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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Apprentices and Masters in Arts, Trades and Disciplines, a classical view of learning and teaching
In many arts, trades and disciplines, the apprentices learn to follow and apply rules and patterns, steps and methods, practices given or demonstrated by masters, and to do so with sufficient care to obtain repeatable, reproducible and verifiable results. The apprentice in the first instance follows the customs and conventions of the master. In this education, rules and patterns, steps and methods or practices may be learnt with or without explanation of why they work or why are they are followed. But over time, the apprentice may see or the master may demonstrate how to combine rules and patterns, steps and methods, or practices to compound them to form further ones in a repeatable, reproducible and hence verifiable manner. That process of combining or compounding lends or creates a hierarchical structure to the learning and mastery of the rules and patterns, steps and methods, or practices. The extent to which the apprentice may meet and obtain an operational command of art, trade or discipline may be subjective - some apprentices will develop abilities beyond that of their master, while others will equal or develop lesser skills. Masters and apprentices alone and in groups may accumulate rules and patterns, steps and methods, or practices to share and extend in a repeatable, reproducible and therefore verifiable manner. Moreover some rules and patterns will fall into disuse or be forgotten while newer ones arise. Each participant in the art, trade or discipline will see practices come and go, and may in time, see why. Intelligence within an art, trade or discipline appears and is used when or while apprentices and masters are carefully to apply their practices, carefully, while learning about the benefits, flexibility, limitations and origins of elementary or compounded (combined) practices. In this, there may some be approximations or some uncertainty known to the master, if not the apprentice, through trial and error, and/or stories of what is feasible or not. The master of an art, trade or discipline in meeting situations that have been handled before or not, will look for practices that work, and try to duplicate or refine them, and if need-be and if-possible, invent new practices to handle new situations. Operational command of a discipline may be enhanced and advanced by stories or theories to explain why rules and patterns work. On the other hand, the latter may also unnecessarily inhibit and limit the operational command. Whence not all is certain. Over years and decades, the customs and conventions of an art, trade or discipline in being refined may become more and more contrived and cease to be immediately obvious to the apprentice. Whence the apprentice needs a guide or a master to show what is possible, and self-instruction is impeded. The operational command of an art, discipline and trade with a history of custom and convention including a hierarchical organization in which some skills and concepts depend on earlier ones in ways that took time and effort to discover and perfect requires guidance from masters of the art, discipline or trade. The apprentice is well-advised to seek that guidance and stand on the knowledge and wisdom of others, including practical and theoretical knowledge of the benefits, limits and origins of steps and methods, to avoid an ad hoc, incomplete and most likely slow, construction or reconstruction of the latter skills and concepts. |
www.whyslopes.com
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