In for a penny, in for a pound
Page Sections:
Writing began in 1990 to report ideas to
local educational authorities for review and refinement in a way that
would count. The bureaucratic impossibility of that, a reflection
of the extreme publish or perish competition in academic life, implied
silence until retirement (about 2020). Instead, self-publication
followed offline in 1994 and online in 1995. Those ideas included
inductive principles
(or standards) for instruction and three starter lessons (three skills for algebra,
two logic puzzles,
why slopes - a
calculus preview ), statistically if not universally effective
1983-89). Those three lessons by themselves could have changes the course
of mathematics instruction in the 1990's. Today Site
how-TOs go
further. They supports the inductive principles and
standards essentially in full. Today, educational authorities need to
re-design secondary mathematics from arithmetic to calculus with the aid
of site innovations and its inductive principles and standards, which
Occam's Razor yet favour over
constructivism.
Common Needs and
High School Mathematics
Secondary and primary school should aim deliberately to
define, reinforce and enlarge the common knowledge of
mathematics. What does TCPITS - The Common Person In The
Street - need?
With the high failure rate in calculus, there should be other ends
for high school mathematics.That should include mastery of
numerical, geometric and algebraic methods and concepts for solving
routine problems and for recognizing when those methods fails. In
particular, a textbook for learning a second language will not only
cover the spelling and grammatical rules and patterns
(exceptions included), it will may also introduce everyday activities to
serve as a stage for introducing or employing words and grammar.
There-in a model for mathematics education. Numbers are everywhere
in street signs, on clocks and in money matters. Geometry too
appears in the use of maps and designs, and in construction
in the home or at work. And algebra in form of the forward and
backward use of the compound interest formula (chapters 14) and of
geometric sums appears in banking, loan, mortgage and annuity matters
(chapters
21-5) Apart from high school mathematics topics required only
in calculus (or in engineering and space travel), preparation for
calculus may serve these other ends while serving the other ends can be a
platform for re-enforcing skills and concepts required by calculus. The
challenge for high school course design is to include routine or
common topics whose potential or importance is clear or can be honestly
and not artificially emphasized, so that each course for the most part
are relevant to common needs. That would be besides i topics
and skills explicitly identified as present due to calculus only or
mostly. In the latter case remarks and digression mentioning
routine calculations in chemistry, physics, biology, accounting, and
local construction or work trades would provide further ends and value to
skill and topic mastery. See critical path analysis
below.
Mathematics Teaching
Methods and Issues: Online Volume Mathematics Curriculum
Notes begins with inductive principles for skill and concept
development and then points to olde flaws and inconsistencies in the
exposition of mathematics which predate and continue in course design
today. Inductive
principles provide standards for course design and delivery,
simple subject-based, results oriented, and with site how-TOs, provide in draft
form at least a full set of values, ends and means for instruction,
direct or indirect, which address or avoid a multitude of content
difficulties - expositional gaps and inconsistencies in modern
mathematics curricula and their successors, diluted or not. Some gaps
were inherited from earlier times. Mathematics education, if was an
empirical art, would collect how-TOs easily understood and repeated in
the classroom, and document them for instruction and self-instruction.
Site pages explore remedies, logically designed but not fully tested in
class - due to qualification insufficient for employment in
education. Clear and precise skill and concept is a must -
technical gaps in exposition should but filled - but they need to be
accompanied by discipline or motivation, so that students will follow
instruction.. Why bother to require students to attend school, year
after year, when motivation & discipline for schooling are
dilute or absent? Calculus is a motivation for high school
mathematics. Yet we need to identify the arithmetic, algebra and
geometry in scenes from daily life and work to say where is the
math and how it helps, otherwise there are motivational gaps.
International Curriculum
Reform
Site how-TOs in
accordance inductive principles for
education. offer the framework for an applied mathematics
program. The program which may be woven into the existing
mixed or applied mathematics curriculum in the UK and elsewhere.
