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Home How to Improve Marks Grades 7 to 12 Guide 1. Arithmetic Lessons 2. Algebra How-TOs 3. More Algebra How-TOs 4. Geometry Steps Into 5. More Geometry 6. Calculus How-TOs 7. Logic in Math 8. Complex Nos Geometrically.
Proper notation & format
makes the hard easier.
Pages For Teachers
Home Notes for Instructors Sec I, Teaching Ideas Sec II , Teaching Ideas Sec III, Possibilities Permissions A Site Map Vol 1, Elements of Reason
Show a student how to learn and that helps one. Show a
teacher or tutor how to make skills and concepts easier for students and
that helps many.
Miscellaneous
Your IP Address & how to use
it
Three Links for Teachers:
(i) First
Year High School Math - Lesson Plans with Fraction Focus
(ii) Second
Year High School Math - Lesson Plans with an algebra focus
(iii) Algebra
Lesson Plans
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.
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YOU are
better than YOU think. Show yourself how:
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Take
greater charge of your work or studies: Read like
a lawyer
for better work & study skills, but do not take
everything literally. |
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particular, two
logic puzzles are keys to site content, and to greater work and study
skills. See if you agree. |
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Parents: Site Area Helping
Your Child or Teen Learn covers 1.
Speaking Skills, 2.
Reading & Writing, 3.
Preparing for Science, 4.
Math Work Books, 5.Books
for Parents, 7.
Having Patience -you'll need ii. Authority: Use it or
lose it. /Parents and teachers need to
say no for little things, otherwise the authority to say
no for big things will vanish.
Welcome. Two of many site reviews
follow.
- Math Forum, 1996 online classification of a dozen math lesson
websites: ... There are appetizers for algebra, arithmetic, logic,
better learning in general, reason, theorem proving and complex numbers.
Strengths here are in Alan's explanation of mathematical concepts using
words and stories: ...
- Magellan, the McKinley Internet Directory,
1996: Mathphobics, this site may ease your fears of the
subject, perhaps even help you enjoy it. The tone of the
little lessons and "appetizers" on math and logic is
unintimidating, sometimes funny and very clear. There are a
number of different angles offered, and you do not need to
follow any linear lesson plan. Just pick and peck. The site
also offers some reflections on teaching, so that teachers can
not only use the site as part of their lesson, but also learn
from it.
Author's Note: Parts of www.whyslopes.com
are well-written and second to none in skill and concept
development. Other parts are not - improvement are planned. .
Visit
visit mathsisfun purplemath, themathpage
and further sites besides to compare and combine different
viewpoints.
For self-instruction, site pages are best for readers15 years to adult.
Younger readers could leave site exploration for later.
Page Contents: [Junior High School
Material][ Senior
High School & Adult Material] [More
Senior High School Maths and Calculus] [College
or Enriched High School Maths] [Math
Education Essay - Where's Maths, Progressive Mastery, Hopes & Conclusions]
For instruction, site pages offer ends, values and methods for skill and concept development not found
elsewhere.
Inductive principles for progressive skill and concept development,
mentioned in the communication of skills chapter 2 of 1995
Volume 1A, Pattern
Based Reason, appear in the foreword to 1996 Volume 1B, Mathematics
Curriculum Notes. The principles give a criteria for judging and
so reshaping course design and delivery. Yet the principles alone are
insufficient. Reasons, or ends and values for mathematics education,
are also need of exposition. Since 1995, visitor reports of
errors small and large has led to site corrections. Professional editing is
still required. In my teaching days, I would repetitively include too
many words in a single sentence in the hope that from word
association students would get the meaning of unfamiliar words and
terms. That repetition in my online books is not to everyone's
liking.
Online Help for Instructors: Many instructors face curriculum
changes or face courses in maths which they have not taught before.
Instructors unfamiliar with maths may be given the task of teaching
it. If someone you know needs help great or small skill and
concept development at the primary or secondary math instruction, suggest
that they join WiZiQ (membership is free)
and then email me a short
(one hundred word say) description of their math teaching assignment and
their background. I will try to use a WiZiQ virtual classroom to discuss and
answer questions, either immediately or after some reflection.
Help Delivery: Here is a quick imperfect demo
of the virtual classroom. If you can see and hear it (excuse the hiss in the
recording) then most likely, your computer system will allow you to attend
the virtual seminars and lectures I may give. The invitation to
them will come in the form of a URL which you follow in web-browser to the
lecture sign-in page - ideally a few minutes before the schedule start of
the seminar. They may a pop-up window in which you have to give flash
(or java) access to your PC speakers, video and microphone. The latter is
optional. I am student of K1-12 maths and logic. I expect or hope that
exposure to the questions or views of others about instruction will improve
my replies - practice make perfect. The service here could fizzle due
to lack of demand. Or it might be overwhelmed by demand.
