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Constructivism and Cognitive Theory, Not UnderstoodSept 18, 2008 Keep it simple. Leave no gaps. Engage students. Let students learn from their own mistakes. Let students construct their own knowledge. Give genuine, authentic and open problems. Let students draw their own conclusions. Use the Socrate's Method All are ringing, appealing, calls for instruction. But constructivism goes further and become empirically absurd in stressing subjectivism. That is in stressing (a) student performance as indicated by subject based test, are not absolute, are in general not repeatable and reproducible, (b) that the workings of a student mind are unreadable and conclusions drawn by students should be respected as internal workings of each person's mind is valid, at least for them; (c) whence teachers should not correct students, not be authoritative, and and let students individually or together learn via their own inquiries and not from teacher given and enforced rules and patterns. (d) consistent with the latter, mastery of rules and patterns, that would include the Euclidean logic in mathematics which has been a model for reason outside as well, is not a sign of intelligence. In empirical arts and disciplines, that is in science and technology, if not business and government, hypotheses regarding whether or not a fact or rule or pattern holds may be partially tested. An observation or test may show that a rule or pattern does not hold. That refutes or implies a difficulty with the proposed rule or pattern. But if test or observation is consistent with the rule, pattern or fact that is not absolute proof. It is a partial confirmation in need of further confirmation until the rule or pattern becomes credible. And where a theory gives or includes an implication not hitherto suspected (there-in lies an element of subjectivity), and the implication holds, that adds credence to theory whereas failure of the implication would have refuted the theory. Empirical arts and sciences in daily life and in research rely on refuted hypotheses as stop or change of direction signs, and on confirmed hypotheses or patterns as indicators of proceed with caution. Absolute knowledge is not possible. Even in Euclidean thought, the strength, reliability or validity of conclusions depends on the acceptance of logic - that is patterns for using implication rules directly or indirectly, alone or in combination; and on the reliability of the original axioms -the explicitly assumed rules or patterns, and perhaps - a situation to be avoided - some tacit ones as well. And in mathematics, a rule, a pattern or a conclusion is tested by providing a direct or indirect chain of reason that implies it. The same might be said for detective work. Empirically, an art or discipline may consist of an expanding collecting and development of rules and patterns, rituals if you like, for use alone or in combination in a repeatable and reproducible manner whose details are observable underway or whose results are observable, and hence verifiable or indicators of the need for a correction. Rules and patterns that may be seen in action or whose results may be observed provide tools which students may manipulate. For example, writing and drawing in arithmetic, algebra, geometry and calculus may record on paper in full detail if possible, the steps that lead to or imply a conclusion in an observable and verifiable manner. Inductive principles and practices for instruction may be followed in course design and delivery. That encourages the use of manuals or how-Tos to make instruction a repeatable and reproducible process. Empirically, an art or discipline consists of a collection of rules and patterns for addressing routine tasks and problems. Instruction can emphasize, verify and correct their use in the hope that students will learn from their mistakes to be careful in the use. Student mastery is tested or proven via formal tests or by instructors monitoring and observing student work without reading their minds. Here some rules and patterns may be learnt by rote and reflex For instance, cooking in the kitchen and driving a car may involve rituals mastered without understanding why they work. They just do. But explaining how rules and patterns may be combined and why introduces thought into education and in manner similar to the boot-up of a computer, may built introduce a thought based development or deductive reason into the mastery of rules and patterns starting with rules and patterns and practices tried and tested, or accepted, without reason. Constructivism appeals to engage and motive students, and the exploration within a constructivist framework of methods for are appealing. Yet the aversion to rule and pattern based reason and the Constructivism position that testing is unreliable imply that constructivism and it followers are pointing teachers and students to an empirically incorrect ideals for the development of skills and concepts beyond Euclidean comprehension. While constructivism depicts direct instruction as a form of rote learning, a depiction that might be accurate if mastery of rules and patterns alone or in combination is not cast as a sign of intelligence. my words above depict constructivism as an educational approach with a foundation and values incompatible with empirical arts and practices. Constructivist standards for instruction in mathematics appear to be more style than substance. Recent calls for open, genuine and authentic problem solving could change course content and substance, while ignoring basic flaws in the inductive or deductive development of algebraic skills and concepts. Constructivism in North America and Europe appears to be a mass movement based on hopes rather reason and inconsistent with the empirical form of the arts and disciplines to which it is applied. |
Mathematics Education Essays etc |
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