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Grades 7 to 12 Guide
1. Arithmetic Lessons
2. Algebra How-TOs
3. More Algebra How-TOs
4.  Geometry Steps Into
5. More Geometry
6. Calculus How-TOs
7. Logic in Math
8. Complex Nos Geometrically.

Proper notation & format 
makes the hard easier.

Pages For Teachers

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Notes for Instructors
Sec I, Teaching Ideas
Sec II , Teaching Ideas
Sec III, Possibilities
Permissions
A Site Map
Vol 1, Elements of Reason

Show a student how to learn and that helps one. Show a teacher or  tutor how to make skills and concepts easier for students and that helps many. 

Miscellaneous

Your IP Address  & how to use it

Three Links for Teachers:
(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

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.

Parents: Site Area Helping Your Child or Teen Learn  covers 1. Speaking Skills, 2. Reading & Writing, 3. Preparing for Science, 4. Math Work Books, 5.Books for Parents, 7. Having Patience -you'll need ii. Authority: Use it or lose it. /Parents and teachers need to say no for little things,  otherwise the authority to say no for big things will vanish.

Welcome. Two of many  site reviews  follow.  

• Math Forum, 1996 online classification of a dozen math lesson websites: ... 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: ...  
• 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. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. 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.
  
  Author's Note: Parts of www.whyslopes.com  are well-written and second to none in skill and concept development.  Other parts are not - improvement are planned. .  Visit visit  mathsisfun  purplemath, themathpage  and further sites besides  to compare and combine different viewpoints. 

For self-instruction, site pages are best for readers15 years to adult. Younger readers could leave site exploration for later. 

Page Contents: [Junior High School Material][ Senior High School & Adult Material] [More Senior High School Maths and Calculus] [College or Enriched High School Maths] [Math Education Essay - Where's Maths, Progressive Mastery, Hopes & Conclusions]

For instruction, site pages offer ends, values and methods for skill and concept development not found elsewhere. 

 Inductive principles for progressive skill and concept development, mentioned in the  communication of skills chapter 2 of  1995 Volume 1A, Pattern Based Reason, appear in the foreword to 1996 Volume 1B, Mathematics Curriculum Notes.  The principles give a criteria for judging and so reshaping course design and delivery.  Yet the principles alone are insufficient.  Reasons, or ends and values for mathematics education, are also need of exposition.  Since 1995, visitor  reports of errors small and large has led to site corrections. Professional editing is still required.  In my teaching days, I would repetitively include too many words in a single sentence in the hope that  from  word association students would get the meaning of unfamiliar words and terms.  That repetition in my online books is not to everyone's liking. 

Online Help for Instructors:  Many instructors face curriculum changes or face courses in maths which they have not taught before.  Instructors unfamiliar with maths may be given the task of teaching it.  If someone you know needs help great or small  skill and concept development at the primary or secondary math instruction, suggest that they join WiZiQ (membership is free) and then email me a short (one hundred word say) description of their math teaching assignment and their background. I will try to use a WiZiQ virtual classroom to discuss and answer questions, either immediately or after some reflection.  

Help Delivery: Here is a quick imperfect demo of the virtual classroom. If you can see and hear it (excuse the hiss in the recording) then most likely, your computer system will allow you to attend the virtual seminars and lectures I may give.   The invitation to them will come in the form of a URL which you follow in web-browser to the lecture sign-in page - ideally a few minutes before the schedule start of the seminar.  They may a pop-up window in which you have to give flash (or java) access to your PC speakers, video and microphone. The latter is optional. I am student of K1-12 maths and logic. I expect or hope that exposure to the questions or views of others about instruction will improve my replies - practice make perfect.  The service here could fizzle due to lack of demand. Or it might be overwhelmed by demand. 

Junior high school  level material: 

 The site math folder Solving Linear Equations  offers a geometric stick diagram  method  which teachers, tutors and parents may use to geometrically introduce the solution of equations of the form ax+b = d or ax+b =cx +d and to algebraically introduce the solution of  systems of equations in essentially one unknown. Most junior high school word problems may be cast in the latter form. Moreover, solving systems of linear equations in essentially one unknown requires a practical if not theoretical mastery of associative and distributive laws for arithmetic. The stick diagram introduction to solving linear equations of the form requires and visually reinforces fraction skill and concepts.  The folder also introduces senior high school topics, that is the solution of triangular and general systems of equations, in manner that may be accessible to junior high school students. Most steps are illustrated with diagrams and animated gifs.  The site folder emphasizes solution verification, so that students may learn  to check solutions and even correct them before submission.  The math folder Solving Linear Equations may be employed for review or self-instruction by senior high school and keen or gifted junior high school students. 

