About Site Volumes

(1) Elements of Reason, its foreword,  introduces site books and  site objectives.

• The first part 1A,  Pattern Based Reason, on striving for objectivity, describes the benefits, origins and limits of rule- and pattern-based thought and methods in general, that is science, technology and society, and also in mathematics. 
  
• The second part  1B, .Mathematics Curriculum Notes describes obstacles to learning and teaching mathematics, and proposes a solution. The foreword of Volume 1B, Mathematics Curriculum Notes, and chapter 2 in Volume 1A, Pattern Based Reason,  points to inductive criteria for completeness of content in course design and delivery. The incompleteness and hints of inconsistencies in the exposition of mathematics are noted too as barriers to learning and teaching.
  
  The two parts together, that is Volume 1 in full,  provide a base for building skills and knowledge, and for  judging and refining educational practices. Cognitive dissonance or confusion in pre-college course design and delivery is implied by the growing practice since say 1990 of  pedagogical principles governing pre-university education  in contradiction with the views of university level  mathematicians and scientists on skills and knowledge in their disciplines. See education, an empirical art for more comments. Intelligence or critical thinking in mathematics and logic is based on the ability to use rules and patterns when they apply in repeatable and reproducible manner.
  
  People who do not yet like mathematics may delay mathematics studies and prepare for success or less misery in mathematics by reading Pattern Based Reason in full, or these  logic extracted from the latter and put at the start of Volume 2 below as preparation for algebra and beyond. Pattern Based Reason,  describes the benefits, origins and limits of rule- and pattern-based thought and methods in general.

(2) Three Skills for Algebra shows how describing or talking about numbers and quantities can become part of the common knowledge of mathematic before and then beside formal ideas in mathematics. Leading logic chapters may improve reading and writing in all subjects, not only mathematics. If you meet difficulties or confusions in studies or work, a remedy for them is to master logic. See if that works.

Equations and formulas may be used forwards and backwards. In the backward use, there are numerical and algebraic solutions.  Talking about forward or direct use of equations and formulas, and talking about numerical and algebraic solutions for the backward or indirect use provides in retrospect, a fourth skill for algebra, and verbalizes the themes or aims of mathematics at the high school and college pre-calculus levels.  Learning to describe or talk about numbers and equations provides words or missing links for understanding and developing mathematical skills and concepts. There clear introduction of the fourth skill  in  Chapter 14 involving compound growth. The development could & should begin with the forward and backward use of formulas for perimeters and areas, and formulas for proportionality relations - See - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions for proportionality relations. The main contribution lies in the greater and clearer use of words to describe and develop an existing theme or thread in secondary school mathematics.

See if chapters 1 to 14 in Volume 2, Three Skills for Algebra give a better understanding of logic and algebra.  See algebra difficulties below and a fourth skill for algebra in Volume 2 - on second thoughts, the volume was misnamed.

(3) Why Slopes and More Math shows how algebraic difficulties can be eased or prevented in and even before calculus begins.

To understand why slopes appear repeatedly in algebra, see the geometric & algebraic calculus previews in the first 6 chapters of Volume 3. The same previews may ease or avoid difficulties (algebra shock)  in the first weeks of calculus and before that in factored polynomial, sign, zero and extrema location. In chapters 14 to 18, the decimal viewpoint or error control introduction of limits, provides a second way to ease or avoid further algebra shock in calculus. The theme, saying how to compute a number or quantity defines it, also provides a perspective to make calculus  more accessible.

Geometric and algebraic previews introduce calculus while providing a context for why slopes and factored polynomials appear in earlier mathematics courses. There-in lies the first way to ease or avoid difficulties in calculus.

Calculus requires key elements of arithmetic,  algebra, geometry and trig at full strength. There-in lies a subject geared standard for instruction, student centered  or not, and for mathematics instruction before calculus to be meaningful and focused. Some drill, repetition, drudgery and correction will be required as students and teachers follow or cover as is or consolidate earlier discovers or inventors of  mathematics to see the benefit of  repeatable, reproducible and hence verifiable answers first, before any emphasis on critical thinking or open problems.

