Appetizers and Lessons for Mathematics and Reason 
www.whyslopes.com - alternative directions for learning and teaching.   Parents: See site area Help Your Child/ Teen Learn 
Français
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||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

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

Volumes 1A is for avid readers. Vols 2 & 3 are for adult, senior high school & college students.
More Site Areas 
1.  Solving Linear Equations  
2. Fractions Ratios Rates Proportions, Units
3. Euclidean Geometry   & complex numbers 
4. Analytic Geometry/Functions 
5. Number Theory
6. Calculus Introduction
7. Complex Numbers (More on)
8.  Real  Analysis 
More Site Areas 
9. Quebec Maths Education  
10. Secondary IV(?) maths
11. Math Education Essays
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
14. Algebra, Odds & Ends, Etc
16  LAMP - Course re Design 
17. Collaborative 
Whiteboard
Math Teacher How-TOs etc
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Multiple Logics in Maths
8. Math Ed. Issues

Next ]



YOU are better than YOU think. Show yourself  how:

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Read  logic chapters 1 to 5  in  Three Skills for Algebra for greater skills & confidence for work  &  study. Learn to read & write with precision,  like a lawyer. 

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 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

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

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

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After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;


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.


Calculus is the college or senior high school mathematics subject required for college or university studies in accounting, business, money matters, science, engineering and health. 

Calculus is  very demanding.  Half the students who take college calculus fail. 

Calculus  employs at full strength  earlier elements of high school mathematics, namely  functions, trig, algebra (including polynomials), mathematical induction, more logic,  geometry and exact arithmetic.  

Get Smart: Would you believe that with mathematical or lawyer  like reading and comprehension skills, you will recognize when or where explanations are clear & complete or in need of improvement not only in mathematics,  but also in studies and work in general.  


Site Reviews:  

  • Magellan, the McKinley Internet Directory, 1996: - Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. ...  The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it
  •   [Math Forum], online 1996 online classifications of the then half dozen sites offering learning and teaching material.. ... There are appetizers for algebra, arithmetic, logic, better learning in general, reason, theorem proving and complex numbers.  Strengths here are in Alan's explanation of mathematical concepts using words and stories: ...

You may be better than you think. Try reading site books and lessons in parallel to your offline courses. Start your self-improvement by learning to read like a lawyer.    

Get Smart:  Would you believe that you do not fully understand a skill or concept until you know how to help others develop an operational command of the skill or concept  

For calculus and for a full-strenth mastery of earlier high school mathematics,  you need to learn how read notes and textbooks like a lawyer so that no nuance, no subtlety and no clause  escapes your attention.  That requires the will and patience to read and understand every word and every step in each calculation and in each story, theory, explanation or  or chains of reason that appears in your notes and textbooks.

Appetizers and Lessons for Mathematics and Reason  was one of the first dozen online mathematics providers in 1995 with a dozen plus webpages and the following introduction:  Collecting and putting first appetizers and lessons easily understood and repeated is a simple way to ease or avoid common fears and difficulties, extend the common knowledge and prepare for further instruction.  
This site since inception  offers advice & directions for self-instruction,  for instruction and course-design.  in accordance with  inductive principles

Site August 24-28th, 2008 how-TOs  indicate  smaller steps, more steps and alternative paths  in arithmetic, algebra, geometry, logic and the start of calculus to aid, speed and enrich skill and concept development in calculus and preparation for calculus.  Site pointers for the greater and clearer use of letters in algebra and site how-TOs  for algebra may aid learning and teaching immediately. They represent a first, outstanding, advance for easing and avoiding difficulties in the introduction of algebra.   Site pages  include a minimal, direct logic only development of Euclidean geometry  sufficient to introduce & develop  trig, vectors and complex numbers in simpler, clearer ways with old-fashioned, Euclidean Geometry, style rigour. Site previews and introduction of calculus further provide means to forestall or avoid shock in calculus varying but full-strength employment of algebraic ways of writing and reasoning. Even in arithmetic, the simple use of a conversion technique in the comparison of decimal provides a logical base for borrowing or conversion operations in subtraction.  All said, the attention to detail in site pages, the provision of thought-based development either aid to instruction or as option for enriched instruction provide a multitude of little steps or little how-TOs for instructors to cover or mention in class as appropriate.  Site how-TOs represent an implementation and a variation of LAMP.

While site pages explore methods of direct and indirect reason, the site coverage of arithmetic, algebra, geometry and calculus relies only on direct reason for the most part and on indirect reason only in the form of the contrapositive use of implication rules. 

Site how-TOs are logically developed and consistent, modulo the applied math assumption of geometric practices - those present in geometrical use of maps and plans with and without the use of coordinates.

Get Smart: Would you believe that a course designer never promises universally success.  The best outcome for any course designer are methods that work most of the time, with recognition that exceptions leave room for thought or worry. 


International Curriculum Reform:
Site how-TOs in accordance  inductive principles for education. offer the framework for  an applied  mathematics program. The program  which may  be woven into the existing mixed or applied  mathematics curriculum in the UK and elsewhere. The program instead of repairing and reinforcing the modern mathematics curricula in North America and Scandinavia of the mid-1950s onward, as initially intended,  gives a logically-consistent successor  . The modern mathematics curricula pretense of following or supporting a context-free development of modern mathematics from set theory stumble and  ceased to be context-free  with the mixed or applied mathematics  introduction of coordinates and drawings in the development of  Calculus, trig and analytic geometry and, if taught, Euclidean Geometry. Secondary-level  modern mathematics curricula was inconsistent  in that arithmetic with the decimal representation of numbers was   avoided in theory - not recognized in axioms  but  required and used in practice. Site how-TOs provide a consistent remedy that may be followed from arithmetic to calculus to set the stage for applied disciplines and the optional study of modern, context-free, mathematics at the undergraduate and university level. 

Testing Site Content: In schools where mathematics teachers have little or no quantitative background or where mathematics teachers have not seen how preparation for calculus provides the core of secondary mathematics, advice and directions for instruction  need to be clear, simple and effective. There-in lies the test of site content, present and future. Site development aimed at compliance with inductive principles for instruction,  and site Reviews suggest the test will be passed in part, if not in full.  But site ideas for instruction have not been tried and tested in the classroom;  Teachers & Tutors  will have to  handle the risks and exceptions in deploying ideas expected  to be statistically but not, as might be hoped, universally effective. 

Get Smart: Would you believe that standard, principles & values without methods and means to strive for them have a weak foundation. 

Remark: I was also amused by the silly TV show, Get Smart, in which the phrase would you believe was always repeated.  If the owners of rights to that show object, the get smart reference above will be removed. 