Senior High School, College and Adult Material:  

Volume 2. Three Skills for Algebra present ideas for logic and algebra mastery. 

Logic chapters show how to use implication rules A if B one at a time and one after another to arrive at conclusions. Logic mastery may improve reading and writing, and ease or avoid difficulties in following instructions at work, at home and in school.  B

The 1995 Volume 2 in talking about three skills for algebra and the forward & backwards use of formulas provides a more wordy, oral dimension and base for introducing and developing algebra. In senior high school mathematics before and including calculus, words have been missing in the introduction of the shorthand role of letters and symbols in mathematics. That role has appeared in silence, given by example. The role has been clear to some, but not all. Talking about three skills for algebra provides a new perspective to make skill and concept developing clearer and richer. That new perspective is enlarged by talking about the forward and backward use of formulas provides a unifying them for high school mathematics.  More generally, most if not all  rules and methods for arriving at conclusion or numerical results in mathematics,  logic and quantitative reason will be used forwards and backwards, or directly and indirectly. Talking about all the foregoing eases or removes the silence in mathematics that stems from the nonverbal element of mathematics. An arithmetic or algebraic expression like a picture is worth thousand words - better seen and grasped in a glance, than described stroke by stroke, or in the case of mathematics, symbol by symbol.  Thus talking about three skills for algebra and the forward & backwards use of formulas provides a more wordy, oral dimension and base for introducing and developing algebra. 

More Senior High School  and Calculus Material:  Calculus previews and  more elements of Calculus  appear in the Volume 3, Why.Slopes.&.More.Math, and in the newer site math folder on Calculus.  

For calculus and senior high school mathematics courses before it, the 1995 site volume 3, Why Slopes and More Math, begins with geometric and algebraic previews of calculus.  The geometric preview explains why slopes are studied in senior high school maths.  The algebraic preview gives the slopes (derivatives) of function y = f(x) as factored polynomials or factored rational functions,  and shows how sign analysis of the factors may locate function high points (maximums) and low points (minimums).    One or both previews in or before calculus provide a context for the study of slopes and polynomial factorization, and also reinforces or develop algebraic reasoning skills. Algebra is required at full strength in calculus. The two previews provide appetizers or starter lessons for that full strength use.

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  employs at full strength  most earlier elements of high school mathematics including  functions,  trig, algebra, mathematical induction, more logic,     geometry and  arithmetic exactly or symbolically.  If you are a high school student, you have to aim for skill and concept development and perfection in the foregoing areas of mathematics and logic.  No detail nor concept should be left unmastered. 

Slopes to straight lines, operations on polynomials, quadratics and functions appear in the analytic geometry site folder. Preparation for calculus requires these topics.   

College or Enriched  High School Maths:  Site folders cover complex numbers, more calculus, number theory and real analysis. (A)  This first  lesson on complex numbers   may be met after mastery of rectangular and polar coordinates, and besides or before trigonometry and the study of vectors in the plane. The site area on complex numbers  (different from the lesson) shows how the college and high school coverage of trig and vectors may be made easier if mastery of complex numbers is put first. (B) The site  number theory folder presents and explores geometric or natural methods to imply or support axioms for real and complex numbers. (C) The site folders and volumes on calculus and  real analysis recall or introduce the decimal base for key theorems and concepts met in courses on the latter. Set theory provides a framework for codifying and developing modern mathematics and its logic.  While it was taken as a model for form course design and delivery in the 1950s, the framework is too technical for most to follow.  Alternative is needed. Site page provide the building blocks for that in a manner that may make the latter framework more accessible. 

Remark:  The site folder on number theory  and the site how-TOs above include or point to the explanation of decimal arithmetic methods for addition, comparison, subtraction, multiplication and long-division.  Site pages  provide or indicate full  thought-based developments of all skills and concepts from first steps in counting to calculus and beyond in an accessible or natural matter.  There is work to be done. Site pages so far indicate some ends, values and methods for mathematics education at the senior  high school and college level.  A base for the latter would be given  by a clarification of ends, values and methods for primary and junior school mathematics. In general, mathematics education will continue to be a mix of learning based on rote and on comprehension. More precisely,  comprehension and justification in mathematics requires the skill to apply and follow methods, step by step, 

Mathematics Education  Essay
June 8, 2009

Content and Delivery Matters and Choices

A. What to Cover and When - Put context first

One of my French textbooks for learning the language introduces words and phrases that an individual or family may meet in everyday activities, say eating in a restaurant;  traveling by car, bus, train or plane; visiting a farm;  and so on.  All provide a context and motivation for word and phrase mastery.  Common activities and common needs in general  may also provide ends, values and a context for developing mathematics or quantitative skills.  At home, at school and at work,  we meet numbers,  calculations, amounts and measures in telling and keeping track of time and dates, in money matters, in using maps and plans.  That being said,  mathematics education in primary and junior high school could be  focused and be explicitly developed most around the requirements of common activities and themes, with some, not much, preparation for future studies.  That focus and development could provide concrete ends, values and paths for skills and concept development in primary and junior high school mathematics.   Later senior high school mathematics could focus on developing the more advanced commonly required skills and concepts, on covering the mathematics needed for in common trades and professions, and on a preparation for calculus and further studies in mathematics.  

