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Adventures in Constructivism
Posted:
Dec 9, 2009 4:00 PM



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 LISPlike language to become a primary feature on MIT's landscape, i.e. Scheme.
MIT's switch to Python for its entrylevel 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 quasicrystals 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 raytracer 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 meshnetworked computing tools (camera, speakers included).
Many of them educate their families and help bring their village into awareness of the global debates, including the socalled 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 knownothing jungledwellers 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 selforganizing around social networking tools  to a point where traditional preInternet campaign strategies no longer apply) we should expect new forms of peertopeer 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 multiuser 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 onetomany 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 postcalculator 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.
Kirby
Exhibits:
About Constructivism: http://coffeeshopsnet.blogspot.com/2009/02/aboutconstructivism.html
Meeting with Alan Kay (wave to Helen): http://worldgame.blogspot.com/2006/04/atsummit.html
Python Enhancement Proposals: http://www.python.org/dev/peps/
Defending Polyhedra: http://mathforum.org/kb/message.jspa?messageID=6918511&tstart=0
 >>> from mars import math http://www.wikieducator.org/Digital_Math