Welcome. If one appetizer or lesson here is not to your liking, try another.  Each one is different.

Site pages  offer pathways  (a) to ease or avoid common fears and difficulties - many should like that; and (b) to give fuller explanations - those who want to understand methods as well as use them will like that.   The ability to read like a lawyer will allow & speed  (a) and  (b).

You may be better than you think if you learn to read like a lawyer.  

Begin or continue site exploration with a page section below

OR with the foreword of Elements of Reason and  further forewords of site books - all online in full with postscripts.    

  • Step Away from Mathematics: For avid readers in school and out, the online site volume  Pattern Based Reason  describes in general  the benefits, origins and limitations of rule and pattern based deeds and thought. Not all is certain in rule and pattern based thought, and that leaves room for thought & worry in many areas of human endeavor, unclear or messy situations included. If an accident of birth leaves people, on the opposite sides of an argument, should the argument continue? Are those arguments small compared to planet-wide environmental problems?  Should we aim for rationing or wait for greater disappointment? Will excess carbon dioxide acidify the oceans?    It is possible to be extremely logical, to be fan of rule and pattern based reason, only to discover yourself in conditions where your logic, rules and patterns do not apply. Read  Pattern Based Reason to understand and appreciate in general the exact and approximate nature of  rule- and pattern- based arts and disciplines. The latter includes mathematics, law, science & technology for better or worse.  In learning and teaching mathematics, would you take advice from one who implies not all certain?
  • Help in Calculus and in Preparation for Calculus: Online volumes Three Skills for Algebra (Chapters 1 to 5, 7 to 11, 14, 16 and 17) and Why Slopes & More Math (Chapters 2 to 6, 14 to 18), and the last chapters in Pattern Based Reason  may change your view of how to meet logic in and outside of mathematics, and how to learn or teach  mathematics from algebra to calculus.  Let these chapters enable, serve as a catalyst for, your learning and teaching.  The initial aim in writing  Three Skills for Algebra   and Why Slopes & More Math  was to make calculus and preparation for it more effective.  Gifted high school students  and avid readers will find Three Skills for Algebra (Chapters 1 to 5, 8 to 11, 14, 16 and 17) an easy read that gives  a head-start and  better performance in their current courses. 

    Many or most  calculus students are surprised (well-shocked) by  calculus use of the algebraic ways of writing and reasoning in very, very challenging ways.  Site pages offer a remedy for students who learn to read like a lawyer  

    Possible Remedy: Two calculus previews aim to ease the shock.  The  first explains why slopes were  met previously - an easy explanation for senior high school students. Calculus is the subject of slope-related calculations for straight lines and curves y = f(x), calculations that may be done forwards and backwards.  Formulas m = f'(x)  for slopes to nonlinear functions y = f(x) are derived from formulas for the function f(x). The second, another easy explanation for senior high school students,  develops algebraic ways of writing and reasoning gradually instead of suddenly, and so should ease or avoid algebra shocks.  In high school studies before calculus, both previews give  motivation and a context for the earlier study of slopes and polynomial factorization, while boosting algebraic reasoning skills.  Chapter 14 in Why Slopes & More Math  and the appendices of the latter,  go further in easing or avoiding algebra shock in calculus and real analysis  by providing a concrete, decimal view of  limits and convergence sufficient for most, while setting the stage for more obtuse epsilon-delta views - necessary because modern mathematics curricula, fifty years ago, did not sanction the use of decimals and so committed a pedagogical error, one with long-term consequences.  There may be more algebra shocks in calculus. The foregoing may ease two of them. 

    Words Before Symbols: The 1995-6 site book Three Skills for Algebra  is misnamed. In retrospect, it covers four skills.  Chapters 8 to 11 speak about the first three. But there is a fourth skill -See Chapter 14 to meet direct and indirect use of formulas in arithmetic and algebraic solutions to problems. Talking about the fourth skill vocalizes a recurring theme from algebra to calculus.  Talking about all four fill a silence in mathematics - the picture-like quality of arithmetic and algebraic expressions & operations, the quality of being better seen in a glance, while being very awkward to read aloud, has blocked the use of words in building skills, comprehension and confidence.  It is time for a remedy.  The calculus previews, geometric and algebraic, and the decimal view of limits, convergence and continuity in the 1995-6 site book Why Slopes & More Math (Chapters 2 to 6, 14 to 18) will ease or remove more algebra shocks and blocks in and before calculus. See too this site geometric development of column methods for polynomial multiplication and addition.  Geometry and a greater, precise & clearer use of words will help introduce and develop algebra.


  • Mathematics Teaching Ideas and Issues: (1) Online Volume Mathematics Curriculum Notes  begins with inductive principles for skill and concept development and then points to olde flaws and inconsistencies in the exposition of mathematics which predate and continue in course design today. Site pages explore remedies, logically designed but not fully tested in class - due to  qualification insufficient for employment in education.  Clear and precise skill and concept is a must -  technical gaps in exposition should but filled - but they need to be accompanied by discipline or motivation, so that students will follow instruction..  Why bother to require students to attend school, year after year, when motivation & discipline for schooling  are dilute or absent?  Calculus is a motivation for  high school mathematics. Yet we need to  identify the arithmetic, algebra and geometry  in scenes from daily life and work to say where is the math and how it helps, otherwise there are motivational gaps. (2)  Site material (see LAMP, July 2008, or Maths Reference &  How-To Pages , August, 2008) ) fills many technical gaps while revising methods to understand and develop skills and concepts more fully, too fully for the average classroom. Some critical path analysis will be required. 

Students:  In telling or developing a story, or solving a problem, learn to write ideas or steps on paper so that you do not forget what you are thinking, so that there is concrete,  observable, path to follow in the story telling or solution development,  for yourself and others to admire or correct.  Good notation and format allows us to develop and express ideas on paper in a legible & coherent manner, so  that mastery of mathematics like mastery of carpentry or sculpture  becomes an observable art for immediate or later  review or criticism by the doer, peers, parents, tutors and teachers.  

Student-Teacher Guide for Grades 7 to 12 
Calculus may appears in Grade 12 or earlier. It is very, very, demanding:

  1. Grades 7 & 8: (A) Verify or develop efficient and exact arithmetic skills with whole numbers and fractions. See Lesson 1 & Lesson 2.  (B) Learning to solve linear equations with stick diagrams will help your fractions skills and show you how algebra requires  efficient and exact arithmetic skills with whole numbers and fractions. (C) Mastering systems of linear equations in essentially one unknown provides a simpler path for solving word problems in which the identity of the essential known stems from the form of the equations in a way that avoids  mental gymnastics.  Students:  You would have difficulty reading and writing if you refused to learn the alphabet. Likewise, if you refuse to master times and addition tables for all pairs of whole numbers from 1 to 10 (or 12), you will have difficulty in arithmetic. Teachers:  Your coverage of probability should use and re-enforce efficient and exact arithmetic skills with fractions.  All: Websites www.purplemath.com and www.mathsisfun.com provide more worked examples to help with (A) and (B).

