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Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com) Entrance Level \|\|Définition d'une variable \|\| Algèbre \|\| Arithmetique \|\| Logique \|\|La raison basée sur les règles et modelés\|\|
Online Volumes 1, Elements of Reason. 1A. Pattern Based Reason 1B. Math Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes & More Math (Optional Book Orders)	More Site Areas 1. Help Your Child or Teen Learn 2. Solving Linear Equations 3. Fractions Ratios Rates Proportions & Units 4. Euclidean Geometry 5. Analytic Geometry/Functions 6. Number Theory. 7. More Calculus	More Site Areas 8. Complex Numbers 9. Qc Maths Education 10. Secondary IV(?) maths 11. Real Analysis 12. LaTeX2HotEqn: 13. Electric Circuits Etc 14. Français 15. Algebra, Odds & Ends, Etc	More Site Areas 16. Math Education Essays 17. Telling & Working with Time 18. Maps, Plans & Drawings 19. Quantitative Skills for home and work, etc, etc 20. Statistics . ** Means Under-construction. Test the Twiddla Whiteboard
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High School and College Mathematics

Preparation for calculus provides the motivation for many skills and topics in high school mathematics courses. Preparation for calculus is good preparation for most, if not all, arts and subjects at work and school that require some mathematics and logic.

Similar Directions: The earlier site preparation for calculus page (written earlier) offers similar directions in three different ways - lean, wordy and very wordy. The words comment on the development of ideas in the classroom or historically.

Computer Games: If you play 3D computer games and want to write your own, you will need a good command of logic, fractions, algebra and geometry. The same advice applies if you want to enter a business, trade or science.

Follow the steps below alone or with help. The review of fractions etc in step 4 should come after steps 2 or 3. Other than that, which step to put first appears to be a matter of taste. Site areas which do not appear in these steps contain further material - optional reading. On first reading, focus on learning how, and leave explanations why for later.

Put logic First (if possible). Read the first logic chapters in Volume 2. Logic mastery will, we hope, ease fears and difficulties, or if you have none, enrich skills and knowledge. Volume 1, Elements of Reason, introduces all site volumes.

Master logic carefully to develop precision reading and writings. Skills and knowledge are easier to obtain when you are able to read precisely what is written, and do not assume too much. Marks in all subjects are base on your written work. Precision reading will help you recognize errors in your written work - does it say precisely what you meant.

Secondary I and II Material

Meet the role of fractions in algebra: Explore the site area Solving Linear Equation with stick diagrams to further develop your algebra skills - those needed for solving problems in one or essentially one unknown, and see how fractions of line segments, the sticks, are combined (added, subtracted, multiplied and divided) exactly in the solution of linear equations.

In solving linear equations, you can check your answers. If the the original equation or equations are not satisfied, look for your mistake in your solution or in your check.

The site area [Solving Linear Equations with fractional operations on Stick Diagrams] develops algebra and fraction skills and sense together in way that can read before or besides the algebra chapters 8 to 14 in Three Skills for Algebra . Teachers & tutors should look at these Effective Algebra Lesson Plans for more material & suggestions for consolidating algebra and fraction skills & sense - a geometric view of the distributive law.

Next read the Chapter 15, solving linear equations, in Three Skills for Algebra, alone or with help. The discussion of general systems is optional for junior high school students.

Test your algebra skills and linear equation problem solving skills.

Remark: Steps 1 to 4 may be covered in junior or senior high school, the sooner the better. The following steps are for senior high school students and older students in college or adult education.

Review or Develop Algebra and Fraction Sense and Skills. Read (i) the algebra chapters 8 to 14 Volume 2, Three Skills for Algebra. Volume 1, Elements of Reason, introduces all site volumes.

The shorthand role of letters and symbols is meaningless for many people in school and out. But the shorthand role is easier to grasp when we first learn to talk about numbers and quantities, and how they may vary, before the use of letters and symbols. Doing that would make algebraic ways of writing and reasoning clearer in calculus and all of high school mathematics.

Chapter 14, Compound Interest, in Three Skills for Algebra, develops algebraic skills with the aid of a calculator. Calculators are useful but success and precision in mathematics requires efficiency with fractions without one.

Alternate Between Steps 3 and 4 if you wish. Each one has a different taste. The addition of animated graphic make Solving Linear Equation with stick diagrams easier than before.

