Original Site Title: Appetizers and Lessons for Mathematics and Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept Development with How-TOs Français: 26 pages
for college students, gifted teens, home-tutoring and K1-12 schooling; and for avid readers in school and out. See site volumes.

Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons. See Site Map

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 12+: 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

Welcome: Site content may develop critical thinking, improve reading and writing, and build mathematics skills. See online chapters on on logic and pattern based reason.

Teachers: This December 2011, 5-phase framework offers a context for mathematics & logic instruction. Phases 1 to 3 focus on skills with actual or potential 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.

Site Review: Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation for high school and college ... mathematics. Read more.

Home < - Volume 1A Pattern Based Reason << Postscript A Story-Telling

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Appendix A. Story Telling
Key Talent

The ability to tell, invent and follow stories is transformed in education and research into the ability to tell, invent and follow explanations and instructions as a means to enrich, advance and provide a context for observable skill development, and a means to share what is the mind. In essence telling stories, one step at a time, and one step after another (in order perhaps) is the key to knowledge of folklore, culture and technical knowledge. We may judge stories as fictional or not, to greater or lessor extent. We judge the consistency of stories in accordance with our knowledge or view of what is possible or allowable. So a good alibi allows a detective to ease or remove suspicions about who committed a action. In writing or inventing a story, an author will attempt to avoid immediate or implied inconsistencies. What Louis Carroll's Alice sees in Wonderland or Through the Looking Glass provides an story to follow, one that is not realistic. Shakespeare and Moliere through their writing create pieces of fiction for us to follow on paper or even to act out. We may judge the characters and plots in stories and plays by our own ideas on what is not possible. But in telling stories, the tellers provide us with a imaginary world to follow and judge. Likewise in telling theories or providing instructions for the operation of a machine or the creation of a product, researchers and teachers are telling us stories, ones that may correspond to and be useful in practice. The composers of a story may adhere to rules and criteria for consistency, rules and criteria that say what is or should be possible or not, in order to decide or conclude what may be in the story or not. Chains of reason may be employed to find what a story implies or requires. The composer may include or not, those implications or requirements in the telling and extension of a story or theory.

The developers of empirical theories in science and technology tell many small technical stories which individually work and are practice in special cases, but not necessarily in all cases. High school and college chemistry for example include alternative theories for the identification of acids, salts and bases. The oxymoron acid-salt reflects the idea that what is an salt in one theory is an acid according to another (a litmus paper test say).

reality, imagination and fiction - exploring ideas, continued

The author of a story in a book or a play creates an imaginary world for us to visit in our minds. The story may or not be consistent with our knowledge of real life. More and less can be suggested in a story than occurs in real-life. Stories can be fictional, half-fictional, approximately true to life or true.

Stories may explain or describe how things came to be. Stories may provide lessons a for reader directly or through the words and interpretations of another. Stories may give us ideas of what to do or not. Stories have plots and chains of events or reasons to follow, real or not. Stories presented on stage as plays may include not only words but also actors and props to make the plot or reenactment easier to follow. Actors have scripts to follow. Actors are defined by their names, costumes and actions.

Most of us, many of us, have the ability to follow a story, its sequence of scenes with words and events, and to recognize what is real or pretend. We can learn stories, invent them and tell them to others via spoken and written words. Stories can be told and retold in ways that are almost repeatable and reproducible. Our knowledge of a culture may come from its stories and myths.

In cooking and construction, plans and recipes give or suggest sequences of steps or actions to take to arrive at results. The steps and the results should be repeatable and reproducible. Technical know-how is based on rules and patterns to follow plus some judgment as to when they can be applied. Trying to apply rules and patterns when items they require are missing usually leads to bad results.

In mathematics, science and technology, as in daily life, there are stories to follow. These stories, normally called theories, describe a situation (say what is what is assumed) and describe as a well assume methods for arriving at results or conclusions in a step by step way. The authors of these stories or theories would like their consistency with reality. A theory is inconsistent with reality if it says two exclusive events occur at the same time or if predictions based on it fail. Unfortunately, the author of theory to say what should happen may capture a pattern in theory that works in some circumstances, but not all. So a theory may be applicable and sufficiently consistent with reality to be useful in some circumstances - those it reflects - while failing in others.

Knowledge in mathematics and science and technology is based on theory and practice. A method or procedure describe in a lab or controlled circumstances how following certain steps will give a result. Those steps and the results, done carefully enough, appear to give repeatable and reproducible independent of the doer. Methods that work in practice may be described and accumulated, and used one at a time and one after another to follow steps and arrive at results, one at a time and one after another. A skilled practitioner may recognize when one method can replace another because it gives the same result or a more convenient result.

Geometry was codified in the works of Euclid, about 300 B. C. The codification consisted of assumptions or definitions about points, straight lines, circles, triangles and the geometric figures composed from the latter. The resulting theory or theories was presented not on stage, but on paper (a prop) with the aid of rulers and compass (more props) to provide construction methods and to suggest and describe results and conclusions one at a time and one after another. Students and teachers and philosophers could follow explanations one at a time and one after another in way that follow some or all of the strands of thought in Euclid's work. The codification provides a mechanical knowledge of geometry because each of us in following the steps should verify that the steps are valid, that the implication rules used in each step are justly applied.

The foregoing gives rule- and pattern-based chain of reasons independent of the followers and authors. All that provides a model for making and arriving at conclusions with rules and patterns not only in geometry, but also in other disciplines where rules and patterns are valued as guides. But this model for reason has its limitations.

Rules and patterns describing what we have observed, drawn from experience, are not absolute. We do not know if they are fully reliable, or we may not precisely when they apply, if at all. When rules and patterns are not reliable, a risk appears. What they suggest, one at a time and one after another, may not be consistent with reality. None the less, recognizing rules and patterns in a subject provides a means for accumulating know-how for arriving at results, and a limited know-why. The latter is given by the chain of reason or suggestion with rules and patterns, reliable or not, that led to a result. (Implication rules and suggestions in a theory may themselves rely on the need for a theory to be consistent. See above).

Teachers & Tutors: Site pages offer better or best practices for providing skills - simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do. Others are welcome to refine or exceed it. Please do.

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

The Logic of Injustice: How Texas sent an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate is not compensation for years or decade of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

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

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.


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Home < - Volume 1A Pattern Based Reason << Postscript A Story-Telling

[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]


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

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