    Three questions to indicate  mathematics is everywhere.  (A) What would your world be like if numbers were to suddenly to disappear?  (B) What would your world be like if circles, triangles, rectangles and even straight lines were not present? (C) What would your world be like if maps and plans were to vanish?  

    Question A, B and C echoes  the topic Un Monde sans chiffres (a world without numbers)  in the French only text for children: Je m'amuse en comptant (Amusing Kids with Counting) ISBN 2-261-0110-5

    The question of what would disappear in society if there were no numbers, no maps and plans (latter represent geometry), or no formulas may lead to a greater study of methods and routines likely to be useful in the daily lives of students and their close ones.

    Exercise: Identify the misuse of the equal sign in the following. 

    People who learn or teach that the foregoing is OK are on the wrong path. 
    Learning & teaching to read and write with precision is a must for easing and 
    avoiding difficulties in mathematics and in all arts and disciplines.  Let site  logic
    chapters test or improve your precision in reading & writing. 

  2. Grades 9 & 10:   (A) Review or follow the advice for grades 7 and 8. (B) This format for formula evaluation (discussion aimed at teachers, readable by students too) shows you how to present your work  Good format and good  notational habit, easily understood and repeated, speed comprehension and reduce errors.  Adopt the format for better marks, for clear communication of comprehension and  reason, and for a solid base for thinking and problem solving. (C) Finish your reading of   site area solving linear equations. (D) Ask your math and/or English (mother tongue) teachers to review logic chapters logic chapters 1 to 5 in site Volume 2, Three Skills for Algebra.   (E) Ask your math  teachers or tutor to cover the key but long algebra chapters 8 to 17. . (F). In particular, ask your teacher or tutor to cover the backward or indirect use of the compound interest formulas, see chapter 14, and to explain the difference between arithmetic and algebraic solution methods. (F) Ask your teacher or tutor to cover the wordy postscript What is a Variable with you. (G) See site treatments of straight lines, polynomials and functions. That includes quadratics,  

    Teachers:  Make mathematics clearer.  In solving problems, identify the forward and backward use of formulas, equations and proportionality relations - item F above numerically and algebraically, Do so along the item B  format for formula evaluation
    Prior to Grade 9 & 10, skill and Confidence in mathematics is initially based on methods (understood or not) - drill and practice - that lead to repeatable, reproducible and hence verifiable results.  Logic too gives methods for arriving at repeatable and reproducible results, modulo the limits & benefits of pattern based reason. 
  3. Grades 11.  (A) Review or follow the advice for grades 7 to 9. (B) When  your teacher mentions complex numbers,  the impossibility of getting square roots of negative numbers, or vectors, ask your teacher or tutor to cover this complex number lesson and java applet with you. Then cover alone or with your teacher/tutor,  the easy consequences of mastering two ways to compute the product of complex numbers - multiply formulas for dot and cross-products, and algebraic (complex number) ways to establish trig identities. (C) Cover alone or with help, the site coverage of ratios, proportionality and the carrying of units in calculations - algebraic or numerical. (D) See the lean site coverage of Euclidean Geometry.  
  4. Grades 12 or Calculus /Precalculus Students:  (A) Review or follow the advice for grades 7 to 9. (B) Ask your teacher or tutor to cover this geometric preview and this algebraic preview (Chapters 2 to 6 in Volume 3) of calculus with you. (C)  In chapters 15 to 18 of Volume 3, see if the notion "Saying how to compute a number directly or in principle defines it" help your understanding of numbers, amounts and quantities defined via the use of limits in calculus and its applications (velocity).  (D) Read the decimal and algebraic discussion of limits etc in Chapter 14 of Volume 3  but only if you are a student who insists on a fuller understanding of limits, convergence and continuity. 

    Calculus mastery in senior high school or college mathematics is the key to many professions in business, health, science and engineering. Preparation for calculus justifies the advice,  directions and standards in the site study guide for grades 7 to 12.  For calculus and senior high school students this why slopes geometric preview of calculus provides a context for the high school study of slopes. This second, algebraic preview of calculus provides an easy introduction and a context for the high school study of factored polynomials and sign analysis of functions.  

Site areas include more material for secondary and college mathematics for learners and teachers to explore. 

Two More Websites: 

  • Website #1: Mathsisfun, cover many skills and topics in a light and easy, junior high school fashion.  
  • Website #2: Purplemath covers mid high school to pre-calculus college  mathematics well. 

Explore these two sites carefully besides this one to prepare for calculus. Look for what you do not know, and master it.

Page Sections: [Top] [Steps & advice to improve marks, performance and comprehension] [Notes for Teachers and Instructors] [Key Appetizers and Lessons

 Exercise for Parents, Students and Teachers - Keep a record of all the math (arithmetic, geometric, algebraic) that you meet during a month (the first four weeks of a school year). That should give you a list of uses and frequencies.


Steps to improve performance (marks) and comprehension

Parents:   Working through these steps alone, with your help, or with the help of a tutor, could be a weekend or out-of-school assignment for your teenage kids.

Step 1 for all (Quick): This format for formula evaluation (discussion aimed at teachers) shows how to present their work  Good format and good  notational habit, easily understood and repeated, speed comprehension and reduce errors.  Adopt the format for better marks, for clear communication of comprehension and  reason, and for a solid base for thinking and problem solving.  Time required: 30 minutes - if you do not read more in the page where this format is given.

A Formatting Theme and Standard: Good format in mathematics aids and speeds skill and concept development. Bad notation  slow understanding and cause errors - undermines performances. Repeat, Repeat, Repeat:   Good notation is a vehicle to develop skills and comprehension.

Step 2 for all (Slow): Test Your Logic Skills, improve your work and study skills, by reading Read  logic chapters 1 to 5  (time required: 3 hours, chapters aimed at avid readers in school and out) from volume Three Skills for Algebra  Then improve your algebra skills by reading chapters 8 to 14  (time required: 7 hours - chapters aimed at avid readers again) - a must tedious in parts  See too ALL of the site area on solving linear equations to check, reinforce or extend algebra and fraction skills and concepts. Finally, try the senior high school and precalculus,  arithmetic skill testing questions with hints of algebra in Chapter 7 ((time required: 2 hours) and

Take a few hours to improve  logic mastery, or to verify your logic mastery is complete.  Logic mastery will speed further learning and problem solving and thus compensate (we hope) for your investment in logic study or review. Invest another three to four hours in Chapters 8 to 14 to improve your understanding or development of algebra - the forward and backward use of formulas, numerically or algebraically, provide unifying themes for organizing algebra.

