Site Material: Key Notes and Themes

1. 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.    
   
   
2. 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. 
   
3. 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.
   
   
4. 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.?)
   
   
5. 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. 
   
6. 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.
   
7. 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.
   
8. 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.  

www.whyslopes.com
Mathematics Education  Essays etc

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4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.

Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
In for a Penny
Constructivism and Cognitive Theory
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I  - Fraction Focus
Sec II -  algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links

Help Me Learn/Teach;

1. Algebra
   words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)
2. Arithmetic
   - exercises
   - with fractions
   - videos on primes, lcm, gcm,lcd, square roots etc
3. Calculus - geometric preview, algebraic preview,
   3 study guides,
   much more
4. Complex numbers
   -starter lesson with java applet - easy consequences for trig & vectors in the plane
5. Education
   - Empirical Course Design & Delivery
6. Fractions
   - alone
   - by rote
   - with algebra
   - videos

7. Functions - introduction
   hindsight - composition aka
   substitution -
8. Geometry, Euclidean - Correspondence of triangles,  Triangle construction,  duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
   
9. Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
10. Logic
    - First Steps -
    Symbols in Logic -
     Occurrence & Truth Tables - Indirect Reason -Indirect Reason More
11. Proportionality
    - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions
12. Real Analysis
    - Decimal View of concepts and of proofs
13. Rules &Patterns in Science, Technology & Society - Pattern Based Reason
14. Mathematical Reasoning, empirical, inductive or deductive
15. Units
    - in rates & slopes & (?) derivatives
    - in ratios & proportions - slopes & rates included
16. Complex Numbers & Vectors & Trig
    -  trig expression for dot & cross - cosine law

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Wordy Site Tour:  Not too Bad ,  Vol 2. Foreword,  1A Foreword,  2  Logic Puzzles,  
What is a Variable  Calculus Starter Guide  Vol 1 Foreword,    
 
Favourite Sites:  The World Clock - Time Zones Around the World  
 BBC News  and mathematics portion of  English National Curriculum  

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