Junior high school level
material:
The
site math folder Solving
Linear Equations offers a geometric stick diagram
method which teachers, tutors and parents may use to
geometrically introduce the solution of equations of the form ax+b = d or
ax+b =cx +d and to algebraically introduce the solution of systems of
equations in essentially one unknown. Most junior high school word problems
may be cast in the latter form. Moreover, solving systems of linear
equations in essentially one unknown requires a practical if not theoretical
mastery of associative and distributive laws for arithmetic. The stick
diagram introduction to solving linear equations of the form requires and
visually reinforces fraction skill and concepts. The folder also
introduces senior high school topics, that is the solution of triangular and
general systems of equations, in manner that may be accessible to junior
high school students. Most steps are illustrated with diagrams and animated
gifs. The site folder emphasizes solution verification, so that
students may learn to check solutions and even correct them before
submission. The math folder Solving
Linear Equations may be employed for review or self-instruction by
senior high school and keen or gifted junior high school students.
Senior High School, College and Adult
Material:
Volume 2. Three
Skills for Algebra present ideas for logic and algebra
mastery.
Logic chapters show how to use implication rules A if B
one at a time and one after another to arrive at conclusions. Logic mastery
may improve reading and writing, and ease or avoid difficulties in following
instructions at work, at home and in school. B
The 1995 Volume 2 in talking about three skills for algebra and the forward
& backwards use of formulas provides a more wordy, oral dimension and base
for introducing and developing algebra. In senior high school mathematics
before and including calculus, words have been missing in the introduction
of the shorthand role of letters and symbols in mathematics. That role has
appeared in silence, given by example. The role has been clear to some,
but not all. Talking about three skills for algebra provides a new
perspective to make skill and concept developing clearer and richer. That
new perspective is enlarged by talking about the forward and backward use
of formulas provides a unifying them for high school mathematics.
More generally, most if not all rules and methods for arriving at
conclusion or numerical results in mathematics, logic and
quantitative reason will be used forwards and backwards, or directly and
indirectly. Talking about all the foregoing eases or removes the silence
in mathematics that stems from the nonverbal element of mathematics. An
arithmetic or algebraic expression like a picture is worth thousand words
- better seen and grasped in a glance, than described stroke by stroke, or
in the case of mathematics, symbol by symbol. Thus talking about three skills for algebra and the forward
& backwards use of formulas provides a more wordy, oral dimension and base
for introducing and developing algebra.
More Senior High School and Calculus Material:
Calculus previews and more elements of Calculus appear in the
Volume 3, Why.Slopes.&.More.Math,
and in the newer site math folder on Calculus.
For calculus and senior high school mathematics courses
before it, the 1995 site volume 3, Why
Slopes and More Math, begins with geometric and algebraic previews of
calculus. The geometric preview explains why slopes are studied in
senior high school maths. The algebraic preview gives the slopes
(derivatives) of function y = f(x) as factored polynomials or factored
rational functions, and shows how sign analysis of the factors may
locate function high points (maximums) and low points
(minimums). One or both previews in or before calculus
provide a context for the study of slopes and polynomial factorization, and
also reinforces or develop algebraic reasoning skills. Algebra is required
at full strength in calculus. The two previews provide appetizers or starter
lessons for that full strength use.
Calculus is the college or senior high school
mathematics subject required for college or university studies in
accounting, business, money matters, science, engineering and health.
Calculus employs at full strength most earlier elements of high
school mathematics including functions, trig, algebra, mathematical
induction, more logic, geometry and
arithmetic exactly or symbolically. If you are a high school student,
you have to aim for skill and concept development and perfection in the
foregoing areas of mathematics and logic. No detail nor concept should
be left unmastered.
Slopes to straight lines, operations on polynomials, quadratics and
functions appear in the analytic
geometry site folder. Preparation for calculus requires these
topics.
College
or Enriched High School Maths:
Site folders cover complex numbers, more calculus, number theory and real
analysis. (A) This first lesson on complex
numbers may be met after mastery of rectangular and polar
coordinates, and besides or before trigonometry and the study of vectors in
the plane. The site area on complex numbers
(different from the lesson) shows how the college and high school coverage
of trig and vectors may be made easier if mastery of complex numbers is put
first. (B) The site number theory
folder presents and explores geometric or natural methods to imply or
support axioms for real and complex numbers. (C) The site folders and
volumes on calculus and real analysis recall or introduce the decimal
base for key theorems and concepts met in courses on the latter. Set theory
provides a framework for codifying and developing modern mathematics and its
logic. While it was taken as a model for form course design and
delivery in the 1950s, the framework is too technical for most to
follow. Alternative is needed. Site page provide the building blocks
for that in a manner that may make the latter framework more
accessible.