The program instead of repairing and reinforcing the modern mathematics
curricula in North America and Scandinavia of the mid-1950s onward, as
initially intended, gives a logically-consistent successor .
The modern mathematics curricula pretense of following or supporting a
context-free development of modern mathematics from set theory stumble
and ceased to be context-free with the mixed or applied
mathematics introduction of coordinates and drawings in the
development of Calculus, trig and analytic geometry and, if taught,
Euclidean Geometry. Secondary-level modern mathematics curricula
was inconsistent in that arithmetic with the decimal representation
of numbers was avoided in theory - not recognized in
axioms but required and used in practice. Site how-TOs provide a
consistent remedy that may be followed from arithmetic to calculus to set
the stage for applied disciplines and the optional study of modern,
context-free, mathematics at the undergraduate and university
level.
Site how-TOs are
logically developed and consistent, modulo the applied math assumption
of geometric practices - those present in geometrical use of maps and
plans with and without the use of coordinates. Occam's Razor may favour
them and inductive
principles for their development and refinement.
Goals and
Methods for
Mathematics Education
Mathematics programs for primary and secondary schools need to be
based on tried and tested methods in accordance with inductive principles.
For instruction with a lean
inductive program to follow, student performance will be a guide to
remedial measures, what steps to retry as is or in expanded form, and
what is next in skill and concept development.
Mathematical induction
indicate how induction in general will fail if steps are too large or not
reachable. Leanness follows from foresight based
on Occam's Razor or critical
path analysis. In that analysis subprograms of study and
effort, will little or no benefit, those not required by later skill and
concept development, may be questioned if not omitted. Finally, there is
a question of motivation for programs of study. The key question that
needs to be clearly answered is as follows. What are the ends to be met?
There may be more than one: (i) an operational mastery of numbers and
mathematical methods needed in daily life with clear and plentiful
examples to span the needs and maintain mastery without too much
duplication; (ii) development of lawyer-like reading abilities for
greater skills and awareness in work, study and citizenship; (iii)
calculus and preparation for it. Site material supports the
last two ends, at least in part.
Sphagetti programming is a term that applies to code that is bloated or
needs replacement. In such code or any code, subprograms that
have input but no output may be removed. CPM is needed to
identify that sphagetti
Transparency is required in course
design. Calculations involving lengths, areas, volumes, weights,
further measures (direct or calculated) can be placed in a context.
Drill and practice with them may be cast in context or situations
fictional or not, but not too specialized. Course design could
present a variety of activities in daily life and identify the
mathematics in them as part of skill and concept development or as a
motivation for the latter. When may before, during or after.
Mathematics instruction may emphasize the applications as ends in
themselves and as tools to develop, refine or consolidate skills and
comprehension that will reappear in further applications and/or the
further development of the subject alone and with others. Each
topic in mathematics should be accompanied by a clear statements in
written and spoken remarks of the one or more of the following: (i)
where it will be needed in daily life (local variety possible), and (ii)
where it will be employed in mathematics or science. Each technique
should be named or have an apt descriptive phrase, for example
compound growth formula, rectangle area calculation, completing
the square, to permit oral and written reference and
discussion, and pointers to where it may be used.
Example: Quadratics are employed in physics to describe
projectile motions.
Quadratics are also required in the further mastery of
polynomials, geometry and calculus,
That motivation would be offered besides
skill and comprehension development and verification in accordance with
inductive principles. Students and teachers become pawns and cogs in a
bureaucratic machine when course design does not mention means, values
and ends. The latter may be short and long-term. The latter may
also require some compromises. The most direct route to the
long-term ends may be too dry and at the expense of early ends, means and
values that could motivate and drive learning and teaching.
Preparation for (a) calculus and (b) for engineering in
secondary school is a long route. Motivation should not be
contrived nor too artificial.
Example: Compound growth and decay formulas, and geometric
sums too, appear in money matters in the description of compound interest and
investment growth/decay, and in loans, mortgages and pension plans
calculations. The same or similar mathematics appears in the
description of population growth, animal husbandry, and
radioactive decay. The forward and backward use of
compound interest or compound growth and decay formulas requires a
knowledge of logarithms,
roots and exponentials.
