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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.
| |
Teaching Tips - Secondary Maths from Fractions to Calculus
Lessons or lesson plans for secondary
mathematics follow.
-
Year of
Fractions - Review and Extension of Primary School Material. The link
here points to a separate page. Next 5 links are internal to this page.
-
Year of Algebra
- Formulas and proportionality relations forwards and backwards, see and
compare arithmetic & numerical solutions for questions.
-
Year of 2D & 3D Geometry - Consolidate algebra and
fraction skills.
-
Year of Proofs, Trig
& Functions - mastery of logic required here or before.
-
Year of Analytic
Geometry - conic sections may appear here
-
Year of Calculus
-
Year of Advance
Calculus (Real Analysis)]. First link is to another page.
Theories (skills and concepts) seen without examples give a
vacuous knowledge. Mathematics mastery in particular further requires numerical
and geometry drawing experience from examples and practice to put theory in
context. Plans for reform given without examples to show how are vacuous in part
and may be hazardous to education - a current complaint. Reforms
have to address why is mathematics and select topics accordingly. Reform focused
on delivery style, reforms which ignores the role of long term objectives in
mathematics education, what should be taught and why, is
bureaucratic. The content issue is key. Calls to engage students
with authentic, realistic and genuine, relevant examples are vacuous when the
callers or advocates do not have a command of the content.
A six year
mathematics program: The first
and second year of this
program, which may label secondary school mathematics, could or should
consolidate fraction skills and sense and introduce algebra, in particular (i)
the solution of linear equations in one or
essentially one unknown, and (ii) the direct and indirect, forward
and backward use of formulas, equations and proportionality relations y = kx
in arithmetic and then in algebraic (literal) manners. While stick diagrams in
item (i) are crutches to develop equation balancing skills and sense, there
cutting, duplication of lengths in the approach develops fractions sense in
the context of line segments. So insist on stick diagrams. The third
year could be the year of examples - a reward for the first two years and
preparation for the next. Then fourth
and fifth years could emphasize logic
and proofs, the Euclidean
and Analytic Geometry of
straight lines (parallelograms & triangles included), similarity and
trigonometry, vectors and complex numbers
(geometric viewpoints), functions (various kinds) and polynomials (some easily
factored). A sixth year may introduce
calculus. In the fifth year or early, this lesson
on complex numbers (a not in the site area on
complex numbers) with field properties given or derived from 2D geometric
assumptions, yields an easier route to trig identities, and to further
material in science and engineering, and mathematics too.
The foregoing program would build a digital
(decimal) and geometric skills and sense which sanctions and extends the
common knowledge of arithmetic, coordinates and maps through explicit
assumptions given by interpolations and extrapolations of numerical and
geometric examples, with set notation and theory used to facilitate
and not dominate the development of mixed or impure mathematics from
numbers to calculus and real analysis. Pure mathematics may build on the
foregoing. The introduction of mathematics needs to depend the
assumption that points and diagrams or sets of points in a 1D, 2D and
3D space are in one to correspondence with coordinates (real numbers,
ordered pairs, triplets). Thus the introduction cannot be pure, and if it
going to be impure, it can serve and extend the common knowledge of
decimals.
The program identifies a lean, fat-free, core sequence
for high school mathematics Fatty additions may include statistics,
perspectives drawing methods for art or construction, Euler Formulas
relating vertices, edges and faces; areas and volume formulas forwards and
backwards, and 2D or 3D geometric transformations, a coordinate
viewpoint after introducing functions and mappings in 1D. In this, advances
for instruction, how to understand and explain matters in smaller more
accessible steps, and emphasis of the verbal description of numbers and
quantities before and then beside symbols could strengthen comprehension and
give alternative routes for instruction, repeatable and reproducible in the
classroom with fewer shortcomings. Students with learning difficulties
should focus on this lean sequence - the first three years might be
sufficient.
Volume 1, Elements
of Reason, introduces all site volumes.
[Online Books and More Site Areas] [Study
Tips] [Directions for High School Mathematics - Calculus Preparation] [Curriculum
Shifts - Shorter, Better, Stronger] [References]
Preparation for calculus provides the motivation for many skills and topics
in high school mathematics courses. Preparation for calculus is good
preparation for most, if not all, arts and subjects at work and school that
require some mathematics and logic.
Similar Directions: The earlier site preparation
for calculus page (written earlier) offers similar directions in
three different ways - lean, wordy and very wordy. The words comment on the
development of ideas in the classroom or historically.
Computer Games: If you play 3D computer games and want
to write your own, you will need a good command of logic, fractions, algebra
and geometry. The same advice applies if you want to enter a business, trade
or science.