Hint 3 for fewer (Slow): Try geometric (15 minutes) and algebraic (2 hours) why-slope-are-studied calculus previews  calculus to consolidate  high school mathematics and ease, if not avoid, the first shock at the full-strength use of fraction and algebra skills in calculus.  The algebraic preview appears in Chapters 2 to 6 on online Volume 3, Why Slopes and More Math. During calculus, see too Chapters 14 to 19 to ease or avoid further algebra shocks. Calculus requires mastery of algebra and also of arithmetic with the use of calculators in the latter minimized. Mathematics education before calculus need to develop an efficient mastery of exact arithmetic with fractions, roots and p (pi) with a minimal use of calculators. These notes are at the senior high school and calculus level.

Hint 4 for fewer (Slow) : This geometric introduction to Complex Numbers (90 minutes),  its immediate consequences (45 minutes) and this how to add and multiply vectors in the plane java applet (10 to  60 minutes) altogether offer  simple ways to understand and explain complex numbers and employ their  properties them to arrive at trig identities, and trig expressions for dot- and cross-products.  See too chapters 24 in Volume 3.  These notes are at the mid-secondary to college level.

See step 4 to geometrically ease or avoid mysteries surround the introduction of complex numbers and the use of complex numbers in shortening and enriching the the development of trigonometry and calculus.  Step 4 could be followed in all or part before step 3. Clearer and stronger comprehension may follow. Good luck.

A Formatting Theme and Standard: Good format is needed to do and record reasoning and calculation steps on paper.   Further, learning to do arithmetic with whole numbers and fractions exactly and efficiently  provides the foundation - the very work habits - needed to understand and develop the home and business use of mathematics, and to prepare for college mathematics. Implement this standard when and where there are calls for the development of communication and reasoning skills in mathematics. 

More Advice and Directions

  1. Develop Better Study and Work Skills: See if  you or your students can digest and  enjoy  logic chapters 1 to 5 in all or part in Volume 2, Three Skills for Algebra.
  2. Make Algebra Easier: See Chapters 8 to 14 in Volume 2, the postscript What is a Variable, and the site area Solving Linear Equation.  Meet a vocalization of a hitherto silent themes, namely the use of equations and proportional relations forwards, backwards and sideways, numerically and algebraically.  Words and vocalization of skills and concepts having been missing or not clearly used in mathematics after arithmetic? Blame that on arithmetic and arithmetic expressions and laws too complicated to read aloud term by term. Volume 2 and further site areas provide a remedy. Hip, Hip, Hip, Hooray. See too the adjacent steps for improving performance (marks) and comprehension.
  3. Make Calculus Easier: See the innovations, fresh and recycled, in this  why-are-slopes studied,  calculus preview and in Volume 3, Chapters 2 to 6 and 11 to 18 (12 optional).
  4. Make Complex Numbers and Unit Circle Trig Easier:  Meet a geometric introduction of complex numbers with links to the law of sign, with the Galilean relativity, origins of the distributive law (an innovation), with the Pythagorean theorem, and with trig formulas for dot- and cross-products, all as easy consequences.  See the complex number starter lesson and site area.
  5. Bad News for Many - higher standards required to avoid a waste of time: Arithmetic skill with decimals and fractions, once a goal of primary school instruction, is becoming or has become a college level, remedial subject in North America and (?) the UK.  Students need to acquire and maintain efficient and precise arithmetic skills with decimals and fractions. Junior, mid and senior secondary students need to meet and master exact and efficient arithmetic skills with decimals and fractions. The Fractional operations on line segments (stick diagrams) in Solving Linear Equations may develop algebra and fraction skills together, and show students that fraction skills are part of algebra. So That being said, senior high school students may follow the links in (A) and (B) above, or steps (2) and (3) below, before reviewing or developing fraction skills.

Aims, Ends and Values for Mathematics Mastery

  • For basic home and business use, primary and secondary students need to be shown directly and clearly  how to measure, calculate and use costs, rates of change, distance, area, volume, density, velocity, interest and taxes, mark-ups, discounts, commissions, wages, salary, annuities (geometric sums) and  intervals of time. Good notation or good formatting habits will help. Figuring well (do arithmetic exactly and efficiently with fractions where numerator and denominators are less than 112 = 12) shows the ability to follow step-by-step methods carefully and wisely. 
  • For work in design, planning and construction, students need to know about arithmetic exactly & approximately, how to measure, about maps and plans drawn to scale in good proportions, or distorted; about basic Euclidean Geometry. Studies in electricity etc that involve phasors or trig would benefit from site coverage of complex numbers.
  • For college mathematics and calculus, students need to be shown directly and clearly  how to do arithmetic efficiently with fractions without a calculator, solve linear equations, working with proportionality and units in calculations, use formulas forwards, backwards and sideway, numerically and algebraically;  right triangle trigonometry, unit circle and complex number trigonometry, logic, Euclidean Geometry, Analytic Geometry and Functions (polynomials, quadratics, straight lines). What requires calculus and beyond? Answer: business and investment calculation; science and engineering; high school math instruction, training future high school math teachers;  health careers.  The requirement is sometimes for fuller comprehension of the mathematics met in a subject and sometimes for filtering - the identification of ability needed in demanding professions. 
Calculus requires high school mathematics (arithmetic, algebra, geometry, trig and functions) at full strength.  If you are in calculus or know that you will be taking it, see chapters 1 to 14, 16, 17, 22-5 in Three Skills for Algebra;  chapters 1 to 5 and 14 to 19 in Why Slopes and More Math, and  the last logic chapters in Pattern Based Reason.  Look for different ways to understand and explain key skills and concepts, nuances and subtleties in site books and areas. 

Where is the mathematics? Quantitative skills, arithmetic and even algebra are everywhere in city, village and even agricultural societies - telling time; describing how long an event or work took in minutes, hours, days, months, years; counting the days until ...; weighing & measuring in buying, selling & cooking; deciding how much to make and what that will require (proportional reasoning); figuring chances of success & failure for each path of action;  handling budgets, rent, debts, annuities, insurance; paying taxes (ouch); counting assets and debts;  following trades that require arithmetic, geometry and algebra - carpentry, metalwork, surveying, navigating, driving, cloth-making, real-estate buying and selling, banking services,  shop keeping, accounting, car maintenance, healthcare (charts and doses). 

