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Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials? In short, we know and should agree that the long-division "gazinta'' (goes into, as in four "goes into'' 31 seven times ... ) algorithm and its computational cousins are obsolete in light of everyday societal realities.He claims that the requirement to be able to do math on pencil and paper has been rendered meaningless by the calculator:
Today, real people in real situations regularly put finger to button and make critical decisions about which buttons to press, not where and how to carry threes into hundreds columns. We understand that this change is on the order of magnitude of the outhouse to indoor plumbing in terms of comfort and convenience, and of the sundial to digital timepieces in terms of accuracy and accessibility.And so, in spite of Leinwand's accusation (in the same paper) that school districts make changes only in geological time, we are currently engaged in a huge cultural experiment testing his theory that kids can gain a knowledge of math without having to put pencil to paper (although, as I mentioned in this post, in some of the constructivist curricula kids are spending much more time learning to multiply than they would have in a classical curriculum). But putting pencil to paper is part of what I would call the craft of mathematics. I think you just don't get intimate enough with numbers and symbols by just watching them flash by on the computer or calculator. I've seen it over and over in students at the college level; the more they've relied on their calculator, the less of a feel they have for numbers and mathematics, and the less able they are at problem solving. They may feel that they understand you while you lecture, but when it comes to actually doing math, to getting the answers themselves, they can't do it; they're impotent. I may be misreading the situation. It may be that the same kids who have trouble with problem-solving at the college level had trouble learning the standard algorithms for computation, and therefore rely more heavily on their calculators. But we do have parallels in the employment world -- experienced engineers who find that junior employees rely too heavily on the answers given by their computer models, and designers who find that their juniors who have used CAD software have a weakened sense of design. We don't know whether the new tools are enabling people to enter these fields who wouldn't otherwise have cut the mustard, or whether the tools are actually weakening the skills of able people. Until we understand why kids who have relied too heavily on calculators for basic computation can't do math, and what is truly essential about the process of teaching kids to do math, we would be wise to continue making huge curricular changes in geological time. Afternote: in the spirit of knowing your opposition, here is a link to the Leinwand paper. You have to register at the Edweek site to access the article.
swoop and swoop
notes on integer, subtraction, & absolute value study sheet
Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory
KUMON & hands-on math
More Singapore math
Pencil and paper
The craft of math
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Steve Leinwand was part of the group that took part in AIR's study of Singapore Math. Some of you have talked about the study on these pages. The study essentially comes out very strongly in favor of Singapore Math (SM) and takes some serious shots at Everyday Math (EM). Yet, infused throughout the report is the window dressing of political spin that any such report is bound to have. Statements that say the U.S is doing things right, that the NCTM standards are good, etc. These statements don't amount to much, given the main thrust of the report, and it seems apparent that maybe Leinwand's role in the process was to get such statements in there. I know a mathematician (on our side of the fight) who decided to meet with Leinwand and found him engaging and friendly, and readily said good things about SM. But he also said good things about EM, so he's a rather strange duck. My assessment is that as political winds change, people go with the flow. It's called "follow the money". People who are involved in piloting SM, when speaking about it publicly in mixed crowds which contain adherents of NCTM's philosophy, will characterize SM and how it is taught as "constructivist". Similarly, NCTM can watch videotapes of how math is taught in Japanese classrooms and say their methods embody the philosophy embodied by NCTM's standards (they don't). In the end, if they come out in favor of the right program, let them save face. If NSF suddenly announced a new grant program for math texts that are "content-based" and emphasize standard algorithms and follow the Californai standards, you would see a switch of philosophy overnight, just like in 1984, when one minute Oceania was always at war with Eastasia, and Eurasia was our ally. Then with a change in policy, the revisionists went to work and now Eastasia was an ally and we had always been at war with Eurasia. Money is the Ministry of Truth in this country. -- BarryGarelick - 25 Jun 2005
The jaw-dropping point of that study is exactly what you say--that number 1 would have anything to learn from number 16, except how to be 16th. It doesn't take a math genius to see the ridiculousness of that conclusion. My only wish is that their children be taught with these curriculums, but without their help or intervention. That should be the first test for any of these textbooks. Your kids first. -- SusanS - 25 Jun 2005
Ralph Raimi described his encounter with Steve Leinwand, it's a fun read: http://www.math.rochester.edu/people/faculty/rarm/nctm_meeting.html -- BeckyC - 25 Jun 2005