Follow the steps below alone or with help. The review of
fractions etc in step 4 should come after steps 2 or 3. Other than that, which
step to put first appears to be a matter of taste. Site areas which do not
appear in these steps contain further material - optional reading. On first
reading, focus on learning how, and leave explanations why for later.
-
Put logic First (if possible). Read the first logic
chapters in Volume 2. Logic mastery will, we hope, ease fears
and difficulties, or if you have none, enrich skills and
knowledge.
Master logic carefully
to develop precision reading and writings. Skills and knowledge
are easier to obtain when you are able to read precisely what is written,
and do not assume too much. Marks in all subjects are base on your
written work. Precision reading will help you recognize errors in your
written work through the question: does it, your written work, say
precisely what you meant?
Secondary I and II Material
-
Meet the
role of fractions in algebra: Explore the site area Solving
Linear Equation with stick diagrams to further develop your
algebra skills - those needed for solving problems in one or essentially one
unknown, and see how fractions of line segments, the sticks, are combined
(added, subtracted, multiplied and divided) exactly in the solution of
linear equations.
Next read the Chapter 15, solving
linear equations, in Three Skills for Algebra, alone or with help. The
discussion of general systems is optional for junior high school students.
Test your algebra skills
and linear
equation problem solving skills.
Remark: Steps 1 to 4 may be covered in junior or senior high
school, the sooner the better. The following steps are for senior high
school students and older students in college or adult education.
-
Review or Develop Algebra and Fraction Sense and Skills.
Read (i) the algebra
chapters 8 to 14 Volume 2, Three Skills for Algebra.
The shorthand role of letters and symbols is meaningless for many people in
school and out. But the shorthand role is easier to grasp
when we first learn to talk
about numbers and quantities, and how
they may vary, before the use of letters and symbols. Doing that
would make algebraic ways of writing and reasoning clearer in calculus and
all of high school mathematics.
Chapter 14, Compound
Interest, in Three Skills for Algebra, develops algebraic skills with
the aid of a calculator. Calculators are useful but success and precision in
mathematics requires efficiency with fractions without one. --- Beside
talking about numbers and quantities, there is a fourth skill for algebra
in Three Skills for
Algebra, namely a development of the ability to talk about or describe
the numerical and algebraic use of formulas and equations with short
descriptive phrases: (i) forward and backward use (or direct and indirect
use) and (ii) algebraic and arithmetic (numerical) solutions. These
phrases appear in Chapter
14. can be used through out high school mathematics to identify
recurring themes - key objectives - and to provide another fresh perspective
on the algebraic way of writing and reasoning.
Alternate Between Steps 3 and 4 if you wish. Each
one has a different taste. The addition of animated graphic make Solving
Linear Equation with stick diagrams easier than before.
If you spend grades 1 to 11 or 12 in mathematics classes
without mastering fractions sense and skills properly and efficiently, you
have been cheated - several hundred or thousand hours of your time has been
wasted.
-
Optional but Recommended: (i) Visit the fraction
pages in the site area, Fractions,
Ratios, Rates, Proportions & Units, to check your fraction sense
(step 4 could have helped in here) and to see the justification of methods
for adding, subtracting, dividing, multiplying and comparing fractions. (ii)
Develop an algebraic view of problem solving with units and with rates and
proportions, binary or multiple, direct, joint or inverse. (iii) learn how
to carry units through solutions in a way that relies more on mechanical
skill in algebra than on thought. Here is an algebraic perspective and
clarification of skills and concepts in junior high school mathematics,
which may be read after steps 1 to 4 above.
The site area Fractions,
Ratios, Rates, Proportions & Units view of junior high school
concepts may help teachers & tutors develop skills and concepts. Senior
high school students may explore this area to review and reform their
understanding. Area material needs to be rewritten to make it readable for
junior high school students. Writing is an iterative process in which the
first draft is not always best.
Fractions are needed for algebra and beyond. In modern times, that is
today, we see and will see more and more cognitive experts and
curriculum advisors suggest the replacement of fractions and algebra
skills and sense development with calculator push-button
exercises in which the intellectual component of mathematics
instruction is eliminated to provide a child- and technology- centered
learning environment. Yet arithmetic mastery was and remains a sign of
intelligence in work and study.
-
Check & Consolidate your Arithmetic Skills. Do
asap, the first set of
arithmetic problems, chapter 7 of Volume 2, Three
Skills for Algebra, See too Simplification
of square roots. Logic mastery
asap is recommended for greatest benefit from site pages.
In doing exact arithmetic, if your result is not the same as that of
another, one of you has made an error. Learning how to follow
methods so that you obtain repeatable, reproducible and thus verifiable
results is a must, not always emphasized, for work, school and home.
