> However, beyond such material, I get thoroughly confused by an
> inability to distinguish proven knowledge, accepted wisdom, and pure > pseudo-science. It seems that a lot of educational research is done > by anecdote rather than by controlled blind large group studies. Thanks Mark for initiating this thread. There needs to be more discussion of the pedagogy so that Squeak/etoys can be optimised as a learning tool. For more criticism of Constructivism/ionism you could also read (though I strongly support constructionism): http://scil.stanford.edu/about/staff/bios/PDF/Cog_Effects_Prog (will open with Acrobat) ON THE COGNITIVE EFFECTS OF LEARNING COMPUTER PROGRAMMING ROY D. PEA and D. MIDIAN KURLAND and http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching Paul A. Kirschner Controlled blind large studies are rarely done. This is because the lab rabbits are real kids and there are real ethical concerns. We are stuck with anecdote and assertion for the large part. We need to critically examine all this, as there is little hard evidence. It would be good if we could examine the large amount of teaching with Game Maker. It has been used back to 2002 and, at least in Australia, there are hundreds of schools using it. The pedagogy of Game Maker and etoys is similar. This large body of data has never been properly examined because it is just ad hoc use by teachers which has never been attached to a university research program Tony _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
David-
Re: Constructivism, where to start?
I'm not real clear myself where it came from. My introduction to it was reading up on Kay's vision for educational computing, and watching some presentations of his on the same topic.
Citing some things from the article you referenced. I didn't read the whole thing. I was primarily interested in the section on constructivism:
"Included on the list for decreased attention in the grades K-4 were 'Complex paper-and-pencil computations,' 'Long division,' 'Paper and pencil fraction computation,' 'Use of rounding to estimate,' 'Rote practice,' 'Rote memorization of rules,' and 'Teaching by telling.' For grades 5-8 the Standards were even more radical. The following were included on the list to be de-emphasized: 'Relying on outside authority (teacher or an answer key),' 'Manipulating symbols,' 'Memorizing rules and algorithms,' 'Practicing tedious paper-and-pencil computations,' 'Finding exact forms of answers.' ...
The variant of progressivism favored by the NCTM during this time was called 'constructivism' and the NCTM Standards were promoted under this banner."
Some of what's described here is what I've been hearing about through other venues. The only characteristic that was news to me was the de-emphasizing of "manipulating symbols". Wow. Does this mean they weren't going to teach algebra somewhere in grades 5-8? One of the benefits I've heard about for learning advanced math, even it is by rote learning of algorithms or rules, even if the student never uses it again, is it at least teaches symbol manipulation and logical thinking.
I get the sense with the way that various reform efforts have played out though is they've thrown the baby out with the bathwater, or maybe the "baby" wasn't there to begin with and the reform methods just expose that more. There was a fairly recent video I saw by M. J. McDermott, a meteorologist, expressing her concern about the state of math education in Washington State's public schools (at http://www.youtube.com/v/Tr1qee-bTZI). She said that when she went back to school to become a meteorologist she noticed a lot of the freshmen were struggling with the math that was being used. She also heard complaints from professors about this. She attributed it to a lack of basic mathematical knowledge, and made her own determination that it was the "new math" they were learning in the public schools. She said that all they knew how to do was use calculators, and work in groups. She spends most of the video demonstrating the new metho
ds they're using for multiplication and division, both the "reasoning" and algorithmic methods. I agree with the "reasoning" cluster problems approach, because I've developed similar methods for "calculating in my head", but the algorithmic ones seemed overly complicated. Personally I'd prefer the standard algorithms to the new ones. I get the sense she missed the boat in her emphasis on these methods. I think the real deficiency she was noticing was the lack of ability in symbolic computation, which she only mentions. She said the students lacked understanding of algebra and trigonometry. This is probably because they just let their calculators do it. The ones that most students have access to now can even solve Calculus problems, I think.
I happened to look at the comments for this video on YouTube and I found that people were calling what McDermott was describing "constructivist" education, though she doesn't use this label. Whether that's an accurate label or not, that's people's impression of it.
As I researched constructivism some more, it began to sound like teaching methods that have been discredited, such as "whole reading" and "whole math", particularly the discovery aspect. I happened upon a couple of old articles by Lynne Cheney (yes, the wife of the current Vice President), back when she was Chair of the Endowment for the Humanities. She harshly criticized constructivism, associating it with these methods. I can't find the original article now, but I remember one where she talked about an instance in a "whole math" course where the kids were supposed to be learning about the Pythagorean Theorem. They were expected to create a right-triangle out of 3 rectangles of graph paper of different sizes (the triangle is the space created by the rectangles placed at different angles to each other). The goal was to get the kids to see the relationship between the rectangles, representing "squared" values. A few would understand that the combination (sum) of two o
f the smaller squares was equal to the area of the largest square, realizing the proof of the theorem: a^2 + b^2 = c^2. The teachers thought they were not supposed to say that any student's result was wrong for fear of damaging their self-esteem, and since this was the case there was no follow-up for those who didn't get it. What this appeared to create was classrooms with lazy teachers, because they barely played a role in the students' education. She said that in that particular classroom, when the students were later tested on their knowledge of this theorem, most didn't understand it.