In many arts and disciplines, it is absurd to  ask an  apprentice to solve open or difficult problems before mastery of standard methods and routines.  Most arts, trades, professions and disciplines in the work place and in college or university studies demand and prize the careful mastery of routine rules and patterns, one at a time, one after another, alone or in combination, all in a repeatable, reproducible, observable and correctable manner. Within each discipline, critical thinking or maximum benefit and least harm demands a knowledge of the origins and limitations of rules and patterns, practices, steps and methods. Mathematics is a discipline in which the ability to  carefully use and combine of rules and patterns, one at a time and one after another, alone or in combination needs to be written carefully on paper to demonstrate and record skill and comprehension. 

In general for pleasure and for applications, we may record, develop and read our thoughts and reasons by writing and drawing words, symbols and diagrams on paper or alternative media.  Spelling and writing stems from our ability to draw. The detailed and deliberate record of our thoughts and reasons may be read or seen or heard later to restore or share them. There-in an extension of our memories and reasoning faculties, singly and collectively. While mind reading is not possible, we may project our thoughts and visions into words and diagrams to communicate and reason alone or in company, and thus to generate or solve problems. That projection may be seen and corrected by ourselves and peers.  Education needs to develop and maintain projection habits and provide it food for thought and, for better and  methods for decision making, for arriving at conclusion, if that possible, along with a knowledge of their benefits, origins and limitations of reason or decision making.  See Volume 1A.

Page Sections: [Top] [Notes for Teachers and Instructors] [Steps & advice to improve marks, performance and comprehension[Key Appetizers and Lessons

Ends, Values and Means: 

  • The will and ability to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause  escapes  attention is an end, values and means for mathematics education in the training of students and in course design and delivery. 
  • Skill and confidence in  mathematics, a written art or discipline,  may follow from care and precision in following and recording the steps in methods and routines in a well-formatted, readable, reproducible and repeatable manner for verification or correction. Here care and precision is another end, value and means for skill and concept development. 
  • Mastery of  thought-based paths for understanding and developing skills, concepts and comprehension is a further end, value and means for mathematics education, and may be considered an extension of the care and precision, required so that no subtlety, no clause and no detail escape attention..

Ideas that cannot be expressed on paper with diagrams, words and symbols are not part of observable skill & comprehension.  Compare and contrast that view with the Allegory of the Cave in Plato's work The Republic where knowledge is based on shadow interpretation. Compare and contrast that view with the dominant constructivist theory of skill and concept  learning, in which mastery is a subjective affair, not for observation nor correction in an objective manner; and in which changes in delivery style in a shadowy manner was suppose to lead to a subjective (anarchistic) view of knowledge, one that in retrospect resembles the state of knowledge before striving for objectivity was the norm in science and  technology, if not law. 

Key Appetizers and Lessons

Page Sections: [Top] [Steps & advice to improve marks, performance and comprehension] [Notes for Teachers and Instructors] [Key Appetizers and Lessons] 

The following appetizers and lessons, online chapters included,  can be read or seen separately. Each one is different. Altogether, they provide  technical themes and content-oriented standards for senior high school and college mathematics studies and instruction to meet or exceed.

  1. If you are able to read logic chapters 1 to 5  in online volume Three Skills for Algebra, you are not too young nor to old for site material and directions, a good fit is expected. Logic - Chapters 1 to 5 develop greater precision in reading and writing for work and studies in and apart from mathematics. Improve your skills and confidence. Some chapters are easier than others. Chapters 2 is hardest. Chapters 3, 4 and 5 are easier.

    Online math & logic  jigsaw puzzles. Each appetizer and lesson, each site page, gives a piece of a math and logic education jigsaw puzzle. Look at the pieces, and try to fit them together by trial and error, one at a time and one after another, in pairs and in larger groups.  Putting the pieces together takes time. If a piece does not fit, try another and another. Each art and discipline, and each problem in daily life, is like a jigsaw puzzle - one or more. You need to find  the pieces and check that  they all present, and put them together by trial and error, with time and labour, starting with the easier parts - the straight edges.  In mathematics, the straight edges are provided by  mastery  of linear chains of reason in Logic - See chapters 1 to 5.

  2. Fraction Starter Lessons:  point to an efficient, operational command of exact arithmetic with whole numbers and fractions. There-in lies a first standard and a must for all of secondary school mathematics.

  3. Solving Linear Equations - begins with a geometric way to visualize and solve linear equations, and then introduce ideas to make word problems and simultaneous easier to understand and explain. Older students can read the examples here in sequence to review and understand how to solve linear equations, how to present solutions (appearance is everything after content). Teachers can use the examples or similar one here to introduce the topic and to reinforce fraction skills and sense. (If I was to redo this, I would choose coefficients to ensure whole number solutions while working with the stick diagrams here.) Pay attention to the format of the solutions here. Errors in format lead to errors in finding or calculating solutions. Here-in lies a second standard and goal for all of secondary mathematics.

  4. Three Skills for Algebra,  See how to use words before and besides symbols. See how to talk about numbers and quantities, and how to describe calculations, and see a hint of the 4th skill. Here we are filling some gaps in your education - indeed in the education of all who have learnt and taught algebra, or written books about it. There is a missing link here. We are providing words that been missing not in the doing, but  in the discussion and explanation of algebra. Read all about. BREAKING THE SlLENCE with a Better and Greater Use of Words in learning and teaching mathematics.  Here-in lies a goal for secondary II mathematics.

  5. Using the Compound Interest Formula forwards and Backwards - there-in lies a 4th skill for algebra, The backward use of equations and fomulas has been a silent part of high school and college courses. The first innovation here is to break the silence by describing that practive with words, that is the phrase "Forwards and Backwards" or "Directly and Indirectly". BREAKING THE SlLENCE  Continued.  The second innovation here is to name ,  illustrate and contrast the concetp of numerical and algebraic solutions of equations. Be satisfied if you can solve the backward use problems numerically - the arithmetic approach. Be estatic if you understand the algebraic approach and its greater power. Here in lies a connecting theme, goal and standard for mid to senior high school mathematics and science. 

  6. Arithmetic,  Watch these videos to perfect skills and comprehension of whole numbers and fractions, etc, etc.  You may think that arithmetic mastery is for primary school students, and further studies in mathematics should not demand skill in computations with a calculator.  That may be true when you go shopping, but the ability to do arithmetic in an efficient, repeatable and reproducible manner, no errors please, is a must for senior high school mathematics and calculus.  As student, you have master the basics - learn to walk, before you run. That being said, if you are adverse to arithmetic, most of the topics can be understood without a great command of arithmetic

  7.  Euclidean Geometry,  Here a lean treatment that will connect construction and duplication of triangles with isometry, parallel lines and how to recognize parallelograms. The ability to follow short Chains of reason developed or asked for in  site logic chapters apart from mathematics appear in connection with geometry. 