Remark: The site folder on number
theory and the site how-TOs above include or point to the
explanation of decimal arithmetic methods for addition, comparison,
subtraction, multiplication and long-division. Site pages
provide or indicate full thought-based developments of all skills and
concepts from first steps in counting to calculus and beyond in an
accessible or natural matter. There is work to be done. Site pages so
far indicate some ends, values and methods for mathematics education at the
senior high school and college level. A base for the latter
would be given by a clarification of ends, values and methods for
primary and junior school mathematics. In general, mathematics education
will continue to be a mix of learning based on rote and on comprehension.
More precisely, comprehension and justification in mathematics
requires the skill to apply and follow methods, step by step,
Content and Delivery Matters and Choices
A. What to Cover and When - Put context first
One of my French textbooks for learning the language introduces words
and phrases that an individual or family may meet in everyday activities,
say eating in a
restaurant; traveling by car, bus, train or plane; visiting a
farm; and so on. All provide a context and motivation for word and phrase
mastery. Common activities and common needs in general may also provide ends,
values and a context for developing mathematics or quantitative skills. At home, at school and at
work, we meet numbers, calculations, amounts and measures in telling and keeping track of
time and dates, in money matters, in using maps and plans. That being
said, mathematics education in primary and junior high school could be
focused and be explicitly developed most around the requirements of common activities
and themes, with some, not much, preparation for future studies. That
focus and
development could provide concrete ends, values and paths for skills and
concept development in primary and junior high school mathematics.
Later senior high school mathematics could focus on developing the
more advanced commonly required skills and concepts, on covering the
mathematics needed for in common trades and professions, and on a
preparation for calculus and further studies in mathematics.
The foregoing focus and sequence of skills and topics in
accordance first with common needs and then less common needs should be provide
motivation. That motivation may be stronger
initially, where the utility is most apparent and then slowly dwindle, but
still convey some motivation. Compare and contrast
that with present day instruction where teachers, parents and students learn
and teach because that is a formal requirement, and not my immediate
reason.
Forgotten Values: The ability to apply a method and record or draw it
steps in sufficient detail for the student or others to follow and verify
later is a key value for mathematics education. The ability to
arrive at repeatable, reproducible and observable results is another value
for mathematics education and quantitative skill development at home, in
school and at work. In arithmetic, figuring well used to be a sign of
intelligence.
B. Progressive, Observable, Development
Volume 1B, Mathematics
Curriculum Notes, begins with an
inductive criteria for course design and for instruction. From
first steps to research level, mathematics consists of observable and concrete
elements in arithmetic, geometry, algebra and logic, etc. Mastery of
mathematics is seen when students write or read decimals, do arithmetic with
decimals and fractions, use formulas forwards and backwards, draw or make
geometric figures, record the forward and backward use of implication rules A
IF B in arriving at conclusions. With mastery of mathematics and quantitative
skills in general cast as a collection of observable, and verifiable skills in
real life or on paper, instructors may provide correction and directions for
for the development of these skills, one at a time, one after another.
Instructors of music and physical activities employ
progressive skill development in which movements are decomposed into simpler
and more accessible ones for practice and mastery, one at a time, one after
another, in sequence. When difficulty with one skill or concept is
seen, familiarity with the progression of skills and movements allows
instructors to diagnose the likely source of student difficulty in order for
students alone or in groups, to be taken back before the likely source to
rebuild confidence and to tackle or circumvent the difficulty. Different
course design and different instructors may be judged by the extent in which
the course designs facilitates progressive skill development.
Volume 1B, Mathematics
Curriculum Notes, after the presentation of a inductive
principles for progressive skill and concept in general and in mathematics,
describes the difficulties and shortcomings of earlier course designs.
In retrospect, first the technical development of numbers
skill and sense from primary school to college level needs to be continuous
and consistent - the decimal command of numbers and arithmetic needs to be
recognized and valued and sanctioned from primary school to calculus,
without throwing out set skills and concepts that may enhance the senior
high school and calculus development of mathematics. Further, the
algebraic shorthand way of writing and reasoning needs to be developed in a
clearer and more accessible manner. How to do so is shown in chapters
8 to 14 of Volume 2, Three
Skills for Algebra, and in site folder Solving
Linear Equations. See the comments above. Fourth, negative
numbers may be introduces as coordinates, and arithmetic with negative
numbers made easier - these notes
on integers (best viewed with Internet Explorer) show how - the
how with rational numbers would be similar. Fifth, the development of
trigonometry and vectors in two dimensions may be simplified and accelerated
by showing students how to add and multiply
points in the plane using (i) rectangular Cartesian
coordinates. and (ii) polar coordinates. Sixth, the introduction
of calculus may be made easier via the use of geometric and algebraic
previews. Within calculus, the elementary to advanced theory of
limits, continuity and convergence may be made more concrete and thus
accessible by stating
and continuing
with development a decimal, error-control viewpoint.
