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"Would you tell me, please, which way I ought to go from here?"
"That depends a good deal on where you want to get to," said the Cat.
"I don't much care where--" said Alice.
"Then it doesn't matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an explanation.
"Oh, you're sure to do that," said the Cat, "if you only walk long
enough."
-- Alice's Adventures in Wonderland, Chapter 6 --
The answer to Alice's or your question about which way to go depends on
your ends and values, and also what steps or routes are available. We may
want all we want, but to be practical, the wants have to be feasible, clearly accessible -
by available routes or steps. Starting education reforms before all steps for
each reform have been clearly identified and described - seeing is believing -
may put ideology first, practice second. Or, we may recall failure to plan is planning for
failure. The foregoing sets forth a standard which is respected in site material -
to this mathematician, each not documented in site material seems self-evident. Where
not, readers may pose questions.
Words Before or Beside Symbols
The oral dimension of mathematics has been missing or weak
as arithmetic and algebraic expressions, axioms included, are
better seen and read in silence. If you read site algebrs
chapters and steps, logic chapters too, you will learn how to use
more words and so add this missing dimension to your learning or
teaching of mathematics. In particular, talking about using
tables, rules, patterns and formulas forwards and backwards will
become a verbal unifying theme from arithmetic to calculus and in
logic. Most formulas in daily life and of technical use are used
forwards and backwards. Instead of apologizing for doing things
backwards, we will mention and prized it from the backward use of
addition tables for subtraction to backward use in calculus of
differentiation rules for anit-differentiation. Talking about
numbers and quantities, known or not, variables included;
describing calculation and arithmetic practices with words; and
naming rules and formulas - or identifying them with short
descriptive phrases, altogether provide define a greater role for
words in understanding and explaining skills and
concepts.
Unique: Site material is counter-trend. Leading
mathematics educators have invested heavily in pyschological
views of how students may learn, but fundamental difficulties in
content presentation and motivation have been overlooked. That is,
the algebraic way of writing and reasoning appears in mathematics
as an immediate or eventual natural talent for some and as
mystery for most. Since 1966 in secondary school, I have
suspected and become convinced that steps too large or missing
explain the mystery - why algebra is harder than need-be. Site smaller and extra gap-filling step could fit into course
design as is to remedy that.
Since my graduate school day, I have thought about the difficulties or inconsistencies that stem from the
1960's insertion of modern set-based mathematics into general
college and pre-college mathematics education. The implementation
I met as students hinted at a full pure deductive path, one that
in practice had small and large inconsistencies or flaws. The largest one was that quantitative needs in
daily and adult life, and in science subjects besides mathematics were
not fully supported by the form and content of the modern
mathematics curricula. The form and content persist today in many
English language mathematics curricula as is or weakened. After many
twists or turns, a five phase framework for mathematics and language-logic
skill development has been developed. The first three phases
in putting the needs
of adult and daily life first may leave a good impression while also
providing a firm base for calculus.
As a mathematician, I have been looking for
consistency and best practices. To that end, I collected
50 or so mathematics local booklets and texts for ages 4 to 13 to
understand what came before the secondary mathematics and how.
Beside I studied the secondary education
possibilities put forth by leading mathematicians in North American and
Western Europe in the early 1960s, before the modern mathematics curricula was implemented
in North America. That study provides a fuller
technical picture, and the perspective that the
form and content of the latter curricula were designed for
college-bound students - those with good preparation or natural aptitude. The needs
of others were not addressed. Cold-war society urgently wanted
engineers, scientists and mathematicians in various forms at a
various levels.
The needs of others were not addressed because
course content and form as not designed for the majority of
students who would stop. While the contributors to the discussion
of mathematics education for college bound acknowledged that,
over the years, college-oriented mathematics courses were given
to more and more in the name of equal opportunity. But the resulting
college implied topics in mathematics appearing more and more formally in
secondary mathematics. Today, for most students in Quebec, if not the rest
of North American, and even elsewhere, the only context year after year
in secondary school and some college courses being preparation for the next final examination.
That context is re-inforced by secondary mathematics teachers and even university education
professors be assigned to teach mathematics or trained future mathematics teachers without
having a mastery of calculus, or senior high school mathematics. I am aware of university
teacher certification programs employing mathematics education professor without a clear
mastery of both. That
This context was
undermined by steps too large or missing in the introduction of
algebra leading to troubles, troubles largely accepted as normal, too normal
in my perspective. . The framework
combined with site steps for re-arranging and improving skill development
point to a unique path, one that most in mathematics and
mathematics education will not have anticipated, but may
eventually sanction. Time will tell.
