Robert Hansen (RH) posted Jun 29, 2014 12:00 AM (http://mathforum.org/kb/message.jspa?messageID=9505693) - GSC' remarks interspersed: > > On Jun 27, 2014, at 10:28 PM, GS Chandy > <email@example.com> wrote: > > > (GSC): I am in full and total agreement with the idea > > that it is indeed 'in the Buddha-nature of things' > > that some learners would like one thing, and that > > other learners would like other things. No hassle or > > disagreement there at all! The 'the Buddha-nature of > > things' must always be preserved. > > > > The underlying issue is only: How to overcome this > > general 'fear and/or loathing' that the great > > majority of learners feel for math, which is surely > > as important reading and writing in the modern world? > > > (RH): Are you talking about arithmetic and fractions or > algebra and calculus? > i am talking about the fact that, in the 'math teaching system' that's in place today, the great majority of students on graduating from school 'fear and loathe' math (or "math lessons", as you'd have it) and have no desire whatsoever to do the intellectually sometimes quite arduous and demanding work of gaining adequate proficiency in math to be able to handle issues in their daily lives thereafter. If the 'math education system' cannot deliver that much, then the system has failed (IMHO).
I really am not too worried about whether it's arithmetic or geometry or algebra, about whether the student is 'mathy' or otherwise.
An effective 'math education system' should be able to ensure that the majority of its students do not come out of school 'fearing and loathing' math (or "math lessons").
As stated earlier, I am NOT a teacher at any level and I do not plan to be one (except for showing people how to apply the 'One Page Management System' (OPMS) to issues of concern to them. The concern could be "To ensure that my students do not leave school 'fearing and loathing' math". Doing that, I claim they would most probably also be in a position to ensure that those students acquire at least 'adequate' proficiency in arithmetic, geometry, algebra, or whatever is to be included in that "3rd R". > > I stated earlier that we can and should expect every > student to be successful with arithmetic and > fractions. It is the math equivalent of reading and > writing. > Fair enough, and your perception here is most probably correct (to an extent) - if it accords with the ideas of other proficient teachers of math at the levels we are discussing. > > But later, during adolescence, wide > variations in interest appear. All students like > mathematics to some degree and for some period of > their development. > I believe you must be correct. If so, the fact that 'most students fear and/or loathe math by the time they pass out from school' is clear evidence, to my mind, that the 'math education system' has failed (and this failure is a whole lot more serious then the failures of the students in math). . And I do believe that we should at least work on overcoming that 'fear and loathing'. All else that's needed for math education would probably follow in a very natural way. > > least get them through arithmetic and fractions. I am > pretty certain I can. But I don?t think we can > continue much more past that, into algebra, geometry, > trig and calculus. > > When I say that I can get (almost) any child through > arithmetic, including fractions, I mean that I can > teach it without reliance on mathematical reasoning. > I am not saying that I will not throw reasoning into > the lessons, but when that fails many students, and > it will, I still know how to teach arithmetic as a > consistent body of rules that just about any student > can master and consistently answer problems correctly > and gain the necessary confidence with the ability. > They will even get good at it. They might not like > mathematical reasoning but they will like their > ability with arithmetic. And this ability, even > though not reasoned, is quite useful, like reading. > I have long been struck by your emphasis on 'algebra' as [possibly] representing the 'initiation', as it were, of 'mathematical thinking' by a learner. This is, I believe, often a characteristic error made by those who 'came to math' via engineering. (I myself am one such, and for quite some time I had believed that 'mathematical thinking' really started with algebra).
As a matter of fact, I believe there is much sound and tested scientific evidence for the view that true 'mathematical thinking' is actually seen in/ demonstrated by a learner when he/she studies geometry, or patterns perceived in nature, or symmetries seen in many of the objects (natural and humanmade) around him/ her, or even when he/she as an infant studies his/her fingers with some profound questions in the mind (about which science as of today knows very little indeed!) I believe - but am unable to provide clear-cut evidence - that 'mathematical thinking' may well be demonstrated by human children almost from the time they are born, when they open their eyes, and so on.
This is an area that may well be worth exploring, but I lack the needed background in human psychology and cognition to justify such ideas - and it would probably take me far away from my prime interest, OPMS.