And who is this person to make this life changing decision for our children? When all the mathematicians and engineers and people in other technical fields are saying that the children must learn the algorithims to do technical work, his decision might lock all American children out of these fields. -- AnneDwyer - 25 Jun 2005
Anne, that's what concerns me most; that kids are getting locked out of fields that might begin to interest them only when they are older. Becky, I read the Ralph Raimi encounter with Leinwand. It was funny. Especially when Raimi suggested he should sell real estate in Florida. I would have taken exception to that myself (but I've lived in Florida so I know exactly what it means!!). The Prentice Hall book I'm currently using is anything but 'gray with problems'. It has pictures and photos and insets and a bazillion different colors of text on every page. It's very distracting to look at, absolutely a disaster if you have trouble focusing. I don't think that's the solution; I don't think Leinwand is even addressing the right problem. -- CarolynJohnston - 25 Jun 2005
I agree with Barry. After the recent AIR report, I didn't know what to think of Leinwand. I now tend to agree that he is a strange duck that stays connected and follows the money. However, no matter what Leinwand says, reading the AIR report will lead you to only one conclusion - which is not a good assessment of Everyday Math. -- SteveH - 26 Jun 2005
" ... experienced engineers who find that junior employees rely too heavily on the answers given by their computer models, and designers who find that their juniors who have used CAD software have a weakened sense of design." I have been dealing with this computer/paper issue for 30 years. Computers can be wonderful tools for learning, but they can also be distracting. Do you really have to do Simpson's Rule by hand to appreciate integration? I think the problem is that schools end up focusing on and wasting time on computers, software, and other technology, rather than the actual subject they are studying. The student might end up being a fantastic computer jockey, but know little about the actual design of a building. (Architects are the professionals; computer jockeys are not.) However, the computer can be a fantastic learning tool when used for parametric studies - how results vary when you change one variable at a time. The university engineering school I went to specifically did not give credit for learning how to use a computer. They assigned the homework, say a structural problem, and you were supposed to learn how to use the Finite Element Modeler on your own time. The focus was on the subject, not the technology. How many writing courses out there spend too much time learning how to use a word processor (all those exciting fonts!) and not enough time learning how to write. Computers aren't bad, they are just very distracting. Having said all of that, I see little need for computers in grades K-8. It's great to have a "special" class where kids learn to use computers - word processing, touch typing, etc., but for all other classes, the focus should be on the subject, not the technology. In math, there is even less of a need for calculators and computers. As I have said elsewhere, calculators and computers should make learning more difficult (expect more), rather than make learning easier (avoidance). -- SteveH - 26 Jun 2005
Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials? Oh my gosh--how did I miss this one??? Unbelievable. -- CatherineJohnson - 26 Jun 2005
I second Barry's comment on the Singapore Math report, which I've read closely. I didn't know who Steve Leinwand was at the time, and I kept being brought up short by the constant references to 'what the United States is doing right' (stem and leaf charts, apparently). -- CatherineJohnson - 26 Jun 2005
MoreSingaporeMath (on Leinwand) -- CatherineJohnson - 26 Jun 2005
An aside about Leinwand. On Joanne Jacob's blog (a wonderful blog for those of you who haven't checked it out), she ran a story about the AIR report and some people commented on Leinwand's involvement. I put my two cents in, and said that his stripes appear to have changed, but maybe there's more money in traditional math now, or words to that effect. I also made comments about EM. I didn't realize it, but Leinwand apparently reads Joanne's blog and he wrote me a to-the-point email that said among other things: "As for money, I continue to find that the money on the side of long-overdue change (including some of the very positive features of the Singapore materials) is still much greater and much more honestly earned than fighting a losing battle to retain a mathematics program better suited to the 19th century. "As for EM, if it was my kid, I'd take EM over Scott or Saxon in a second, fully confident that even in the hands of a mediocre teacher, both kid and teacher would be way ahead. But if you give me the opportunity to conduct the professional development we need and free me from the constraints of poorly designed high-stakes state assessments, bring on the Singapore materials even faster and rest assured I argue for skipping about the same percent of material I find skippable in any other program." I wrote him an apology for my snide remarks, and said I would be glad to agree to disagree with him in a more civil manner. He wrote me a nice note back saying he would welcome that. Whew! -- BarryGarelick - 27 Jun 2005
Yoiks! Don't you hate it when that happens? Leinwand really genuinely feels that what he calls '19th century math' is bad. I think it's people's feelings about their own schooling and math training, and about math itself, that determines where they come down in this debate. We see learning the algorithms as a necessary and temporary step in a kid's education. He sees it as being an impediment to a kid's learning what he really needs to know. He thinks it's the bathwater -- and I think it's part of the baby. -- CarolynJohnston - 27 Jun 2005
