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 2 Three Skills For Algebra << Appendix D. What to do in School and Why

[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]


Appendix D. What to do in School and Why

Volume 2, Three Skills for Algebra

A Why Go or Attend

Education is for both sons and daughters. The assumption that someone else will provide home and shelter for us is too often false. So you should get the most out of your schooling and education. Your ability to get a job, your future earnings and your future joys and companions may all depend on what you learn in and after school. All this may also depend on factors beyond your control (wars, social conditions, the economy or parents). The suggestions provided below are for some, not all eventualities.

School can be a place where you learn to master rule and pattern-based thought (logic) and where you master further skills and knowledge. You are in school to learn about the wide range of human knowledge and behavior. Try to understand whatever you might be asked to study or do. Think for yourself and ask why you or others do this or that.

In school, you should look for the ideas new to you and for the ideas worth repeating to others. That is what makes a class or a subject worthwhile. Again, ideas which you have already seen are reassuring and comforting - not much work is needed here, but you only learn from the ideas new to you. Look and search for them.

What you see in school and in books represents the experience of others. When you are observant, you can learn from the experience, skills and mistakes of others instead of your own.

History courses, English (or literature) courses, science courses and some realistic novels offer this experience secondhand. Secondhand knowledge of hardship is preferable to first hand. Ask your teachers, relatives and the others about the joys and the difficulties they met or foresee. Learn from their experience. Ask for opinions. Guidance from others requires the statement of opinions. Get two or more opinions even if you liked the first. Seek and politely allow opinions different from your own. Different points of view may sharpen or change yours.

B Health and Social Skills

You are also in school to meet people and to learn how to mix or socialize with others. To this end, join a club or group activity. See how people, including yourself behave in groups. After school, the opportunities to mix may decline - or you may not develop the habit of mixing and socializing. Choose activities you like. Try one, two or several, but leave enough time for your studies and for special events. Suggestion: each week, get three or more hours of physical exercise. This exercise could come from physical labor. Or, you may find a sport or activity which you can do now and later. This exercise should build your health without risking it and without damaging it.4

4I saw in a university soccer match or practice in 1984, a player with a small cast on his leg. I thought he was risking permanent damage.

In 1990, after twenty years of cross-country skiing for exercise etc, I began to ski in colder and colder conditions without fear. The eventual result was a deep frostbite to my cheeks, an area difficult to protect. Then for five years, exposure to the cold was an unpleasant experience followed by hours of pain or discomfort - a distraction from work and play that is not recommended.

C Suggestions for Learning

By law you are required to attend school. Make sure your time is not wasted. Make sure that some of your courses are with helpful, hard-driving, teachers. Ask them for advice on what to do or what could be useful to you. Further advice follows. It repeats in part advice given in previous appendices.

  1. Look for the ideas new to you and for ideas worth repeating. When you are preparing for a test or for a future lesson, your studying is done when you can find no ideas new to you.
  2. Try to remember the names of places, people and ideas. You can use the names in conversations, essays and tests later.
  3. Learn to read precisely what is written. This skill will serve you well. It gives you more independence both in class and when you leave school. It may allow you to learn at your own pace.
  4. Learn to take notes. When no textbook is present, note-taking skills will be needed. When a textbook is present, look at it first. (Reading it in advance may allow you to take fewer notes and understand lessons better.)
  5. Learn to type. Today, computers are used in all areas of office work bureaucracy and technology. These computers are controlled by keyboards. Accurate, if not fast, typing skills will make your exposure to computers and report writing more pleasant.5

    5This advice is valid now. Advances in computer technology - the introduction of voice-controlled dictation/computer systems - may make part of this advice stale or obsolete.
  6. Get careful thinking skills. That is, master the use of rules and patterns. Every area of skill and knowledge offers rules and patterns which you might follow. Learn to read exactly what each says. To follow, to agree or to disagree with rules, you need to understand exactly what they say and exactly what they don't say.
  7. In high school take courses which provide immediate job skills such as auto-mechanics, typing, metalwork, woodwork, drafting etc. Master arithmetic and learn to read and write carefully. Employers want skilled workers. They are easier to train and worth keeping. Even if you are planning a college education, practical job skills could get you a summer job. They may allow you to work and pay for part of a college education. Care is required to take the best and avoid the worst of the academic and non-academic courses in your school.
  8. Take English or master another language of your choice, well. This includes reading, writing, speaking and reasoning. When you write, tell a story, describe what is, or present an opinion or defend one. Watching for ideas worth repeating, will help.
  9. Take history courses. Courses with ideas new to you are worth taking. If possible, avoid history courses which only promote the group, state or country in which you live. History courses tell us about the experiences and mistakes of past, if not present generations.
  10. Read newspapers which do not (always) glorify the nation or group to which you belong. Contrary opinions make us think. So look for and read newspapers with views you occasionally find disagreeable.
  11. A little uncertainty in the words of a teacher leaves room for thought and the practice of thinking skills.


Appendices with (repetitive) advice for Students: [ B How to Learn ] [ C. How to Read ] [ D. What to do in School ] [ PS. Study Tips ] [ PS: Time and Effort ] [ E. How to Study Math and Why ]

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.


Return to Page Top

Home < Volume 2 Three Skills For Algebra << Appendix D. What to do in School and Why

[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]


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.