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*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Jay Lemke <jaylemke@umich.edu>*Date*: Wed, 8 Jul 2009 17:58:14 +0200*Delivered-to*: xmca@weber.ucsd.edu*In-reply-to*: <BLU125-W157E9DF29A90D7EAE825B9A1290@phx.gbl>*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*: <OF8F505A15.EAB579A6-ON862575EC.0049A0FC-862575EC.0049B87B@spps.org> <BLU125-W157E9DF29A90D7EAE825B9A1290@phx.gbl>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

JAY. Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke On Jul 8, 2009, at 5:24 PM, larry smolucha wrote:

Thoughts from Francine Smolucha on teaching math: Learning math occurs within a discourse on "Why do I have to know this?" Like all discourse this is a situated discourse, occuring within a classroom or as a one to one interaction (with either a live person or with a media intermediary such as a book or computer tutorial.) A discourse analysis would reveal these typical lines of discourse: Question: Why do I have to know this?" (1) Because some authority figure said so/ because I said so/ because the school board said so/ because the state mandates it/ because the oppressive colonial/capitalist school system mandates it . . . . (2) Because it is on the test/ the unit test/the final exam/ the state achievement test/the college entrance exam/ the scholarship qualifying exam (3) Because this is what kids your age are expected to know/ in our school/school district/state or in other countries. (4) Because you need to know this in order to get a good job. (Do you really? Will you ever need to know this as an adult?) (5) Because you will use it in everyday life? (Really?????) (6) Because you are exercising your brain/math is mental exercise for your brain just like push-ups build up your muscles/ math is a type of mental gymnastics.(7) Because I learned it, your father/mother learned it/ yourgrandfatherlearned it/that is how our family bettered itself (economically).(8) Because you will receive a token/a cookie/a smiley facesticker/ . . . ..(or because failure will be punished with a poor grade, beinggrounded, etc.)One key factor here is whether the discourse takes place between significant others. If the teacher is not a significant other for the learner, the teacher's words are tuned out. If the learner is not a significant other for the teacher, the teacher is mouthing platitudes. Some of these arguments for learning math turn into slippery slopes. (1) If the state is imposing this on us let us rise up andover throw the oppressive government system. Then we don't have tolearn math.(4) If I need to know math in order to get a job or a good job, howcome peoplewho know a lot of math are out of work (can't find a job/a decentjob)?Usually, we are talking about people who the learner knows personally like Dad or Mom or an Uncle.(5) Not even all elementary school math is used in the everyday lifeof anadult, much less a child.Making the Case for Math is crucial in learning math because itmotivatesthe learner to put continual effort into a challenging course of study (a course of study that does not often provide immediate tangiblerewards.) A formal part of teacher education in math should behelping the futureteacher develop a effective/versatile discourse style for Making theCase for Math.How about Making the Case for Learning History? Making the Case forLearning Grammar?As well as, Making the Case for Learning Science. Making the Case for Learning to Read and Write is another matter.If the cultural system values these academic skills, the teacher'sjob is much simpler.If the cultural system undermines the development of cognitiveability, theteacher has to create a sub-culture, counter-culture, an alternativeculture.In Western cultures, for example, the incessant mindless rhythms ofpop/hip-hop music and videogames, and the hyper arousal cycle associated with them, underminesthe brain'sability to concentrate and be able to think.To: xmca@weber.ucsd.edu Subject: RE: [xmca] a minus times a plus From: ERIC.RAMBERG@spps.org Date: Tue, 7 Jul 2009 08:25:18 -0500 Those interested in this discussion please take the time to read what Sylvia Scribner's research has to offer. http://lchc.ucsd.edu/Histarch/jaap84v6n1-2.PDF Hope people find this as helpful as I have. eric "A.Bakker" <A.Bakker@fi.uu.nl> Sent by: xmca-bounces@weber.ucsd.edu 07/07/2009 06:59 AM Please respond to "eXtended Mind, Culture, Activity"To: <ablunden@mira.net>, "'eXtended Mind,Culture,Activity'"<xmca@weber.ucsd.edu> cc: Subject: RE: [xmca] a minus times a plusInteresting discussion! Let me dwell on two projects in response toJayand Andy. 1. what kind of math do we need at work? We have analyzed the mathematical knowledge required in 239intermediate-level professions (think of service engineering,florist,baker, low level analyst in science labs, builders, car mechanics,salaryadministration, secretarial work, hairdresser etc). Some of thesedo nothave to do any calculations at all (butcher in a factory justselectinggoodand bad parts of meat), but the vast majority of professions facesimplearithmetic, geometry (area, volume), data handling and risk, andsometimesformulas. Even at the lowest level of education, lab analysts facesomehigh-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 analysisgives amore nuanced image. Moreover, there is more than math at work and in daily life: math required for higher-level education. Vocational students without enoughmathematical and scientific baggage have trouble getting throughtheirhigher vocational education (nursing, teaching, management etc).however,I should note that our Dutch school system differs drastically from theAmerican one because our vocational education is big (60% of thestudents)and starts early (pre-vocational education at age 12). 