Key Notes and Themes

Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com) \|\|Définition d'une variable \|\| Algèbre \|\| Arithmetique \|\| Logique \|\|La raison basée sur les règles et modelés\|\|
Online Volumes 1, Elements of Reason. 1A. Pattern Based Reason 1B. Math Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes & More Math (Optional Book Orders)	More Site Areas 1. Help Your Child or Teen Learn 2. Solving Linear Equations 3. Fractions Ratios Rates Proportions & Units 4. Euclidean Geometry 5. Analytic Geometry/Functions 6. Number Theory. 7. More Calculus	More Site Areas 8. Complex Numbers 9. Qc Maths Education 10. Secondary IV(?) maths 11. Real Analysis 12. LaTeX2HotEqn: 13. Electric Circuits Etc 14. Français 15. Algebra, Odds & Ends, Etc	More Site Areas 16. Math Education Essays 17. Telling & Working with Time 18. Maps, Plans & Drawings 19. Quantitative Skills for home, shopping and work 20. Statistics Useful, or Not. Test the Twiddla Whiteboard
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Site Material: Key Notes and Themes

Online chapters on logic and pattern based reason may entertain and inform. Precision reading, writing and speaking are useful in work and studies. The logic chapters may lead to them. Good luck.

logic chapters

Words have missing in algebra from the first use of formulas to calculus. Online Chapters 8 to 14 in Volume 2, Three Skills for Algebra, and its online postscript what is a variable show how and doing so enrich, clarify and extend skills and concepts for students and teachers, novice to expert. Chapter 14 in introducing the direct and indirect use of formulas, and presenting, comparing and contrasting arithmetic and algebraic solutions for the indirect or backward use of formulas verbalizes, hitherto unifying themes in secondary and college level mathematics. Teachers: The determination of proportionality constants for direct, inverse and joint variation etc would provide an occasion for the annunciation of these themes.

Fraction skills are a must for algebra. Words problems can be difficult. Solving linear equations in one or several unknowns may be difficult. The site area solving linear equations digested in full may be used to ease or avoid phobias and enrich or extend skills and conceptsvery early in secondary school if not in primary school. Recognition that words problems in secondary I and II mathematics which require the writing of one equation in one unknown are equivalent to a system of equations in essentially one unknown will avoid the absurdity of doing or requiring mentally, operations best done with algebra on paper.

For calculus, a geometric preview, and online chapters 2 to 6 plus 11 to 18 in Why Slopes and More Math may speed studies and give motivation or a context for the study of slopes and factored polynomials before calculus. This material shows students and teachers how to make the full-strength use of algebra more accessible! (Question: Where is the modern mathematics curricula which introduced similar ideas in all or part.?)

The law of signs and the existence and properties of complex numbers may be learnt without comprehension in secondary and college mathematics. Yet in Euclidean plane, a definition of addition of points with rectangular coordinates and a definition of multiplication via polar coordinates would lead to a geometric comprehension.

What comes first, the chicken or the egg? Before modern mathematics hatched, matters were met in a less formal manner, but still understood. Can the egg reappear in primary instruction? Modern mathematics and modern mathematics curricula may build or derive algebra and geometry from assumed patterns or axioms for real numbers (or sets) and the codification of geometry via coordinates. Before this chicken hatched, that is the codification, visual geometric arguments and tacit counting principles suggested manipulatively or hands-on, the properties of numbers whole to complex. There-in lies the egg. This site treatment of number theory points to a high level development of the chicken from the egg. account. Yet in retrospect, the counting, geometric and decimal strands of primary school school might be organized and rephrased so that hands-on experience with manipulatives, a primary school representation of the egg, leads to a thought-based development of the axioms. Poincare might appreciate that. The that may provide the substance of a forthcoming site area.

In mass education, the ends of mathematics instruction are obscure, not yet fully transparent. The ends of mathematics instruction need to be defined and clearly explained, so there more to learning and teaching than preparing for the next final examination. Calculus, the key to the comprehension of methods and formulas in accounting, engineering, science and technology, provides one end. But development of practical numerical and quantitative skills and illustration od reason, inductive to deductive, provides a few further ends in societies where numerical and quantitative skills and concepts for better or worse appear in the home, in buying and selling, in technical trades, accounting, technology, engineering and sciences. Mathematics itself may be out of context in societies where formal measurement systems for distance, time and quantity are recent encounters. Apart from that in pollution-age societies, students en mass may be best served by a lean path preparing for calculus, which weaves in or also emphasizes practical skills and the mastery of skills and concepts, one at a time and one after another, alone or in combination, while eliminating artifacts (evolutionary appendices) inherited from before and developing skills and concepts in a spiral, yet just in time manner. That being said, the form and content of course design from counting to calculus could be revisited, Different paths or expositions compared and contrasted to make the hard easier, to see the benefits and limitations of different paths, and to take into account physical and mental difficulties. That will require many heads.

Making the hard easier may lead to the return in leaner form of topics deemed to be too hard for student egos. In my high school days 1966-9, I suspected difficulties in mathematics came from steps too large and words missing in the introduction of algebra. Then, a decade and a half later, in fall 1983 as an instructor, I invented three lessons three skills for algebra, why slopes and two logic puzzles to make algebra alone & in calculus simpler to understand and explain; to strengthen reading, writing & reasoning; and to hint at the role of logic in mathematics. Those lessons and further site ideas stem from inductive principles for course design and delivery met in 1981 outside in mathematics; and from the earlier example of guest speakers, mathematicians and one physicist 1975-80 at McGill University. Those speakers made what was hard, easier.