Appetizers and Lessons for Mathematics & Reason Français: 26 pages
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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

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What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

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Site Review: Mathphobics, this site may ease your fears of the subject, perhaps even help you njoy it. ... unintimidating, sometimes funny and very clear. ... . Read all. Continue with Volume 2, Three Skill for Algebra.

Site Review. Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation ... Read all. See site books as well.

Teachers & Tutors: Site material uniquely explains common troubles in terms of steps too large or missing. Plus, 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.

Location: Site Entrance < Archives < Mathematics Education Essays << mathematics curriculum shifts

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Mathematics Course Design (Curriculum) Shifts

Start with pattern based reason or Logic

[Online Books and More Site Areas] [Study Tips] [Directions for High School Mathematics - Calculus Preparation] [Curriculum Shifts - Shorter, Better, Stronger] [References]

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 Three 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. 

    There is a fourth skill for algebra in Three Skills for Algebra, namely a development of the ability to talk about or describe the numerical and algebraic use of formulas and equations with short descriptive phrases: (i) forward and backward use (or direct and indirect use) and (ii) algebraic and arithmetic (numerical) solutions.  These phrases appear in Chapter 14. can be used through out high school mathematics to identify recurring themes - key objectives - and to provide another fresh perspective on the algebraic way of writing and reasoning.

    In mathematics, I would like to see the first two years of secondary school consolidate arithmetic and introduce algebra skills. Then I would like the third year to be given as a reward. That is, I would like it  to provide applications, one at a time, and one after another, to develop a favorable impression for students who have begun to dislike the subject and may drop out,  but an impression that need not be terminal as it includes motivation for further studies.

  • Third Fourth 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. 

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Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills.

Mathematics Skills For Ages 3 to 14

Skills with take home value

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons


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Location: Site Entrance < Archives < Mathematics Education Essays << mathematics curriculum shifts

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Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

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

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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