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Topic: Adventures in Constructivism
Replies: 1   Last Post: Dec 10, 2009 6:20 PM

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kirby urner

Posts: 3,690
Registered: 11/29/05
Adventures in Constructivism
Posted: Dec 9, 2009 4:00 PM
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We hear a lot about Constructivism in this archive and of course have many
ways to tell its story. The common wisdom is it traces to Piaget and his
work in early childhood development. He concluded that (a) children have
their own natural conceptual development progression and (b) an effective
curriculum will work with this grain, not against it i.e. will help a child
"construct his or her own understanding" in a manner consistent for children
at that stage of mental development.

The lineage need not be this strictly adhered to however. Some followers of
Caleb Gattegno, who popularized the use of Cuisenaire blocks in early math
learning, consider him the paradigm constructivist. In the physics teaching
community, people point back to Robert Karplus.

The math branch of the Constructivist tree eventually spouted what we might
call Constructionism (note the small change) and is the heading under which
MIT began to build muscle, through its Media Lab and so forth. Logo paved
the way for a LISP-like language to become a primary feature on MIT's
landscape, i.e. Scheme.

MIT's switch to Python for its entry-level EE flagship in 2007 is only the
latest chapter and marks the maturation of "cosmopolitan OO" (cleanly object
oriented yet with a heterogenous flavor, very Dutch). Seymour Papert
(Logo), Alan Kay (DynaBook), Guido van Rossum (CP4E)... Kenneth Iverson
(APL), these are among the touch stones on a constructionist's necklace.

The Gattegno bricks were color coded in a strictly consistent manner related
to length, meaning letter symbols, corresponding to colors, could be swapped
into arithmetic expressions. The emphasis was on learning to read algebraic
equations and to solve them, including with fractional solutions, and to do
this far earlier than the traditional, more plodding arithmetic sequence,
which went operation by operation (addition, subtraction, multiplication,
division). Having these colored bricks as a part of ones visual vocabulary,
as "right brain" graphics, was considered a feature, a way of giving
Constructivist teachers an edge over their peers.

Fast forward to our present time and spatial geometry has become the defacto
standard for computer imaging and molecular modeling. Ever since Linus
Pauling, chemistry has been "off the plane" with the discovery of buckyballs
in the 1980s a dramatic reminder of this trend. The icosahedral structure
of the virus and existence of quasi-crystals likewise catapulted spatial
geometry into the public eye. Nanotechnology is all about lattices,
relative frequencies (impurities), molecular structures.

The Constructivist assault on spatial geometry has been through its
constructionist branch. This stands to reason because "mechanical drawing"
(as the subject used to be labeled) requires many hours of practice in the
absence of computer software. Learning to draw in perspective and to render
the human body, artifacts, natural settings, remains a goal of many students
and needs to be satisfied. However when it comes to polyhedra in
particular, we're more likely to retrieve our vectorial coordinates from a
database and run them through a ray-tracer or OpenGL. We also like making
these things with our hands, and/or by folding origami. Make: Magazine (by
O'Reilly) may be considered another Constructivist Tool (like how Forbes was
a Capitalist Tool).

What changed the equations such that constructivists would pour into spatial
geometry in a big way? Are we suggesting a GeometryFirst curriculum to
compete with the Gattegno AlgebraFirst approach?

The most important trend was the plummeting cost of software and hardware
and the sudden relevance of these technologies to both big and small
business thanks to two revolutions, the personal computer (PC) and the open
source revolution.

A second important trend has been the rising importance of biotechnology and
the biological sciences, which are ravenous for computer services.

A third trend: MIT and OpenCourseWare, a commitment to putting fully
developed curricula into the commons, not just as slides or lecture notes,
but as full length lectures, detailed web sites with plenty of primary
source materials (Stanford's implementation of the Buckminster Fuller
Archive is a good example).

Put another way: given the rising need for spatially fluent technology
workers and the falling barriers to entry in terms of both physical and
metaphysical goods, put pressure on mathematics programs to deliver a more
mature approach to math learning that took advantage of our new tools.

This convergence of forces resulted in a surge in Constructivism and the
implementation of new digital mathematics curricula throughout the land (not
all branded the same way of course, YMMV).

I should probably weave One Laptop per Child into the above narrative, as
it's obviously in this lineage. However, given the futuristic, some might
even say alien flavor of the XO (especially with the Sugar UI), we might
want to let Piaget off the hook. We don't really have the answers regarding
what children are capable of, left somewhat to their own devices with such
mesh-networked computing tools (camera, speakers included).

Many of them educate their families and help bring their village into
awareness of the global debates, including the so-called math wars (of
interest to math teachers). Literate self schoolers crop up in various
necks of the woods flexing leadership qualities and seek each other out over
the Web. So lets not carry around this stereotype of know-nothing
jungle-dwellers versus urban elites. The playing field is already a lot
more level than it has been and it's getting harder to get away with stuff
simply because one knows how to obfuscate.

Based on what's happening more generally among the young (a lot of
self-organizing around social networking tools -- to a point where
traditional pre-Internet campaign strategies no longer apply) we should
expect new forms of peer-to-peer teaching (and networking) to continue
emerging. Mashing existing tools together to positive synergetic effect is
one way to add value, and is what many schools are encouraging (including
our own Saturday Academy and LEP High). The multi-user game genre is
likewise a collaboration genre, per 2nd Life, although not all 2.0 tools
have this avatar aspect.

Those of higher skill may enter these domains as "teachers" (role models,
experienced players) but there's no traditional classroom structure and
therefore perhaps no obvious way to judge the student:teacher ratio. Common
sense might suggest no school experimenting with these modalities could
receive government accreditation, but that'd be to forget what century we're
in: such experiments are already well underway, especially in the military.

In sum, I think a lot of the constructivist literature is still premised on
a one-to-many relationship between a teacher and a set of students in
physical proximity. With Math 2.0 tools, this may no longer be assumed.
Peer groups are pioneering their own collaboration methods and spreading
these globally through subcultural networks. For example, Python's
engineers have a system of PEPs (Python Enhancement Proposals) whereas other
parts of Cyberia work with RFCs (requests for comments).

Published, shared standards form the backbone of any such culture of
collaboration and mathematics is distinguished for having some of the most
accepted of all standards (has a lot of pithy backbone).

DIgital mathematics involves appreciating the role of standards even if we
don't dig into them too deeply at first. For example, we've been talking
about extended precision Decimal and Integer arithmetic on this list
recently and its relevance in a post-calculator context. These algorithms
trace to published IEEE/ANSI standards and sharing some of this lore (in the
form of storytelling) helps students better understand what we mean by
"algorithm" in the first place (as in "Euclid's Algorithm")).

Anyway, I'm somewhat skeptical that we want to keep calling this
Constructivism, or even Constructionism, but am open to proposals to guide
the semantics in that direction. Who wants to champion Constructionism in
the 21st century? Or shall we call this something else? Note that I'm
asking about mathematics in particular, although these same pressures and
trends apply across the board.



About Constructivism:

Meeting with Alan Kay (wave to Helen):

Python Enhancement Proposals:

Defending Polyhedra:

>>> from mars import math

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