|
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
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. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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.
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
| |
Chapters 1
Keys to Success
Previous: 1 Lowering
Barriers
The deductive and algebraic examples described above could be keys to the
success of intermediate level instruction. They are easily described and
repeated. They offer a wide and accessible image of mathematics. With such an
extension of the common knowledge of mathematics, students reaching the more
deductive expositions of advanced courses will be aware or know what concepts
are being axiomatically codified and replaced. Two coexisting lines of thought
can be mastered, one common and easily explained, and one refined and more
deductive. Masters of mathematical exposition may give one and then the other
line of thought in their presentation of a topic.
Both students and teachers can recognize that there is an informal
presentation of mathematics and pattern-based thought, in which ease of
exposition is the guide, and that there is a separate codifications and
framework for the rigorous and formal development of mathematical thought.
Currently putting the latter first provides the barrier to any extension of the
common knowledge. Putting the informal description first may extend the common
knowledge and provide a context or background for the more rigorous exposition.
Here informal mixing of algebraic, geometric and physical reasoning departs from
the pure deductive exposition of mathematics from symbolically and algebraically
stated axioms or assumptions about sets or numbers. But the deductive exposition
itself may become more understandable and accessible in the context for it
provided by the departure. Within the framework provided by the departure,
advanced mathematics is given a context for its rigour and an abstraction from
the now extended, common knowledge.
The above approach parallels that found in the historical development of
analytical or algebraic thought. Examples and strands of reasons were developed
first and then threaded together in a more and more rigorous fashion. Elementary
and intermediate instruction can provide examples or informal strands of
mathematical reasoning – an operational command and preview of the
subject – which higher instruction can gradually codify. That is, higher
instruction may fit this operational command into the axiomatic development and
exposition of mathematics. Instructors of higher mathematics may say to their
students: here is the common or easily understood representation of this idea
which you have met before, and here is how it can be deduced from axioms.
More Chapter 1 Sections: [ Up ] [ 1 Two Barriers ] [ 1 Lowering Barriers ] [ 1. Keys to Success ] [ 1 Units & Decimals ] [ 1 Chapter Guide ]
Next: 1 Units &
Decimals - Missing Links
| |
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.
|