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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. |
Evaluation of the North American Mathematics Curriculum - Hook, Line and SinkersThe modern mathematics curricula, say 1955-80 inconsistently introduced ideas from higher level mathematics but provided a nearly expert, discipline-based, discipline-centered approach to course design and delivery, with a few awkward elements. The olde problem of too many symbols and not enough words in the introduction of algebra was not recognized and so persisted. The decimal-free nature of modern mathematics - its lack of dependence on the decimal representation of real numbers - meant the common use of decimals, required in high school arithmetic, was not sanctioned and implied the decimal viewpoint of error control and continuity, a view that lingers with the study of scientific notation a * 10k for measurements (with 0.1 < a < 1), was otherwise avoided. The discussion of ratios a :b and multiple a:b:c also continued in an awkward manner. The sprit of the modern mathematics curricula was not child-centered. It was discipline centered. It focused on the elements of mathematics which appeared, which would be needed for comprehension of high level mathematics in a context-independent matter. That focus provided a hard route to follow due to the lack of a clear introduction of algebraic concepts and due to the avoidance of decimals - the sanction of their use in daily life (weights, measures and calculations) and the absence of any dependence in the high school & college development of mathematics. That focus made learning and teaching harder than need-be. The new fashioned (context-free) description of functions as sets of ordered pairs that satisfy a vertical line property appeared too suddenly and too absolutely. The companion concepts of - how one number depended on others - and function notation y = f(x) should be emphasized first. The modern mathematics curricula selection and introduction of skills and concepts was not optimal. Its introduction was nearly expert, but not expert enough - too much enthusiam, not enough thought. In recent decades, factors outside of the discipline led to curriculum reforms 1989 onward that have ignored and compounded the earlier difficulties in course design and delivery First, the end of streaming in course design and delivery, the merging of course content for enriched instruction into general instruction added topics not essential into the high school education of every student. Second, the rich treatment of Euclidean Geometry was judged too hard, too intimidating for the general student, so it was dropped - site pages indicate a leaner, minimal treatment of Euclidean Geometry, one that depends on direct use of logic. Third, arithmetic drill, practice and correction was considered not important and so de-emphasized in North America and UK(?) schools in favour of calculator use and spreadsheet use. But students need to have an automatics, efficient command of exact arithmetic with whole numbers and fractions, one that does not require them to reach for a calculator for every simple calculation, if they are to master algebra, trig, functions and calculus That is a discipline-based view. Anything less delays or dilutes high school and college level mathematics - changes the discipline in a way that earlier masters would not understand - and so undermines any reason for the study of mathematics, year after year in high school. So course content needs to be maintained and protected by discipline experts. Mastery of the skills and concepts through their ability to do calculations and follow rules and patterns in a repeatable, reproducible and hence verifiable manner. That requires care and precision. It can be a struggle to understand precisely the chains of reason, verbal and symbolic, in a mathematics text due to steps to large, Not every one has the patience for it. The high school and college exposition of mathematics from algebra to calculus may make that mastery harder than need-be with algebraic skills and concepts introduced awkwardly. Site pages point to a remedy for that.
But factors not expert in mathematics, the soft science in the form of psychology and theories of learning may call for critical thinking and independent judgment but oppose the mastery of rules and patterns, alone and in sequence, for the sake of repeatable, reproducible, verifiable and reproducible results. That points to a conflict or inconsistency between expert views of mathematics and hard sciences - how university professors in the hard sciences and mathematics might value and define their disciplines - and the anti-rule, anti-bureaucratic but still bureaucratic applied and developed theories and practices for education reform. Factors who are not expert in mathematics may influence and control course design and delivery Mathematics course design and delivery should identify what skills and concepts are essential to provide a curricula which is learn but effective. The advance in site pages for the exposition of the mathematics suggest how. Those advances and the question
provides mathematics education committees in schools and colleges with opportunities to make learning and teaching simpler and more effective. Course design and delivery with some variations may be built on the collection and development of appetizers and lessons easily understood and repeated by teachers and effective in the classroom. |
www.whyslopes.com
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