  8. Complex Numbers - a simple geometric approach If you have mastered polar coordinates, this visual and geometric approach will complete your earlier understanding. Students of electricity, engineering and physics knowlingly or not, employ complex number ideas in the basic or advanced concepts and calculations. If high schools mathematics introduced simple geometric approach to saying how to add and multiply points in the plane, many difficulties would disappear. This geometric approach or its easy consequences simplifies learning and teaching of the law of signs, of unit circle trigonometry and of vectors.

    Remark: The site area on   Euclidean Geometry shows how the latter implies the distributive law for complex numbers

  9. Calculus - Geometric Preview  and  Calculus - Algebraic Preview. The first preview explains why slopes or rates of change are studied in senior high school mathematics. The second one goes further into slope interpretation and shows how factored polynomials and sign analysis thereof helps in saying where a function y  = f(x) is increasing or decreasing. Calculus is the door-opener for studies in science, engineering, nursing (for some reason), medicine, accounting and on on. Calculus is the subject which requires skill and concept development and mastery in arithmetic, algebra, logic,  geometry and trigonometry. The full strength requirement for the latter is a shock for student who did not know about those requirements and also for those who know.  The calculus previews develop and motivate geometric and algebraic skills and understanding before or during calculus.

Site advice and directions for learning and teaching mathematics will take time to understand and follow. Follow  closely, but not too closely - site advice and directions are approximately correct, for some circumstances, not all.

Where is Mathematics? Keep your ears and eyes open. At school, at home, in going out, watch for the occurrence of measurements and calculations, and how they are done, and why.  The result may be some questions to motivate your mathematics studies. The assignment here is to collect questions.  The forthcoming site areas in preparation (preparation postponed)

17**. Telling & Working with Time
18**. Maps, Plans & Drawings
19**. Quantitative Skills for  home and  work, etc, etc 
** Means Planned - Here are descriptions for teachers, not students.

aim to develop skills and concepts in context. Give  methods to use and apply in a repeatable and  reproducible manner in common situations involving mathematics. One of the textbooks I am reading for learning French organizes its lessons around themes: going to a restaurant or theatre, riding on a bus or train or plane, visiting a shop, working as a carpenter, and so on. The lesson then provides the words or vocabulary useful in each setting or activity. Mathematics courses may describe similar visits or activities,  and connect the latter to mathematics.  There-in lies a value-providing ends and context for meeting rules and methods of mathematics.

Students and teachers do not have to see a thought-based development of all rules and patterns,  if the rules and practices lead to repeatable, reproducible and hence verifiable (right or wrong) results. But a thought-based development helps in understanding the benefits, origins and limitations of rules and patterns, customs and practices, so that the latter can be applied carefully and not mis-applied.

The thinking part of an art or discipline:

The  thinking part of an art or discipline comes after the assumption & careful  mastery of some rules and patterns, steps and methods, practices and conventions.  Careful mastery means you can use the latter to arrive at results in a repeatable, reproducible and hence verifiably right or wrong manner.  The thinking part of a subject begins when you start to combine rules and patterns, steps and methods, practices and conventions, to obtain new ones in a repeatable, reproducible and hence verifiable manner. Thinking or critical thinking within an art or discipline continues through recognizes the benefits, origins and limitations of rules and patterns, steps and methods, practices and conventions, so that the approximations in the application of the latter are known or avoided.  The combination of rule and patterns, customs and practices, steps and methods, one after another, may lead to short parallel strands of reason and hence a thought-based development of an art or discipline beside the empirical mastery of rules and patterns etc with confidence building results that should be  repeatable,  reproducible and hence verifiable. Once the ability to form or follow strands of reasons within an art or discipline is present and respected or appreciated, fuller and fuller thought-based developments can be offered, if not in class, then in print. The first phase of education could be based on rote - here are the facts and methods - learn to use them in a repeatable, reproducible and hence verifiable right or wrong manner.  Later phases may then build on that via a mix of deductive and rote mastery of further rules and patterns.

Page Sections: [Top] [Steps & advice to improve marks, performance and comprehension[Key Appetizers and Lessons[Notes for Teachers and Instructors]

Notes & Remarks for Teachers and Instructors  

For Methods and a focus Pre-Secondary, Junior Secondary and Remedial  College Instruction, see (i) First Year High School Math - Lesson Plans with Fraction Focus (ii) Second Year High School Math - Lesson Plans with an algebra focus (iii) Algebra Lesson Plans.  Upper primary school instructor may read site lesson plans for secondary I and II to know what their students will meet, and to prepare for it.

Top-Down versus Bottom Up.  Site how-TOs and steps for skill and concept are based on inductive principles for instruction and  the collection of starter and further lessons likely to be effective in class, statistically if not universally,  and also easily understood and repeated by instructors. The explicit statement of how-TOs (alone or with exercises and exercise correction guides to come) directly and clearly endeavor to provide teachers & tutors without a quantitative background, a simple and complete path for instruction, or if you insist a lower bound for instruction in an inductive manner, a lower bound or standard to exceed.  Collecting how-TOs and refining existing ones with repeatable and reproducible results in the classroom could raise that lower bound. Here top-down approaches for instruction which begin with principles and standards are practical when and only when  prior knowledge, that prior  how-TOs, are sufficient in number and quality to meet the standard and follow the principles in style. The question where are your how-TOs need to be asked when standards and principles for learning and teaching, and for cognition, originating outside of mathematics,  point to  and demand instructional paths or styles. 

Top Down: My writing would have ended in the early 1990s if the 1989 standards of the National Council of Teachers of Mathematics (NCTM) had included how-TOs to make constructivism easily understood and repeated by teachers in the high school classroom, teachers who in the majority had little or no quantitative background in mathematics, or a mathematical background without a knowledge of calculus and what it demands. I am waiting for the NCTM to go beyond illuminations and provides how-TOs for indirect instruction, easily understood and repeated, with an indication of whether or not preparation for calculus is to be explicitly supported.  I am also waiting for the NCTM to shift its stand on the constructivism, and in particular that the absent of mind reading means that mathematics  is a subjective art, not for correction.  Arguments for that shift may be found in site content.  In particular, the master of mathematics, elementary to advanced, knows how to record and develop thoughts on paper in way that does not require mind reading, in mechanical ways that are observable and hence  verifiable, or correctable. So feedback is possible.  So communication, reason and problem solving are observable arts in which striving for objectivity is a possibility, modulo the limits of rule and pattern based thought and processes on paper and in practice. . 

