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

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

Ages 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
Forewords + leading chapters give original reasons, still valid, for site content & growth.

About: Site material shows how common troubles stem from steps too large or missing. Site material may develop critical thinking, improve reading and writing, and build mathematics and pattern based reasoning skills. Online Volumes 1, 1A and 2 give avid readers in school and out the best places to begin. If one site element is not to your liking, try another. Each is different. Many are unique

Teachers & Tutors: This December 2011, 5-phase framework offers a context for mathematics & logic education. Phases 1 to 3 may focus on skills with actual or potential local value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus. Reform: look before you leap - plan all in detail first.

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.

Site Foreword

Welcome.

If you are an avid reader or older student, you may find the forewords and leading chapters of site volumes to be informative and entertaining. The forewords and chapters in question indicates the context and motivation for site growth since book composition 1995-6. Early site reviews reflect portions of site books then online when the aim to sell the books. Since then the books have been completed and posted online in full along with further pages that may explain critical thinking, improve reading and writing, and provide a good base for learning and teaching mathematics from arithmetic to calculus. When I taught college, two adult students followed my advice to read the first chapters in the course textbook. Each said they like different parts. Likes and dislikes vary. In the case of this site, if one site element or view is not to your liking, try another. Each one is different. Buried between a few lemons, there are a few gems.

If you are an instructor or would be skill provider not fully versed in mathematics but required to teach it, you may appreciate the presentation here of clearer ends, values and methods for skill development not found elsewhere. That includes site starter and further lessons almost ready for immediate use as is or woven into your courses- just add exercises. In the rough edges of this site, you will find ideas for lessons instead of lessons ready to use or refine. If your standing as a teacher depends on how well you prepare students for tests and final examinations, site material may - not all is certain here - show you how while filling gaps in your own command of mathematics.

If you are an instructors well-versed in the discipline, you will find starting points for skill development, mathematically correct, but different in small and large ways from what you have met as a student or employed as a teacher. Two examples, the first one small and the second one large, follow.

• Saying a whole number if prime if it not the product of two SMALLER whole numbers includes the extra work SMALLER. College level number theory would not use it. But in elementary and early secondary mathematics, this extra word implies the numbers 11 and 13 are prime because they do not appear as a product inside the 10 and 12 times table respectively. Likewise, the numbers 7, 5, 3 and 2 are prime because they do not appear inside the 6, 4, 2 and "1" times tables. Different starting points harmless to the overall development of skills and concepts easier for pupils and teachers.

• The oral dimension of the exposition of mathematics and logic skills and concepts needs improvement. Both arithmetic and algebraic expressions, formulas and axioms included, are usually better read in silence than read aloud, term by term. Naming formulas and axioms helps. The names or descriptive phrases can be used to talk about formulas and computations. Beyond that, before and besides the shorthand role of letters and symbols in coding and extending mathematical thought on paper, we can talk about numbers, amounts and quantities. That enlarges the verbal dimensison. In addition from addition tables to calculus, we can talk about the backward use of rules, patterns and formulas. Here the forward use is expected, but the backward use appears very often but in silence.

  Site algebra starter chapters and lessons expand the oral dimension of algebra by showing how all formulas can be used forwards and backwards, numerically and literally/algebraically. The oral dimension of arithmetic and hence algebra can be further extended by showing and emphasizing how counts, totals and products can be obtained in many ways by forming and adding or multiplying subcounts, subtotals and subproducts. The resulting practices are consistent with and implicitly extend axioms for arithmetic with whole, integer, rational and real numbers. In doing so, they make the secondary school development of prime factorizations and polynomial operations easier to justify in class. The general goal here is not to impose a minimal set of axioms on students, but a functional, generally accessible, consistent set that implies an operational command of mathematics without distracting from it or overwhelming students with technical concerns. Those few students who study higher level mathematics may enjoy the leaner and more rigourous development of skill and concepts.