I found a counterpoint to Cheney's article at http://www.mathforum.org/kb/plaintext.jspa?messageID=712380 from Tim Craine, an education professor who taught geometry. My read of his analysis is it's likely the problem with the classroom Cheney described was due to the teachers not following through. He doesn't say it, but I think what he means is it takes a teacher who's fluent in math and cares about educating kids to really make this exercise work. He agrees with something I've thought about, which is once a student has made their discovery, the teacher can come in and help the student analyze it, so they can see the implications. The phrase that came to mind is, "Once you've got something, understand what you really have."
Quoting from Craine:
"These discoveries, however, cannot be left to chance, but are more likely to take place under the guidance of a teacher. Contrary to Gardner's assertion, cooperative learning does not imply that children receive 'no help' from the teacher, nor that there are 'winners' who discover what they are supposed to and 'losers' who don't. A skilled teacher guides the entire class in a discussion of what they have observed and in formulating the appropriate generalizations."
It's looking like what is probably also a problem is something Alan Kay has talked about just generally with math and science teachers, which is that many are not fluent in math or science, and further have an aversion to it. I had the thought today that perhaps one reason why rote methods are used is that the teachers don't have to understand it to present it and make it look like they're teaching something. And maybe some students get the material anyway, though they'd probably do just as well not having such a teacher and just reading the math text.
What's apparent to me as I've done a little more research into this to write my response here is it appears constructivism can be misapplied in unskilled hands.
Personally I agree with what I've heard about constructivism from Kay, since I always learned best through doing. I remember that I always did better in science classes where we frequently did labs. I think that's one reason why I liked computer programming and later computer science so much. I spent some time thinking and theorizing about computing, but it was mostly hands-on work.
Re: It seems like education is largely done by anecdote, rather than scientific research
In terms of scientific education studies, I don't know of any either. There may be some scientific studies on cognition, which would relate well to education. I imagine they would be more general, just focusing on how the mind understands concepts, learns them, thinks about them, etc., not specific to certain areas like math and science. There may be some that are specific to language, for example.
---Mark
[hidden email] >Date: Wed, 21 Nov 2007 17:32:37 +0000
>From: "David Corking" <[hidden email]> >Subject: Re: [Squeakland] Panel discussion: Can the American Mind be > Opened? >To: [hidden email] >Message-ID: > <[hidden email]> >Content-Type: text/plain; charset=ISO-8859-1 > >Mark wrote: > >> Re: attempts with constructivism >> >> I hope you're right. I have heard criticisms of constructivism, based on >> anecdotes, but I've always wondered whether what's been evaluated is >> actually constructivism or just some group's ideological interpretation of >> it (the group that says they're implementing the pedagogy, that is). I >> haven't studied it in detail, but the ideas behind it, as presented by Kay, >> make sense to me. > >I think it is worth studying in detail, but I am not sure where to >start. First I think we need to learn to distinguish among > >1. constructivism the psychological hypothesis - as proposed by Piaget >as I understand >2. constructivism the pedagogy >3. constructionism - another pedagogy - and a word coined by Seymour >Papert. Note the 3rd syllable. > >(There is also constructivism the epistemology, which I can't even >spell, that also originates with Piaget.) > >I recently read this unsympathetic 2003 article on the US history of >constructivist pedagogy in maths >http://www.csun.edu/~vcmth00m/AHistory.html >But it is largely anecdotal (which is fine for a historian, but not >when we are responsible for the education of the next generation.) > >inability to distinguish proven knowledge, accepted wisdom, and pure >pseudo-science. It seems that a lot of educational research is done >by anecdote rather than by controlled blind large group studies. Any >pointers to the good stuff? Or tips to help a natural scientist to >understand the research methods of the social sciences? _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by forster@ozonline.com.au
Tony Forster wrote:
> Controlled blind large studies are rarely done. This is because the lab > rabbits are real kids and there are real ethical concerns. We are stuck with > anecdote and assertion for the large part. We need to critically examine all > this, as there is little hard evidence. For better or for worse, our society uses real kids for blind (and even double blind) trials of medical treatments. The ethics of a pendulum swinging from 'new math' to 'new new math' to 'back to basics' and on, based each time on anecdote, are, to the naive observer, as great a cause for concern as giving two matched groups of children differing curricula for a couple of years. Perhaps saying that ruins my chances of influencing education, but instead of advocating such trials, and dismissing current research methods, my next step is to understand how, as a society, we should interpret an anecdotal study. What are the benchmarks a study must meet to be considered good evidence to support making a change (to the learning environment, the learning methods, and even the learning objectives, or even just to an individual lesson plan?) Educators like yourself work hard on these studies to get them through peer review, or incorporated in government policy, and often aim for to be utterly dispassionate. So, how should a concerned parent (or administrator or politician) work with teachers in their community to separate the wheat from the chaff. _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
Here's what we did for City Building, Playground, and then Etoys
