YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
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
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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 www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
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.
Mathematics Ed. or
Instruction in General
I. Three Goals to Set for students.
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A. Master the rules, methods or patterns in arithmetic, algebra, trig and calculus so that in your hands, they lead to the same results as others - repeatable, reproducible and hence correct results. If you belong to a group of students whose results differ after using the same method to arrive at them, you or the group have problem. More drill and practice will be required alone or with help.
First Sign of Intelligence: The patience and ability to figure well, to follow multistep method in arithmetic carefully, precisely, to a right answer, points to and develops the ability to read and write clearly, with precision and not with confusion in many arts and disciplines.
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B. Watch for the use or combination of rules and patterns, one at a time and one after another, in sequence, which gives or suggest new rules and patterns.
Second Sign of Intelligence: If you see how rules and patterns can be combined to get results or more rules and patterns, you have found the key to the thought-based development of more skills and concepts, those to come and if you like, those you have seen.
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C. Watch for the origins of rules and patterns to understand their benefits and limitations.
Third Sign of Intelligence: If you develop ability and interest to see and know the limitations of rule and pattern based reasoning in theory and practice at home, at school, at work and in society in general, you are becoming a critical thinker. Good luck. This third sign of intelligence is not always appreciated.
In arithmetic and beyond, students need to learn to apply rule and patterns one at a time and then in combination, one after another, in repeatable, reproducible and hence verifiable manner. In days gone by, precision figuring skills were taken as a sign of intelligence or potential to follow, if not bend, rules and methods, with precision to meet the needs at hand. Rules and patterns with repeatable, reproducible and therefore verifiable results literally provide a base for society to function, but there is a caution. Rules and patterns once found or given need not be fair, nor sustainable in the long-term. Their assumption always involves some risk. The knowledge of how to use rule and patterns in a precise, repeatable and reproducible way, and the knowledge of the benefits, origin and limitations are both needed for critical thinking and intelligent problem solving at many levels.
II. Supporting Aims A and B
The thought-based development of mathematics is both inductive and deductive. Key patterns (algebraically described?) can be suggested and illustrated say by drawings and by numerical examples, that is the inductive part, and then assumed for use in further chain of reason. That is the deductive part. A balance is needed. Too much deductive detail will alienate students. Ease of exposition or comprehension should be the guide. Where derivation is too long, assumptions (axioms or assumed patterns) should be explicitly stated for the sake of transparency. The aim is an operational command of mathematics with a knowledge that an inductive-deductive account of the discipline is possible. References for fuller thought-based development should available for students who want to go farther.
After arithmetic, an operational command of quantitative skills sufficient for mathematics to pre-calculus level may follow from lessening algebraic difficulties as indicated the site entrance, from mastering logic and from the assumption and geometric interpretation of the properties of real and complex numbers, from the easier consequences of those properties, and from the assumption of that all real numbers have decimal expansions, finite or infinite. The geometric introduction of complex numbers only requires and involves the junior high school level familiarity of coordinates, translations, reflections and rotations in the plane, and may be use to develop that familiarity.
See site areas: 2. Linear Equations & Fraction Skills 3. Fractions Ratios Rates Proportions Units 5. Analytic Geometry 8. Complex Numbers 10. Secondary IV(?) math; the online volumes 2. Three Skills for Algebra and 3. Why Slopes (A Calculus Intro) & More Math and the site area 7. More Calculus. The material here can be presented as rules and patterns to use and combine with confidence in results coming from their repeatable, reproducible, and therefore verifiable nature. The coverage of logic here aims to develop precision reading and writing, and an understanding of how implication rules B IF A can be used and combined directly, one at a time and one after another.
The support of aims A and B advocated here may be simple, short and effective enough to allow more students to start calculus while also serving the needs of students heading for business and technical trades (surveyor, plumber, carpenter or electrician) for which calculus is not normally required.
In starting the development of trigonometry after the introduction of complex numbers AND the assumption of the field properties of complex numbers, turns the development into an algebraic exercise and so make trigonometry and the properties of vectors in the plane easier and more accessible. . Here students are meeting and using rules and patterns that are easily understood or assumed, and easily combined to arrive at further rules and patterns. Ease of exposition is the guide. Harder routes in which less is assumed can be left for students of mathematics and/or the hard sciences.
III. Supporting Aims B and C
Axioms and postulates in mathematics are labels for rules or patterns that have been assumed in order to secure a base for deduction. Further rules and patterns are then tested in mathematics by looking for direct or indirect chains of reason (arguments) that imply them. That provides a proof. Rules and patterns thus proven may then being used in further tests or proofs. The weakness of this deductive (more rigorous) style of reason lies in the choice of initial axioms and postulates. Chains of reason provide stories to follow one at a time and one after another. But these stories, rigorously or deductive put together through chains of reason, become works of fiction when the initial axioms or postulate are not true. On the other hand, if there are elements of truth in the original axioms and postulates, these stories may be non-fictional.
The mixed mathematics development of synthetic (coordinate-free)
4. Euclidean
Geometry in site pages inductively suggests and clearly identifies geometric
rules and patterns, those assumed for use in deductive reasoning. There is
motivation here for the indirect statement of the parallel postulate as given by
Euclid, namely the assumption that two lines segments extended will meet on side
of a transversal will if the interior angles on that side of the
transversal sum to less than two right angles. This coverage of Euclidean
geometry with the selection of a unit length and assumptions about
coordinates and their decimal representation to imply the field properties of
real and complex numbers taken as assumptions in the support for aims A and B
above.
Most, if not all, of the deductive chains of reason offered here will be direct. Ease of exposition, making the ideas more accessible, is the objective here. That being said, in the development of Euclidean and then Analytic Geometry here, there is focus on the possible origins of assumptions - how they can be suggested by and extrapolated from experience. Besides this, there is a focus on deductively deriving the consequences.