In too many schools or college, it is impolite to suggest that better methods for instruction exist. Any suggestion implicitly criticizes the classroom habits and formation of fellow instructors. But an environment in which sharing and offering fellow instructors ideas for instruction is impolite slows effective reform in instruction, reform based on a free sharing and refinement of what works or could work better.  Allow yourself the freedom to consider and discuss site suggestions for better instruction without being offended.  Bon Appetite.

Logic: The site introduction of logic is math-free. It is aimed at improving work and study skills while hinting at the role of logic (implication rules) in mathematical proof and definitions.  The site introduction of logic showing the need for greater precision in reading and writing may lead readers to cultivate that precision. The math-free aspect may allow development of logic skills and concepts in parallel in and outside of maths. 

The solution of jigsaw puzzles where pieces are inspected and fitted together in a persistent trial and error fashion until a picture emerges and puzzled is solved provides a model for combinatorial, opportunistic and thinking-out-of the box problem solving in and outside mathematics - whatever works.  Calls for problem solving in mathematics and other disciplines may be shaped and refined by showing showing students how build comprehension and how to reason by combing rules and patterns, reliable and ethical ones preferred, in a repeatable, reproducible, verifiable and hopefully ethical manner. 

The site coverage of logic ends with thoughts, not definitive, on indirect methods of proof and reason.  See the last logic chapters and postscripts in Volume 1A, Pattern Based Reason.  Volume 2, Three Skills for algebra ends with duplicates of the same chapters but omits the postscripts. Indirect reason could be illustrated in detective and mystery stories. But the few stories (fiction) I have read end in sudden revelations and  of how the main character solved the problems, revelations involving clues and evidence not previously available to the reader. Remedies would be welcome.

Algebra: The site introduction of  algebra, what is a variable, solving linear equations and operations on polynomials clarifies nuances and subtleties while providing a clearer and greater role for words and geometry in its mastery and exposition. Thus, it builds and sets new and higher standards. There is one pre- or co-requisite to the mastery of algebra, namely efficient, calculator-free arithmetic skills with whole numbers and fractions.  The lack of drill and practice to develop and maintain the latter undermines high school instruction from algebra to calculus. Mastery of figuring skills with whole numbers and fractions should be kept and polished in K5-12. Parents may hope for a sound development of figuring skills, but should trust verify as well.  Centralized and bureaucratized design and implementation of mathematics education may eventually lower standards. 

Arithmetic and algebraic expressions where order or operations are implied by position and/or parenthesis are best seen and understood in silences, non-verbally, like pictures and diagrams. Site words on three or four skills for algebra and on what is a variable compensate for that silent aspect of mathematics.  Learning to talk about numbers and quantities, easily and precisely provides a striking advance for the development and comprehension of mathematical disciplines.

The site use of fractional operations on stick diagrams to visually and geometrically introduce the algebraic methods for  solving linear equations employs letters to denote unknown lengths - a concept easier to grasp, more concrete, that asking students to let a letter or symbol denote an unknown number (or a variable). Starting algebra with letters x to represent lengths may be an accidental return to the role of geometry in algebra, a role hinted at by reading x2 as x-squared and x3 as x-cubed.  Fractional operations on sticks (line segments0  by themselves may consolidate comprehension of fractions and illustrate the exact role of fractions in algebra. That being said, with hindsight, I would start with equations that have whole number solutions instead of fractional ones.  

Remark: For example, the site stick diagram method is a geometric path, effective for most,  not all, invented in spring 2005 to introduce algebraic solution methods for linear equations.. One pupil had difficulty with the transition from geometry  to algebra.  In applying the method, you need to control or handle the risk.  The latter student need special treatment. With him, if I had had him for more one hour, I would have taught the algebraic form by rote, so he could obtain and check solutions, even if he did not understand the attempted geometric development. 

After the introduction of algebraic methods for solving linear equations, the site area on solving linear equations introduces (i) triangular or permuted triangular systems of equations; and (ii)  system of equations in essentially one unknown - the solution of the latter requires use of  (a) the associative law for multiplication and (b) the distributive law.  The verification of answers (an important part) forces students to look for mistakes between the start of their solution and the end of their check. An in all the foregoing, require students to format their work so that the sequence of steps in their figuring or reasoning process is recorded in a clear, legible and sequential manner.  

Suggestion:
Have students avoid in place operations in which the sequence of those operation is unclear. In place of quantity, seek quality and for that require or give marks for clarity and format in student presentation of their solution steps and solution verification steps. In general, written solutions should be and become stand-alone and self-sufficient units in the notes of a student which record and communicate the use of mathematical methods. Implement this standard when and where there are calls for the development of communication and reasoning skills in mathematics. 

Calculus: The site introduction of calculus shows why slopes and factored polynomials are studied in high school. This two part introduction eases or avoids algebra shock in calculus begins by providing geometric and algebraic calculus previews which are understandable, skill and confidence building, presentable even before calculus begins. There lies a further advance for the development of algebraic skills in and before calculus. Putting these previews at the start of a calculus in essence gives a simple, easy to understand, preview of the derivative-based max-min analysis in differential calculus, one that develops algebraic skills slowly and systematically and gently instead of sudden  The site introduction of calculus goes on by adopting and sanctioning a  decimal,  error control viewpoint of limits, convergence and continuity. The site coverage of real analysis   goes further in providing decimal-based proofs of the theorems of calculus, usually given without proof in first courses on calculus.  Calling for the return of decimals and their explicit sanction in course design and delivery contradicts the 1950`s and 1960`s modern mathematics curricula, curricula continuing today in diluted form, but it should ease or avoid algebra shock in both calculus and university level courses on real analysis. 

Complex Numbers: The site introduction of  complex numbers provides a simple, visual, geometric introduction of the addition and multiplication of points, arrows and vectors in the plane in a manner that might be enjoyed in college level instruction today in science, engineering and mathematics, in the present-day training of electricians; and in junior high school courses where rectangular and polar coordinates are mentioned.  Easy consequence senior high school and college level consequences include trig formulas for dot and cross-products, and yet another proof of the Pythagorean Theorem.  The site introduction revamped could also be a basis for a leaner high school curriculum in which the role of signed numbers as coordinates appears before the law of signs.  The late physicist R. Feynman described his subject as a the addition and multiplication of arrows in the plane. Secondary mathematics too could be described in the same way.