In accordance with my earlier formation in modern mathematics and the empirical sciences, I hold that skills needs to be visble to both learnt and taught, to be repeatable, reproducible and verifiable. The Western educational physchological notion that true knowledge is a product of student reflection located in their minds, apart from reliable testing and observation is good for spiritual retreats from this material world. But in cooking, carpentry, construction work, car mechanics, mathematics and science, the demonstration of observable skills provides standards for skill mastery and may further show how later skills follow from earlier ones. Not all skills in daily life are learnt with multiple levels of comprehension, often in complete. No one truly knows how the digestive tract works, but we can generally describe it in mechanically and chemically.

In mathematics, counting, measuring, drawing, figuring and reasoning most if not all abilities can all be observed mechanically in space or mechanically on paper. So visible, repeatable and reproducible skills and practices can be presented to student to copy, master and refine alone and in longer and longer sequences. In general, skill concept development should take small, gradual steps to make mastery more or very likely. While students may have learning difficulties, those difficulties should not stem nor be compounded by from steps missed or too large in skill and concept development. A step is too large when alternative steps succeed in easing or avoiding common troubles. See site content for examples. Where talent in say mathematics is seen as natural, one that can be directly nor surely provided, there-in a challenge for course designers and textbook writers to accept, not ignore, in their exposition of ends, values and methods for skill and concept provision. In my secondary school days, I saw a problem in the exposition of algebra, did not think I would be one to address it, but over time I saw and collected piecemeal, more and more ideas and starter lessons to address the challenge. When I could not bear my own silence on this problem and further sensed if not clearly seen in the exposition of mathematics and logic, I began to write.

My current position in mathematics education has four parts. First, steps too large or missing explain a great many skill and concept development. Second, different start points may make the hard easier. Third, later skills depend on earlier ones. Fourth, the site five phase framework for mathematics education offers context and motivation for instruction in communities entering, in and/or leaving the demands of present-day: counting, measuring and figuring with money, amounts, lengths and time directly or in proportions. In the framework, secondary schooling which aims to prepare student for calculus-based college programs is delayed and re-arranged to collect and put first skills and concepts with take-home value for adult and daily worth mentionning. Before skills san take-home value worth mentionning are covered, the aim is to leave a good impression and context on children and adolescents, 15 and under, so that the only clear reason for secondary mathematics is not just preparation for the next test or final examination.

The forewords and leading chapters of site volumes indicate the initial motives for site material and growth. Given that, the current site exposition of skill development methods and the five-phase framework for context, if not motivation, essentially ends my private research into methods and then ends and values for instruction, save for some small finishing touches.

Alan Selby
December 28, 2011

Unique and Perhaps Better Skill Development Methods

See site Geometry and Calculus material, and site books too, for more.

Arithmetic Steps

Some good practices for skill development in arithmetic appear in site arithmetic and number theory steps. Where these good practices are not best, say so.

1. Definition of Primes, Simplified : A simpler definition of prime numbers which takes advantage of the 12 times tables to identify small primes upto 13. In particular, a whole number is said to be prime if it is not the product of two smaller whole numbers. With the word smaller in the definition, the whole numbers 11 and 13 are prime because it is not given by a product inside the 10 and 11 times table. [This was the small example given above]

2. Quick Prime Factorization of Small Whole Numbers: Emphasis of a square or square root rule to provide QUICK prime factorization skills for whole numbers less than 169 = 13². In particular, a whole number less 169 is prime if and only is it is not a multiple of the primes 2, 3, 5, 7 and 11 less than 13. Simple divisibility rules and calculators (an overkill) here may be used to recognize multiples of 2, 3, 5 and 11. Quick prime factorization of whole numbers is a key to exact and efficient fraction practices employed in mathematics from algebra to calculus. There is no escape.

