Panel discussion: Can the American Mind be Opened?

> 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


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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.

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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:

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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

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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
of Viewpoints. 

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
how the powerful 'progressive' and 'back to basics' movements could be
rationally compared with alternatives.

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
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?

"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
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?

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
models, for example, is a necessary condition of learning 'real math'
fluency?

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
prove my assertion, or, even better, could a reasoned argument prove
my assertion?"

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
that 'back to basics' and 'progressivism' are equally as inadequate?

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
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?

If you like multiple choice tests, then (e) all of the above.

Cheers,

Alan

----------

David

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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

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David:

Further, but perhaps drifting off topic for squeakland, is it provable
that 'back to basics' and 'progressivism' are equally as inadequate?

Alan:


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:


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/


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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.

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Hi David --

At 02:25 AM 11/24/2007, David Corking wrote:
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.


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


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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:

Further, but perhaps drifting off topic for squeakland, is it provable
that 'back to basics' and 'progressivism' are equally as inadequate?

Alan:
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/

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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:

Further, but perhaps drifting off topic for squeakland, is it provable
that 'back to basics' and 'progressivism' are equally as inadequate?

Alan:
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/

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On Sat, 24 Nov 2007 06:31:15 -0800, Alan Kay <[hidden email]> wrote:

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.

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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:

Some form of effective kitchen maths needs to come before algorithms.

At this stage I'm left with more questions than answers.

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Bill Kerr
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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 :-)

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:

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:

Further, but perhaps drifting off topic for squeakland, is it provable

that 'back to basics' and 'progressivism' are equally as inadequate?

Alan:

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/

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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

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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

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subbukk wrote: So like in programming, early optimization is a no-no. My experience
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

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On Mon, Nov 26, 2007 at 11:38:30PM +0530, subbukk wrote:
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


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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

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