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Have your gifted students read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
tell students to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes their attention. |
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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
Tell students that 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. In Volume 2, Three
Skills for Algebra, a 4th skill for algebra appears in Chapter 14. It
provides a unifying theme for high school mathematics - equations and formulas
may be used forwards and backwards, directly and indirectly, numerically in arithmetic
solutions & with literals in algebraic solutions.
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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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Foreword
Four principles offer an inductive philosophy for the explanation and
comprehension of math and reasoning skills. Three of the principles were met in
a course on how to teach Nordic, that is cross-country skiing. The course was
taught one weekend early in 1981, by an instructor-trainer from CANSKI, the
CANadian association for Nordic SKIing in Flin Flon, Manitoba. Nordic ski
instruction may begin with a lesson on how to put on the boots and attach them
to the ski and also how to hold the ski poles – to be precise one holds not
the poles, but their straps in way that will guide the poles.
There is a technique here, one that is not obvious. The course gave minute
attention to the details which novice and even experienced skiers might not
know. In this course on ski instruction, the more complicated movements or
skills were deliberately preceded by simpler motions. Each of which was easy to
describe, master and/or review separately. This course turned Nordic ski
instruction into an art. The four principles follow.
1. Each discipline needs to be presented, so that students understand what
they are learning and why. Without a knowledge or an opinion of why, students
may lose interest and not go further. The why could be approximate — a
little uncertainty leaves room for thought.
2. Pathways through easily described and repeated ideas may extend
knowledge of any discipline, area of thought or belief. One or more paths
through easily described and easily repeated topics may allow those who travel
further to tell others willing to listen, what to expect and again possibly
why. Of course, differences of opinion exist on which disciplines should be
taught or what pathways in them should be followed.
3. Awkwardness with an idea or skill often signals difficulty with previous
ones. It may indicate at least one earlier skill has been missed or forgotten.
When an awkwardness is felt or seen, learners should go or be taken back to
practice the missing skills, more precisely the ones just before them. This
retreat aims to restore confidence and build skills, so that the learner can
go further. This requires a diagnostic skill – a knowledge of or opinion on
how the topics in question can be organized and taught. Here again opinions
may differ.
4. Each collection of mental and physical skills should be organized into a
ladder-like sequences of steps with the basic ones first and the more advanced
ones second. Learning in any subject stumbles when a first or succeeding step
is not easily reachable from those before them. [1] To
climb a ladder, the initial steps must be reachable, and each further step
must be reachable from the one or ones before it, else failure occurs.
Explanations should follow chains of reasons or persuasion which begin at the
level of the student.
In mathematics education there are two barriers to comprehension to be
lowered or removed. First, the algebraic or symbolic way of writing and thinking
is better seen and read silently than read aloud or spoken. This has been an
obstacle to the comprehension and communication of mathematical thought. Second,
the deductive nature of formal mathematics exposition with its long chains of
reason and preparation implies that concepts appearing at the end of a course
are not comprehensible to students in the middle of the course nor at its
beginning. Mathematics beyond the last concept mastered may seem impenetrable
and mysterious.
To lower both barriers, students may be given lessons, easily described and
repeated, which require a minimal formal comprehension of mathematics and logic
while presenting ideas essential to deductive and to algebraic or symbolic
thought. Recognizing, collecting and offering first such lessons may extend the
common knowledge of mathematics beyond the mastery of arithmetic, counting and
simple formulas that should be obtained in elementary school. This work
identifies such lessons and indicates ideas for math and logic instruction from
primary school to the start of college. Some of the ideas may be worth reading,
repeating or refining – the three Rs that this author hopes for.
Alan Selby
Montreal 1996
Postscript: The jumpmath
program unconnected to whyslopes.com
says the following: One feature distinguishes
our workbooks from regular math textbooks, however: in the JUMP workbooks,
teachers are consistently shown how to help students who are having trouble
moving forward by breaking mathematical concepts and operations into the most
basic elements of understanding and perception on page 2 of the Jump Teacher
Manual - Fractions, one of three plus pdf files in the Jump publication
page. The quote italized here complies with inductive principles given
above.
Next: Three Remarks (Postscripts)
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www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,
Foreword + Chapters 1 to 10 + 12
Book Entrance
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5 Books & Articles to Read
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links
Chapter 11: Primary School Mathematics
11 Primary Math
11 Cue Cards
11 Counting
11 Decimals - Addition
11 Decimals -Times
11 Decimals & Subtraction
11 Fractions and Division
11 Notational Conflict
11 Reciprocals Etc
11 Decimals - Ratios
11 Size Comparison
11 Numbers, +ve or -ve
11 Rename < Sign
11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's
Area: Help Your Child or Teen
Learn
Most students in high school are not heading for calculus,
but most topics in high school mathematics are present due to calculus.
Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math
Education Essays. Site areas and pages provide pieces of the a Mathematics
Education, Jigsaw Puzzle, in formation.
-Inductive
principles for course design & delivery require a clear
description of where and how skills and concepts may rest on earlier ones, so
that difficulties may be explained and remedied by looking for what was
missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation
and comprehension need to be identified and corrected. In
reading, spelling and writing, students have to learn all the letters in the
alphabet, not just some. and memorize spelling. Anything less implies
difficulty.
Likewise in mathematics, students have to master key skills
and concepts, one at a time and one after another. Anything less implies
difficulty.
Modern mathematics curricula introduced an inconsistency
into course design and delivery. They did not sanction the use of decimals nor
the use of diagrams in skill and concept development but decimal arithmetic
and diagrams are needed for student comprehension and for an operational
mastery of quantitative skills. That implies the need for an mixed-math
curricula based on a systematic development of operational skills, sufficient
for applications and sufficient to provide a base & context
for the very optional study of pure mathematis.
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