(I've written about this before, but I don't think on this thread). "City Building" is a wonderful (at that time non-computer) curriculum designed by Doreen Nelson that is very rich and has been used successfully for many age ranges - in our case we implemented it with Doreen's help for 3rd graders - which was the youngest group tried up to that point. Google Doreen and "City Building" for a wealth of info on this terrific curriculum design. Playground was a different way to do Etoys (similar graphics model and a different programming model). This was implemented in a grade 4-5 classroom (the school didn't have grades by age, but "clusters" by developmental level - which works a lot better). Doreen helped in every step of introducing "City Building" to very willing "3rd grade" teachers. Still, it took 3 years before the deep quality in the curriculum was manifest in the classroom and in the students and what they did and how they did it. Photographs of each of the three years would not reveal much visible difference. It was what the children were concerned with, how they talked about it, and how they went about the processes that changed profoundly. Trying to trace all this back into "what happened?" we came to the inescapable (and not too surprising) conclusion that the teachers had also changed -- they had learned much more about design and systems over the three years, and this was manifested in a "well above threshold" assessment from Doreen and the rest of us in the 3rd year. It's worth noting that assessments of fluency do not require control groups because what is being judged is not a teaching method or a curriculum per se, but results. Were the children doing deep "City Building"? No for the first two tries, Yes from the 3rd try onwards. Similarly, "are the children doing real math and real science or not?" Questions like these are easily answered by people who can tell the difference (just as musicians and coaches can assess their learners for degrees of fluency). The City Building experience and our long stay in this school allowed us to try the same multiple year assessment for Playground programming and its curriculum (with similar results). Basically, there are just a lot of things that don't get normalized in single trials of even worthwhile curriculum ideas that get smoothed out over a few years. The teacher gets more knowledgeable and confident. The curriculum is improved from some of the bugs found. The software often requires tons of work over the three years before it is above threshold, etc. When we started on Etoys 10 years ago, we had the three year trial in mind, and decided that all the initial curriculum would be tested over three years before we wrote it up (the substance of Kim's and BJ's book "Powerful Ideas in the Classroom" is about a dozen projects, each of which was tested over three years). What we don't know from this methodology is whether there are better ways to teach Etoys and the math and science powerful ideas in these examples. And we don't know whether the choices of the math and science examples are the most appropriate. But what we do know is that the processes of their book are highly likely to result in more than 90% of a class of children getting fluent in what's in the book, and that includes strong elements of differential vector geometry, acceleration and Galilean gravity, etc. This leads to interesting arguments, especially wrt young children, of the kind "if you can get 10-11 year olds to do real math and real science, then it doesn't much matter what the specific subject matter is". And "if the specific subject matter can be strongly related to adult uses and thinking about real math and real science, then all the better". This bypasses the much more difficult problems of taking a given theory of subject matter (school maths, etc.) and trying to contrast different ways of teaching it. We do not do that at all, and the Etoys work was done as part of "science time" in these classrooms (a great place to teach real math given the difficulties with the school math goals and processes). The main point here is that above threshold fluency for 90%+ of the children is one of the most important benchmarks -- and it can be done a little more easily than trying to use specific control groups if the subject matter is very different from school theories, yet still recognizable by experts. A side comment. The reactions against "the new" take partial form in demands for "super scientific studies", and most of these are simply not feasible, if our "three years for a good experiment" is valid. But the largest most devastating studies in the US are the "whole country" results that show beyond a shadow of a doubt that the existing educational process is not resulting in more than a small percentage of children getting above acceptable thresholds in reading, writing, math and science (and thinking). This is the problem they don't want to even discuss. Contrastive studies are not interesting unless both are above threshold. If neither are, back to the drawing board. If one is, then a more detailed contrast is of little value. Cheers, Alan -------------------- At 05:29 PM 11/22/2007, David Corking wrote: >Tony Forster wrote: > > > Controlled blind large studies are rarely done. This is because the lab > > rabbits are real kids and there are real ethical concerns. We are > stuck with > > anecdote and assertion for the large part. We need to critically > examine all > > this, as there is little hard evidence. > >For better or for worse, our society uses real kids for blind (and >even double blind) trials of medical treatments. > >The ethics of a pendulum swinging from 'new math' to 'new new math' to >'back to basics' and on, based each time on anecdote, are, to the >naive observer, as great a cause for concern as giving two matched >groups of children differing curricula for a couple of years. Perhaps >saying that ruins my chances of influencing education, but instead of >advocating such trials, and dismissing current research methods, my >next step is to understand how, as a society, we should interpret an >anecdotal study. > >What are the benchmarks a study must meet to be considered good >evidence to support making a change (to the learning environment, the >learning methods, and even the learning objectives, or even just to an >individual lesson plan?) Educators like yourself work hard on these >studies to get them through peer review, or incorporated in government >policy, and often aim for to be utterly dispassionate. So, how >should a concerned parent (or administrator or politician) work with >teachers in their community to separate the wheat from the chaff. > >_______________________________________________ >Squeakland mailing list >[hidden email] >http://squeakland.org/mailman/listinfo/squeakland _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
It was not my intention earlier in this thread to challenge the work
