tag:blogger.com,1999:blog-16984363.post113054728735499140..comments2015-08-20T11:40:50.122-07:00Comments on Siva's Glob of Thoughts: Long division? Ugh!D. Sivakumarhttp://www.blogger.com/profile/05750992965116762335noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-16984363.post-41042956658257647502009-03-20T04:44:00.000-07:002009-03-20T04:44:00.000-07:00Well, it would be a little better if we weren't go...Well, it would be a little better if we weren't going on people's gut feelings here, and there were actually some research as to which elementary math education is the most effective in preparing students for mathematical reasoning later on.<BR/><BR/>Perhaps long division - which isn't that difficult, and is the first kind of inverse problem students usually run into - is bad training for learning number sense, and perhaps it's good. Perhaps number sense can be instilled with calculator use, and perhaps a calculator acts as a crutch so that it's even more difficult to teach the proper lessons. Perhaps the best lesson from long division is not <BR/>number sense, but how to neatly organize a problem. Perhaps it's like practicing scales in music, where one is learning agility - but here it's mental agility as opposed to finger agility. Perhaps it's a waste of time. I'd rather see some hard data right here www.forumvadisi.comAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-16984363.post-50042695206390694952009-03-20T04:43:00.000-07:002009-03-20T04:43:00.000-07:00Well, it would be a little better if we weren't go...Well, it would be a little better if we weren't going on people's gut feelings here, and there were actually some research as to which elementary math education is the most effective in preparing students for mathematical reasoning later on.<BR/><BR/>Perhaps long division - which isn't that difficult, and is the first kind of inverse problem students usually run into - is bad training for learning number sense, and perhaps it's good. Perhaps number sense can be instilled with calculator use, and perhaps a calculator acts as a crutch so that it's even more difficult to teach the proper lessons. Perhaps the best lesson from long division is not <BR/>number sense, but how to neatly organize a problem. Perhaps it's like practicing scales in music, where one is learning agility - but here it's mental agility as opposed to finger agility. Perhaps it's a waste of time. I'd rather see some hard data right here www.forumvadisi.comAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1171971049748367462007-02-20T03:30:00.000-08:002007-02-20T03:30:00.000-08:00I'm afraid you don't know what you're talking abou...I'm afraid you don't know what you're talking about. Please read this http://www.shearonforschools.com/why_long_division.htmHanshttps://www.blogger.com/profile/14931678321077968154noreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1132356925126671032005-11-18T15:35:00.000-08:002005-11-18T15:35:00.000-08:00By the same token, decimal long division gets a 1 ...By the same token, decimal long division gets a 1 digit under-approximation to the right answer very quickly. These days, estimation is a much bigger component of things than it used to be.<BR/><BR/>Maybe the real reason we teach PPA is to prepare students for the notion of an algorithm/working on an assembly line/etc. Seeing a variety of algorithms: addition, multiplication, long division does give a better gut-level understanding of the notion of executing an algorithm. Maybe we can replace long division with Sachertorte recipes and get the same effect...<BR/><BR/>On a related note, when I was in school we also learned a pencil and paper algorithm for computing square roots. (I remember it only vaguely but it was similar to long division.) If long division is so important why don't we teach square root algorithms, too?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130755045255492252005-10-31T02:37:00.002-08:002005-10-31T02:37:00.002-08:00This comment has been removed by a blog administrator.Luchttps://www.blogger.com/profile/14516675619999076244noreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130755059161268802005-10-31T02:37:00.001-08:002005-10-31T02:37:00.001-08:00This comment has been removed by a blog administrator.Luchttps://www.blogger.com/profile/14516675619999076244noreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130755025786979512005-10-31T02:37:00.000-08:002005-10-31T02:37:00.000-08:00This method looks like the same as yours, but much...This method looks like the same as yours, but much more simpler: http://www.doubledivision.org/Luchttps://www.blogger.com/profile/14516675619999076244noreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130707081934914022005-10-30T13:18:00.000-08:002005-10-30T13:18:00.000-08:00Claire wrote: One motivation for long division is ...Claire wrote: <I>One motivation for long division is that it leads one to notice phenomena such as periodicity of rational numbers and understand why that occurs.</I><BR/><BR/>Excellent point. The question is, which subset of the student population (and at what level) need to learn this? One could argue that the notion of successive approximation is much more basic than the long division algorithm, whose correctness is likely less obvious to a pre-highschooler. In other words, should we treat long division a little bit like faster (than n^2 time) algorithms for multiplication, that is, as "advanced material"? Perhaps it is an excellent algorithm to teach in introductory discrete mathematics (like <A HREF="http://mathworld.wolfram.com/KaratsubaMultiplication.html" REL="nofollow"><BR/>Karatsuba's multiplication algorithm</A>). Perhaps it would be a good idea to teach such "simple" algorithms for multiplication, division, sorting, string matching, etc., in high school, emphasizing their proof of correctness.D. Sivakumarhttps://www.blogger.com/profile/04674739621423058308noreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130701283583007582005-10-30T11:41:00.000-08:002005-10-30T11:41:00.000-08:00Well, my son already understands of course how to ...Well, my son already understands of course how to do divisions by successive approximations. When he is free to do it in his own way, he does it a little differently from the usual algorithm, though, approximating either above and below, whichever appears more convenient: for example, to divide 179 by 6, he would first approximate by 30, then divide 1 by 6 and substract the result from 30.<BR/><BR/>One motivation for long division is that it leads one to notice phenomena such as periodicity of rational numbers and understand why that occurs.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130579276040805062005-10-29T02:47:00.000-07:002005-10-29T02:47:00.000-07:00Well, it would be a little better if we weren't go...Well, it would be a little better if we weren't going on people's gut feelings here, and there were actually some research as to which elementary math education is the most effective in preparing students for mathematical reasoning later on.<BR/><BR/>Perhaps long division - which isn't that difficult, and is the first kind of inverse problem students usually run into - is bad training for learning number sense, and perhaps it's good. Perhaps number sense can be instilled with calculator use, and perhaps a calculator acts as a crutch so that it's even more difficult to teach the proper lessons. Perhaps the best lesson from long division is not <BR/>number sense, but how to neatly organize a problem. Perhaps it's like practicing scales in music, where one is learning agility - but here it's mental agility as opposed to finger agility. Perhaps it's a waste of time. I'd rather see some hard data.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130572606628879262005-10-29T00:56:00.000-07:002005-10-29T00:56:00.000-07:00The more important point of it was to realize that...The more important point of it was to realize that division was distributing something as evenly as possible into some number of portions, and that it could be done by successive approximations -- a way to reason about it so that the correctness of the algorithm is "obvious" to a 6-yr old. As opposed to following a mechanical recipe which is what happens in much of early education.D. Sivakumarhttps://www.blogger.com/profile/04674739621423058308noreply@blogger.comtag:blogger.com,1999:blog-16984363.post-1130569373560582862005-10-29T00:02:00.000-07:002005-10-29T00:02:00.000-07:00"A few years ago, I taught my then-six-year-old to..."A few years ago, I taught my then-six-year-old to do division, essentially by binary search."<BR/><BR/>You realize that what you were teaching him was in fact long division, just in binary?<BR/>And doing it with powers of 10 is just long division in base 10, with an inefficient method for finding the next digit in the quotient.Anonymousnoreply@blogger.com