A Word about Mathematics Education. Reason, communication and problem solving need to be based on a skill and concept development and some perfection in reading, writing and arithmetic. In elementary school or before, children may learn the or an alphabet, say 26 letters, and the digits 0 to 9. Some children may object to meeting and mastering the alphabet - too many letters, why bother.  If we said to these children, let make reading and writing simpler. You only have learn 20 letters, not all 26.  That would cause problems in reading and writing, and understanding words and their meaning. In mathematics students need to master the use of decimals to efficiently represent and efficiently do arithmetic on whole numbers and fractions. That mastery is useful or should be useful with weights, measures and calculation in daily life, albeit education that goes on and on beyond the age of 14 delays and hides  that usefulness, that is a problem to address.  The situation today where students are taught mathematics from primary school to college without knowledge or efficient mastery of arithmetic is similar to students study language without being equipped with a knowledge of the alphabet and a mastery of spelling and grammar. In particular, we would not tell a child or teen that a complete knowledge of the alphabet is optional.  Schools, colleges, teachers and parents should not be telling and should not be allowing students to think that mathematics can learnt or taught without efficient figuring skills with whole numbers and fractions.

In mathematics and logic, rules and methods  for solving routine problems need to be mastered well. When rules and methods for solving routine problems are approximate, for some circumstances, not all, students need to learn that as well.   Meeting and mastering rules and methods, exact or approximate, for solving routine problem provides students a model for tackling non-routine or authentic problems.  

Site  lessons and lesson plans focus on the technical development of skills and concepts with what may be a repeatable, reproducible and verifiable methods for building skills, comprehension and confidence.  

This technical development may be dry and the hints of applications in that development to abstract or remote. 

Thus there is a need for detailed examples of mathematics in practice and context, culturally dependent, say in buying and selling, in construction and production at home and at work, in paying taxes (ouch), in keeping records of income and expenses, and in further common trades and practices of a society or culture. The examples might show or emphasis how mathematical operations seen in or learnt in one context help in another.  For example, in teaching people to speak and write French, the textbooks I am reading offer scenes or scenarios (eating at home or in a restaurant, buying and preparing food,  traveling on a train, driving an car, visiting a hospital) to introduce and develop vocabulary and language skills in context and in a comprehensive manner for that context. Mathematics education would benefit not only from site technical innovation, fresh or recycled, standing on the work of others, but also from a series of spiraling and expanding vignettes introducing or developing the mathematics employed in common place activities, trades,  professions and school subjects, elementary to advanced. 

Cultural Note: What examples are appropriate or their selection will need to reflect and expand upon the cultural and economic history or origin of students and their parents, and which activities are common place or dreamt of.  In particular, the expansion pollution age  industrial, agricultural and resource based societies provides a context, common place examples of arithmetic and geometry, that may be absent from others societies coming into contact with or surrounded by that expansion. That for better or worse raises the question of how the other societies may adapt or react to that contact and the changes it wrought. For examples first nation societies, indigenous people,  may express and describe numbers and quantities differently  and not have the language, the words, to directly describe  examples of arithmetic and geometry present in the  industrial, agricultural and resource based societies which today, for better or worse, dominate the planet, Malthusian style. That clash of cultures is least demanding on larger societies  (save for the advent of change due technology or resource exhaustion) with their greater inertia and is most on smaller societies with less inertia.  

Ends and Values:  In many arts and disciplines, there are practices, customs and values  to be met and mastered one at a time, one after another, alone or in combination.  Customs have developed over many years, decades and even centuries in ways a student cannot fully anticipate. So explanations, clear and direct, not confused, are needed to communicate the evolved and often less than obvious, customs and practices. As part of the teaching process, students may be given situations or puzzles to extend or challenge their skills and knowledge, and to set the stage for mastery of a custom or practice. But at the end of the day, the instructor should describe the custom or practice, and encourage its mastery in a repeatable and reproducible manner. And in that customs and practices may be learnt by rote or with some explanation, preferable full and as much as the student can grasp. There are some arts, trades and disciplines in which  mastery of rules and patterns in a repeatable and reproducible manner is more important than and may serve in place of a thought-based development. That being said,  critical thinking within an art, trade disciplines about when to apply or follow a step or method depends on an knowledge of the benefits, origins and limitations of rules and patterns to avoid errors or mis-application. There-in lies a justification for a thought-based development and a general discussion of rule- and pattern-based reason.

 High school mathematics beside calculus implied content should include methods and routines useful to student and their close ones in the task and problems of daily life, including money matters, weights, amounts and measures.  Such methods and routines could be taught in way that would engage students and also aid the basic math and logic skills needed for calculus. Basic skills include the ability to write and thus show the steps in a calculation for verification or correction.

Mathematics education reform in secondary and primary school needs to strike a balance between the standards implied by calculus for mathematics mastery and the standards implied by social theories of learning. Diluting the primary and secondary school  mastery of mathematics needed for calculus,  to make mathematics more engaging for social reasons,  is similar to inviting students to swim in the deep-end of a swimming hole while limiting their instruction to learning how to wade.  Yet insisting that all students be strongly prepared for calculus is also impractical,  even with site innovations for making that preparation easier to learn and teach.  How to strike the balance is a question left to another day.   Course design in  secondary and/or primary school needs to explicitly identify the mathematics needed for calculus and the full-strength mastery of that material, while also preparing students for the frequent appearance in of elementary mathematics (arithmetic, geometry and algebra) - the question of what would disappear from daily life if there were no knowledge of numbers,  geometry and algebra might guide the formation of students in an engaging manner while emphasizing an operational command of every day mathematics in ways that appear to be reliable in a repeatable, reproducible, observable manner.  Balance in mathematics education might provide students with all the foregoing while striving to do so gently in a manner that serves the needs of calculus.    

 Education reform in secondary and primary schools appears to be in  rebellion against the demands of calculus for a full strength mastery of key skills and concepts.  Course design cannot properly cover nor prepare for the part of secondary school mathematics needed for calculus, while moving to an more engaging manner that mathematics that does not meet the dryer and more technical demands of calculus.  Diluting coverage of the part needed for calculus in secondary school, and the preparation for that part in primary school, appears to compound the fears and difficulties that stem from formally covering that part or formally preparing for it.   

Mathematics is an art and discipline in which the apprentice has to learn how to follow routines and methods in standard,  repeatable and reproducible well-formatted, manner to provide a base for the communication and record of reasoning in a manner the apprentice & others may check and verify. The notes and written work of the apprentice must show mastery of routines and methods in each topic, before tackling any further dependent topic. While solving difficult problems may be part of research in many arts and disciplines, students need to see the benefits, origins and limits of routine problem solving methods to recognize when non-routine methods are required.  The apprentice needs to see the benefits and limitations of the master methods before transcending them. 

 

 

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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 twiddla.com to set up whiteboards to work with the webpage of your choice. It works better with some browsers than others.

Precalculus sites mathsisfunwww.purplemath.com  might be appealing than this one.  Do not go. 

 

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