2. basing science units on authentic practicesIndeed, many math and science problems at school are not veryrealistic.Itis in fact quite hard to design good ones. Over the past years wehavetriedto base educational units on authentic practices in which scienceor mathis used (with activity theory in mind as well). We have 'translated' authenticgoals to learner goals, adapted ways of working and knowledgerequired tobe 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 somesuccess, there are still many challenges in designing good units -even ifwe allow the learning goals to be drastically different from theStandards(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 thenwe havea 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 schoolsituations. Hedescribes a contradiction that cannot be resolved completely:educationpromises to prepare kids for their future and for society. At thesametime,education cannot really fulfil its promise. What students learn isoftensomething that teachers can easily test. Chevallard argues that themainreason 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 gradesand canbe tested in objective ways. A lot of things that are very usefulto learndo not make it to the curricula, simply because they are so hard toteachand test (medicine, psychology, sociology, social skills etc.) Arthur Bakker-----Original Message-----From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu]OnBehalf 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 reteachingmath-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 theycanthen be used as tools to think with ... or whether that usuallydoesn'twork, puts kids off from the subject, and it's better to letconceptsappear more naturally in the context of real-world problems,issues,activities which are not about math or science, but in whichmath-usingand science-using activities and practices can play a helpful part. The academic, and intellectual answer, as part of a cultural andinstitutional tradition, is that we cheat students out of thepower ofmath and science if we don't give them the systems of abstractconcepts,and that other approaches tend to degenerate into second-ratepracticalism that avoids theory, critique, alternatives,creativity,etc. My own view, after a long time participating in, observing,andtrying to analyze the teaching of science, and to a lesser degree mathematics, is that the powerful systematic conceptual tools are averyadvanced 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 thatit isnot just a failure to find the magic method of teaching that is theproblem with math-as-math and science-as-science in thecurriculum. Itis the content itself. Or, really, the lack of content, the lack ofengagement with real life activities that are meaningful andimportantto the students, in the modern math and science curricula. And I donotsee the solution as inventing clever artificial problems and topicsthatseem to be relevant to real-life, but which are in fact justexcusestodo 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, insomedetail and depth, several domains and examples or projects in whichtheconcepts have been highlighted for their usefulness (and thatincludesusefulness for critical thinking, for imagining alternatives ---notjust for engineering practical constructions or solutions). Butthiscomes at the end of a long learning process, and almost as a kindofside-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, inmountainclimbing, in raising a pet or growing some food, in figuring thecostofbetter garbage collection in the neighborhood, in organizing ablockparty, in understanding when to go to the hospital or what countsasevidence in a court case. It might be better to say that there areissues of quantity and degree, of probability and risk, ofnutritionandcause and effect in all these domains and phenonena, and that theworkarounds and tricks and mnemonics and practical methodsaccumulatedacross them all tend to implicate some more general strategies ---whichwe could just tell you, but then the odds are you wouldn'tunderstandorremember 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 thepoint at which interest and possibility tend to focus studentstowardsome preferred specialization. Then the balance shifts, again notallthe way toward abstract disciplines (as, for example, medicaleducationhas struggled to sort out for a long time now), but a bit moretowardthe justification of more emphasis on theoretical learning, aspart ofmembership in a specialist community of knowers/doers. What are the practical situations in which you need to multiply aminustimes a plus? not textbook imaginaries, but for real? If you hadsomebroad and in-depth knowledge about such a situation, would itthen besohard to make sense of how signed numbers multiply there? And howfar astep is it, and how necessary a step for all to take, from aninductionfrom several such well-understood situations to the puremathematicians'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 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 ofthem,invite to this study, persons of about 18 years, who have already studiedthenecessary sciences. Relieve the parents of taxes and grant thescholarssufficient means." According to my brief cyber-sphere search, these are the words ofEmperorConstantine.So, anyway, we all know what road that empire took. I doubt itwaslack ofengineers though :) So, given the very similar verbiagespilling outof NSF these days, I agree with Jay, perhaps slowing down and taking aminuteortwo to rethink this wouldn't be bad at all.If I read you correctly Jay, one big worry you have is that wedon'tend upreifying mathematics (in the sense Constantine seems to be doingwithengineering) in the frustration we experience with our almostcompletefailure in teaching it. It reminds me of mountain-climbing. For me at least, this is onehellof a difficult sport, and the few times I've ever participated, it hasbeenareal big struggle to get to the top. And we're talking Mt.Washington,ameasly ~6000 ft peek. Anyway, I struggle, sweat, almost pass-out,andfinally I'm there. It is AWESOME, the joy is overwhelming. 