Wasn't the 19th century the age of reason? Maybe he's inadvertently plugging a good thing. -- BarryGarelick - 27 Jun 2005
I think it's people's feelings about their own schooling and math training, and about math itself, that determines where they come down in this debate. There's more to it than that, since this debate goes back over one hundred years now. But I don't understand it well. Ed speculates that classic progressive ed philosophy, which has been dominant for over a century, may have merged with postmodernism (semiotics, postructuralism, etc.) He thinks that may be one reason why we are constantly told that math is a language. I feel quite a bit less sympathy for a person like Steve Leinwand than I might when he makes an observation like, 'If it were my kid I'd take Everyday Math over Saxon.' We're not talking about 'his kid,' we're talking about my kid. I consistently see a masked power motive and agression in the language and stance of constructivists. Their language frequently brings Orwell to mind. Classically, they use the language of tolerance and openness to express a highly intolerant stance, e.g. there are 'many different ways of solving problems,' but the Saxon way is not one of them. And their contempt for parents is open. To say, at all times and in all forums, that parents object to constructivism because 'it's not what they're used to from their own childhoods' is, frankly, offensive and insulting. But back to masked power: here in our district, we parents were not given a choice. We were not even told that the program being selected was a constructivist program. Nor was any non-constructivist program evaluated or even examined. The document laying out criteria for selecting a program states that the program must be constructivist. The only question was which one. We are told, consistently, by school board members as well as administrators, that the administration looked at 'many' different curricula, and that parents 'had a chance' to express their views at school board meetings. Well, of course, when parents don't know what fuzzy math is, they haven't had a chance to express their views. I believe that parents should have a choice. Parents who want Everyday Math can have Everyday Math. But the rest of us should be able to have Saxon or Singapore. -- CatherineJohnson - 27 Jun 2005
"... than fighting a losing battle to retain a mathematics program better suited to the 19th century." Ah yes, this old nugget. He better explain exactly what this means. I am sick and tired of hearing it tossed out there as if there is something other than really old-fashioned algebra that is important. Why not throw out that what we all want is to have math just like when we were growing up. These arguments are so bad as to be laughable! "As for EM, if it was my kid, I'd take EM over Scott or Saxon in a second, fully confident that even in the hands of a mediocre teacher, both kid and teacher would be way ahead." Go ahead with your child, but not mine! My son uses EM at school (supplemented at school with mastery practice! Apparently, even the school doesn't think that EM is good enough by itself.) and I supplement with Singapore Math at home. I look at the two books side-by-side. There is no comparison. I don't know the details of Scott or Saxon, but I am sure there are those here who can comment on their levels of expectation. "...both kid and teacher would be way ahead." Perhaps he thinks that Scott or Saxon is just too hard and that kids will make better progress with an easier program. Is it OK that lower grades use curricula that do not prepare kids for a full course in algebra by eighth grade? "... But if you give me the opportunity to conduct the professional development we need and free me from the constraints of poorly designed high-stakes state assessments, bring on the Singapore materials even faster and rest assured I argue for skipping about the same percent of material I find skippable in any other program." Me, me, me. If only. More time, more money, more experiments to come up with a "balanced math" approach just like we have a "balanced literacy" approach. Sounds like job security to me. Once again, they argue with generalities, but then they want to define the details. No Thank You! He is a co-author of a report that basically says how wonderful Singapore Math is and how poor EM is, but we can't use it just yet. It still needs work and he is the one to do that work. We have to modify it to meet the needs of 21st Century Math. Sarcastic? You're darn right! If he wants to be in a position of deciding the math curricula and futures for millions of kids, then he better be prepared to defend his position with details. This is all about who is in control of curricula and who gets to decide. I surely do not Mr. Leinwand or the education community dictating what my son needs for 21st Century Math. No more If Only. There are textbooks and curricula that work right now. Let's use them. -- SteveH - 28 Jun 2005
On the maths as a language: I see maths as like a language in that you have to learn the 'vocab words' i.e. algorithims etc, before you can make fully understandable sentences. Getting rid of the basics is as sensible as trying to 'teach' a child to talk and have them not use any nouns like "Juice!" rather than "I would like some juice". Obviously maths is not exactly the same because children will learn their first language(s) easily and without explict instruction - so maybe a good comparison is teaching a second language. -- SamanthaRawson - 04 Nov 2005