YOU are better than YOU think. Show yourself  how:  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Welcome.  The ends, means and values of mathematics education represent a jigsaw puzzle with some pieces missing, or as yet undefined. In this site area is a depository for some of the pieces, and ideas on how to find the missing ones.

For navigation, click on the next or back buttons, or return to this area entrance to select another essay. Bon Appetite.

1. Site History and Content - through site reviews 1995 onward.
2. Site Eurekas - Site Highlights, an old view
3. How this Site Differs
4. Reactions to Site Material - comments & questions, good and bad.
5. Site Origins
6. About Site Lesson Plans - Another tour of Site Content

Challenges for Education Reform

1. Managing Reform - Assigning Responsibilities. (Should anyone be responsible? Should anyone be in charge? Is reform headless?)
2. Mathematics in Context - What Context?
3. What Should be Learnt and When?
4. Grouping Students - Streaming?
5. Learning Takes Time and Effort
6. Making the Hard Easier but Ignoring how and so missing the Point
7. Hook, Line and Sinker - Mathematics Education Inconsistencies Reform in North America
8. More on Mathematics Education: Covers: For a leaner curriculum, Education an empirical art,  More on testing, Constructivism versus Empirical Methods.
9. Four Skeptical Essays on Constructivism Revisited - Incompleteness
10. Euclidean Model for Development. Damage Reversal
11. Educational Follies - Learning By Discovery incomplete, cannot work, compound difficulties.
12. An Educational Inconsistency.
13. Modern Education

But teaching by indirect instruction requires not only a knowledge of what can be taught directly, but also a knowledge of how to explain all elements indirectly. Anything less invites or compound difficulties. Ouch.

Ideas and Principles For Instruction and Educational Reform

1. Inductive Principles For Instruction - systematic skill and concept development.
2. Fairness in Education - requires systematic development of all skills and concepts.
   
   Can education be fair if students are tested on natural talents instead of developed ones?  Mastery of a skill,  say the algebraic way of writing and reasoning, is regarded as a natural talent when and only when  we do not know how to systematically develop that skill or concept. Site material reduces the number of natural talents required in the mastery of mathematics.  Find the  four skills for algebra in chapters 8 to 14 of Volume 2, Three Skills for Algebra, to see how to artificially and artfully develop the algebraic way of writing and reasoning, and thus make mathematics fairer.
   
3. Apprenticeship in art, trades and disciplines, a classical view.
4. Education is an Empirical Art
5. Key Notes and Themes
6. Three Remarks
7. For a Leaner Mathematics Curriculum
8. Need for a Mixed Mathematics Curricula
9. Extent and Need for Quantitative Skills depends on your society
10. Ways to be a Better Instructor - Ideas and Methods - try with caution
11. Four Ways to Improve Education Reform, and avoid disaster.

Lesson Plans, Aims and Goals (Ends, Values and Means?)

1. Three Aims for Students - Ends and Values
2. Three Goals for Mathematics Education, etc - Ends, Values, Unifying Themes
3. Lessening or Avoiding Algebra Difficulties
4. Algebra Lesson Plans
5. Algebra, Geometrically
6. Mathematics Curriculum Shifts
7. Advice and Suggestions for Course Design and Delivery
8. Teaching Tips - from fractions to Calculus
9. Math Education Perils (Arithmetic, Algebra, Calculus)
10. Talk the algebra talk
11. First Year High School Math - Lesson Plans with Fraction Focus
12. Second Year High School Math - Lesson Plans with an algebra focus
13. Third Year High School Math - Lesson Plans with a Focus on Slopes
14. Math Wall Posters
15. How Letters Appear in Mathematics
16. Map, Plans and Drawings, a multi-year project

Links

1. Links - Just a few.
2. Activities to Engage Students - links to explore

Logic and Reason in Mathematics

Mixing Rote & Thought-Based Development

1. Cultivating Intelligence - Why value careful mastery of rules and patterns, steps and methods, practices, in a repeatable and reproducible manner.
2. Multiply Kinds of  Reason in mathematics - Essay I
3. Multiply Kinds of Reason in Mathematic- Essay IIs - On the hierarchical development of rules and patterns, steps and methods, and practices in pure and applied mathematics (mixed mathematics). What is proof? What options are there for a thought-based development and verification of college and pre-college mathematics?
4. Theory of Knowledge - Stories, Longer and longer
5. Formal or Informal Peer Review
6. Education in Mathematics, Science and Technology - All based on empirical verification and empirical skill development and verification. But in mathematics we can offer a full thought-based development while in science and technology, we can introduce the scientific method and introduce lab equipment, but can only provide a full-thought based development through visits to the lab and library. The lab alone is insufficient. 
7. Maths Instruction in General - Three Goals A B and C to Set for Student, Supporting those goals and why rewrite the curriculum
8. Operational Viewpoint - Aim for an Operational Command of Mathematics First.- For students with no immediate interest in the know-why, a focus on the practice, an operational command of key skills and concepts may make comprehension later of the know-why easier and more appealing. The calculus teacher may says to students - learn to do now and to understand later.
9. How to Set Standards for textbooks and course materials -  Need for Inspection by University Domain experts outside of Education Faculties to ensure bureaucratic course design and textbook composition does not lead to nonsense in mathematics education.

Teacher Training

1. Teacher Certification Issues and Cautions
2. Math Ed. Professors - Training, Education of

www.whyslopes.com
Mathematics Education Essays
57 or so 

Area Entrance & Hub
Ideas for Better Instruction
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
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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