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In mathematics education, the ends and values, and steps are a great
mystery for many students and teachers: Theirs being but to learn and
teach without understanding why. The words below try to describe ends
and values and steps, for mathematics and logic-skill development in
schools and colleges, subject to the constraint of what is feasible in
skill provision. The underlying premise is that site material from
counting to calculus base for clear effective skill development from
counting to calculus, a base in which previous technical troubles are
handled by smaller and extra steps well-put or improveable in site pages.
While the depth and variety of learning difficulties implies skill
mastery cannot be guaranteed, The site framework for skill development
outline below would be incomplete without the smaller and extra steps in
site pages to make key skills and concepts easier to learn and teach.
While society may direct mathematics education to prepare and select
students for say calculus- and statistic-based college programs, most
students in school do not reach or pass those programs. Many leave high
school before graduation. But graduating or not, all primary and
secondary students would benefit from putting mathematics for practical
use first - by rote if need-be, so that they do miss the take-home value
of learning to handle routine problems by counting, measuring and
figuring with numbers, with maps-plans-diagrams drawn to scale, with
measuring and drawing devices; and with formula evaluation in the early
years of instruction. The mid-years of instruction may add mastery of
algebra for extending basic skills and providing more background
knowledge and know-how. In that, student time is too precious for us to
devise methods for them to form and master skill and concepts their way,
while they still need to master routine skills and concepts. Near the end
of the latter, more and more deliberate preparation for college studies
may slowly switch from emphasizing skills with actual or potential
take-home value to emphasizing skills and comprehension with technical
value for further studies not only in colleges but also in pre-college
trades and professions. So multiple work and academic ends can be served.
Phase 1 ==> Primary School Mathematics
In the first instance, five to six years of primary education may provide
mastery in reading, writing and arithmetic.
In that, the child who complains that there many letters in the alphabet
to learn will learn or be told that learning to recognize and write all
letters is a must reading, writing and spelling. In reading and writing,
rules and patterns for grammar and punctuation may lead to greater skills
of practical value for adult and daily life.
Arithmetic skills may begin with learning to gradually count, measure
and figure with small and than larger and larger whole numbers alone and
with fractions, all with the aid of decimal notation. There counting and
arithmetic may be suggested by playing or manipulating actual, drawn or
imagined objects alone and in sets. In daily life, numbers and measure will
be met
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in telling and tracking time in years, months, weeks, days,
hours, minutes and seconds;
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in describing how many or how much, exactly or
approximately.
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may be met in finding lengths, areas, volumes, capacity,
weight or mass;
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in handling money: counting, saving and spending in whole
units and pennies - hundredths;
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in describing chance or likelihood, and choosing what chances
to take or avoid;
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in using maps, plans and diagrams drawn to scale to describe
angle, lengths, locations and paths or routes;
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calculating with distance, time and speed, or calculating with
per unit costs.
The associated basic skills in counting, measuring and figuring may be
developed by playing games and activities that entertain and/or may seen
as having practical value for adult or daily life. All is part of growing
up. Apart from a few technical innovations to clarify and strengthen the
introduction counting and figuring with maps-plans-diagrams drawn to
scale, and with whole numbers, fractions, decimals and signs basic
mathematics skill and concept is fine.
Remark: Sighted people have the ability to recognize and name like
or similar shapes in daily life. In reading and writing, that ability
allows the recognition of letters, digits and further symbols. In maps,
plans and diagrams drawn to scale that ability allows students and
teachers to talk about like or similar shapers before any formal
definition or codification of the term similar in later studies. In this
case a common ability put practice first, and theory or the later
codification of mathematics, second. From an empirical perspective, that
is acceptable.
Phases 2, 3 and 4 ==> Secondary school or Late Primary School
Presently, secondary mathematics in arithmetic, algebra, geometry, logic,
calculus and even statistics focuses on the needs of college programs
that are calculus or statistics. The programs may span commerce or money
matters, and STEM, the latter being an acronym for Science, Technology,
Engineering and Mathematics. Insurance, accounting and financing are very
mathematical. Mathematics provides a language for the expression of many
ideas in college level programs. Like any other language it can be used
for fiction, speculation and non-fiction. Calculus and calculus-based
programs require arithmetic, algebra, geometry, logic at full-strength in
secondary or early college mathematics. Some school systems are faster
than others. Where common use of the phrase "you are doing that
backwards" is intended as a criticism, the carefull, full strength
forward and backward use of rules and patterns in mathematics and logic
is or should be a deliberate objective of secondary and college courses
in mathematics, science, technology, money matters and law. The ability
to understand a skill or practice forwards is a strength, one to strive
for.
