Logic mastery strengthens comprehension and so improves home, work & study abilities .
Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits

Ages 14+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles
Forewords + leading chapters give original reasons, still valid, for site content & growth.

A New Mathematics Curriculum

Volume 1B. Math Curriculum Notes, 1996, begins with progressive or inductive principles for observable and verifiable skill development, identifies barriers to skill development and then offers ideas to lower the barriers and to make skill development easier and richer, ideas implemented in site pages.

Volume 1B, Mathematics Curriculum Notes, begins with inductive principles for the step by steps in arts and discipline where skill mastery is the aim.   The leading chapters in 1B describe and reflect on mathematics education difficulties - nuances or small inconsistencies - that affected the modern mathematics curricula 1960 to 1990 in Canada and the USA.  

• Mathematics education world wide may be progressive in part, but not full. In particular, from first use of formulas to calculus, the shorthand algebraic roles of letters and symbols is a great mystery to  more than need-be. The fault lies with conventions that avoid or do not use words in the introduction and rationalization of that shorthand role. This algebra guide points to remedy, one that is continued in the site ideas for senior high school mathematics and calculus. 

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• Since the advent 1990 onward in North America of skill-adverse educational theory , theories that say true knowledge is located in the mind apart from any skill development or focus,  the exact and efficient command of arithmetic is not provide in most primary schools and not emphasized in secondary schools. The site area Help your Child/Teen Learn identifies 18 Booklets for preschool to grade 3, and grade 4 to 8, to help parents how to check and if need-be develop primary school mathematics skills, arithmetic included, in a lean and efficient manner.  The same booklets show primary teachers what skills and concepts they need to include. That should provide a lower bound for primary school mathematics. There would be harm in completing the grade 7 or 8 booklets before the start of grade 7 or 8. 
  
  For primary and earlier mathematics instruction in the process of trying to identify what should be taught and when,  I found 18 books,  eleven for preschool to grade 3, and another seven for grades 4 to 8 which provide a lean but sufficient preparation  for high school mathematics in progressive manner - many students should be able to cover booklet by the end of grade 6 or 8 before the site program for junior high school mathematics begins.

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• The algebraic way of writing and reasoning requires a coherent and efficient command of arithmetic with whole numbers, fractions and rational numbers.  The site  arithmetic guide includes most of the usual skills that requires, but also adds a few methods or skills that will complete the command or its development. 

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• Late primary school and early secondary  mathematics may introduce the use maps and plans without and then with coordinate systems for squares and then points.  Showing students how to draw and use figures and plans drawn to scale on a map for the measurement of angles and distances, and areas provides a practical introduction to geometry.  Except for the few instances where students may survey and measure real-objects, geometry for the most part is done on paper.  The methods of drawing figures or plans to scale provides a very practical method for solving problems even before any formal mention of similarity theory or trig functions.  To learn more, see the site  geometry guide, and the first third of the site section Maps, Plans,  Similarity & Trig,  

Goals: Mathematics and Logic Skills with Take Home Value) Logic and money  mastery is a must for consumer self-defense.   Arithmetic skill development  in showing students that a mistake in one step of a calculation or process leads to further errors implies the care or  attention to detail needed  to ease to avoid many difficulties in work and studies. Arithmetic skill development done well introduces or provides  ends, values and methods  for work and study.   Contracts, agreements and instructions at home, at work and in studies require precision in reading and writing. Many contracts involve money matters.  The in-school mastery of arithmetic alone or with some algebra, and  too,  would have immediate and long-term take home value for students for understanding and negotiating or avoiding agreements.  Logic  and arithmetic, skill development together should lead learners to care and greater  precision in reading and writing. Thus logic and arithmetic skill development has take-home value.  Arithmetic mastery by rote if need-be is possible in the last years of primary school and the first years of high school.  Arithmetic mastery should be pushed in secondary school to include all likely money matters students and their families meet now or later.    With logic mastery there is no harm in trying to develop it in all or part  as early as possible  as long as that is done with the understanding that what is hard at nine years of age  should be simpler at fourteen or sixteen years.  For self-defense and sharper skills,  there is also no harm and some benefit in understanding the origins, benefits and limitations or rule and pattern based reason. Reference:  Volumes 1A and 2 and this arithmetic guide.

