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 On Jul 7, 2009, at 1:59 PM, A.Bakker wrote:
Interesting discussion! Let me dwell on two projects in response to Jay andAndy. 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 somehigh-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 enoughmathematical and scientific baggage have trouble getting through theirhigher vocational education (nursing, teaching, management etc). however, Ishould note that our Dutch school system differs drastically from theAmerican 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 practicesIndeed, 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 bemanageable to students (grade 10-12). The idea was to use meaningfulrelationships between goals, tools, knowledge etc in outside-school practiceas sources of inspiration for school units. Although we have had somesuccess, 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, cellbiology...).So I agree with Jay that content is a major problem, but even then we have alot 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 teachand test (medicine, psychology, sociology, social skills etc.) Arthur Bakker-----Original Message-----From: email@example.com [mailto:xmca- firstname.lastname@example.org] 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 re teachingmath-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 canthen 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 conceptsappear more naturally in the context of real-world problems, issues,activities which are not about math or science, but in which math- 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 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 degreemathematics, is that the powerful systematic conceptual tools are a veryadvanced 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 isnot just a failure to find the magic method of teaching that is theproblem with math-as-math and science-as-science in the curriculum. Itis the content itself. Or, really, the lack of content, the lack ofengagement 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 todo 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 ofabstract 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 includesusefulness 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 mountainclimbing, in raising a pet or growing some food, in figuring the cost ofbetter 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 areissues of quantity and degree, of probability and risk, of nutrition andcause and effect in all these domains and phenonena, and that theworkarounds 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 orremember 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 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 ofmembership 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 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,inviteto this study, persons of about 18 years, who have already studied the necessary sciences. Relieve the parents of taxes and grant the scholarssufficient 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 it was lack ofengineers though :) So, given the very similar verbiage spilling outof NSFthese days, I agree with Jay, perhaps slowing down and taking a minuteortwo to rethink this wouldn't be bad at all.If I read you correctly Jay, one big worry you have is that we don'tend upreifying mathematics (in the sense Constantine seems to be doing with engineering) in the frustration we experience with our almost completefailure in teaching it.It reminds me of mountain-climbing. For me at least, this is one hellof adifficult sport, and the few times I've ever participated, it has beenareal big struggle to get to the top. And we're talking Mt. Washington,ameasly ~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 staredownthe thing and feel a creeping dread, even if the way down is many timeseasier than the way up.This story can go in many directions from here, as many as there arepeoplewho have made it (oh, God, this is cheesy) mountain-top. They are notuniversally happy stories however. I DO think it is useful to know some mathematics and have a host ofscientific concepts to think with and through at our disposal. Noneof 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 attemptingto create ZPDs that can instill basic arithmetic in kids whose daily(andarguably 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,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 thatspecific motor activity can not only presage basic arithmetic, but caneven 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 <email@example.com>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 performancelast 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 gettinghomework 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 cause of failureto correct her maths problems and the teachers giving her homework shedoesn't understand. She is now 15 and her maths homework is also beyond her father! :) and thecrisis of the transition from childhood to adulthood around this age,makesit impossible for the father to get Marissa talk about it to him, orengageMarissa in games of 20 Questions or something to lead her to the joysofasking 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 sobut then we are getting together. This is a problem that just may betractable, theoretically interesting for sure, attractive ofexperiencecollaborators, and god knows, of practrical importance to lots of kids. mikeOn Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <firstname.lastname@example.org> wrote:Yes, Mike and F.K., these are very disturbing issues. Both that whatwethink we want to teach seems to depend on deeper (e.g. 4000-yeardeep)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 knowwhereto start, especially for those we have most systematically failed.We do indeed need to not give up. But we also need, I think, toadmitthatit's time to seriously re-think the whole of the what, why, and howofeducation. Math is a nice place to focus because at least some of itseems universally agreed to be useful by almost everyone, because professionalmathematicians and most people, including teachers and mathematicseducators, seem to hold radically different views about what the subject is,and because success in teaching it, measured in almost any way, ispretty near the bottom of the heap.Yes, we can find somewhat better ways to teach the same stuff, butmaybe 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 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 withAlgebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay andothersbelieve his method of introducing youngesters to math has some extrapower. As I understand it, others on xmca are dubious and look to other sources ofdifficulty. Karen Fuson, in her article on "developing mathematicalpower 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 "coming back 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 unbridgable gap thechildren 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 to subtract8from 15 and know enough to treat thesecond zero as a 9. But the child, even understanding that the taskthe teacher is focused on isdisabled because when asked 15-8 the answer =3 and only painstaking attention to the problem set up with fingers and subtracting one byone, 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 <email@example.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 justthe 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 ofmathematical understanding into an easily digestible four minutesfor 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 <firstname.lastname@example.org>:I am currently reading article by Fuson suggestion by Anna Sfard onwholenumber operations. I also need to study Anna's paper with exactlythisexample in it. Not sure what moment of despair at deeperunderstandinghitme. Now that I am done teaching and have a whole day to communicatethings are looking up!! Apologies for doubting I could have deep understanding of why minus x minus = plus and minus x plus = minus. At present myunderstanding remains somewhat bifurcated. The former is negation ofanegation as david kel long ago suggested, linking his suggestion toAnna'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 email@example.com http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________xmca mailing list firstname.lastname@example.org 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 email@example.com http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ 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-- ------------------------------------------------------------------------ Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/ Orders: http://www.erythrospress.com/store/main.html#books _______________________________________________ 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