Essential readings for learning about learning . . .
The Geometry of Wholemovement
by Bradford Hansen-Smith
Hansen-Smith begins by rejecting the faith-based axioms and postulates of Euclid, and replaces them with the reality of the sphere, from which the circle is derived through compression. This foundation, he argues, is more in harmony with the physical reality of the universe (cf. Kepler's Harmonies of the World). This entire book is based on circles--paper plate circles! Every discovery-based construction in the book is based on folding and attaching paper plates--no cutting or gluing allowed. He takes the reader-constructor through the creation of lines, polygons, polyhedra (even Archimedean), spirals, and more.
The Geometry of Wholemovement could be, and should be, an essential text for any geometry class. Unfortunately, however, today's standardized test-driven curriculum leaves no room whatsoever for the type of insight-oriented learning that Hansen-Smith provides.
by Bradford Hansen-Smith
Hansen-Smith begins by rejecting the faith-based axioms and postulates of Euclid, and replaces them with the reality of the sphere, from which the circle is derived through compression. This foundation, he argues, is more in harmony with the physical reality of the universe (cf. Kepler's Harmonies of the World). This entire book is based on circles--paper plate circles! Every discovery-based construction in the book is based on folding and attaching paper plates--no cutting or gluing allowed. He takes the reader-constructor through the creation of lines, polygons, polyhedra (even Archimedean), spirals, and more.
The Geometry of Wholemovement could be, and should be, an essential text for any geometry class. Unfortunately, however, today's standardized test-driven curriculum leaves no room whatsoever for the type of insight-oriented learning that Hansen-Smith provides.
Organizing and Memorizing: Studies in the Psychology of Learning and Teaching
by George Katona
Katona begins by stating that the association (behavioral) model of psychology prevails in education, which he finds objectionable—not because it might be wrong, but because sole adherence to it has prevented other possible types of learning from being studied. He therefore took it upon himself to conduct a multitude of studies that tested alternatives to the association model. It even ended up being the case that many of the studies directly contradicted associationist theory—especially that espoused by Thorndike. The book reports Katona’s major findings on learning, with a special emphasis on memory, since implicit in effective learning is long-term retention. Here is a brief summary of his results:
1. Rather than being “taught” a principle, it is better that a principle is discovered without aid, as it results in better retention.
2. The opposite of rote learning is learning by understanding, which occurs when material is grouped so as to reveal inner relationships—an organization conducted by the learner.
3. Learning is more effectively transferred (applied to other tasks) when generalization occurs.
4. Direct practice, which is the shortest route to achieving a task goal, results in poor transfer of learning. It is better to study a multitude of examples and infer general principles than to quickly learn how to solve a problem.
5. Detours, or seemingly circuitous routes, prove to be more efficient than direct teaching. Spending extra time getting at structural features of given content results in better retention and transfer.
In short, Katona found that the immediate goal in education should not be to see how much “memorizing and cramming can be eliminated” (259), so that students no longer “give back experience in the form in which they had received it,” but “learn to learn” (260).
Science and Hypothesis
by Henri Poincaré
What is science? What is the nature of scientific discovery? Poincaré, dealing with the fundamentals of science in four broad categories (Number and Magnitude, Space, Force, and Nature), analyzes the breakthroughs occurring during his time. Rather, though, than being a treatise on the most recent trends in his field, Poincaré lets the ephemeral nature of scientific theory serve as a foil to epistemology and metaphysics.
The themes that run throughout are, first, the tension caused by the specialized conventions of science. The controversy with non-Euclidean geometry, for example, is compared to a debate over whether to use the metric or imperial systems of measurement. Both suffice, being merely tools for a higher purpose. Secondly, the role of relations in science supersede any other so-called objective systems. Whether one adheres to Fresnel or Maxwell in studying optics, for example, the fact is that the relational aspects, as discovered through experiment, remain the same. Poincaré correctly predicts, in fact, that the concept of the luminiferous ether would eventually become obsolete, as it seemed to be a convenient tool for relating the various properties of space, light, and electricity. The final, overarching theme in Science and Hypothesis is the hermeneutical circle between the simple and complex. Hypotheses steer science toward the simple (unification), but experimental phenomena can tend toward the complex. This is, claims Poincaré, a fruitful tension that is justified by probability theory.
by Henri Poincaré
What is science? What is the nature of scientific discovery? Poincaré, dealing with the fundamentals of science in four broad categories (Number and Magnitude, Space, Force, and Nature), analyzes the breakthroughs occurring during his time. Rather, though, than being a treatise on the most recent trends in his field, Poincaré lets the ephemeral nature of scientific theory serve as a foil to epistemology and metaphysics.