In most secondary mathematics courses today, the long-term goal of
preparing for calculus- and statistic-based programs serves the needs of
society for skilled people. Unfortunately, in most secondary courses as
taught today, that long-term goal is too distant from the experience of
student, many parents and many teachers. Year after year in secondary
school, the key question why master this or that has the answer:
preparation for the next final examination. With more than half the
teachers in North America assigned to mathematics instruction without
training in it nor a mathematical discipline, most students and most
teachers do not see the value of mathematics mastery beyond this
preparing for final examination target and beyond the hear-say that
mathematics mastery is a ign of intelligence. That can be demoralizing.
In some Canadian and US primary and secondary schools, students may be
pushed promoted to next grade for the sake of their "self-esteem" even
if they have not mastered skills and concepts needed in the next grade.
That makes the next grade more difficult. That promotion without being
prepared and the employment of teachers half-trained or untrained in
mathematics happens despite or because of the importance of mathematics
in the preparation of students for high school mathematics. The result
may be a lack of skill and confidence upon graduating or leaving school
before that.
Remedies for the context and motivation
problems are prescribed below.
In my secondary school days 1965-69, I sense that there were gaps in the
introduction of algebra. I sensed that words were missing to explain and
justify or rationalize the shorthand role of letters and symbols in
algebra after formula evaluation. That obstacle to understanding and
developing algebra continues today. But arithmetic mastery is a
prerequisite. That represents a further obstacled, mentioned in the 1950s
by the NCTM, that compounds the first How later skills and practices
depend on earlier ones need to be clearly stated in the text and comments
of course design and materials. Failure to do that allows gaps in
learning and teaching to persist unchallenged.
Today, instructors, students and home-tutoring parents will find remedies
for most technical troubles in college-oriented, secondary mathematics
courses today in site chapters and steps. But the cumulative structure of
skill and concepts, that is how later ones depend on earlier one still
needs to be remembered and accepted. That said, site coverage of
arithmetic, algebra, logic, geometry and calculusincludes a few
refinements or nuances to make skill and concept development more
accessible and clearer.
Secondary Mathematics Programs
Remedies for Context and Motivation Problems.
The question of context and motivation for secondary mathematics remains.
To serve the needs of all students, and not just the few who enter
college programs in commerce and STEM, the early years of secondary
mathematics may mix mathematics for practical use - that is develop
skills and methods with actual or potential value for adult- and
daily-life in modern society - with technical preparation for
calculus-based programs. A mathematics-free development of logic-language
methods useful in adult and daily life, and technically useful may
included in this mix. I imagine future secondary mathematics and calculus
course design and delivery may demonstrate and verify skills and
practices in phases 2 to 5 below, the first phase being met in primary
school.
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Phase 2 in a practice first, explanation where not overwhelming
manner, shows how counting, measuring and figuring with
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numbers - whole, fractional and decimals;
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maps-plans-diagrams drawn to scale; and
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algebra limited to formula evaluation
provides a mastery of mathematics for practical use. The aim here is
to provide and extend the common know-how of mathematics and even
math-free logic to greatest possible, so the value for daily and
adult life in handling routine situations is clear to students,
parents and teachers. All can be taught in scout-like, be prepared,
perspective. In this common formulas methods useful for money
matters, for distance-time-speed problems, for unit cost problems,
and for calculations in geometry and measurement are given
illustrately directly.
We may emphasize count computation by forming and adding subcounts,
sum or total computation by by forming and adding subsums or
subtotals with unsigned and signed amounts; and the calculation of
products by forming and multiplingly subproducts. The latter will be
useful in prime factorization, a must or plus for the technical
development of efficient fraction skills. In the study of say
distance, time and speed, formulas for each quantity will be given
for the sake of take-home value. Then in the later Phase 3
development of algebra skills, how each of the formulas leads to the
others will covered.
The aim of phase 2 is skill mastery in a practice first manner, for
the sake of take-home value. Students will be shown how to do and
record mathematical steps in ways that can be seen for checking as
done or later, so that the domino effects of care and mistakes become
clear. Doing work in steps that can be seen and checked provides an
end, value and tool, a skill in itself, for further skill development
in and beyond mathematics.
