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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. |
Three Goals to Set for students.
In arithmetic and beyond, students need to learn to apply rule and patterns one at a time and then in combination, one after another, in repeatable, reproducible and hence verifiable manner. In days gone by, precision figuring skills were taken as a sign of intelligence or potential to follow, if not bend, rules and methods, with precision to meet the needs at hand. Rules and patterns with repeatable, reproducible and therefore verifiable results literally provide a base for society to function, but there is a caution. Rules and patterns once found or given need not be fair, nor sustainable in the long-term. Their assumption always involves some risk. The knowledge of how to use rule and patterns in a precise, repeatable and reproducible way, and the knowledge of the benefits, origin and limitations are both needed for critical thinking and intelligent problem solving at many levels. II. Supporting Aims A and BAfter arithmetic, an operational command of quantitative skills sufficient for mathematics to pre-calculus level may follow from lessening algebraic difficulties as indicated the site entrance, from mastering logic and from the assumption and geometric interpretation of the properties of real and complex numbers, from the easier consequences of those properties, and from the assumption of that all real numbers have decimal expansions, finite or infinite. The geometric introduction of complex numbers only requires and involves the junior high school level familiarity of coordinates, translations, reflections and rotations in the plane, and may be use to develop that familiarity.
The support of aims A and B advocated here may be simple, short and effective enough to allow more students to start calculus while also serving the needs of students heading for business and technical trades (surveyor, plumber, carpenter or electrician) for which calculus is not normally required.
III. Supporting Aims B and C
The mixed mathematics development of synthetic (coordinate-free) 4. Euclidean Geometry in site pages inductively suggests and clearly identifies geometric rules and patterns, those assumed for use in deductive reasoning. There is motivation here for the indirect statement of the parallel postulate as given by Euclid, namely the assumption that two lines segments extended will meet on side of a transversal will if the interior angles on that side of the transversal sum to less than two right angles. This coverage of Euclidean geometry with the selection of a unit length and assumptions about coordinates and their decimal representation to imply the field properties of real and complex numbers taken as assumptions in the support for aims A and B above. Most, if not all, of the deductive chains of reason offered here will be direct. Ease of exposition, making the ideas more accessible, is the objective here. That being said, in the development of Euclidean and then Analytic Geometry here, there is focus on the possible origins of assumptions - how they can be suggested by and extrapolated from experience. Besides this, there is a focus on deductively deriving the consequences. IV. Modern Set-Based MathematicsModern mathematics with its set-based description of axioms for real numbers given (or derived from assumptions about sets or natural numbers) provides a geometry-free, model for understanding, describing and developing the properties of real and complex numbers, and also properties of functions which appear in calculus, all apart from the use or mention of decimals. Geometry-free means there is no dependence on diagrams or suggestive reason. But there is a great dependence on direct and indirect deductive reason, and on the shorthand role of letters and symbols to codify, record and develop concepts and results. In North America, if not elsewhere, modern mathematics curricula from birth in the late 1950s to abandonment in the late 1970s or 1980s with their set-based presentation of axioms for real numbers aimed to prepare students for the more rigorous treatment met in university programs of study in pure mathematics. But the introduction of modern mathematics curricula used geometry to introduce and illustrate the properties of real numbers, avoided all mention of decimals in the representation and properties of real numbers, and required arithmetic skills with decimals in examples involving calculations or coordinates, and used geometric diagrams to introduce functions. There are was also no systematic development of the algebraic way of writing and reasoning. It was just assumed. So the modern mathematics curricula while preparing for pure, geometry-free and decimal-free, mathematics did so inconsistently and awkwardly. |
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