3. Fraction Operations Explained: A thought-based development of addition, comparison, subtraction, multiplication and division operations starting with simpler cases where operations are easily explained, and continuing on to general cases where all operations are justified by raising terms. In higher mathematics, if not elementary mathematics, comprehension of why methods work is highly valued, it is part of the spirt of mathematics mastery. Understanding how and why operations are justified should move you away from learning by rote. Reference: fraction operations by raising terms

4. Arithmetic and Fractions With Units: Figuring with denominate numbers, that is multiples of units of measure for physical quantities and units of value for monetary quantities. This practical value for calculations involving speed, rates in general and associated proportionality constants in daily life and also in practical and applied arts and sciences. (In algebra taught by rote, you may see similar figuring with multiples and powers of variables in products and quotients. The path here has more meaning and is very practical)

5. Oral Dimension of Arithmetic: Verbal description and extension of common practices for finding counts, totals and products by forming and adding or multiplying subcounts, subtotals and subproducts. Here calculation practices are introduced and described orally instead of symbollically, the latter being harder for many to grasp. For many, how to calculate averages and how to calculate perimeters of polygon are best described with words, the use of letter or symbols being to complicated to understand in the first instance. Mastery of common practices for counting, totaling and multiplying do not have to wait for their algebraic description. Instead, the verbal forms can be given. [These arithmetic notes expand on part of the big example given above.]

6. Place Value Revisited: An exposition of place value in decimals with places before an after the decimal point in groups of three may amuse and inform. In it, students in North America may learn how to read aloud and write on paper the decimal

   6,571,045,375,905,333,034,412.450,033,870

   as 6 sextrillions, 571 quintrillons, 45 quadrillionths, 375 trillionths, 905 billionths, 333 millions, 34 thousands, 421 ones, 450 thousandths, 33 millionths and 870 billionths. In contrast, students elsewhere may use the following "SI" (system international) method how to read aloud and write on paper the decimal form of 6 zettaunits, 571 exaunits, 45 petaunits, 375 teraunits, 905 giga-units, 333 megaunit, 34 kilounits 421 ones, 450 milliunits, 33 microunits and 870 nanounits.

7. Addition, comparison, subtraction, multiplication and division of decimals: The site development may covered more lightly than presented. The development of place value methods for all but long division is thought-based. Why methods work are both indicated. Long division method is given without justification, but with a method to check results. In all methods, students will meet the domino effects of care and mistakes. Avoiding the latter provides an end, value and tool for skill mastery, an echo of the old fashion idea that figuring well is a sign of practical intelligence.

8. Signed Numbers: The site description of arithmetic with integers and arithmetic with signed numbers is not bad. The site objective so far has not been to cover everything in mathematics, but to develop and express ideas on how mathematics should be learnt or taught. That being done, a clearer account of arithmetic with signed numbers is due.

9. More Steps To Elaborate - not in site material: Talk about scientific notation, arithmetic with, and arithmetic with mixed decimals - that is, decimals with multiple places before and after the decimal point. Relate foregoing to fraction skills and practices. Explain the comparison, addition and subtraction of scientific in terms of of finding a common factor or denominator.

Algebra Starter Steps

Again, later skill skills depend on earlier ones. Full mastery of algebra requires an efficient and exact command of arithmetic with decimals, fractions, sign numbers and prime factorization. But if your mastery of arithmetic is not yet exact and efficient, you might find Volume 2, Three Skills for Algebra, Chapters 1 to 14, for the most part to be an easy reads. You could put those chapters before or after a review and mastery of the arithmetic and number theory practices described above. Bon Appetit.

Some good practices for skill development in algebra appear in site arithmetic and number theory steps. Where these good practices are not best, say so. [These algebra notes expand on most of the big example given above.]

1. Write it, Right: Written work formats which direct students to do and record evaluation steps for formulas and arithmetic expressions in ways that can be seen as done or later for confirmation or checking. The recommended format's mechanical benefit lies in making the domino effects of care and mistakes clearer. Avoiding the domino effect of mistakes provides an end, value and tool for skill mastery in and outside of mathematics at home, at school and at work.