of Viewpoints. Instead I wanted to get a foothold into understanding how the powerful 'progressive' and 'back to basics' movements could be rationally compared with alternatives. Thank you for taking my question as a provocation - it is very illuminating to read the work of Rose, Kay et al justified from this perspective. I need to confess now that I have read 'Mindstorms' but not yet 'Powerful Ideas' - does the book address whether or not there is a 'Hawthorne effect' in the trials? In other words, could simply the intensive attention of all involved, coupled with the novelty, willingness to persevere for the second and third year, and the involvement of real subject matter experts, have been sufficient in itself to produce a fluency result that is well above acceptable threshold? Is it provable(*) that the student creation of computer models, for example, is a necessary condition of learning 'real math' fluency? * By 'provable', I mean: "could a future experiment be designed to prove my assertion, or, even better, could a reasoned argument prove my assertion?" Further, but perhaps drifting off topic for squeakland, is it provable that 'back to basics' and 'progressivism' are equally as inadequate? Or is the poor performance of public education in some countries a consequence, not of the learning theory nor curriculum, but caused by the 'received wisdom' not being applied properly, or even some external factors, such as low resources, attitudes to authority, or the currently fashionable complaint of students' learning styles not being catered for? David _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
Hi David --
At 05:29 PM 11/23/2007, David Corking wrote: It was not my intention earlier in this thread to challenge the work I certainly didn't take it that way - in part because we claim almost nothing. What we have been interested in is whether 90% of the children we've worked with -- taught by a teacher, not by us -- gain real fluency in what we are trying to teach them. We found that it took 3 years to introduce each new curriculum element (as described in my last post). Instead I wanted to get a foothold into understanding I disagree with the simplistic versions of both of these. If "progressive" means what it meant long ago - "Dewey education" - then I am very much in favor of what he was trying to do and what he wrote about. If "back to basics" means "Bennet or E.D. Hirsh", then I'm very much in disagreement with what they are trying to do, and their general view of "education". Subjects like real math and real science, with a goal to help children get fluent, are best assessed by real mathematicians and real scientists. Separate issues are: what parts of the real stuff should be taught to children, how should the teaching be done, etc. This is very important in its own right - recall the very bad choices made by real mathematicians when they chose set theory, numerals as short-hand for polynomials, etc. during the "new math" debacle. This is why Seymour Papert was so impressive -- he was that rarity, a first class mathematician who both cared about and understood important principles of how children think. He chose real math that was both deep and in rhythm with how children think about relationships. Thank you for taking my question as a provocation I didn't - it is very "Powerful Ideas" is written to help teachers teach a dozen or so projects in real math and real science, using Etoys. It makes no claims and leaves a tiny bit of philosophy to the Afterword. http://www.vpri.org/pdf/human_condition.pdf In other words, could simply the Schools should be all about the Hawthorne Effect. The ones that aren't should be closed. I think you misunderstood one part of my description of the process. The 3 years is with the same teacher but with three different groups of children. Each group deals with the materials and process for the same amount of time. The other part of your question wasn't asked or answered by what we did (since we wanted the children to express the math and science they learned in terms of working Etoy models). That's what we tried to do, and that's what we assessed. If the "it takes 3 years" story seems reasonable to you, then imagine what it would take to do a real longitudinal transfer experiment using control groups (about 7 years). We have never been able to find a funder that is willing to fund what it really takes. Is it provable(*) that the student creation of computer It's provable that it isn't (people have been learning "real math fluency" for thousands of years without computers). The important thing (Papert again) is what math and when? Computers make a huge difference here for pretty much everyone. Also, see the Afterword in the book for what science learning is really about (hint: computers are not at all required, but they allow more rich choices in the world of the child). I've used many analogies to music in the past. You don't need musical instruments to teach music, they just help (and in no small part because there are lots of different kinds). A child who is not that interesting in singing might be very interested in learning the guitar, one that is not interested in guitar might be interested in a sax, etc. Different learners need lots of different entry points. Computers can provide many different entry points, and can be the medium for the kinds of mathematics that science uses. A pretty good combination. * By 'provable', I mean: "could a future experiment be designed to No. But something might be done with a goal of more than 90% fluency -- computers could almost be indispensable here ... Further, but perhaps drifting off topic for squeakland, is it provable I said above that the simplistic versions of both are quite wrongheaded in my opinion. If you don't understand mathematics, then it doesn't matter what your educational persuasion might be -- the odds are greatly in favor that it will be quite misinterpreted. Or is the poor performance of public education in some countries a If you like multiple choice tests, then (e) all of the above. Cheers, Alan ---------- David _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
Hi Alan, You digressed into 'new math' and I disagree
