key notes and themes

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.

To improve your work and study skills, start with with math-free logic chapters. Read them in any order you like. Logic mastery may teach you to read and write more carefully. That care will ease or avoid difficulties and confusion in studies and work. The logic chapters also hint of the role of logic (rule-based thought) in connecting and organizing mathematical skills and concepts.

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.

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills.

Mathematics Skills For Ages 3 to 14

Skills with take home value

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.