Appetizers and Lessons for Mathematics & Reason Français: 26 pages
A 1100+ page site with math-free logic chapters and wordy algebra chapters.
For comprehension, study site chapters and steps. Go beyond rote learning.

Logic mastery strengthens comprehension and so improves home, work & study abilities .
Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits

Ages 14+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math
, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles
Forewords + leading chapters give original reasons, still valid, for site content & growth.

Site Review: Mathphobics, this site may ease your fears of the subject, perhaps even help you njoy it. ... unintimidating, sometimes funny and very clear. ... . Read all. Continue with Volume 2, Three Skill for Algebra.

Site Review. Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation ... Read all. See site books as well.

Teachers & Tutors: Site material uniquely explains common troubles in terms of steps too large or missing. Plus, this December 2011, 5-phase framework offers a context for mathematics & logic education. Phases 1 to 3 may focus on skills with actual or potential local value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus.

Location: Site Entrance < Archives < Mathematics Education Essays << education an empirical art

[1] [2] [3] [4][5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64]


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.

 

Bookmark this page

Road Safety Messages. First Question: When and why should you face traffic?

More Site Folders and Pages

Parents: Help your child or teen learn:

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.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

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.

More Algebra

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.

70 Calculus Starter Lessons


Return to Page Top

Location: Site Entrance < Archives < Mathematics Education Essays << education an empirical art

[1] [2] [3] [4][5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64]


Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

All trademarks and copyrights in this are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest © 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved --- Skype or Email to contact.