Volume 1B, Mathematics
Curriculum Notes, also describes simply yet precisely, the role of rule-based reason, that is
logic, in providing a thought-based framework and codification for
mathematical thought. While modern course begins with axioms for
real numbers stated in an algebra- and set-based manner, those axioms
and the necessary mastery of algebra for their comprehension may implied by
refining and recasting the primary and junior high school development of
number skills and sense, and by more carefully and less silently
introducing in junior or early high school studies, the shorthand
algebraic way of writing and reasoning in mathematics. Skill and concept
development has to be progressive.
Ideas which are easily repeated and understood may
provide a common knowledge of mathematics and the rule-based
reason sufficient for a more formal and rigorous comprehension.
The modern mathematics program of the 1950's and 60s' had logical elements,
but it assumed the algebraic shorthand role of letters and symbols in the
mathematics instead of developing the latter. Mastery of the latter
shorthand role was not a natural talent for everyone. The axioms
themselves provided an artificial base for skill and concept
development. But they did not sanction the common use of decimals.
Volume 1B was written in the first instance to identify those strikes and to
pose the question of how to remedy them. The remedy, much to my
surprises, appears to be lie in the form of developing number theory, and the
aforementioned axioms, based on the counting principles hidden and not mention
in primary school arithmetic and based on accessible, easily accepted,
geometric practices designed to introduce and represent signed coordinates and
complex numbers too. The foregoing yields a semi-formal, thought-based
foundation for developing number sense and implying the properties of numbers,
whole to complex, that are take in all or part to be axioms in modern
mathematics curricula.
From the mid-1950s to the present year 2009, to
the best of my knowledge, the mathematics course design and delivery in North
America first implemented and fine-tuned the modern math program for three and
half decades before non-technical concerns dominated mathematics
instruction. Mathematics instruction and quantitative skill development
at home, in school and in the workplace needs a clarification of logical
dependencies in skill and concept development, and beyond that critical path
analysis of what can be taught be taught and when. The development
itself needs to be optimized for the sake of the latter and for the sake of
lean and effective curriculum.
C. Summary, Conclusions and Hopes
Site material offers technical support for mathematics and
logic education from primary school to college level. Adult,
college and senior high school students may explore site material for
self-instruction. Site material follows from a concrete viewpoint of
education in which the latter offers progressive skill and concept development
in arts and disciplines based on and defined by observable skill and concept
development. Whence site viewpoints may be countertrend while site
technical contributions will be timeless. For the technical development
of mathematics, talking about three skills for algebra and the forward
and backward use of formulas, rules and methods in mathematics and in logic
provides and clarifies the oral dimension in mathematics and logic education
from algebra to calculus. Talking about and clarifying the ends, values and
methods of instruction, coupled with critical path analysis, may lead to clearer course design and delivery with
objectives understandable to parents, instructors and
students.
D. Observable and Objective Skill Development
Versus Constructivism and
Subjectivism
Striving for Objectivity: The past development and study of law and
the hard sciences was based on the hope for greater objectivity. The
rule of law should be independent of both judges and the state or status the
accused. The application of law should be based on precedent and on
written laws. The peer review process in the hard sciences emphasizes
repeatable, reproducible and hence verifiable results. In mathematics, a
claim is verifiable if its author shows or implies on paper how to obtain the
result from earlier well-known results and methods. Practices in modern
society depend on equipment or methods that give repeatable and reproducible
results, alone or in combination. But large industrial, urban or
rural scale repetition may compound small harmful effects of those results,
effects not noticeable on a small scale. So repeatable and reproducible does
not imply safe. Observable arts and discipline are based on methods and tools
which give repeatable and reproducible results. In pre-constructivism
instruction would cover those methods and tools, and aim to lead students to
employ and perform with them in a way that leads to repeatable,
reproducible and hence verifiable results. But not all is certain - Site
Pattern Based
Reason, naively describes the hopes, origins and limitations of rule- and
pattern-based thought and methods.
- England: The mathematics part of National
Curriculum of England describes performance levels for students in four
stages from ages 5 to 17 years. The sequence or progression of skill
and concept development is clear. Yet it might benefit from a
critical path analysis which covers and emphasizes dependencies of later
skills on earlier ones for the sake of remedial to enriched
instruction. Parents and teachers, not necessarily strong in
mathematics, given a knowledge of the dependencies would be able to better
deliver remedial to enriched instruction. I say England
and not the UK here as Scotland has it own curriculum. The Scottish
and Irish National Curriculums may overlap and differ. Other models
for instruction may be noteworthy.
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Help Me Learn/Teach;
- Algebra
words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
-new Arithmetic
Folder
- 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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