I'm afraid I don't quite 'get' most of your argument in your above-quoted paragraph. I observe that this is NOT on account of my not having studied ('American') English poetry - I do assure you that I'm pretty well able to follow the subtleties of most argument put forth in English (any variant of it) - except when it's argued ineffectively. In any case, I believe much of whatever argument is contained within that would be for teachers to agree with or argue against. (This is notwithstanding Wayne Bishop's dislike of the term "stakeholders"). > > As I have said before, this world is as full of > stories with numbers as it is with stories with words > and you need arithmetic to read the stories with > numbers. > Again, I'm not qualified to arrive at any expert judgment about the above. > > Is what I said the same as teaching *rote*? > I don't know, and I'm unable to tell. To find out, it would probably be necessary to test out some of students who may have learned from you. > > Not > exactly. Arithmetic is simple enough and consistent > enough that just about any student can master it > without mastering the theory behind it. In fact, all > of us who went to school before they stopped teaching > arithmetic mastered it that way and got to the theory > later. Arithmetic is coherent just through its basic > consistency without any formal thought. Add in > spreadsheets and the student will be doing as well as > 90% of the people currently using any form of > mathematics in their lives. > > Why this won?t work with algebra and later is that > algebraic problem solving requires algebraic > reasoning. You cannot close the chapter on algebra > with just a few consistent operations or identities. > Without that reasoning there is no algebra. Sure, I > can tutor a student so that they pass a particular > algebra test, but that is only because the test is > finite and I know how to deliver the finite set of > rules for that finite test such that the student > retains enough of them to pass it. But that isn?t > algebra and the student and I are on the same page > that we are ONLY trying to pass ONE exam, not learn > algebra. The student wants the pain to be over and I > will make the pain be over. And this is not an > engineering student, it is a student majoring in > social work or nursing, who shouldn?t be required to > pass algebra in the first place, although I am fine > with the exposure. And this only works because the > exam itself is fake. It can be passed without > algebraic reasoning. The studen! > t just doesn?t have enough test smarts to know this. > And then there are cases where the student is almost > getting it and wants to get it and that is a another > type of tutoring. In that case you are teaching it > for real. Khan academy (self) attendees are usually > those types. > > It takes a particular kine of *like* to get through > algebra. When a student asks ?What is this for?? or > ?What are we going to use this for?? they are saying > that they don?t like it. > On the other hand, that very student may well be seeking to to *do something* with what he/she may have learned. Such a question may well be evidence of a 'question-asking frame of mind', which I consider to be the sine qua non for 'learning' and 'problem solving'. > >A student that likes a > subject doesn?t care what it is for. > See above. I on the other hand would tend to believe that a student 'who likes a thing' may well be most interested to *do* something with what he/she has learned. i believe this is an unsubstantiated generalisation of yours. > >I think that is > the only version of *like* that can get a student > through algebra and leave them *owning* it in the > end. I don?t think it is possible for us to teach > that kind of *like*. All we can do is to give the > student honest exposure to the subject in an > environment (without anxiety) and see if they like > it. > Well, ... I can only state that I do not understand your claims above - and I do assure you that that lack of understanding is NOT connected with my lack of exposure to ('American') English poetry. > > One method I proposed awhile back is to give two > grades for algebra 1. One grade would represent > whether you at least tried while the other grade > would represent how successful you were (exams) and > whether you are ready for algebra 2. The one that > counts for graduation is the one that shows that you > tried while the one that counts on your transcript > for subsequent courses is the exam based grade. This > would expose all students ! > to REAL algebra while alleviating the moral hazard in > teaching and grading. Teachers, including me, do not > like to fail students that try. This way, we are not > failing them, and, while they can?t take algebra 2 > yet, they can, if they desire, take algebra 1 again > and if the do well enough, then take algebra 2. Or > they can go on and find what they really like. > Broadly, I *think* I'd agree with your ideas about the transition from Algebra 1 to Algebra 2, and so on, but these are pretty huge areas of 'human cognition'. > This would? > > 1. Expose all students to real algebra, because you > can?t like algebra if the exposure is fake. > 2. Not require them to like it, which removes a lot > of the anxiety. > 3. Not overwhelm the system with a bunch of students > that don?t like it. > 4. Allow fewer and better mathematics teachers to > focus on and tutor those that like it or think they > like it. > 5. Allow all students the opportunity to find what > they like. > I can agree with many of your ideas above - but i believe most of them would require a considerable amount of further exploration before we can make such definitive statements.
(Sent off in a rush, as I have to conduct a quick OPMS workshop for a visitor who is leaving town tonight, therefore, E&OE).