You wrote: > > Separate issues are: what parts of the real stuff should be taught to > children, how should the teaching be done, etc. This is very important in > its own right - recall the very bad choices made by real mathematicians when > they chose set theory, numerals as short-hand for polynomials, etc. during > the "new math" debacle. When I was 16 I moved schools, and joined a cohort who had been educated in the Schools Mathematics Project, a English incarnation of 'new math'. I had kind of a traditional classical math education up to that point, and I felt like a fish out of water for a few weeks. My first impression was that my new classmates thought much more like real mathematicians, and at first that seemed like a pointless stuffy homage to academia. Later I learned to enjoy the math for its own sake, but I had another surprise a couple of years later. The SMP kids seemed much better equipped for the world of applied math at university and technical college. Set theory and number theory are vital for computer scientists (as I understand), matrix algebra and numerical methods for engineers. So when I got to college (to study engineering), I was glad to have had a chance to try my hand at real nineteenth century math in high school. By the way, I never learned, even today, any kind of general algebra or shorthand for polynomials, so I cannot comment on that. It didn't hurt that in those days, most math teachers in England were math major graduates (so perhaps an example of the benefits of the Hawthorne effect we discussed.) By the way, the SMP still exists in a cut down form: http://www.smpmaths.org.uk/ It didn't go down in a public fireball like 'new math' in the US, but instead seems to have been quietly squashed by the all powerful National Curriculum steamroller. Best, David _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by Alan Kay-4
David:
Further, but perhaps drifting off topic for squeakland, is it provableAlan:
I said above that the simplistic versions of both are quite wrongheaded
in my opinion. If you don't understand mathematics, then it doesn't
matter what your educational persuasion might be -- the odds are greatly
in favor that it will be quite misinterpreted. David, I read the original maths history http://www.csun.edu/~vcmth00m/AHistory.html that prompted your initial questions about constructivism and agree that it critiques the cluster of overlapping outlooks that go under the names of progressivism / discovery learning / constructivism - fuzzy descriptors But more importantly IMO it also takes the position that the dichotomy b/w "back to basics" and "conceptual understandings" is a bogus one. ie. that you need a solid foundation to build conceptual understandings. The problem here is that some people in the name of constructivism have argued that some basics are not accessible to children. (refer to the H Wu paper cited at the bottom of this post) I think the issue is that real mathematicians who also understand children development ought to be the ones working out the curriculum guidelines. This would exclude those who understand children development in some other field but who are not real mathematicians and would also exclude those who understand maths deeply but not children development. This has not been our experience in Australia. I cited a book in an earlier discussion by 2 outstanding maths educators documenting how their input into curriculum development was sidelined. National Curriculum Debacle by Clements and Ellerton http://squeakland.org/pipermail/squeakland/2007-August/003741.html For some reason the way curriculum is written excludes the people who would be able to write a good curriculum -> those with both subject and child development expertise For me the key section of the history was this: "Sifting through the claims and counterclaims, journalists of the 1990s
tended to portray the math wars as an extended disagreement between those
who wanted basic skills versus those who favored conceptual understanding
of mathematics. The parents and mathematicians who criticized the NCTM
aligned curricula were portrayed as proponents of basic skills, while educational
administrators, professors of education, and other defenders of these programs,
were portrayed as proponents of conceptual understanding, and sometimes
even "higher order thinking." This dichotomy is implausible. The parents
leading the opposition to the NCTM Standards, as discussed below, had considerable
expertise in mathematics, generally exceeding that of the education professionals.
This was even more the case of the large number of mathematicians who criticized
these programs. Among them were some of the world's most distinguished
mathematicians, in some cases with mathematical capabilities near the very
limits of human ability. By contrast, many of the education professionals
who spoke of "conceptual understanding" lacked even a rudimentary knowledge
of mathematics.
More fundamentally, the separation of conceptual understanding from
basic skills in mathematics is misguided. It is not possible to teach conceptual
understanding in mathematics without the supporting basic skills, and basic
skills are weakened by a lack of understanding. The essential connection
between basic skills and understanding of concepts in mathematics was perhaps
most eloquently explained by U.C. Berkeley mathematician Hung-Hsi Wu in
his paper, Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy
in Mathematics Education.75" Papert is also critical of NCTM but is clearly both a good mathematician and someone who understands child development - and has put himself into the constructivist / constructionist group I followed that link in the history to this paper which is a more direct and concrete critique of discovery learning taken too far, with well explained examples of different approaches: http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING A Bogus Dichotomy in Mathematics Education BY H. WU cheers, -- Bill Kerr http://billkerr2.blogspot.com/ _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
Thanks Bill. You wrote:
> > http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf > > BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING > A Bogus Dichotomy in Mathematics Education > BY H. WU Great reading! It is amazing to me that the unambitious proposals in that paper here might be considered radical or heretical by the establishment. _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by dcorking
Hi David --
At 02:25 AM 11/24/2007, David Corking wrote: >Hi Alan, You digressed into 'new math' and I disagree > >You wrote: > > > > Separate issues are: what parts of the real stuff should be taught to > > children, how should the teaching be done, etc. This is very important in > > its own right - recall the very bad choices made by real > mathematicians when > > they chose set theory, numerals as short-hand for polynomials, etc. during > > the "new math" debacle. > >When I was 16 I moved schools, and joined a cohort who had been >educated in the Schools Mathematics Project, a English incarnation of >'new math'. I had kind of a traditional classical math education up >to that point, and I felt like a fish out of water for a few weeks. >My first impression was that my new classmates thought much more like >real mathematicians, and at first that seemed like a pointless stuffy >homage to academia. Of course, I was referring to elementary school new math in the US, which tried to teach arithmetic via set theory and polynomial bases for different numeral systems. It would not be at all surprising if the SMP were better. The point is not about the worth of set theory and number theory (both good topics for high school) but about whether they are appropriate for younger children. I have degrees in both pure math and molecular biology, and I agree very strongly with Papert's view that various kinds of geometrical thinking, especially incremental, are better set up for children's minds, and also allow deeper mathematical thinking to be started much earlier in life. One way to think about this is that "mathematical thinking" (like musical thinking) is somewhat separate from particular topics - so the idea is to choose the most felicitous ones. >Later I learned to enjoy the math for its own sake, but I had another >surprise a couple of years later. The SMP kids seemed much better >equipped for the world of applied math at university and technical >college. Set theory and number theory are vital for computer >scientists (as I understand), matrix algebra and numerical methods for >engineers. So when I got to college (to study engineering), I was >glad to have had a chance to try my hand at real nineteenth century >math in high school. > >By the way, I never learned, even today, any kind of general algebra >or shorthand for polynomials, so I cannot comment on that. I think you did, since "356" and all other numeral forms of numbers (whatever their base) are shorthands for polynomials (the 3, 4, and 6 are the coefficients for polynomials of powers of ten in this case). >It didn't hurt that in those days, most math teachers in England were >math major graduates (so perhaps an example of the benefits of the >Hawthorne effect we discussed.) Why call this Hawthorne? I don't think this is what you mean here. Cheers, Alan >By the way, the SMP still exists in a cut down form: >http://www.smpmaths.org.uk/ >It didn't go down in a public fireball like 'new math' in the US, but >instead seems to have been quietly squashed by the all powerful >National Curriculum steamroller. > >Best, David _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by Bill Kerr
Thanks Bill --
I think you make the central point about all this. Cheers, Alan ---------- At 05:53 AM 11/24/2007, Bill Kerr wrote: David: _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by Bill Kerr
Hi Bill --
I just read Professor Wu's paper. I agree in the large with his assertion that the dichotomy is bogus, but I worry a lot about his arguments, assumptions and examples. There are some close analogies here to some of the mistakes that professional musicians make when they try to teach beginners -- for example, what can a beginner handle, and especially, how does a young beginner think? Young children are very good at learning individual operations, but they are not well set up for chains of reasoning/operations. Take a look at the chains of reasoning that Wu thinks 4th and 5th graders should be able to do. Another thing that stands out (that Wu as a mathematician is very well aware of at some level) is that while people of all ages traditionally have problems with "invert and multiply", the actual tricky relationship for fractions is the multiplicative one a/b * c/d = (a * c)/(b * d) which in normal 2D notation, looks quite natural. However, it was one of the triumphs of Greek mathematics to puzzle this out (they thought about this a little differently: as comeasuration, which is perhaps a more interesting way to approach the problem). A few years ago I did a bunch of iconic derivations for fractions and made Etoys that tried to lead (adults mostly) through the reasoning. One of the best things about the divide one is that it doesn't need the multiplication relationship but is able to go directly to the formula. So these could be used in the 5th grade. But why?, when there are much deeper and more important relationships and thinking strategies that can be learned? What is the actual point of "official fractions" in 5th grade? There are many other ways to approach fractional thinking and computation. I like teaching math with understanding, and this particular topic at this time - and provided as a "law" that children have to memorize - seems really misplaced and wrong. Etc. Cheers, Alan At 05:53 AM 11/24/2007, Bill Kerr wrote: David: _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by Alan Kay-4
On Sat, 24 Nov 2007 06:31:15 -0800, Alan Kay <[hidden email]> wrote:
> Hi David -- > > Of course, I was referring to elementary school new math in the US, > which tried to teach arithmetic via set theory and polynomial bases for > different numeral systems. It would not be at all surprising if the SMP > were better. > > The point is not about the worth of set theory and number theory (both > good topics for high school) but about whether they are appropriate for > younger children. I have degrees in both pure math and molecular > biology, and I agree very strongly with Papert's view that various kinds > of geometrical thinking, especially incremental, are better set up for > children's minds, and also allow deeper mathematical thinking to be > started much earlier in life. Random data point: I had "new math", though in the 10-12 years age group. I'm pretty sure I'm the only one who got it and I am, admittedly, something of an outlier. (I immediately started working through different bases, including base 11, which made hex and binary easier the following years when I started programming.) I haven't been able to teach it to a younger kid, unless that kid has "instant" math, in which case it's not really teaching so much as a brief introduction. _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by Alan Kay-4
Good discussion :-)
To be honest I've never been certain about the best way to teach "number" and have tended to try a smorgasboard in practice Perhaps Alan is correct David? Professor Wu (good mathematician) is making a brave attempt to make the teaching of algorithms to young children more concrete but his approach still puts too many demands on most children. From my experience of teaching of maths I feel that for disadvantaged students too many eyes would glaze over for some of the steps. It might work for his children but not for 90% of children. I still feel that he makes some valid points and criticisms. I like the transparency and open-ness of his paper, as well as the conceptual position put by its title. Another paper by Ellerton and Clements identifies the main issue as this: "... many children who correctly answered pencil-and-paper fraction questions such as 5/11 x 792 = q could not pour out one-third of a glass of water, and of those who could, only a small proportion had any idea of what fraction of the original full glass of the original full glass of water remained"
- Fractions: A Weeping Sore in Mathematics Education http://www.aare.edu.au/92pap/ellen92208.txt Some form of effective kitchen maths needs to come before algorithms. At this stage I'm left with more questions than answers. -- Bill Kerr http://billkerr2.blogspot.com/
