4 August 2023

The ways in which we represent problems, domains, the world, directly affect our ability to understand, learn, and transform them. Thus, finding better representations is central to finding better ways to interact with the world.

In their paper *Restructurations: Reformulating Knowledge
Disciplines through New Representational Forms*, Uri Wilensky
and Seymour Papert^{1} coin the terms
*structuration* and *restructuration* to refer to the
relationship between knowledge representation, and the “representational
infrastructure” used to represent that knowledge. Specifically,

**Structuration**: the encoding of the knowledge in a domain as a function of the representational infrastructure used to express the knowledge.**Restructuration**: a change from one structuration of a domain to another resulting from a change in representational infrastructure.

Wilensky and Papert give beautiful examples of how
*restructuration* can profoundly change our ability to understand
and manipulate knowledge.

The first example they give is that of the shift from Roman to
Hindu-Arabic representation of numbers, and the impact this change had
on humans’ ability to both understand and perform arithmetic operations.
No amount of pedagogical improvements in the context of the Roman system
could have even approached the improvement made by the
*restructuration* brought about by the new representational
infrastructure of the Hinu-Arabic system.

As a second example, they note how Andrea A DiSessa^{2} describes the historical
restructuration of simple kinematics from a text-based to an algebraic
representation; without being able to appeal to algebraic notations such
as d=vt, Galileo struggled to handle a
problem involving the relationship between distance, time and velocity.
In this example, “algebra becomes the epistemological entity capable of
transforming a complex and difficult idea into a form that is within the
intellectual grasp of a high school student.”

As a third example they present different ways to define a circle. For Euclid, a circle is defined by the fact that all its points are at the same distance from a reference point called its center. The second view of the circle comes from a major restructuration of geometry by representing geometric entities in algebraic form. Here, the circle is defined by an equation such as x^2 + y^2 = K. The third view uses a computational object known as “the turtle”, which is an entity that has two “state properties”: a position and its heading.

A turtle in motion has two velocities: its linear velocity is the rate of change of its position; its angular velocity is the rate of change of its heading. […] If a turtle moves with both velocities constant it will draw a circle! What is remarkable about this is that the turtle draws the circle without reference to any external entity such as Euclid’s “center” or Descartes “coordinate axes.”

A fourth and final view of the circle is that of an *emergent*
property of a system of independent *agents*.

Place a large number of turtles at the same place. Give each one a random heading. Make them all move forward […] by the same amount. They will form a circle.

Here, the circle emerges from the behaviors of a large number of
independent agents.^{3}

Now, how do we know if a restructuration yields a superior structuration? Wilensky and Papert identify five dimensions to evaluate restructurations:

*Power properties*: a restructuration of a domain must be able to do what could be done before and preferably more as well.*Cognitive properties*: a restructuration should be more easily learned.*Affective properties*: a restructuration can make the knowledge more engaging.*Social properties*: “restructurations generate memes that can have varying evolutionary fitness in the social landscape.”*Diversity properties*: “[…] we look for restructuration of domains to match people’s learning styles.”

The last three are important from a pedagogical perspective (which is one of the authors’ primarily concerns), but for the purpose of modeling a problem to be solved by computers and computer programmers, the first two seem to be the more important ones.

Restructurations go beyond simple changes of representation. They
represent a more foundational change (the representational
*infrastructure*) with a potentially broad and deep impact, such
that multiple kinds of representations are potentially enabled or
affected.

How do we search and find restructurations for our problems or
domains of interest? Unfortunately this is not something the paper
discusses, so we’ll have to search for answers elsewhere. Eric Steven
Raymond^{4}, however, does provide us
with something of a clue in his *Rule of Representation*, which
says: *Fold knowledge into data, so program logic can be stupid and
robust.*

Even the simplest procedural logic is hard for humans to verify, but quite complex data structures are fairly easy to model and reason about. […] Data is more tractable than program logic.

I’ll come back to this later. For now, let us just acknowledge the importance of finding better representation. Even small representation changes with modest improvements are of value. To start, let’s keep an open mind and a watchful eye in search of better ways to encode our world.

“Restructurations: Reformulations of Knowledge Disciplines Through New Representational Forms,”

*Constructionism*17, no. 2010 (2010): 1–15.↩︎*Changing Minds: Computers, Learning, and Literacy*(Mit Press, 2000).↩︎At this point the authors introduce

*agent-based modeling*. In contrast to more traditional*equational modeling*, where we try to develop an equation that fits the observed data, agent-based modeling makes use of simple and local computational rules that apply to individual agents. Global properties emerge from the interactions of the simple localized agents. I won’t go into agent-based modeling as the focus here is on*restructuration*in general.↩︎“The Art of Unix Programming” (Boston: Addison-Wesley, 2004).↩︎