Explanations as to why topics are covered
should be brief, simple and non-fictional, or if fiction or nearly so,
presented as food for thought. While rates and
proportions, probability and averages are be useful in daily
activities or pleasures, the study of quadratics,
further polynomials is motivated in all or almost all by calculus.
Economic models and calculations involving quadratic functions in
particular provide food for thought, and should be presented as
such, rather than as great application and great motivation for the
study of quadratic and further polynomials. Preparation for
calculus provides the simplest explanation, even if that be dry and
boring. Offering motivation is fine, but artificial motivation
should be presented as such - food for thought and/or exercises to
develop or verify mastery.
Meeting the demands and requirements of
calculus would be a long term goal. In preparation for calculus,
instruction may emphasize the value of being able to apply arithmetic,
algebraic and geometrical methods in a show-work, repeatable,
reproducible and verifiable or correctable manner. The ability to
apply rules and patterns one at a time, one after another, alone or in
combination, with or without comprehension of why those rules and
patterns work, is a sign of care and diligence and a prerequisite
to intelligence in such applications.
The ability to follow rules and patterns
in one subject is a sign of specialized intelligence. With regrets, it
is not a guarantee of general intelligence. C'est la
vie.
The thought-based development of mathematics
is based on rules and patterns drawn from experience (or not) and
then assumed. An axiom in mathematics is simply an assumed
pattern.
Exercise for Parents, Students and Teachers - Keep a record of
all the out of school mathematics (arithmetic, geometric, algebraic)
that you meet That should give you a list of uses and
frequencies.
The student of calculus or before may be
advised to learn to do, and worry about the why later. That push
towards rote learning has merit. While a comprehension of the origins and
motivations for rules and patterns in mathematics is desirable, the
thought-based development of mathematics is based on the careful and even
mechanical application of rules and patterns, alone or in sequence and in
combination, to provide results or further rules and patterns to follow.
Comprehension in mathematics and its thought-based development is based
on the ability to see, follow and apply rules and patterns alone or in
combination, one at a time, one after another. Once that ability
appears, the student who has learnt by rules and patterns by
rote, in a plug and play manner, may return to their study to
replace their rote mastery by a thought-based mastery.
For instant in developing site pages, I
realized I had learnt decimal methods for arithmetic by rote and not
understood their origin. Site pages provide a remedy. Site pages
endeavor to provide a thought-based development and combination of the
rules and patterns required in calculus itself, and in what the latter
demands from arithmetic, algebra, geometry and logic. Those
pages, quickly written, are presently in need of polishing. But
for that, they are done.
The thinking part of an art or
discipline:
(repetition of preceeding theme)
The deductive thinking part of an art or discipline comes
after the assumption & careful mastery of some rules
and patterns, steps and methods, practices and conventions.
Careful mastery means you can use the rules and patterns etc to
arrive at results in a repeatable, reproducible and, if hence
verifiably right or wrong manner. The thinking part of a
subject begins when you start to combine rules and patterns,
steps and methods, practices and conventions, to obtain new ones
in a repeatable, reproducible and hence verifiable manner.
Thinking or critical thinking within an art or discipline
continues through recognizing the benefits, origins and
limitations of rules and patterns, steps and methods, practices
and conventions, so that the approximations in the application of
the latter are known or avoided. The combination of rule
and patterns, customs and practices, steps and methods, one after
another, may lead to short parallel strands of reason and hence a
thought-based development of an art or discipline besides and
even on the empirical mastery of rules and patterns etc with
confidence building results that should be
repeatable, reproducible and hence verifiable.
Once the ability to form or follow strands of reasons within an
art or discipline is present and respected or appreciated, fuller
and fuller thought-based developments can be offered, if not in
class, then in print. The first phase of education could be based
on rote - here are the facts and methods - learn to use them in a
repeatable, reproducible and hence verifiable right or wrong
manner. Later phases may then build on that via a mix of
deductive and rote mastery of further rules and patterns. That is
to say deductive reason need not be explicitly axiomatic.