On Nov 25, 2007 3:28 AM, Alan Kay <[hidden email]> wrote:
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Hi Bill --
I think the main thing in teaching "number" is to distinguish it from "name" or "numeral" -- and I think the rush towards teaching "base 10 numerals" too early is one of the big problems in early elementary mathematics. Numbers are ordered ideas that can be put in correspondence and taken apart and recombined at will. Names and numerals are symbols for these ideas that have varying degrees of usefulness for different purposes. So one of the biggest questions any math educator should ask is: what symbols should I initially employ for numbers to help children understand "number" most throughly? Most "child math" experts - like Mary Laycock, Julia Nishijima, etc. - would argue that a wide variety of analog (both unary and continuous) representations should be employed together (bundles of sticks, bags of objects, lengths of stuff, etc.), and each of these can have several labels attached ("one", 1, etc.). These can stay in use for much longer than is usually done in school. E.g. Some really great "adding slide rules" can be made from rulers, and then the children can make some really detailed large ones (even using their playground for baselines). These adding slide rules can add any two numbers together very accurately, whether "fractional" or not, and they can have scale changes to reveal what is invariant about two numbers (their ratio), etc. This can be used to make multiplication machines, etc. Another use of number that uses names in a non-destructive way is the "equality game" of "how many ways can you make a number". First graders are very good at this an even though they don't know what "1000000" stands for (except it is large) they understand that they can make this or any other number many ways by a combination of additions and subtractions that add up to zero. This is a way to start algebraic thinking without needing variables. And so forth. Wu actually makes a point against himself when he argues that phonics decoding is a good idea, even though no fluent reader decodes. This is similar to how sight reading is taught, especially for keyboards. Eventually the pattern results in a direct hand shape and mental "image" of the sound (or for text reading, a mental image of the idea). The question is how to get there, and teaching how to decode seems to help a little in early stages (maybe even just for morale purposes) rather than trying to teach either like Chinese characters. It takes 2-5 years to get fluent at such learning, so there usually need to be other supporting mechanisms (not the least is material that can be dealt with successfully after a few months or a year). So, what Wu should be asking is "what framework do children need to get started in number and mathematical thinking about number?". Another interesting example of what is not happening came out in a Mary Laycock workshop in which I was a "floor guy" (literally since I was on the floor with the children). One of Mary's games was to hand out a series of sheets of 10 by 10 squares, each divided in regions, with the question, "how many squares are in each region?" The 4th-5th grade children start by counting the squares in the regions. As the regions got more complicated, the children did not see that they could switch over to geometrical reasoning -- to see what fraction of the whole was occupied by each region and then divide -- instead they kept on trying to count the little squares and fractions of squares. Children who had learned to think mathematically would have had a strategy to look for the best representations for the problems, and these children had not acquired many (if any) math meta-skills. To bring up a musical analogy again ... one of the best collections of advice about how to teach children to play the keyboard is in Francois Couperin's 1720 treatise "The Art of Playing the Harpsichord". First, he says, keep the children away from the harpsichord because it isn't musically expressive enough. And keep away from sheet music because it "isn't music". Instead, take them to the clavichord (loud, soft, and pitch modulation -- more expressive than a piano) and teach them how to play some of their favorite songs that they like to sing, and help them be as expressive in their playing as their singing is. This is music. Play duets with the children, etc. After they have done this for a sufficient time (from 6 months to several years), then you can introduce them to the initially less expressive harpsichord (which, like the organ, can only be expressive through phrasing). But they will have learned to phrase very naturally from their clavichord experience and this will start to come out in their harpsichord playing. Finally, now that they have learned to "talk" (my metaphor), they can learn to read. Now they can be shown the written down forms of what they have been playing. And now they can start to learn to sight read music. When I was teaching guitar long ago, I used this basic scheme as much as possible, because "real guitar" has to be both music and "attention out" (so that you can mesh musically in a conversation with other musicians). Also, the guitar has some serious physical problems which have to be addressed gradually over weeks and months. Getting the students to play real stuff while all this is going on makes a foundation for the next level of much harder work. Learning to play patterns by ear allows the player to concentrate on their musicality and accuracy. Then they can be shown the patterns as both shapes and as decoded mappings in members of a key, etc. The egregiously misunderstood Suzuki violin method also follows these ideas. (It isn't mechanical -- read his books.) Couperin's essay is a pretty good set of distinctions concerning the general confusions between art and technique, and between ideas and media. You eventually have to get to all of these, but leading with art and ideas tends to preserve art and ideas, and leading with technique and media tends to kill art and ideas. I think it is really that simple. Cheers, Alan At 03:47 AM 11/25/2007, Bill Kerr wrote: Good discussion :-) _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
On Monday 26 November 2007 8:03 pm, Alan Kay wrote:
