[from: Concept-Rich Mathematics Instruction]
Teacher: Very good. Now, look at this drawing and explain what you see. [Draws.] Debora: It's a pie with three pieces. Teacher: Tell us about the pieces. Debora: Three thirds. Teachers: What is the difference among the pieces? Debora: This is the largest third, and here is the smallest . . . Sound familiar? Have you ever wondered why students often understand mathematics in a very rudimentary and prototypical way, why even rich and exciting hands-on types of active learning do not always result in "real" learning of new concepts? From the psycho-educational perspective, these are the critical questions. In other words, epistemology is valuable to the extent that it helps us find ways to enable students who come with preconceived and misconceived ideas to understand a framework of scientific and mathematical concepts. Constructivism: A New Perspective At the dawn of behaviorism, constructivism became the most dominant epistemology in education. The purest forms of this philosophy profess that knowledge is not passively received either through the senses or by way of communication, just as meaning is not explicitly out there for grabs. Rather, constructivists generally agree that knowledge is actively built up by a "cognizing" human who needs to adapt to what is fit and viable (von Glasersfeld, 1995). Thus, there is no dispute among constructivists over the premise that one's knowledge is in a constant state of flux because humans are subject to an ever-changing reality (Jaworski, 1994, p. 16). Although constructivists generally regard understanding as the outcome of an active process, constructivists still argue over the nature of the process of knowing. Is knowing simply a matter of recall? Does learning new concepts reflect additive or structural cognitive changes? Is the process of knowing concepts built from the "bottom up," or can it be a "top-down" process? How does new conceptual knowledge depend on experience? How does conceptual knowledge relate to procedural knowledge? And, can teachers mediate conceptual development? | Concept-Rich Mathematics Instruction Is Learning New Concepts Simply a Mechanism of Memorization and Recall? Science and mathematics educators have become increasingly aware that our understanding of conceptual change is at least as important as the analysis of the concepts themselves. In fact, a plethora of research has established that concepts are mental structures of intellectual relationships, not simply a subject matter. The research indicates that the mental structures of intellectual relationships that make up mental concepts organize human experiences and human memory (Bartsch, 1998). Therefore, conceptual changes represent structural cognitive changes, not simply additive changes. Based on the research in cognitive psychology, the attention of research in education has been shifting from the content (e.g., mathematical concepts) to the mental predicates, language, and preconcepts. Despite the research, many teachers continue to approach new concepts as if they were simply addons to their students' existing knowledge-a subject of memorization and recall. This practice may well be one of the causes of misconceptions in mathematics. Structural Cognitive Change The notion of structural cognitive change, or schematic change, was first introduced in the field of psychology (by Bartlett, who studied memory in the 1930s). It became one of the basic tenets of constructivism. Researchers in mathematics education picked up on this term and have been leaning heavily on it since the 1960s, following Skemp (1962), Minsky (1975), and Davis (1984). The generally accepted idea among researchers in the field, as stated by Skemp (1986, p. 43), is that in mathematics, "to understand something is to assimilate it into an appropriate schema." A structural cognitive change is not merely an appendage. It involves the whole network of interrelated operational and conceptual schemata. Structural changes are pervasive, central, and permanent. The first characteristic of structural change refers to its pervasive nature. That is, new experiences do not have a limited effect, but cause the entire cognitive structure to rearrange itself. Vygotsky (1986, p. 167) argued, It was shown and proved experimentally that mental development does not coincide with the development of separate psychological functions, but rather depends on changing relations between them. The development of each function, in turn, depends upon the progress in the development of the interfunctional system. From: Jim Bromer Sent: Monday, August 09, 2010 11:11 PM To: agi Subject: [agi] Compressed Cross-Indexed Concepts On Mon, Aug 9, 2010 at 4:57 PM, John G. Rose <johnr...@polyplexic.com> wrote: > -----Original Message----- > From: Jim Bromer [mailto:jimbro...@gmail.com] > > how would these diverse examples > be woven into highly compressed and heavily cross-indexed pieces of > knowledge that could be accessed quickly and reliably, especially for the > most common examples that the person is familiar with. This is a big part of it and for me the most exciting. And I don't think that this "subsystem" would take up millions of lines of code either. It's just that it is a *very* sophisticated and dynamic mathematical structure IMO. John Well, if it was a mathematical structure then we could start developing prototypes using familiar mathematical structures. I think the structure has to involve more ideological relationships than mathematical. For instance you can apply a idea to your own thinking in a such a way that you are capable of (gradually) changing how you think about something. This means that an idea can be a compression of some greater change in your own programming. While the idea in this example would be associated with a fairly strong notion of meaning, since you cannot accurately understand the full consequences of the change it would be somewhat vague at first. (It could be a very precise idea capable of having strong effect, but the details of those effects would not be known until the change had progressed.) I think the more important question is how does a general concept be interpreted across a range of different kinds of ideas. Actually this is not so difficult, but what I am getting at is how are sophisticated conceptual interrelations integrated and resolved? 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