The themes that run throughout are, first, the tension caused by the specialized conventions of science. The controversy with non-Euclidean geometry, for example, is compared to a debate over whether to use the metric or imperial systems of measurement. Both suffice, being merely tools for a higher purpose. Secondly, the role of relations in science supersede any other so-called objective systems. Whether one adheres to Fresnel or Maxwell in studying optics, for example, the fact is that the relational aspects, as discovered through experiment, remain the same. Poincaré correctly predicts, in fact, that the concept of the luminiferous ether would eventually become obsolete, as it seemed to be a convenient tool for relating the various properties of space, light, and electricity. The final, overarching theme in Science and Hypothesis is the hermeneutical circle between the simple and complex. Hypotheses steer science toward the simple (unification), but experimental phenomena can tend toward the complex. This is, claims Poincaré, a fruitful tension that is justified by probability theory.
Science and Method
by Henri Poincaré
In the heyday of the pure logical systems of Hilbert, Russell, et. al., Poincaré composed a work that demonstrated, step by step, the necessary bankruptcy of such systems. Poincaré was especially aware of the damage caused by the logico-deductive method in the field of education, through his observations at the École Polytechnique. In particular, as he pondered on the elegance of great mathematical proofs, Poincaré wondered why such proofs were often incomprehensible to students. He came to the conclusion that it was because the long chain of arguments in a proof leave out “that indefinable something that constitutes the unity of the demonstration” (126). In other words, students failed to see the whole picture and thus could not grasp the proof.
To many of Poincaré’s colleagues, this condition of the student mind was seen as a defect. In return Poincaré asked poignantly, “Should we constrain young people to change the nature of their minds?” (126). He naturally answered in the negative and insisted it is the instructor who must adapt. In this capacity, the duty of the instructor is to treat the student like an embryo, which is said to pass through the stages of evolutionary history as it develops. “The educator must make the child pass through all that his fathers have passed through, more rapidly, but without missing a stage” (127). In fact, Poincaré’s advice for finding the best way to teach something is to find out how that thing was originally discovered, and to attempt to find the original, true nature of the discovery, and then to cause that same discovery to occur in the mind of the student. All else is a mere relating of discrete facts or an imposition of arbitrary logic. Logic, however, is not seen by Poincaré as being bad in itself. It is necessary for providing proofs of discoveries; but the actual discovery occurs through the process of intuition.
Productive Thinking
by Max Wertheimer
This seminal work of the pioneer of gestalt theory, Max Wertheimer, lays out how thought processes (toward discovery) work, which, in effect, provides a framework for how learning occurs. In contrast to utilizing blind repetition, arbitrary trial and error, given formulae, deductive/inductive logic, or associationist stimulus-response behavior, a genuine learning experience is said by Wertheimer to have the following characteristics:
1. sense of a problem that emerges from a gap, disturbance, or “trouble” within a given structure; i.e., a problem that is dynamically driven by the whole of the structure itself
2. a grasping toward structural truth (as opposed to piecemeal truth) through which possible solutions are guided by the structural requirements of a situation
3. a consistency of development that remains “sensible” within the requirements demanded by the structure
4. moment(s) of insight in which proposed solutions are seen to naturally fit the situation
Wertheimer illustrated these processes in action through various examples that he himself had witnessed over the years. Some examples were with grade school students, others with adult subjects, and the rest include figures such as Galileo, Einstein (a personal friend of Wertheimer), and Wertheimer himself. Through problem situations such as finding the area of a parallelogram, discovering relativity, describing a workplace hierarchy, etc., Wertheimer detailed, step by step, how each of these problems were solved (or, in some cases, not solved) by the subject. They involved processes that cannot be explained merely through the prevailing theories of associationism and traditional logic. In fact, even though great discoveries are often handed down to us in logical terms, such logic (which is useful for explanation) is imposed on the discovery after the fact for the sake of stating it in clear terms. Einstein, for instance, did not produce his theory of relativity through a series of axioms and postulates resulting in a syllogistic, stepwise proof. The process, rather, deeply involved the steps mentioned above, and only afterwards could be stated in deductive terms. Thus, as Wertheimer asserted, any mode of teaching which does not recreate the steps of discovery required by a given problem situation serves to “cut to pieces living thinking processes,” and “dissect them, and thus show a dead picture stripped of all that is alive in them” (237).
