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.
How to Lessen Algebra Difficulties
Three remedies for algebra difficulties likely to be effective and involving a greater and clearer use of words are proposed for digestion and refinement. The remedies (a) describe and illustrate three skills for algebra; (b) describe and illustrate the forward & backward use of formulas and equations, proportionality relations included; and (c) re-introduces the concept of what is a variable with words that can be understood first before & then besides the shorthand roles of letters and symbols. The three skills and the two equivalent phrases "Forward and Backward Use" and "Direct and Indirect Use" vocalizes a unifying and previously unspoken themes in the use of formulas and proportionality relations. Examples showing how appear in chapters 8 to 14 in Volume 2, Three Skills for Algebra. See too site areas on solving linear equations and on fractions.
Before or besides the simple use of formulas in primary school and of the mastery via numerical examples of methods for addition, subtraction, multiplication and division of fractions, the algebraic description of these operations on fractions (rules for them) may give a taste of the shorthand role to come of letters and symbols in describing associative, commutative and distributive properties for arithmetic with real and complex numbers.
The unavoidable occurrence of arithmetic and algebraic expressions, better seen in silence, too awkward to read aloud, has been a huge barrier to the role of words in understanding and describing algebraic skills and concepts. Formulas and equations like pictures are worth a thousand words. So the exposition or development of skills and concepts has been too quiet. Site pages provide a more vocal, a more audible path.
Silence in the exposition or explanation of algebraic skills and concepts has twisted or complicated mathematics from first use of simple formulas to full-strength use of algebra in advanced calculus. In retrospect, there has been a domino effect.
While introducing more words into the exposition of mathematics, to lower one barrier and not raise another, we need to remember that skill development and perfection requires some drill and practice - not too much but enough. The algebraic way of writing and reasoning before or during its development. requires mastery of arithmetic with fractions without a calculator - arithmetic should be repeatable, reproducible, verifiable and automatic. Calculators and spreadsheets may remove the burden of arithmetic in complicated situations, handling data, but a careful command of exact arithmetic with fractions remains a base for operations in algebra, trigonometry and calculus. Calculations with units of measurement and measurements themselves should also be developed and maintained.
Mathematicians: The student need to understand and have an operational command of what is a variable before any mention of functions and sets in their instruction. The neat function views may be a set-based codification of a concept that needs to be understood & mastered before the codification begins.
For secondary mathematics, the easily understood and used remedies above lead to a greater use of words in development of algebraic skills and concepts, and in that, recognition of a unifying theme - the forward and backward use of equations.
The above vocal development of skills and concept can and should be helped by geometric perspectives. See the site starter lessons for (i) solving linear equations, (ii) the distributive law and multiplication of both decimals and polynomials and (iii) complex numbers. See too the algebraic and geometric previews of calculus. Calculus is the subject which requires the algebraic way of writing and reasoning at full strength. The previews prepare for that full strength use, may be met in pre-calculus courses, and aim to ease or avoid algebra shock in calculus. (iv) A decimal viewpoint of limits, or error control in the evaluation of functions and limits, also helps avoid algebra shock.
www.whyslopes.com
Mathematics Education Essays
57 or so
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