2. Extending the Shorthand Role of Letters: In formulas, all but one letters and symbols stand for numbers or measures that will be given, so that the one can be computed. In equations, the challenge is to find the missing value of letters, one or more, in given calculations and equalities. The site coverage of stick diagrams allows letters to denote lengths, visible lengths, whose value is unknown. This geometric role of the letters hopes to make the more general role of letters in denoting unknowns to be found clearer to students. The stick diagram methods employs fractional operations on line segments that parallel and may be easier to understand in the first instance, than the corresponding operations on linear equations in one unknown. A three column stick diagram method here aims to make solving linear equations easier to solve while reinforcing fraction skills and sense. But the use of stick diagrams is temporary. The plan here is for students to master the solution of linear equations in one unknown without the use of stick diagrams. The third column in the method shows how. The only trouble with this method is that in large groups, some students will be quicker than others in making the transition through the stick diagram method to solving equations without their use. That being said, all students may be kept busy learning to solve linear equations with and without sticks, by showing them formats for doing and recording solutions and for checking solutions, all in the context that skill has been seen in order to be believed. Mastering the formats is part of the skill being developed here. From formulas for perimeters and areas, students are familar with the role of letters in denoting a visible length or quantity. Introducing the solution of linear equations with stick diagrams allows unknowns and the letters representing them to be identified first with lengths that can drawn and seen as part of a gradual but deliberate step to extending the role of letters to denote an unknown with no physical significant in a linear equation. The step will be serve some well, be neither positive nor negative for others, and too much of a distraction for others. There is no pleasing everyone.

   After mastery or development of stick-free methods to solve linear equations in one unknown, the next steps is to master the solution of systems of equations which are essentially triangular, or have essentially only one unknown. Most words problems in junior high school mathematics may be cast as triangular systems or as systems of equations in essentially one unknown. Doing so avoids the difficulty in such word problems of mentally identifying the essential unknown in one's head. Writing the system makes that identification clearer - more mechanical. The great advantage of good notation and good format in decimal arithmetic and then algebra is the simplication of calculations or reasoning processes. That needs to be emphasized.

3. From Equivalent Computation Rules to Distributive, Associative and Commutative Laws: In arithmetic skill development, verbal description and extension of common practices for finding counts, totals and products by forming and adding or multiplying subcounts, subtotals and subproducts have a take-home value for adult and daily life too important to wait for the mastery of algebra. That being said, students often have difficulty in understanding and explaining algebraically stated and encoded properties of numbers, whole to real. But with and without the use of calculators, students may understand the role of formulas in providing computation rules. After learning that the order of operations is key to obtaining repeatable and reproducible results in the evaluation of arithmetic expressions, formulas included, students may be shown that different computation rules may give the same result. That observation sets the stage for introduction and comprehension of distributive, commutative and associative laws. In particular, the two computation rules f(a,b,c) = a(b+c) and g(a,b,c) = ab +ac clearly represent different sequences of arithmetic operations on the values of a, b and c. However, geometric motivations for the distributive law implies when the two computation rules should give the same result, that is be equivalent. Site starter steps introduce this equivalent computation rule perspective to make the algebraic shorthand roles of letters and symbols in this instance easier to learn and teach.

4. Expanding the Oral Dimension of Mathematics: Arithmetic expressions with decimals and fractions, and complicate algebraic formulas are often best seen and grasped in a glance than read aloud, term by term or group by group. The difficulty of reading algebra and arithmetic expressions aloud alone or as part of equations and laws leads to a great silence in mathematics skill provision. Verbal description common practices for finding counts, totals and products by forming and adding or multiplying subcounts, subtotals and subproducts, plus verbal description of simple calculations breaks that silence. But more can be done. Before and besides the introduction of letters or symbols to denote counts and measures, we can talk about the latter being known or not, visible or not, constant or not, variable or not, and how. That represents the first skill for algebra in site Volume 2, Three Skills for Algebra. In particular, the key concept of what is a variable can be understood before and then the use of letters to denote a number. Besides talking about numbers and measures, we may also name mathematical methods and formulas. Names are powerful. While pictures and formulas may be worth a thousand words, naming them extends the power and reach of words. Who has heard of the Mona Lisa? In the case of mathematics, we may name formulas, methods and laws, or identify them with short descriptive phrases. That adds more words to its exposition and lessens the silence. Beyond that, we may also talk about rules for using letters and symbols alone or compounded as in subscript & superscript notation to denote pure and denominate numbers. The silence can be broken in simple ways.

