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YOU are better than YOU think. Show
yourself how:
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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.
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Goals for Mathematics Education
The story of the race in which an overconfident hare is
beaten to the finish line by a slower but non-stop plodding tortoise gives a
lesson on determination for all students, slow or gifted.
See For Greater Clarity in
Mathematics Course Design after the three goals.
1. From Logic to Precision Reading and Writing
For the first goal, and for the greatest help from site pages to your
work and studies, start with and see if you enjoy logic
chapters 2 to 5 in the online book Three
Skills for Algebra
Logic
chapters 2 to 5 aim to ease or avoid difficulties in school and at
work by making you aware of the need for care and precision in
reading. Once you can read exactly or precisely, you will have a
better chance of seeing the mistakes in your writing - the difference between
what you meant to say and what you said. Precision writing allows one to
state the maximum possible without becoming inexact or wrong. Parents:
If your teen has difficulties in learning, once a year, try to guide your
teen, through these logic
chapters, as many as possible, or ask your teen's school to test and
check logic skills.
For literate math-phobics, reading online logic
chapters 2 to 5 may postpone mathematics studies for a few hours while
building skills and confidence for work and studies in general and for
mathematics education too
Mathematics studies may be further postponed by
reading Pattern Based Reason,
an online book, the source of the logic chapters. The question of what rules
or patterns are reliable and which ones apply, if any, leaves room for thought
and worry.
2. From missing words or links, to unifying themes
For the second goal,
learning how to describe numbers and equations with words to make algebra
mastery stronger and easier to obtain
site lessons will change how mathematics is understood and explained.
There has been a silence, a lack of
words, in mathematics because arithmetic and algebraic expressions are
difficult to read aloud in a way that listeners understand or follow the order
of operations. So expressions of many kinds are better read in silence
and understood nonverbally in a glance. There-in lies the silence.
With the passage of time, the following program for
reducing the silence, or adding words to lessen its effect, may be
improved. But here is the first draft, likely to be immediately effective
in college and high school instruction.
Online algebra chapters
8 to 14 in site volume Three
Skills for Algebra show (i) how to describe numbers with words apart
from and besides symbols, that is the first skill; and show (ii) how to
use describe formula usage as direct or indirect, and describe indirect
use as a numerical or algebraic solution. Item (ii) in retrospect
points to a fourth skill for algebra in the volume. The second skill
is given by our ability to describe calculations with words and
formulas. The third skills is given by ability to change the description of
how numbers and quantities are computed via replacement or substitution
operations. Learning the four skills or emphasizing as unifying themes
in high school and college will change mathematics instruction. The online
essay What is a Variable provides a word-based and word-only explanation, and
so illustrates the first skills for algebra, our ability to describe numbers,
amounts and quantities with words alone, before any formalism of modern
mathematics, indeed before the use of any symbol.
Recognizing our ability to describe numbers, amounts and
quantities, and to describe operations on formulas or equations provides themes
to unify mathematics education from algebra to calculus. I kid you not.
To lessen the silence, we use names, descriptive phrases and
more words in mathematics by naming equations and formulas (old hat), by
learning how to describe numbers, amounts and quantities before or besides the
use of letters to stand for them (a first site contribution to mathematics
education), and by learning short descriptive phrases for common operations on
equations and formulas - direct and indirect use, numerical or algebraic
solution. There-in lies another site contribution to mathematics
education.
3. From Geometric Quantities to Algebraic Sense
For the third goal, we observe people have greater comfort
and less panic in algebra when letters are used in geometric formulas to stand
for lengths, areas and volumes which can be drawn or sketched. Those geometric
formulas can be evaluated when the geometric quantities in them are known, given
or measured. But the formulas are understandable, they provide arithmetic
recipes to follow, even when the geometric quantities or numbers appear in
drawings, but with values not yet given or measured. There in lies a start
for algebra - several steps to provide an geometric start for algebraic skills
and concepts.
-
The site three column method for solving
linear equations with stick diagrams to go from use of letters denoting
a length in formulas for area and perimeters to acceptance we hope
of a letter denoting a number in equation.
-
Geometric
(area) views of the distributive law leads to methods for
multiplying and adding polynomials and also the decimals representation of
whole numbers.
-
This complex number
starter lesson represents the last of several efforts in site pages, the
first appears in Volume 3, Why Slopes and More Math, to provide a concrete
coordinate-based development of complex numbers. Easy consequence imply trig
formulas for dot- and cross-product, and algebraic proof of the cosine law.
-
Geometric
& algebraic previews of calculus to explain why slopes &
factored polynomials appear, and doing so provide a gradual, rather than
sudden start to the full strength use of algebra in calculus.
Thinking
Part of Mathematics and Logic: There are three kinds of rule-based
intelligence in mathematics, logic and most pattern-based subjects. The
first kind met in primary school arithmetic consists of skills with
repeatable, reproducible and therefore verifiable results - results that are
then considered right or wrong. The second kind also met in primary school
consists of pattern or rule recognition. The development or exploitation of
the ability to recognize or suggest simply patterns in order to predict the
next element in a sequence. If the prediction fails, another pattern is
required. The third kind appears after inductive mastery of logic, that
is mastery of implication rules If A then B and their use. The second kind
follows the use of implication rules and definitions and assumptions, one at a
time and one after another, to arrive at logical conclusion in a repeatable,
reproducible and therefore verifiable manner.
Pure modern mathematics depends on or assumes the ability of students to
think logically and algebraically. Pure modern mathematics does not require
geometric diagrams nor physical assumptions for its thought-based development
from axioms (assumed patterns, algebraically described) about sets and numbers.
To develop student ability to think logically and algebraically, and to
introduce the geometric concepts in trig and calculus, concrete geometric
diagrams if not physical assumptions are required. And the modern mathematics
high school and college mathematics of the last half of the 20th Century while
inspired by the axiomatic structure of pure mathematics followed in practice,
impure mathematics programs of study, with some contortions or routes more
complicated than need-be in course design. The latter was due to the conflict,
not then fully understood, between the demands of pure mathematics and the
demands of mixed mathematics, say trigonometry, calculus and pre-coordinate
synthetic, Euclidean geometry. That was the situation outside of the
United Kingdom in other parts of Europe and also in Canada and the rest of
North America.
Greater clarity in course design from primary school to college calculus
level could follow by using manipulatives, physical objects and geometric
suggestions and assumptions to develop an accessible, mixed mathematics
curricula which leads to results in a repeatable, reproducible and therefore
verifiable manner, in an empirical, if not pure manner.
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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
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