>... So one of the biggest questions > any math educator should ask is: what symbols should I initially > employ for numbers to help children understand "number" most throughly? Coming from a culture steeped in oral tradition, I find 'sounds' better than 'symbols' when doing math 'in the head'. The way I learnt to handle numbers (thanks to my dad) is to think of them as a phrase. 324+648 would be sounded out like "three hundreds two tens and four and six hundreds and four tens and eight. three hundreds and six hundreds makes nine hundreds, two tens and four tens make six tens and four and eight makes one ten and two, giving me a total of nine hundreds seven tens and two". Subtraction was done using complements. So 93-25 would be sounded out as "five more to three tens, six tens more to nine tens and then three more, making a total of six tens and eight'. The technique works for any radix - 0x3c would be "three sixteens and twelve'. In India, many illiterate shopkeepers and waiters in village restaurants use these techniques to total prices and hand out change. No written bills. The advantage with sounds is that tones/stress/volume can be used to decorate numbers. With pencil and paper, changing colors, sizes or weights would be impractical. Subbu _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by Bill Kerr
On Saturday 24 November 2007 7:23 pm, Bill Kerr wrote:
> I followed that link in the history to this paper which is a more direct > and concrete critique of discovery learning taken too far, with well > explained examples of different approaches: > > http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf > > BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING > A Bogus Dichotomy in Mathematics Education > BY H. WU Prof. Wu does well to call the bluff in treating skills vs. understanding as a zero sum game. However, I find some of his claims run counter to my own observations of how children learn. The claim "children welcome any suggestions that save labor" is simply not true. On encountering a concept for the first time, children tend to repeat it many times even though the process is quite tedious. It is only after many repetitions that they become receptive to suggestions to shortcuts. Either they discover the pattern by themselves or can be nudged gently towards the Aha discovery either by the teacher or by their peers. The issue that I have with algorithms being taught in schools is that they are introduced too early in the learning curve and are often introduced as "the method". I have seen many untutored people learn to do additions left to right. They would tie themselves into knots if asked to use the conventional right to left method. Subbu _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
subbukk wrote:
> On Saturday 24 November 2007 7:23 pm, Bill Kerr wrote: > >> I followed that link in the history to this paper which is a more direct >> and concrete critique of discovery learning taken too far, with well >> explained examples of different approaches: >> >> http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf >> >> BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING >> A Bogus Dichotomy in Mathematics Education >> BY H. WU >> > Prof. Wu does well to call the bluff in treating skills vs. understanding as a > zero sum game. However, I find some of his claims run counter to my own > observations of how children learn. The claim "children welcome any > suggestions that save labor" is simply not true. On encountering a concept > for the first time, children tend to repeat it many times even though the > process is quite tedious. It is only after many repetitions that they become > receptive to suggestions to shortcuts. Either they discover the pattern by > themselves or can be nudged gently towards the Aha discovery either by the > teacher or by their peers. > with learning is that I get introduced to something, then I have a period of grinding before I 'get it', then I can expand on that knowledge. Karl > The issue that I have with algorithms being taught in schools is that they are > introduced too early in the learning curve and are often introduced as "the > method". I have seen many untutored people learn to do additions left to > right. They would tie themselves into knots if asked to use the conventional > right to left method. > > Subbu > > _______________________________________________ > Squeakland mailing list > [hidden email] > http://squeakland.org/mailman/listinfo/squeakland > > _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by K. K. Subramaniam
On Mon, Nov 26, 2007 at 11:38:30PM +0530, subbukk wrote:
> On Monday 26 November 2007 8:03 pm, Alan Kay wrote: > >... So one of the biggest questions > > any math educator should ask is: what symbols should I initially > > employ for numbers to help children understand "number" most throughly? > Coming from a culture steeped in oral tradition, I find 'sounds' better > than 'symbols' when doing math 'in the head'. The way I learnt to handle > numbers (thanks to my dad) is to think of them as a phrase. 324+648 would be > sounded out like "three hundreds two tens and four and six hundreds and four > tens and eight. three hundreds and six hundreds makes nine hundreds, two tens > and four tens make six tens and four and eight makes one ten and two, giving > me a total of nine hundreds seven tens and two". Subtraction was done using > complements. So 93-25 would be sounded out as "five more to three tens, six > tens more to nine tens and then three more, making a total of six tens and > eight'. The technique works for any radix - 0x3c would be "three sixteens and > twelve'. > > In India, many illiterate shopkeepers and waiters in village restaurants use > these techniques to total prices and hand out change. No written bills. > > The advantage with sounds is that tones/stress/volume can be used to decorate > numbers. With pencil and paper, changing colors, sizes or weights would be > impractical. Subbu, Thanks for sharing this. I think that it is very interesting that sound and oral skills can be a basis for mathematical thinking. My cultural background is less oral, so I did not even think of this as a possibility. It seems that music and mathematics are somehow connected, but I never thought to extend this to verbal types of music. Dave _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
In reply to this post by K. K. Subramaniam
On Nov 26, subbukk wrote:
> So 93-25 would be sounded out as "five more to three tens, six > tens more to nine tens and then three more, making a total of six tens and > eight'. The technique works for any radix - 0x3c would be "three sixteens and > twelve'. I imagine this would work very well for systems with multiple bases, like pounds-shillings-pence, days-hours-minutes-seconds. Does it? My mental model for arithmetic is a clock face, which is very clumsy in base 10 (and worse in base 16!) Thanks for sharing this subbuk Best, David _______________________________________________ Squeakland mailing list [hidden email] http://squeakland.org/mailman/listinfo/squeakland |
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