The foregoing material accompanied by these  ends, values and methods for work and study provides a very solid base for senior high school mathematics.  Logic mastery itself may be part of senior high school mathematics - all depends on the school system and student ability.

Senior High School  or First Year  College Mathematics -  three ends or three bases for further instruction  A first common, base part gives a natural stopping point for students who would like to would end their mathematics, with some topics and skill that have take-home value - serve common need - while a quick view of the role of logic in mathematics. There is more to mathematics than being given a method and data to use in it; and  a base for further studies for students who plan to pursue intermediate or advance studies in mathematics, science, engineering and commerce at an intermediate or advanced level. This second middle part  gives preparation for a light form of calculus. a light form of calculus sufficient as end in itself, or as an  appetizer for those going on to the strong form This third and last part    (incomplete) gives or will give Calculus with proofs  preparation for calculus with proofs. This part includes rigorous proofs (elementary for mathematicians at least) of all the usual theorems from a decimal viewpoint.

The site introduction to logic in chapters 1 to 5 of Volume 2, Three Skills for Algebra, is mathematics-free except for a wordy introduction to mathematical induction - the coverage of which can be left for later. Avid readers in school and out may enjoy the exposition. What is hard at eleven may be easier at fifteen and easier still at seventeen.  The foregoing logic chapters may be covering in the development of reading and writing skills. These could provide a firmer base for senior high school mathematics before or after the latter begins.  

In recent years, North American course design has omitted Euclidean-Geometry  from senior high school mathematics, nominally because it is too difficulty.  The site coverage of this topic employs only direct use of implication rules and develops the topic in a lean manner, that sufficient to support a geometric development of trigonometry and complex numbers. To give students a light sense of the role of logic in mathematics, this treatment of Euclidean Geometry may be included in the common part  for all, or the middle part for fewer. I would prefer the former.  Strictly speaking, the operational assumptions tacit in the use of maps and plans with coordinates provide an alternative base for the introduction of trigonometry and the development of complex numbers.  The Chinese square dissection proof the Pythagorean avoids the need for harder Euclidean Geometry style proofs.

The site introduction of complex numbers gives a visual or geometric introduction of complex numbers in which the commutative, associative and distributive laws or properties are obtained from corresponding properties of real numbers and the observation that image under a rotation of the midpoint of a line segment between two endpoints is the mid-point of the line segment between the image of the two endpoints. This introduction can be done at the high school level of mathematics rigour before the development of periodic trigonometric functions.  The early and trigonometric free development of complex numbers shortens the exposition of periodic trigonometry functions and immediately implies trigonometric formulas for the dot- and cross-products of points or vectors in the plane.  It is bit of mystery to me why this shorter route did not appear in the mathematics curricula, earlier. 

The third and last part is presently incomplete.   The site coverage of functions and proofs for the theorem have yet to be embedded into third part.  Sets and set notation remains to be used. Set notation is useful in the description of intervals and functions. It is useful too in the development of counting methods in combinatorics or the for sake of probability calculations.  It useful in the exact or precise description of probability skills and concepts.  The operations of intersection, union and complements correspond to the operators AND, OR and NOT in logic. Set theory was introduced with great keenness in the implementation of modern mathematics curricula.  Presently I am satisfied with the arithmetic, algebra and geometric guides of the lower bound for mathematics course design described above.  They prepare well for senior high school mathematics. But the optimal form of the senior high school mathematics is still open to debate. While the mathematicians involved in the 1950 to 1960 discussion of education focused on the preparation of students headed for university studies in mathematics, science and technology - the cold war had started, the question of how mathematics education may serve common needs was not addressed.  Mathematics instruction should serve common need to the greatest extent possible, and put the service of common needs before preparation for university mathematics.  The question of how to end mathematics education  in manner that its leaves a good impression and does not become a source of discomfort is remains to be addressed.  

 Lamp an earlier program, these Mathematics education essays,  and site volumes all provide prequels to the above effort.  All being said,  for the last four decades, the professor of mathematics due to the competitive environment in which they live and are formed spend five minutes or less thinking about the content of mathematics.  It remains to be seen whether or not the technical changes proposed here will be of interest to university professors and within the reach of mathematics education professors. 

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science