20minuteslater, as my muscles cool down and my adrenaline levels-off, Istaredownthe thing and feel a creeping dread, even if the way down is manytimeseasier than the way up.This story can go in many directions from here, as many as therearepeople who have made it (oh, God, this is cheesy) mountain-top. They arenotuniversally happy stories however.I DO think it is useful to know some mathematics and have a hostofscientific concepts to think with and through at our disposal.Noneof thisis Bad (or Good for that matter) in and of itself. The Purpose,ofcourse, is what is at issue. ZPDs are value agnostic. Mike and his team at LCHC are currently attemptingto create ZPDs that can instill basic arithmetic in kids whosedaily(andarguably far stronger) ZPDs pull them in many other (sometimesdirectlyopposite) directions. Some of those ZPDs, however, are not indirectconflict with math. That is my hunch, or assumption. The task,then,isperhaps 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, butcanevenaid in its acquisition. So, might not Jay's concern (if I readhimright) 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 havemissedimportant lessons while on holiday - may explain her poorperformancelastyear in maths, even though maths has always been her weaksubject..She has caught up a bit but she is still badly behind.It seems that the issue Mike has raised also applies: she isgettinghomework 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 understandsomething.I presume the hesitancy about speaking up is probably the causeoffailureto correct her maths problems and the teachers giving herhomeworkshedoesn't understand.She is now 15 and her maths homework is also beyond herfather! :)and the crisis of the transition from childhood to adulthood around thisage,makesit impossible for the father to get Marissa talk about it tohim, orengage Marissa in games of 20 Questions or something to lead her to thejoysofasking others. Discussion over the dinner table is apparentlyalsounconducive 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 sobut then we are getting together. This is a problem that justmaybetractable, theoretically interesting for sure, attractive ofexperiencecollaborators, 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 thatwhatwethink we want to teach seems to depend on deeper (e.g. 4000-yeardeep)knowledge than it's realistic to expect most people to learn(orwant 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 toknowwhereto start, especially for those we have most systematicallyfailed.We do indeed need to not give up. But we also need, I think, toadmitthat it's time to seriously re-think the whole of the what, why, andhowofeducation. Math is a nice place to focus because at leastsome ofitseems universally agreed to be useful by almost everyone, because professionalmathematicians and most people, including teachers andmathematicseducators, seem to hold radically different views about whatthesubject is,and because success in teaching it, measured in almost anyway, ispretty near the bottom of the heap.Yes, we can find somewhat better ways to teach the samestuff, butmaybeit's the stuff itself (the content of the curriculum, viewednotjust 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 ofpluses,you still get a minus. JAY. Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke 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!Andyour question re math seems to merelevant to other areas of knowledge as well when you ask,"Can wecondensefour 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 Moxhayandothers believe his method of introducing youngesters to math has someextrapower.As I understand it, others on xmca are dubious and look toothersources of difficulty. Karen Fuson, in her article on "developingmathematicalpower ins whole number operations" focuses on introducing numberoperationsthrough very simple, familiar, imaginable, events where exchange is involved.Its odd to me experiencing the cycle of time, the "comingback tothebeginning and recognizing it for the first time" that is happening for me right now witharithmeticand early algebra. The sourceis quite practical with social significance: the unbridgablegapthechildren I work with face betweenwhat their teachers are teaching about (say) subtraction(2005-118is my current keystone example) trying to get their kids to learn that the first step is tosubtract8from 15 and know enough to treat the second zero as a 9. But the child, even understanding that thetaskthe teacher is focused on is disabled because when asked 15-8 the answer =3 and onlypainstakingattention to the problem set up with fingers and subtractingonebyone, 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. mikeOn 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-mathspecialists and struggling neophyte learners of algebra. in order to pull off such a proof, or even just to understand justthe few lines of proof, you almost need to be a mentalmasochist.Who likes to go through mental torture? Can we condense four thousand years of human development ofmathematical understanding into an easily digestible fourminutesfor 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 AnnaSfardonwholenumber operations. I also need to study Anna's paper withexactlythisexample in it. Not sure what moment of despair at deeperunderstandinghit me. Now that I am done teaching and have a whole day tocommunicatethings are looking up!! Apologies for doubting I could have deep understanding ofwhy minus x minus = plus and minus x plus = minus. At presentmyunderstanding remains somewhat bifurcated. The former isnegationofanegation as david kel long ago suggested, linking hissuggestiontoAnna's comognitionapproach. The second I think more of in terms of number lineandmultiplication 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_________________________________________________________________ Windows Live™ SkyDrive™: Get 25 GB of free online storage. http://windowslive.com/online/skydrive?ocid=TXT_TAGLM_WL_SD_25GB_062009_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

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**References**:**RE: [xmca] a minus times a plus***From:*ERIC.RAMBERG@spps.org

**RE: [xmca] a minus times a plus***From:*larry smolucha <lsmolucha@hotmail.com>

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