time" that is happening for me right now with >>>>>>>>> >>>>>>>> arithmetic >>>> >>>>> and >>>>>>>>> early algebra. The source >>>>>>>>> is quite practical with social significance: the unbridgable gap >>>>>>>>> the >>>>>>>>> children I work with face between >>>>>>>>> what their teachers are teaching about (say) subtraction (2005-118 >>>>>>>>> is my >>>>>>>>> current keystone example) >>>>>>>>> trying to get their kids to learn that the first step is to >>>>>>>>> subtract >>>>>>>>> >>>>>>>> 8 >>>> >>>>> from >>>>>>>>> 15 and know enough to treat the >>>>>>>>> second zero as a 9. But the child, even understanding that the task >>>>>>>>> the >>>>>>>>> teacher is focused on is >>>>>>>>> disabled because when asked 15-8 the answer =3 and only painstaking >>>>>>>>> attention to the problem set up with fingers and subtracting one by >>>>>>>>> one, >>>>>>>>> with full compliance and even eagerness by the child, brings >>>>>>>>> her to 7. >>>>>>>>> >>>>>>>>> Now suppose this phenomenon is ubiquitous, affects 100's of >>>>>>>>> thousands of >>>>>>>>> children, and is heavily correlated with social class. >>>>>>>>> >>>>>>>>> Then .... ??? .... >>>>>>>>> I think my frustration is probably equivalent to yourse in >>>>>>>>> intensity, but >>>>>>>>> the quality is of a somewhat different nature. >>>>>>>>> mike >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Sat, Jun 6, 2009 at 3:11 AM, Ng Foo Keong <lefouque@gmail.com> >>>>>>>>> wrote: >>>>>>>>> >>>>>>>>> I was trained in mathematics at the University of Cambridge (UK) >>>>>>>>> >>>>>>>>> for my undergraduate studies, concentrating more on pure >>>>>>>>> >>>>>>>>> mathematics (including algebra). I am able to roll out a >>>>>>>>> >>>>>>>>> rigorous abstract proof of why "minus times minus" is a "plus", >>>>>>>>> >>>>>>>>> using only the basic axioms of real numbers (actually you only >>>>>>>>> >>>>>>>>> need a few of those axioms). >>>>>>>>> >>>>>>>>> >>>>>>>>> However, abstract proofs aren't likely to be useful for non-math >>>>>>>>> >>>>>>>>> specialists and struggling neophyte learners of algebra. in >>>>>>>>> >>>>>>>>> order to pull off such a proof, or even just to understand just >>>>>>>>> >>>>>>>>> the few lines of proof, you almost need to be a mental masochist. >>>>>>>>> >>>>>>>>> Who likes to go through mental torture? >>>>>>>>> >>>>>>>>> >>>>>>>>> Can we condense four thousand years of human development of >>>>>>>>> >>>>>>>>> mathematical understanding into an easily digestible four minutes >>>>>>>>> >>>>>>>>> for learners? >>>>>>>>> >>>>>>>>> >>>>>>>>> thus the huge gulf of understanding still persists. that's why >>>>>>>>> >>>>>>>>> as an educator, i feel so useless being unable to help other >>>>>>>>> >>>>>>>>> people. :-( >>>>>>>>> >>>>>>>>> >>>>>>>>> F.K. >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> 2009/6/4 Mike Cole <lchcmike@gmail.com>: >>>>>>>>> >>>>>>>>> I am currently reading article by Fuson suggestion by Anna Sfard on >>>>>>>>> whole >>>>>>>>> >>>>>>>>> number operations. I also need to study Anna's paper with exactly >>>>>>>>> >>>>>>>> this >>>> >>>>> >>>>>>>>> example in it. Not sure what moment of despair at deeper >>>>>>>>> >>>>>>>> understanding >>>> >>>>> >>>>>>>>> hit >>>>>>>>> >>>>>>>>> me. Now that I am done teaching and have a whole day to communicate >>>>>>>>> >>>>>>>>> things >>>>>>>>> >>>>>>>>> are looking up!! Apologies for doubting I could have deep >>>>>>>>> understanding >>>>>>>>> >>>>>>>>> of >>>>>>>>> >>>>>>>>> why minus x minus = plus and minus x plus = minus. At present my >>>>>>>>> >>>>>>>>> understanding remains somewhat bifurcated. The former is negation >>>>>>>>> of >>>>>>>>> >>>>>>>> a >>>> >>>>> >>>>>>>>> negation as david kel long ago suggested, linking his suggestion to >>>>>>>>> >>>>>>>>> Anna's >>>>>>>>> >>>>>>>>> comognition >>>>>>>>> >>>>>>>>> approach. The second I think more of in terms of number line and >>>>>>>>> >>>>>>>>> multiplication as repeated addition. >>>>>>>>> >>>>>>>>> Perhaps the two will coalesce under your combined tutelage. >>>>>>>>> >>>>>>>>> mike >>>>>>>>> >>>>>>>>> >>>>>>>>> And member book links are coming in. Nice. >>>>>>>>> >>>>>>>>> mike >>>>>>>>> >>>>>>>>> >>>>>>>>> _______________________________________________ >>>>>>>>> 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 >>>>>>>> >>>>>>>> >>>>>>>> -- >>>>>>> ---------------------------------------------------------------------- >>>>>>> >>>>>>> >>>>>> -- >>>> >>>>> Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/ >>>>>>> Orders: http://www.erythrospress.com/store/main.html#books >>>>>>> >>>>>>> _______________________________________________ >>>>>>> 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 >>>>>> >>>>>> >>>>>> >>>>> _______________________________________________ >>>>> xmca mailing list >>>>> xmca@weber.ucsd.edu >>>>> http://dss.ucsd.edu/mailman/listinfo/xmca >>>>> >>>>> >>>> -- >>>> ------------------------------------------------------------------------ >>>> Andy Blunden (Erythrós Press and Media) >>>> http://www.erythrospress.com/ >>>> Orders: http://www.erythrospress.com/store/main.html#books >>>> >>>> _______________________________________________ >>>> 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 >>> >>> >>> >> _______________________________________________ >> xmca mailing list >> xmca@weber.ucsd.edu >> http://dss.ucsd.edu/mailman/listinfo/xmca >> >> > -- > ------------------------------------------------------------------------ > > Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/ > Orders: http://www.erythrospress.com/store/main.html#books > > _______________________________________________ > 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

**Follow-Ups**:**RE: [xmca] a minus times a plus***From:*"Esther Goody" <eg100@hermes.cam.ac.uk>

**References**:**RE: [xmca] a minus times a plus***From:*"A.Bakker" <A.Bakker@fi.uu.nl>

**Re: [xmca] a minus times a plus***From:*Jay Lemke <jaylemke@umich.edu>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

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