The foregoing focus and sequence of skills and topics in accordance first with common needs  and then less common needs should be provide motivation.   That motivation may be stronger initially, where the utility is most apparent and then slowly dwindle, but still convey some motivation.  Compare and contrast that with present day instruction where teachers, parents and students learn and teach because that is a formal requirement, and not my immediate reason.  

Forgotten Values:  The ability to apply a method and record or draw it steps in sufficient detail for the student or others to follow and verify later is a key value for mathematics education.   The ability to arrive at repeatable, reproducible and observable results is another value for mathematics education and quantitative skill development at home, in school and at work.  In arithmetic, figuring well used to be a sign of intelligence. 

B. Progressive, Observable, Development

Volume 1B, Mathematics Curriculum Notes,  begins with an inductive criteria for course design and for instruction.  From first steps to research level, mathematics consists of observable and concrete elements in arithmetic, geometry, algebra and logic, etc.  Mastery of mathematics is seen when students write or read decimals, do arithmetic with decimals and fractions, use formulas forwards and backwards, draw or make geometric figures, record the forward and backward use of implication rules A IF B in arriving at conclusions. With  mastery of mathematics and quantitative skills in general cast as a collection of observable, and verifiable skills in real life or on paper, instructors may provide correction and directions for for the development of these skills, one at a time, one after another. 

Instructors of music and physical activities employ progressive skill development in which movements are decomposed into simpler and more accessible ones for practice and mastery, one at a time, one after another, in sequence.  When difficulty with one skill or concept is seen, familiarity with the progression of skills and movements allows instructors to diagnose the likely source of student difficulty in order for students alone or in groups, to be taken back before the likely source to rebuild confidence and to tackle or circumvent the difficulty. Different course design and different instructors may be judged by the extent in which the course designs facilitates progressive skill development.  

Volume 1B, Mathematics Curriculum Notes,  after the presentation of a inductive principles for progressive skill and concept in general and in mathematics, describes the difficulties and shortcomings of earlier course designs. 

In retrospect, first the technical development of numbers skill and sense from primary school to college level needs to be continuous and consistent - the decimal command of numbers and arithmetic needs to be recognized and valued and  sanctioned from primary school to calculus, without throwing out set skills and concepts that may enhance the senior high school and calculus development of mathematics.  Further, the algebraic shorthand way of writing and reasoning needs to be developed in a clearer and more accessible manner.  How to do so is shown in chapters 8 to 14 of Volume 2, Three Skills for Algebra, and in site folder Solving Linear Equations. See the comments above.  Fourth, negative numbers may be introduces as coordinates, and arithmetic with negative numbers made easier - these notes on integers (best viewed  with Internet Explorer) show how - the how with rational numbers would be similar. Fifth,  the development of trigonometry and vectors in two dimensions may be simplified and accelerated by showing students how to add and multiply points in the plane using  (i) rectangular Cartesian coordinates.  and (ii) polar coordinates.  Sixth, the introduction of calculus may be made easier via the use of geometric and algebraic previews. Within calculus, the elementary to advanced  theory of limits, continuity and convergence may be made more concrete and thus accessible by stating and  continuing  with development a decimal, error-control viewpoint. 

Volume 1B, Mathematics Curriculum Notes, also describes simply yet precisely, the role of rule-based reason, that is logic, in providing a thought-based framework and codification for mathematical thought. While modern course begins with axioms for real numbers stated in an algebra- and set-based manner,  those axioms and the necessary mastery of algebra for their comprehension may implied by refining and recasting the primary and junior high school development of number skills and  sense, and by more carefully and less silently introducing in junior  or early high school studies, the shorthand algebraic way of writing and reasoning in mathematics. Skill and concept development has to be progressive.  

Ideas which are easily repeated and understood may provide a common knowledge  of mathematics and the rule-based reason sufficient for a more formal and rigorous comprehension.