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Phase 3 will introduce algebra in systematics steps that will bypass
common troubles. See site algebra starter lessons. The introduction
will show students how to explain and rationalize the shorthand roles
of letters and symbols in solving equations, in describing the
properties of numbers, and beyond that in calculations that have some
take-home value. The forward and backward use of formulas and
proportionality relations, numerically and literally - algebraically
that is, will be emphasized here as a unifying theme in mathematics
beyond arithmetic and formula evalution. With that, formulas given
earlier will be used forwards and backwards. Perhaps, operations with
fractions will be described algebraically.
In the context of money matters, formulas for compound interest or
growth and formulas for geometric sum will be introduced numerically
and used forwards and backwards, algebraically and numerically.
Earlier use of likelihood in deciding whether or not to take a chance
will be algebraically extended here into set-based probability
theory. Besides the math-free mastery of logic, a simple account of
Euclidean Geometry based on the direct use of implication rules will
provide a stand-alone-body of knowledge and prepare the way for the
further study of geometry and trigonometry in this phase or the next.
Again in this phase or the next, the natural logarithm and its
inverse or antilog, the natural exponential functions may be
introduced numerically with an algebraic description of their
fundamental properties. Then their properties may be used to show how
to compute radical, powers and more exponentials using the natural
logarithm and its inverse. Site material shows how. Saying how to do
a calculation defines it.
In essence, the this phase will be more algebraic and more technical
than the first. Moreover, it will include islands and bodies of
skills and practices with some practical to nominal value for adult
and daily-life, with intellectual value in showing the role of logic
in geometry, or with intellectual value in providing a mastery of the
connections between logarithms, their backward use or inverses, and
the the calculation of roots and exponentials. In this instructors
are invited to emphasise algebra for practical use as much possible
while systematically providing a mastery of algebraic thinking
skills, using site methods to ease or avoid common troubles.
Remark: The Modern mathematics curricula for mathematics
introduced axioms for algebra that assumed the existence of numbers
and employed their decimal represenation without explicitly
sanctioning the latter and arithmetic operations with decimals. So
axioms consistent with the decimal-free formulation of higher level
pure mathematics, modern style, were disconnected from the common use
of decimals in daily and adult life, and in other secondary school
subjects. But in secondary mathematics and calculus taught as service
subjects, the lack of sanction and formal mention of decimals
represents a gap between the needs of other subjects and everyday and
the in-class theorectical development of mathematics. The remedy for
that is to clearly and explicitly assumed numbers have decimal
expansions in secondary school mathematics before calculus, and in
calculus itself, the remedy is to employ a decimal based development
of limits and continuity in the context of decimal error control of
approximations and calculations. The remedy recommended here will
make theory in calculus and secondary mathematics consistent with
numerical practice and with numerical perspective of concepts. The
remedy here will also make algebra skill development before calculus
and in calculus upto its decimal-free epsilon-delta view of limits
and continuity easier to grasp. See site volumes 2 in this phase or
the next, and site Volume 3 in phase 5.
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Phase 4 will cover technical skills and concepts required for
calculus. Most if not not all other skill and concepts required for
calculus which have some take-home or intellectual value for adult
and daily life, value worth mentioning, will appear in earlier phases
2 and 3 say. Some of thes technical skills have cross-curricula
value. Those students heading for college programs in STEM will or
should see quadratics, logarithms, conic sections and systems of
equations forwards and backwards in advanced secondary studies of
biology, chemistry and physics. But commerce or business oriented
students who avoid the latter may not appreciate the cross-curricula
value in full, even though some question in mathematics may hint at
it. The earlier phases may provide students with the will, confidence
and study habits needed to master the technical material here. To
what extent remains to be seen. The discussion of slopes in Phase 3
or this phase, and the discussion of factored polynomials may be
accompanied by light calculus previews for the sake of motivation and
for the sake of algebra skill development.
The skills and practices to be covered here are technical. In that,
real-life or authentic problems apart from mathematics will distract
students and teachers from seeing and mastery the technical theory or
story which underlies and explains the theory and practice. Here
mastery of theory is important for college programs. Here
trigonometry may be introduce as a way to use sketches in place of
diagrams drawn to scale for solving geometric problems. Given that
most secondary mathematics courses in North America today introduce
trigonometric skills and practices by rote, site steps in geometry
provide a clearer and quicker thought-based development with a few
innovations to make the hard easier. The proof is in the details.
While present day courses introduce a zoo of functions met in
calculus in a technical manner, I wonder whether or not the
properties of function in general and those in this zoo could be
included in a just-in time manner as part of a first or second course
in calculus.