See too these Real Player arithmetic
webvideos - a few a day, not all at once.
Aim for a logic-based mastery of mathematics after arithmetic. That
being said, arithmetic can be learnt by rote, know-how without the
know-why, provided you put aside your calculator and learn the times and
addition tables and learn to do arithmetic with fractions and
decmals (add, multiply, divide and subtract) in an objective,
efficient and automatic manner - arithmetic results should be
repeatable and reproducible, and you should know that an error in one step
makes all the rest wrong. Once you have a logic-based mastery
of mathematics after arithmetic, you can if you want retreat to
develop a deeper, logic-based understanding of arithmetic, a retreat that
could become easier, and a retreat that can be woven in to the explanation
of further mathematics for skill perfection and enrichment.
Secondary IV and V Material
-
Master Geometry without and with coordinates: Site
areas on Euclidean
Geometry and Analytic
Geometry offer senior high school students and teachers lean
logic-based development and connections of plane geometry, plane
trigonometry and functions of one variable. The site coverage of Analytic
Geometry does not include all that calculus requires, but is a start,
and the missing material can be found elsewhere.)
| Remark A: The treatment of Euclidean
Geometry is not full, but it is enough to provide a logic-based
consolidation of the skills and concepts seen in junior and high
school mathematics, those needed to develop analytic geometry and
calculus. The treatment of Analytic
Geometry assumes results of the site treatment Euclidean
Geometry with the assumption that real numbers alone or in ordered
pairs may provide coordinates for lines and planes in space. The
result is a logical, coordinate based, development of the key skills
and concepts in analytic geometry, plane trigonometry and functions.
The reliance seen here on geometric diagrams can be replaced and will
be in studies of modern pure mathematics. Or, we could use the
alternate route in Remark B. |
Remark B: Step 6 follows the traditional path of
defining trigonometric functions for acute angles with the aid of
similarity postulates before defining them for all angles. This complex
numbers introduction leads to trigonometry in general for all
angles, with right-angle triangle, similarity based, trigonometry
coming last. For the brave, that gives faster route for
developing the senior high school mathematics which calculus and
electrical studies requires. This route is leaner in that
its reduces the need for Euclidean
Geometry to a discussion of similarity
principles. |
| Remark C: In the modern mathematics
curricula of the late 1950s and 1960s, sputnik inspired, there is a
fuller treatment of coordinate-free Euclidean geometry along side a
general emphasis on logic. Geometric proofs were challenging - not
student friendly. So Geometry was eliminated. But Euclidean Geometry
was the traditional place for the emphasis of logic and Euclidean
model for reason. Site logic
and Pattern Based
Reason chapters present the Euclidean model in a math-free way and
do so to develop better study skills - or the precision reading and
writing better work and study skills demand. |
-
Test your arithmetic and Algebraic Skills: Try the remaining
problem sets in Chapter 7 of Volume 2. Get someone to identify all
errors in your answers in notation and comprehension, so you can learn from
your mistakes.
-
Optional: Explore the Number
Theory Site Area. Here is a mix of easy and challenging lessons, some in
sequence. If one lesson or sequence is not to your liking, try another.
Secondary VI & VII Material
-
Meet or Revisit Calculus: Begins with the why slopes geometric
preview before the more algebraic
why slopes preview chapters in Volume 3. Then explore more of the site Calculus
Introduction.
Remark: The introduction points to simpler ways to cover the first
steps in calculus. Those simpler ways are for all. The algebraic way of
writing and reasoning is usually required suddenly in calculus. The previews
here and the latter decimal view of limits, continuity and convergence
provides a more accessible and less algebraic demanding or shocking approach
to calculus.Then the introduction includes enriched material - the proofs
that are often omitted in first courses. Innovations here make the proofs
easier to understand, but not simple. The enriched material is for people
who do not like to accept mathematical methods without proof. The site area Real-Analysis-Decimal-View
(advance calculus) and the calculus introduction at this site emphasize an
error-control decimal view of limits, continuity, convergence.
Remark The Modern Mathematics movement of the 1950s and 60s made
calculus algebraically hard or inaccessible need-be by following a
decimal-free view prevalent in pure mathematics. Here is a correction
sufficient for students outside of pure mathematics that may provide a
stepping stone and context for the decimal-free, epsilon-delta view of pure
mathematics.
Remark: Steps 5 onward can be followed or explored in
any order you like.
Learners at all levels need someone to review their written
work for mistakes in notation and comprehension in order to learn from their
mistakes. Every time someone (on your side) identifies a mistake, say thank
you because now you know not to make that mistake again. Do not worry,
your helper will be employed in identifying further mistakes. It is a
win-win situation.
| |
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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