See too in chapters 14 to 18 the decimal viewpoint or error control introduction of limits for a second way to ease or avoid difficulties, and also to  meet  the theme, saying how to compute a number or quantity defines it. Those perspectives may  make calculus & beyond more accessible.

The More Site Areas

1. Helping Your Child or Teen Learn offers parents advice and directions, approximately correct, for some circumstances, not all.
   

2. (Ages 14+)  [Solving Linear Equations via fractional operations on Stick Diagrams] Explore these lessons consolidate fraction skills and sense in learning or teaching algebra.  This site area can be combined with chapters 8 to 12 and 14 to 16 in Volume 2 to provide junior high school, senior high school and adult students with a solid base in algebra.
   

3. (Ages 14+)  [Fractions, Ratios, Rates, Proportions & Units] - a precise reference for  instructors and for students with gifted or stubborn reading skills.  For an operational command of fractions, master simplification, cross-cancellation in multiplication (an exercise in simplification), division of fractions (another exercise in efficient multiplication and simplification), and then addition and subtraction with least common denominators and more simplification. Simplification may employ rules for recognizing multiples of 2, 3, 5 and 10, and exploit or emphasize 10 or 12 times table. Instructors: (A) The fraction part of this site area can be combined with the solution of some linear equations with fractional operations on stick diagrams to consolidate and extend fraction skills and sense. (B) The discussion of ratios, rates, proportions and units, because of its algebraic nature may be best digested after the mastery of [Solving Linear Equations in all or part, and after chapters 8 to 12 and 14 to 16 in Volume 2.
   

4. (Ages 14+)  [Euclidean Geometry] - correspondence, isometry, bisection,  perpendiculars, properties of parallelograms, parallel lines and triangles, emphasis on definitions and proofs. Here is a self-contained minimal treatment, that needed for analytic geometry and trig, a treatment which employs logic in a simple fashion.  See too  logic chapters 2 to 5  in online Volume 2, Three Skills for algebra.
   

5. (Ages 16+)  [Analytic Geometry, Vectors, Functions] - a collection of senior high school material, mostly needed for calculus. The collection is not yet complete, but what is here may still help.
   

6. (Ages 15+)  [Complex Numbers] - optional reading besides trig, calculus, phasors, roots of negative numbers and vectors,  nominally for college  yet simple enough for senior high school studies or  technical trades. Gifted students 14 plus may read as well - see what is not understood now, and leave the rest for later.  This site area is best explore after this   Complex Numbers starter lesson. The starter lessons includes an applet to show how to add and multiply vectors and complex numbers in the plane.
   

7. Ages 16+)  [Number Theory]  -a full theoretical development from tally marks to real numbers. Includes a thought-based development of numbers & their properties with and without decimals.  Includes justification for methods  that might be met in high school mathematics, methods  given without proof.
   

8. (Ages 16+)  [Calculus Intro] - support for  a first course on calculus appears here. See how different ways to introduce ideas may ease difficulties AND enrich knowledge.  The first chapters of Volume 3 can be read first.  Three annotated guides to calculus are available too.
   

9. Secondary IV Mathematics - this site area offers some support for the Quebec secondary IV mathematics course 436. The support is as yet incomplete, and some parts remain to be rewritten or refined.
   

10. Real Analysis:  Here a decimal viewpoint of real analysis to provide a context for the decimal free viewpoint and to make the latter more accessible.
    

11. Quebec Maths Education - Scathing Notes on. 
     

12. LaTeX2HotEqn;  The HotEqn applet provides a means to present LaTeX encoded mathematics expressions online. This site area provides an applet to automate the process of converting some LaTeX documents into webpages.
    

13. DC Electric Circuits:  The lessons provide an enriched mathematical viewpoint of the electric circuits theory that appear in Quebec 416-436 physical science course. Quebec students should explore this part for enrichment only.
    

14. Francais: ||Définition d'une variable || Algèbre || Arithmetique || Logique || La raison basée sur les règles et modelés||
    

15. Teacher's Corner - 55+  Essays on Education - Concerns and Ideas for Course design and Delivery

Mathematics is a discipline  given by rules and patterns or skills and concepts which  have to be met and mastered one at a time and one after another.