# Cognitive Process Instruction: Research on Teaching Thinking Skills

By Jack Lochhead; John Clement | Go to book overview
intuitive disposition one wants to take advantage of, not do battle with. Local causality is only a formal property of variational geodesics; it is manifest in the local and constructive turtle definition. (The local-global dichotomy is one of the important heuristic themes from computation which play a central role in turtle geometry and which seem quite valuable in many other areas as well.)Along the same lines it is very easy to relate a turtle geodesic to such things in everyday experience as the path of a car with wheels straight (each wheel turning at the same speed) or a jet airplane with rudder straight and wing engines running equally fast. The formal elegance of avoiding the question of local construction through a variational definition leaves out these important experiential ties.6Having bypassed formal axiomatics with appropriately active definitions, it is not hard to take this turtle quite far in nonflat geometries, ( diSessa, 1975) far enough in fact to bring high school students in contact with many of the most important ideas in mathematics--the concept of transformations and in- variants, continuity, the importance of topological considerations, and Stokes- like theorems.7 It is one of the prime advantages of informal presentations that students can begin developing feelings for and even the ability to use some of these extremely valuable and broadly applicable ideas long before their formal abilities are up to very general and/or precise formulations. Planting the seeds for understanding "powerful ideas" allows time to nurture notions of purpose and use which can keep a student's head above water in the rising tide of details necessary later for true mathematical integrity.
SUMMARY
I have argued that axiomatics or other formal systems may be useful models of "good" representation of knowledge for certain purposes, but they are not sufficient as pedagogical models. Other kinds of more informal presentations have a great number of advantages in being able to take into account the specific abilities and knowledge students acquire from everyday experience.The argument has been organized around the computational metaphor, which has two parts.
 1 Human thinking and knowing is process. It is complex but exhibits organization of a type which is hardly of the logical formalist type. A particularly important outgrowth of this is a concern for knowledge which is self-directing, organizational--in short, control knowledge.
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6
A line as an abstract entity with certain properties has been replaced by a line as the process which draws it! One may wish to generalize this comparison of turtle geometry with standard geometry by contrasting mathematics of construction to mathematics of constraint. The latter defines entities by a series of constraints (e.g. axioms), does not deal with the vagaries of models, and does not bother to tell the student either (a) that the entity of concern is a suitable generalization for everything he knows about, or (b) that there are no known examples of such a thing.
7
By the latter I mean any of the group of theorems which compute the totality of something spreading over a region by computing something else on the boundary of that region. Important examples are the fundamental theorem of calculus, the calculus of residues in complex analysis, Gauss's theorem in electrostatics and gravitation, Stokes' Theorem in electrostatics, any conservation law for flowing substances, the concept of state function in thermodynamics, and the existence of potential (e.g. energy) functions.

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