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Appetizers and Lessons for Mathematics &
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Location: Site Entrance << Mathematics Skills Year by Year |
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Primary and Junior High School MathematicsFolder webpages essentially provide a year-by-year model of what might be taught and when. The key idea is that earlier provide a base for later ones. Respecting that dependence, and not rushing instruction since that may overwhelm students appears to tbe main constraint. In each year of arithmetic and other areas, the accumulation of skills implies more variation in content is possible in the following years. The distribution of skills among age groups is an estimate based on a reading of other course designs and common work exercise booklets available in Montreal and Toronto. Hopefully the model will do no harm. Since my teenager years, I have been attracted by the deductive and axiomatic development of mathematics or its subjects. But in retrospect, my education in primary school mathematics was incomplete in that I met and learnt methods without fully understanding why they worked. That attraction, the idea that mathematics instruction should be deductive motivated the thought-based development and exploration of mathematics in site pages. Until recently, whenever I entered the mathematics classroom, I felt my instruction was incomplete if I did not explain why the mathematical methods met in a course held. But mathematics can be mastered at two levels. The higher level consists of comprehension, in all or part, of why mathematical work. The base level consists of a mechanical mastery of methods, by rote, in a way that leads to calculations and figuring that can be done and recorded step by step for confirmation or correction. Before any efficient, deductive and algebraic Euclidean style, axiomatic development of mathematics, students have to master deduction and algebra. Thus an Euclidean account is inaccessible without mastery and appreciation of both deduction and algebra. The first several years of mathematics instruction may focus on skill and practice mastery, with full comprehension optionals, but with mastery being observable and when applied without errors, leading to repeatable and reproducible steps and results. In this skill and practice development, methods to follow may be instroduced or shown by example. Over time, the inter-relationships between methods may be seen, with some implying others. In general, an operational and empirical command of a subject does not require a lean axiomatization of the subject, with skill and concept derived from a minimal set of rules and patterns. For common needs and the common person in the street, when the aim is an operational command of a subject, instruction and course design may aim to present a web of consistent rules and patterns, a web clearly described and easily understood and repeated, a web that does not have to be derived from a minimal collection of assumed rules and patterns. Such derivation may be left to those students willing to adopt some of the ends and values of pure matheamtics. The web we want to weave in mathematics will however set the stage for such a derivation while focusing on common needs first, and others needs such as preparation for college studies, second. The axiomization of an art or discipline with prior empirically developed rule, patterns or practices, may elevate some practices to the state of axioms for the sake of an deductive codification and arrangement of islands and bodies of knowledge within a web or subweb.
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Location: Site Entrance << Mathematics Skills Year by Year
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