Remark: While the site set-of-ordered-pairs treatment of
functions includes a few expositional simplifications and twist to
make the underlying concepts easier, the ordered-pair representaton
of function, a key but artificial element of modern pure mathematics,
may de-emphasized here in favour an equivalent computation rule
perspective. In that, the vertical line rule becomes a critiria for a
set of order pairs to graphically give a computation rule for a
function. That step away from the theory and its formalities may make
comprehension of functions and their inverses easier.
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Phase 5 would consist of single-variable calculus. At this level, all
site innovations in arithmetic, algebra, geometry and logic would
make differential calculus easier to learn and teach. One option here
is to give a light calculus course in which functions and integrands
are restricted to polynomials and rational functions. Then most
properties and theorems of calculus on limits, continuity,
differentiation and integration may be developed and illustrated more
simply with polynomials and rational functions. with function
composition with limited to the outer function be polynomial, the
chain rule would be easily implied and explained using mathematical
induction. That approach, a detour, would circumvent student
difficulties with a more direct approach to function composition and
the chain rule. This light approach would leave mastery of calculus
with trigonometry and inverse trigonometric functions to a second
course in differential and integral calculus. Giving calculus lightly
besides secondary school courses on mechanics would show how the
assumption of constant acceleration and/or inverse square laws for
gravatitation leads to equations of motion in mechanics. That would
be a cross curricula benefit of a calculus light option in this phase
or the previous. Sans or with the calculus light option, site
material includes a few innovations in Volume 2 and 3, and in its
calculus lessons to make the hard easier. Innovations includes
calculus previews and decimal error control perspective of limits and
continuity in Volume 3 and its appendices.
The five phases above include probability concepts lightly before phase 3
and thoroughly in phase 3. However, I am not enthused about the inclusion
of statistical methods in mathematics courses that have take-home value
for adult or daily life, nor in mathematics courses that prepare for
calculus-based college studies. Statistical methods appear to be
introduced and employed by rote in college programs in the social
sciences. Those who want statistics in the high school or college
programs should devise courses apart from mathematics for its
introduction and development.
Summary or Conclusions
Talking about five phases for mathematics education from counting to
calculus provides a direction skill development clearer to parents and
teachers. In the North America, in the 1960s, the introduction of the
modern mathematics programs for primary and secondary mathematics
introduce the notion that society knows best in the domain of mathematics
education to the extent that parents with technical backgrounds were not
expected to understand the change of direction and scope. The policy of
lack of clarity in course design has continued to the present day. The
site presentation and summary of five phase program is noteworthy for its
brevity and use of plain language to make skill development ends, value
and methods clearer to parents and teachers with some or a good mastery
of secondary mathematics and calculus.
Site Origins
Writing since the first days of 1991 has been an iterative and uncertain
affair. The skill development principles and standards outlined in Volume
1B, Mathematics Curriculum Notes, were met outside of mathematics in
1981, and were then well-known before I met them. Those principles and
standards crystallized my view that the algebra skills were too much
assumed and not adequately explained in the development of mathematics.
And technical following those standards and principles was also
inadequate. Learners might be expected to bring their own motivation and
keenness to a physical activity, one that provide pleasure. But in
compulsory mathematics education, students do not bring that enthusiam, a
lack of enthusiam easily compounded by gaps or difficulties in the
exposition of mathematics that made it harder than need-be.
By fall 2007, all the technical difficulties I had seen in mathematics
and logic education were resolved. But two issues remained. The first was
motivational - what ends and values, what context, can we provide for
skill and concept development? The result was a few incomplete attempts
to define alternative programs LAMP and POMME for mathematics education.
The question of ends and values for instruction led to a general pause in
site content development. In fall 2010, work began on a new content
management systems with the aim of making site content and organization
clearer for web surfers and easier to change and maintain. That being
done, attention return to the question of ends, values and feasible paths
for skill and concept development in mathematics and logic.
The online description of the five phase framework for logic and
quantitative skill development essentially addresses the technical and
motivational issues facing that development which I have seen and noted
since my own secondary school days. Site remedies began offline in 1983
where just before the granting of my doctoral degree, I presented three
lessons spanning two logic puzzles, three skills for algebra and why
slopes - a calculus preview - to small group of calculus or precalculus
students. Those three lessons and ideas for step by step skill
development met outside of mathematics in 1981 led to site volumes 2, 3
and 1B. The foreword and leading chapters of the latter set the stage for
this five phase framework, one more phase than the older and more
primitive four phase approach in 1B. Writing is an iterative affair.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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