5. Introducing The Algebraic or Literal Way of Reasoning: The algebraic or literal way of reasoning in mathematics may be introduced by giving numerical examples in which certain numbers of coefficients vary between the examples, but their role or operations on them are repetitive. Then a numerical pattern may be described algebraically. The leading section of Chapter 15 in online Volume 2, Three Skills for Algebra, gives a clear example. In this volume called Three Skills for Algebra, Chapter 14 introduces the forward and backward use of the compound interest formula in arithemtic and algebra examples. The repetitive nature of the examples is intended to make the algebraic or literal way of writing and reasoning clearer or self-evident. The silence in mathematics due to expressions and operations being too awkward to read aloud is further broken here by offering a unifying theme for the mastery of algebra in secondary mathematics and science. In them, every formula and proportionality relation will be used forwards and backwards, numerically and literally. Before that, in basic mathematics, addtion and times tables may be used backwards in subtracting and division operations. After that, in calculus, the backward use of differentiation methods gives integration methods. And in logic at the level of calculus or better yet in earlier secondary mathematics, we may regard the contrapositive form of an implication rule as a backward use of the implication rule. Talking about and emphasizing the forward and backward use of rules, methods and patterns provides a unifying theme in the exposition of mathematics, logic and science, if not further subjects. Talking and emphasizing will lead students to anticipate and look for the backward use of all the rules and patterns provided to them.

Euclidean Logic - Mathematics Free Introduction

The following chapters develop thinking and reasoning skills needed in daily life. They provide a standard or model for arriving at conclusions and making decisions: how to argue politely if you must. They also strengthen basic skills needed in mathematics, science, technology, writing, persuasion and communication. Reason and persuasion touch all skills and all disciplines.

1. The chapter Implication Rules presents two logic puzzles. Each consisting of a rule and five questions. Answers are also provided. The puzzles show the difference between one- and two-way implication rules.

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2. The chapter Chains of Reason describes how to directly use rules one at-a-time or chained together, one-after another, for arriving at conclusions and judgments.

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3. The chapter Longer Chains of Reason starts to indicate the special role of reason in mathematics. It describes, in a very non-mathematical fashion, the concept of induction, a method used in mathematics to arrive at conclusions. This concept of induction is an example of a method of reason employed mainly or only in mathematical subjects. Reading is optional before the coverage of mathematical induction in calculus or senior high school mathematics.

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4. The chapter Islands and Divisions of Knowledge describes how rule and pattern-based bodies of thought may be organized. Here different starting points, first principles or assumptions, may lead to the same body of rule-based knowledge.

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   In mathematics, Euclid's logical or rule based arrangement of geometry provided a model for reason. The above chapter with words and images apart from geometry describes the model and the variations possibly within it.

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In 1989, a calculus student attended a general lecture of mine covering the difference between saying A IF B and saying A IF and ONLY IF B. He said he enjoyed the lecture, but that did not see what logic had to do with the mastery of mathematics. It seems his mathematics education had not emphasized nor yet met the role of logic in the development of methods and concepts - definitions and theorems. I do not know if Euclidean Geometry was then still being taught in local high schools. That being said, the above site chapters in explaining the difference, in showing how to chain implication rules A IF B together, one a time, one after another, and in describing islands and bodies of implication rule based knowledge, may lead to greater precision in reading, writing and talking; and so may ease or avoid difficulties in work and studies. The same chapters also capture with words part of the role of logic in mathematics.

More ways to Change Skill Development

The above steps extend and refine some in the essay Multiple Ways to Improve Mathematics Skill Development. See too the introductions to each site step and substep in arithmetic, algebra, geometry and calculus, and the foreword to site volumes. Site material is not optimally put, but site innovations are enough to provide better practices for instruction at multiple levels.

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