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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. |
Operational ViewpointStudents need an operational command of fractions, logic, algebra, geometry, trig and calculus. Seeing how to follow multi-step methods to obtain and present results in a repeatable, reproducible, readable and therefore verifiable (right or wrong) manner may be source of confidence in the reliability of mathematics and a source of abilities or a sign of intelligence for further work and studies. For students with no immediate interest in the know-why, a focus on the practice, an operational command of key skills and concepts may make comprehension later of the know-why easier and more appealing. For high school mathematics and calculus, fraction sense and an efficient command of arithmetic with fractions is more important than full comprehension of why the calculation methods work, but seeing the why, depending on the student, may help with the practice. Logic or chains of reason appear in many places in mathematics. A mix of logic and rote learning may be optimal.
The thought-based development of applied mathematics present in site pages may stand alone or be seen as platform for further studies in modern mathematics, pure or applied. High school mathematics with its reliance on diagrams and coordinates for its comprehension and development is mixed rather than pure mathematics. The exposition of mathematics, the introduction of algebra or the shorthand roles of letters and symbols has been confusing in the past for many literate students, skilled and intelligent outside of mathematics. Innovations in site material may make existing mathematics courses easier while setting the stage, trust but verify please, for expositional or content changes in high school mathematics and calculus
The implications or scope of site material grew from just a few ideas to submit to educational authorities to a full, self-contained theory for changing and improving mathematics education, all driven by the inductive principles for instruction and by reports of poor results in high school mathematics. Writing began in 1991 to report and develop further fall 1983 starter lessons for logic, algebra and calculus which had been useful in easing or avoiding difficulties in college classrooms 1983-89, lessons which had been motivated by a sense of incompleteness in the introduction of skills and concepts. More:For an operational command of arithmetic, logic, algebra, trig, complex numbers and calculus, students need not see a logical development of geometry. Students may obtain some of the algebraic-deductive maturity needed for calculus with or through site coverage of algebra and logic if the arithmetic (field) properties of complex numbers and the decimal representation of real numbers are assumed. For gifted or interested students, the field properties can derived.
In mixed mathematics curriculum (course design) before the study, if any, of pure mathematics, the full axiomatic development and connection of rules and patterns is not for beginners. Courses instead may focus on a local thought-based development in which some rules and patterns are explicitly assumed or given, and then combined to obtain further rules and patterns. Here the ability to combine rules and patterns is the objective. Where a full thought based development is too ambitious, too many details for students to follow, this local objective provides a more accessible alternative. Mathematics education to the level of advanced calculus may have the role of providing a connected, geometric, physical and thought-based development of skills, concepts and comprehension which is functional or operational, and which provides the algebraic- deductive- geometric maturity sufficient for an operational command of mathematics, and sufficient to allow but not compel students to study a more rigorous, axiomatic development and written codification of mathematics. In this role, ease of comprehension may chosen in the initial development of skills and concepts, so the logic and results are easily understood and repeated. For example the easily understood area-based development of column methods for the multiplication and addition of polynomials is only valid when coefficients and variables are non-negative, but the column methods once learnt, can be explicitly assumed for real- and even complex value coefficients and variables. Advanced mathematics courses can provide the missing rigour. Students may be content in the first instance with mastering the use and combination rules and patterns one at a time and one after another in a repeatable, reproducible and thus verifiable manner apart from technical details that may alienate or impede their comprehension. At the same time, those details should available in appendices or references for students wanting more. In mathematics starting with arithmetic, students need to learn or show how to arrive at results in a repeatable and reproducible manner, and in that learn or show a mistake in step of a method leads to results that nor repeatable and reproducible. The thought-based development of mathematics is possible after the arrival of the skills and patience to needed apply rules and patterns with care and precision, one at a time and then in combination, one after another. The ability to recognize and combine rules and patterns, once seen and learnt, allows for a thought-based development and connection of skills and concepts, those to come and optionally those previously covered. The ability to follow rules and patterns in repeatable, reproducible and therefore objective ways needs to be cultivated along with some caution. That is rules or patterns we apply or follow in this manner may be wrong or off, and in need of correction or abandonment. That is where critical thinking appears.
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