If you spend grades 1 to 11 or 12 in mathematics classes without mastering fractions sense and skills properly and efficiently, you have been cheated - several hundred or thousand hours of your time has been wasted.

Optional but Recommended: (i) Visit the fraction pages in the site area, Fractions, Ratios, Rates, Proportions & Units, to check your fraction sense (step 4 could have helped in here) and to see the justification of methods for adding, subtracting, dividing, multiplying and comparing fractions. (ii) Develop an algebraic view of problem solving with units and with rates and proportions, binary or multiple, direct, joint or inverse. (iii) learn how to carry units through solutions in a way that relies more on mechanical skill in algebra than on thought. Here is an algebraic perspective and clarification of skills and concepts in junior high school mathematics, which may be read after steps 1 to 4 above.

The site area Fractions, Ratios, Rates, Proportions & Units view of junior high school concepts may help teachers & tutors develop skills and concepts. Senior high school students may explore this area to review and reform their understanding. Area material needs to be rewritten to make it readable for junior high school students. Writing is an iterative process in which the first draft is not always best.

Fractions are needed for algebra and beyond. In modern times, that is today, we see and will see more and more cognitive experts and curriculum advisors suggest the replacement of fractions and algebra skills and sense development with calculator push-button exercises in which the intellectual component of mathematics instruction is eliminated to provide a child- and technology- centered learning environment. Yet arithmetic mastery was and remains a sign of intelligence in work and study.

Check & Consolidate your Arithmetic Skills. Do asap, the first set of arithmetic problems, chapter 7 of Volume 2, Three Skills for Algebra, See too Simplification of square roots. Logic mastery asap is recommended for greatest benefit from site pages.

In doing exact arithmetic, if your result is not the same as that of another, one of you has made an error. Learning how to follow methods so that you obtain repeatable, reproducible and thus verifiable results is a must, not always emphasized, for work, school and home.

See too these Real Player arithmetic webvideos - a few a day, not all at once.

Aim for a logic-based mastery of mathematics after arithmetic. That being said, arithmetic can be learnt by rote, know-how without the know-why, provided you put aside your calculator and learn the times and addition tables and learn to do arithmetic with fractions and decmals (add, multiply, divide and subtract) in an objective, efficient and automatic manner - arithmetic results should be repeatable and reproducible, and you should know that an error in one step makes all the rest wrong. Once you have a logic-based mastery of mathematics after arithmetic, you can if you want retreat to develop a deeper, logic-based understanding of arithmetic, a retreat that could become easier, and a retreat that can be woven in to the explanation of further mathematics for skill perfection and enrichment.

Secondary IV and V Material

Master Geometry without and with coordinates: Site areas on Euclidean Geometry and Analytic Geometry offer senior high school students and teachers lean logic-based development and connections of plane geometry, plane trigonometry and functions of one variable. The site coverage of Analytic Geometry does not include all that calculus requires, but is a start, and the missing material can be found elsewhere.)

Remark A: The treatment of Euclidean Geometry is not full, but it is enough to provide a logic-based consolidation of the skills and concepts seen in junior and high school mathematics, those needed to develop analytic geometry and calculus. The treatment of Analytic Geometry assumes results of the site treatment Euclidean Geometry with the assumption that real numbers alone or in ordered pairs may provide coordinates for lines and planes in space. The result is a logical, coordinate based, development of the key skills and concepts in analytic geometry, plane trigonometry and functions. The reliance seen here on geometric diagrams can be replaced and will be in studies of modern pure mathematics. Or, we could use the alternate route in Remark B. Remark B: Step 6 follows the traditional path of defining trigonometric functions for acute angles with the aid of similarity postulates before defining them for all angles. This complex numbers introduction leads to trigonometry in general for all angles, with right-angle triangle, similarity based, trigonometry coming last. For the brave, that gives faster route for developing the senior high school mathematics which calculus and electrical studies requires. This route is leaner in that its reduces the need for Euclidean Geometry to a discussion of similarity principles.

Remark C: In the modern mathematics curricula of the late 1950s and 1960s, sputnik inspired, there is a fuller treatment of coordinate-free Euclidean geometry along side a general emphasis on logic. Geometric proofs were challenging - not student friendly. So Geometry was eliminated. But Euclidean Geometry was the traditional place for the emphasis of logic and Euclidean model for reason. Site logic and Pattern Based Reason chapters present the Euclidean model in a math-free way and do so to develop better study skills - or the precision reading and writing better work and study skills demand.