IV. Why Rewrite the Curriculum
Preparation for calculus and other disciplines still requires to a greater or lesser extent, a good command of arithmetic with decimals and fractions, algebra alone and in further subjects, analytic geometry and trigonometry and logic. For that good command, some drill and practice is required along side a thought-based development of skills and concepts. The support for aims A, B and C above points to a leaner and more accessible development of the necessary skills and concepts for calculus and for most technical subjects that require algebra, trigonometry or complex numbers. This lean program in reaching or developing key skills and concepts directly and systematically is more likely to build skills and confidence necessary to retain students in this program from start to end. Shorter and leaner provides a remedy for longer, more awkward and less focused mathematics programs that seem to continue year after year in a manner in a seeming aimless, ad hoc, manner, an endless manner that may alienate students and be difficult to motivate or justify in the classroom. Less well-chosen would be better.
Modern mathematics with its set-based description of axioms for real numbers given (or derived from assumptions about sets or natural numbers) provides a geometry-free, model for understanding, describing and developing the properties of real and complex numbers, and also properties of functions which appear in calculus, all apart from the use or mention of decimals. Geometry-free means there is no dependence on diagrams or suggestive reason. But there is a great dependence on direct and indirect deductive reason, and on the shorthand role of letters and symbols to codify, record and develop concepts and results.
In North America, if not elsewhere, modern mathematics curricula from birth in the late 1950s to abandonment in the late 1970s or 1980s with their set-based presentation of axioms for real numbers aimed to prepare students for the more rigorous treatment met in university programs of study in pure mathematics. But the introduction of modern mathematics curricula used geometry to introduce and illustrate the properties of real numbers, avoided all mention of decimals in the representation and properties of real numbers, and required arithmetic skills with decimals in examples involving calculations or coordinates, and used geometric diagrams to introduce functions. There are was also no systematic development of the algebraic way of writing and reasoning. It was just assumed. So the modern mathematics curricula while preparing for pure, geometry-free and decimal-free, mathematics did so inconsistently and awkwardly.
The modern mathematics curricula, its course content, has further lingered in course design and delivery, preserved in an ad hoc manner as the sway of pure mathematicians over mathematics course content has been replaced by by the sway of others with a greater knowledge of pedagogy, that is how a master teacher should cover material, and a lesser knowledge of mathematics. Education reform since the 1990's has been based on the assumption that a change of delivery style with less content, and less drill and practice, would make mathematics instruction more effective. The net result is an ad hoc dilution and shift away from the modern mathematics curricula of the late 1950's.
All the foregoing with the inconsistency the modern mathematics curricula from arithmetic to calculus with it nominal aim of preparing for geometric-free development of ideas in pure mathematics provides the freedom here to put aside the awkward elements of the modern mathematics programs, and propose a gentler, mixed mathematics approach likely to provide a good operational command of arithmetic, algebra, complex numbers, trig and some calculus in a more accessible manner, and likely to provide the algebraic, geometric and deductive maturity needed to appreciate pure mathematics. There-in lies a context and a justification for a change of substance and not just delivery style in mathematics instruction. Indirect instruction, constructivism or guided discovery say, when fully developed should be compared and contrasted with direct instruction fully developed in accordance with inductive principles for intruction, and the following theory of knowledge for technology, the hard science and mathematics.
www.whyslopes.com
Mathematics Education Essays
57 or so
Area Entrance & Hub Ideas for Better Instruction 4 Ways to Improve Reform Theory of Knowledge Peer Review The Trouble With Algebra Course Design and Delivery How Letters Appear Sit Down & Study Modern Education Key Notes and Themes Site Lesson Plans How This Site Differs Site Origins Math & Logic Puzzles Comments on site content.
Words For Instructors Inductive Principles Fairness Principles Apprentices & Masters Three Remarks For a Leaner Curriculum Mixed Maths Curricula Cultivating Intelligence Reason - 3 kinds in maths Logic in Mathematics Science Education Maths Instruction in General Operational View & Values Standards Ends and Values Goals & Unifying Themes Algebra Lesson Plans Algebra, Geometrically Mathematics Curriculum Shifts Teaching Tips - Fractions to Calculus Math Ed Perils Talk the algebra talk Sec I - Fraction Focus Sec II - algebra focus Sec III - Focus on Slopes Maps-Plans-Drawings Math Wall Posters Education, Empirical Art Damage Reversal North American Math Curriculum Managing Reform Essay January 2007 Educational Follies Contructivism Incomplete Missing the Point I Mathematics in Context What and When, A Challenge Grouping Students Teacher Certification Education of Math Ed. Professors Site Eurekas Links Help Me Learn/Teach;
- Algebra
words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)- Arithmetic
- exercises
- with fractions
- videos on primes, lcm, gcm,lcd, square roots etc- Calculus - geometric preview, algebraic preview,
3 study guides,
much more- Complex numbers
-starter lesson with java applet - easy consequences for trig & vectors in the plane- Education
- Empirical Course Design & Delivery- Fractions
- alone
- by rote
- with algebra
- videos
- Functions - introduction
hindsight - composition aka
substitution -- Geometry, Euclidean - Correspondence of triangles, Triangle construction, duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
- Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
- Logic
- First Steps -
Symbols in Logic -
Occurrence & Truth Tables - Indirect Reason -Indirect Reason More- Proportionality
- Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions- Real Analysis
- Decimal View of concepts and of proofs- Rules &Patterns in Science, Technology & Society - Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes & (?) derivatives
- in ratios & proportions - slopes & rates included- Complex Numbers & Vectors & Trig
- trig expression for dot & cross - cosine law