The modern mathematics program of the 1950's and 60s' had logical elements, but it assumed the algebraic shorthand role of letters and symbols in the mathematics instead of developing the latter.  Mastery of the latter shorthand role was not a natural talent for everyone.  The axioms themselves provided an artificial base for skill and concept development.  But they did not sanction the common use of decimals.  Volume 1B was written in the first instance to identify those strikes and to pose the question of how to remedy them.  The remedy, much to my surprises, appears to be lie in the form of developing number theory, and the aforementioned axioms, based on the counting principles hidden and not mention in primary school arithmetic and based on accessible, easily accepted,  geometric practices designed to introduce and represent signed coordinates and complex numbers too. The foregoing yields a semi-formal, thought-based foundation for developing number sense and implying the properties of numbers, whole to complex, that are take in all or part to be axioms in modern mathematics curricula.  

From the mid-1950s to the present year 2009, to the best of my knowledge, the mathematics course design and delivery in North America first implemented and fine-tuned the modern math program for three and half decades before non-technical concerns dominated mathematics instruction. Mathematics instruction and quantitative skill development at home, in school and in the workplace needs a clarification of logical dependencies in skill and concept development, and beyond that critical path analysis of what can be taught be taught and when.  The development itself needs to be optimized for the sake of the latter and for the sake of lean and effective curriculum. 

C. Summary, Conclusions and Hopes

Site material offers  technical support for  mathematics and logic education from primary school to  college level.  Adult, college and senior high school students may explore site material for self-instruction.  Site material follows from a concrete viewpoint of education in which the latter offers progressive skill and concept development in arts and disciplines based on and defined by observable skill and concept development.  Whence site viewpoints may be countertrend while site technical contributions will be timeless.  For the technical development of mathematics,  talking about three skills for algebra and the forward and backward use of formulas, rules and methods in mathematics and in logic provides and clarifies the oral dimension in mathematics and logic education from algebra to calculus. Talking about and clarifying the ends, values and methods of instruction, coupled with critical path analysis,  may lead to clearer course design and delivery with objectives understandable to parents, instructors and students.  

D.  Observable and Objective Skill Development 
Versus Constructivism and Subjectivism

Striving for Objectivity:  The past development and study of law and the hard sciences was based on the hope for greater objectivity.  The rule of law should be independent of both judges and the state or status the accused.  The application of law should be based on precedent and on written laws.  The peer review process in the hard sciences emphasizes repeatable, reproducible and hence verifiable results.  In mathematics, a claim is verifiable if its author shows or implies on paper how to obtain the result from earlier well-known results and methods.  Practices in modern society depend on equipment or methods that give repeatable and reproducible results, alone or in combination.   But large industrial, urban or rural scale repetition may compound small harmful effects of those results, effects not noticeable on a small scale. So repeatable and reproducible does not imply safe. Observable arts and discipline are based on methods and tools which give repeatable and reproducible results.  In pre-constructivism instruction would cover those methods and tools, and aim to lead students to employ and perform with them in a way that leads to repeatable,  reproducible and hence verifiable results. But not all is certain -  Site Pattern Based Reason, naively describes the hopes, origins and limitations of rule- and pattern-based thought and methods.

• England: The mathematics part of  National Curriculum of England describes performance levels for students in four stages from ages 5 to 17 years.  The sequence or progression of skill and concept development is clear. Yet it  might benefit from a critical path analysis which covers and emphasizes dependencies of later skills on earlier ones for the sake of remedial to enriched instruction.  Parents and teachers, not necessarily strong in mathematics, given a knowledge of the dependencies would be able to better deliver remedial to enriched instruction.   I say England and not the UK here as Scotland  has it own curriculum. The Scottish and Irish National Curriculums may overlap and differ.  Other models for instruction may be noteworthy.

Help Me Learn/Teach;

1. Algebra
   words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)
2. Arithmetic
   -new Arithmetic Folder
   - exercises
   - with fractions
   - videos on primes, lcm, gcm,lcd, square roots etc
3. Calculus - geometric preview, algebraic preview,
   3 study guides,
   much more
4. Complex numbers
   -starter lesson with java applet - easy consequences for trig & vectors in the plane
5. Education
   - Empirical Course Design & Delivery
6. Fractions
   - alone
   - by rote
   - with algebra
   - videos

7. Functions - introduction
   hindsight - composition aka
   substitution -
8. Geometry, Euclidean - Correspondence of triangles,  Triangle construction,  duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
   
9. Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
10. Logic
    - First Steps -
    Symbols in Logic -
     Occurrence & Truth Tables - Indirect Reason -Indirect Reason More
11. Proportionality
    - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions
12. Real Analysis
    - Decimal View of concepts and of proofs
13. Rules &Patterns in Science, Technology & Society - Pattern Based Reason
14. Mathematical Reasoning, empirical, inductive or deductive
15. Units
    - in rates & slopes & (?) derivatives
    - in ratios & proportions - slopes & rates included
16. Complex Numbers & Vectors & Trig
    -  trig expression for dot & cross - cosine law