Test your arithmetic and Algebraic Skills: Try the remaining problem sets in Chapter 7 of Volume 2. Get someone to identify all errors in your answers in notation and comprehension, so you can learn from your mistakes.

Optional: Explore the Number Theory Site Area. Here is a mix of easy and challenging lessons, some in sequence. If one lesson or sequence is not to your liking, try another.

Secondary VI & VII Material

Meet or Revisit Calculus: Begins with the why slopes geometric preview before the more algebraic why slopes preview chapters in Volume 3. Then explore more of the site Calculus Introduction. Volume 1, Elements of Reason, introduces all site volumes.

Remark: The introduction points to simpler ways to cover the first steps in calculus. Those simpler ways are for all. The algebraic way of writing and reasoning is usually required suddenly in calculus. The previews here and the latter decimal view of limits, continuity and convergence provides a more accessible and less algebraic demanding or shocking approach to calculus.Then the introduction includes enriched material - the proofs that are often omitted in first courses. Innovations here make the proofs easier to understand, but not simple. The enriched material is for people who do not like to accept mathematical methods without proof. The site area Real-Analysis-Decimal-View (advance calculus) and the calculus introduction at this site emphasize an error-control decimal view of limits, continuity, convergence.

Remark The Modern Mathematics movement of the 1950s and 60s made calculus algebraically hard or inaccessible need-be by following a decimal-free view prevalent in pure mathematics. Here is a correction sufficient for students outside of pure mathematics that may provide a stepping stone and context for the decimal-free, epsilon-delta view of pure mathematics.

Remark: Steps 5 onward can be followed or explored in any order you like.

Learners at all levels need someone to review their written work for mistakes in notation and comprehension in order to learn from their mistakes. Every time someone (on your side) identifies a mistake, say thank you because now you know not to make that mistake again. Do not worry, your helper will be employed in identifying further mistakes. It is a win-win situation.

Words For Instructors

Mathematics education reform since 1990 has been focused on delivery style changes - direct instruction versus different styles of indirect instruction and how to engage or motivate students. But site material adds new perspective.

Skill and concept development needs to follow or take small, logical steps, one at a time and one after another. When steps are too large or missing, or their description is unclear, learning and teaching are both harder than need-be. Site pages point to slowly realized remedies for gaps and shortcomings in skill and concept development and definition. There lies a new perspective that needs to be considered along side or even before changes in delivery style. Where gaps or shortcomings have contorted skills and concept development, the remedies point to straighter paths and also to optional, alternatives paths. The site aim since inception in 1995 has been to be collect and put first lessons, easily understood and repeated in class, fresh or recycled, to ease or avoid common difficulties, to extend the common knowledge and to prepare for further learning. Success appears to be in sight. Site ideas will ease or avoid confusions and difficulties while enriching studies and instruction.

As of January 24, 2007, site lessons plans for secondary mathematics and calculus instruction are essentially complete in accordance with inductive principles met in 1981 for skill & concept development. Primary school teachers should master site secondary I and II material to understand what is of most importance in their development of elementary school mathematics.

The online Volume 2. Three Skills for Algebra, introduces and clarifies the use of words to describe numbers & quantities and introduce terms to describe hitherto silent or unspoken themes in the algebraic use of equations and formulas - their direct and indirect, and the possibilities of numerical and algebraic solution for the indirect use problems. It seems that we can add a new verbal dimension to mathematics via a greater use of words or key phrases to describe numbers and quantities, and to describe the direct and indirect use of formulas. The insertion of more words in the introduction of algebraic reading, writing and reasoning will ease or avoid difficulties for novices, and enrich the comprehension and expository skills of students, tutors and instructors in understanding and explaining algebra at many levels.

Site pages also include hints of how geometry may be a source of arithmetic and algebraic skills and concepts upto the level of calculus. I say hints as more elaboration will come here or elsewhere to make the geometric origins clearer - more obvious and useful. Site pages aim is to connect well-known elements of algebra and geometry with words and diagrams focused on developing numerical, algebraic and logic skills and sense in an thought-based manner where ease of comprehension and plausibility is more important than rigour from the introduction of digits for decimals notation to the end of elementary calculus..

