Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

Online Volumes
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

 (Optional Book Orders)
More Site Areas 
1. Help Your Child or Teen Learn 
2. Solving Linear Equations
3. Fractions Ratios Rates Proportions & Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. More Calculus
More Site Areas 
8. Complex Numbers 
9. Qc Maths  Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
14.  Français
15. Algebra, Odds & Ends, Etc
More Site Areas 
16. Math Education Essays
17. Telling & Working with Time
18. Maps, Plans & Drawings
19. Quantitative Skills for  home, shopping and work 
20. Statistics Useful, or Not.

Test the
Twiddla Whiteboard

[Site Entrance & Hub]Back ] Area Entrance & Hub ] Next ][Site Exit]
Education, Empirical Art


YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 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;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
     | |     |
   \             /   
\    =   /

Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

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.

Education, An Empirical Art

In empirical arts, practices with repeatable and reproducible results come first, tested via trial and error. Then theories and principles come later to summarize, to codify, to refine and even enlighten the practices. While practices or sequences of them in some empirical or hands-on arts in science, technology and business, assembly lines included,  may comply with principles and standards, even be connected and organized and designed around said principles and standards,  the forerunner to such organization or optimization should consists of methods that are tried and tested, methods that work in a plug and play manner, methods whose benefits, origins and limitations are described..

Education is an empirical art. We may not read a student's mind, how a student thinks or links together skills and patterns, yet  we can observe and test student performance, skill by skill, concept by concept, and encourage, but not guarantee, mastery of standard calculations and standard arguments or chains of reason in algebra, geometry and beyond. In some disciplines, not all, there are right and wrong answers due to methods that lead to repeatable and reproducible, and thus verifiable results independent of whom-ever applies the method. Learning how to apply and combine methods carefully to obtain reproducible and thus verifiable results is an old sign of intelligence in many old arts and disciplines in business, trades, science, engineering,  technology and bureaucracy. The latter is subject to the limitations of rule and pattern based thought and practices, and the critical knowledge that not all is certain in empirical based thought and practice. 

Critical thinking in science and technology begins with an awareness that what we hope for, dream of or construct in our minds remains speculation or faith IF or WHILE it or its consequence cannot be observe or tested directly to be corroborated if not confirmed. The foregoing is a rebuttal to the constructivist theory of learning, the part which opposes testing, the existence of questions with right or wrong answers, and which says student knowledge, if individually constructed, should not be contradicted.  Empirically sound education must oppose wishful thinking. That being said, constructivist methods for engaging, authentic, genuine material and the development of critical thinking could be incorporated into education as an empirical art.

More on Testing. Knowledge empirically found or tested is relative and not absolute. Instruction which relies on testing skills and concepts can only identify errors in the mastery of the latter while correct responses only confirm, but do not guarantee mastery. But the level of student competence in a discipline defined by skills and concept mastery can be estimated from the degree of difficulty, the unlikelihood of correct responses if skills and concepts have not been mastered,  and comprehensive of a test or series of test. Here individualized testing may be informative that mass testing. Empirical soundness of instruction and testing, the issue of lessons and associated tests with  repeatable and reproducible results locally and beyond, should not be scrutinized in an absolute manner.  Cognitive theory should look at education as an empirical art.

While a teacher can not read the mind of a student, a teacher may see and correct mistakes, minor to major, in the content and style of student writings and further  endeavors or products, so that the student may learn from his or her mistakes, and possibly learn how to make fewer mistakes. In the short span of education, several years or more, the student will meet subjects  in which individual construction or organization of skills and concepts cannot in the first instance replace the early collective and refined products of many minds. 

Instruction is an empirical art with value judgments and decision dependent on the subject  at hand and what students produce - observable behaviors or products only.  Any else is subjective - not repeatable and reproducible. In  particular, the constructivist approach to instruction, despite fine calls for authentic, realistic and engaging material and practices in the classroom, calls that should be heeded and empirically supported as much as possible, in its opposition to the testing and measurement of skills and performance provide vacuous standards for instruction and undermines the sequential nature of learning in which skills and concepts at one level need to be learnt and verified before the next level begins.

 

 

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


[Top of this Page][[Site Exit] Back ] Area Entrance & Hub ] Next ]
[Comments, Reactions, Feedback]
www.whyslopes.com

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,
a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.