Mathematics Curriculum Shifts

Volume 1, Elements of Reason, introduces all site volumes.

Site innovations for mathematics and logic education were initially developed to fill skill and concept gaps and flaws sensed in the high school exposition of modern mathematics curricula prevalent from mid-1950s to the 1980s in schools and colleges. However, exploration and refinement of ideas for learning and teaching points to an alternative thought-based development of high school mathematics (algebra, geometry, trig and functions) needed for calculus. The net result may be fewer but more effectives hours in high school mathematics.

These curriculum shifts could be the basis for a leaner and more effective mathematics instruction.

Two Shifts - clearer and effective ways to develop algebra and fraction skills and sense: The puzzle of how to introduce the algebraic way of writing and reasoning clearly and directly was first met by in high school days 1965-70. Difficulties of fellow students and instructor in understanding and explaining algebra slowed the site author's education. The first algebra chapters in the 1995-6 Volume 2, Three Skills for Algebra, point to a solution - a greater verbalization in mathematics in which the overlooked ability of describing or talking about numbers and quantities is recognized and emphasized. That is before and then besides the introduction of letters and symbols in algebra as placeholders for numbers and quantities in calculations or their description. The spring 2005 site area Solving Linear Equations with fractional operations on stick diagrams also introduces algebra in a parallel approach to the foregoing, which comes first is a matter of taste, while consolidating fraction sense and skills. The two approaches together provide a solid base for algebra for students starting their teenage years, or later remedial instruction. Algebra self-instruction alone or with help allows student to benefit immediately. For self-instruction, the algebra chapters in Volume 2 are recommended first. Volume 1, Elements of Reason, introduces all site volumes.

Third Shift - Complex Numbers & Easy Consequences: Vectors & coordinates, polar & rectangular, are used in a very simple, logical development of complex numbers., one that implies a quick, logic-based development of senior high school mathematics (and the use of complex number methods with eⁱ in technical and engineering schools.)

Technical note: Assumption that the head to tail addition of vector described displacements in the line or plane is independent of our choice of rectangular coordinate systems implies the distributive law for real and complex numbers. In other words the geometric assumption that the coordinate description of sum of displacements gives a new logical development of the properties of real and complex numbers in ways that simplify and provide a base for analytic geometry and trigonometry - that favored in university program without explanation. This logical development based on geometry covariance, an idea that appears in relativity, provides an axiomatic shift for mathematics education with consequence for high school and college studies. See the logic chapter Islands and Divisions of Knowledge for thoughts on multiple starting or entry points in the deductive arrangement of ideas. Self-instruction in complex numbers alone or with help allows student to benefit immediately At the college level in engineering and physics, the properties of complex numbers and benefits for trig via the cis function were often presented as efficient shortcuts without proof. Here is a justification that may accelerate college and high school instruction.

A further shift - calculus re-arranged.: Calculus demands full mastery of logic, fraction skills and sense, algebra, analytic geometry, trig and functions. That demand provide a standard and goal for high school mathematics instruction which needs to be emphasized as the coverage of more and more topics in high school may distracts learning and teaching from the full mastery.. Even with that full mastery, calculus employs the algebraic way of writing and reasoning at full strength. The site calculus introduction employs geometric and algebraic previews, and decimal view of error control in computations, to develop the multiple full strength uses of the algebraic way of writing and reasoning gradually and systematically in ways that should eliminate or avoid some calculus perils, and allow more to go further. Calculus self-instruction alone or with help allows student to benefit immediately. Note in a recently seen discussion of the modern mathematics curricula of the 1960's, there is mention of a slope-oriented analysis which site geometric and algebraic previews may duplicate. If that is the case, site previews are re-inventions and not new.

Expert Instruction (Mastery Learning): In classes, grades of 50%, 65% or 80% in a sequence of assignments and tests say how well you are doing, but do not say what you have missed. If the teacher or marker identifies and correct all mistakes in your answers, you can learn from your mistakes, and you know what you missed. In my classes, I intend to make a checklist of skills and topics, so that I can record which ones have been mastered to report to student a grade - the percentage of skills and topics which appear to be mastered, and to track and report what remains to be reviewed by the student or re-taught. Efficient learning (more gain for less effort) might follow. But I am advocating here what I have yet to do in class, an expert approach to learning and teaching. Tutors too can be hired to follow this approach instead of being hired to improve marks.