The young mathematic student for instance may benefit
from a thought-based mastery of decimal methods for addition,
comparison and subtraction, a comprehension built inductively
from examples without or before the formal deductive IF-THEN use
of implication rules. But the young student in learning
decimal methods for multiplication and long division would not
immediately benefit from an immediate in-class thought-based
explanations of why the methods work. None the less, skill
and confidence in their results might and should follow from
drill and practice, and methods for verifying or checking
results. Later on, arithmetic methods may be employed
as part of deductive arguments with the implicit assumption that
they work - give correct results. That being said, site
pages include and point to the thought-based development or
mastery of decimal methods for multiplication in full and long
division at least in part.
Teachers: Site how-TOs, inductive
principles, and values also give and define a necessary
alternative to constructivism and like Alice in
Wonderland, subjective theories of education. Ideas that
cannot be expressed on paper with diagrams, words and symbols are
not part of observable skill & comprehension. Compare
and contrast that view with the Allegory of the Cave in
Plato's work The Republic where knowledge is based on
shadow interpretation. Compare and contrast that view with the
dominant constructivist theory of skill and concept
learning, in which mastery is a subjective affair, not for
observation nor correction in an objective manner; and in which
changes in delivery style in a shadowy manner was suppose to lead
to a subjective (anarchistic) view of knowledge, one that in
retrospect resembles the state of knowledge before striving for
objectivity was the norm in science and technology, if not
law.
Ends, Values and Means:
- The will and ability to read notes and textbooks like a lawyer,
so that no nuance, no subtlety and no clause escapes
attention is an end, values and means for mathematics education in
the training of students and in course design and delivery.
-
Skill and confidence in mathematics, a written art
or discipline, may follow from care and precision in
following and recording the steps in methods and routines in a
well-formatted, readable, reproducible and repeatable manner for
verification or correction. Here care and precision is another
end, value and means for skill and concept development.
-
Mastery of thought-based paths for understanding and
developing skills, concepts and comprehension is a further end,
value and means for mathematics education, and may be considered
an extension of the care and precision, required so that no
subtlety, no clause and no detail escape attention..
Writing began to support and strengthen the
thought-based development of mathematics along the lines of the
modern mathematics curricula of the mid-1950s onward.
Identification of gaps in exposition and logic in Volume 1B,
Mathematics
Curriculum Notes, did not change that plan. It
indicated more work to do. As a late 1960's student of the
modern mathematics curricula, I was engaged or hooked by its
purported thought- and logic-based development, Euclidean style
of mathematical knowledge. Thus I was against rote
learning.
In retrospect rote learning co-exists with thought-based
development of skills and concepts in mathematics.
(1) At the high level, in the introduction of calculus,
students are given key theorems (rules and patterns) to apply and
told the proofs are too complicated for immediate mastery - site
pages may lessen the level of complication, but the advice to
mastery the key theorems and learn how to apply, but to skip the
proofs still applies. As a student, I did not like the gap that
represented in my skill and concept development.
(2) Precalculus students will learn formulas for areas and
volumes via mix of rote learning and derivation. The formulas that
cannot be derived before calculus can be derived or explained as
applications of calculus in the form of slope calculation reversal
(integration).
(3) Site pages point to the inductive (drawn from experience) and
thought-based development of primary and secondary school
mathematics. But in decimal arithmetic, the easy inductive or
thought-based mastery of place-value addition, comparison and
subtraction aids skill development. But multiplication and long
division methods, must for late primary and early secondary school,
are easily mastered by rote but may be too complicated for a
thought-based development. The latter may come, if it comes at all,
at the senior high school level.
Note: Calculus requires and extends the earlier full
strength use of algebra in secondary mathematics. The latter in
turn requires a full strength mastery of exact and approximate
arithmetic with the decimal format of numbers alone or in
fractions. That mastery begins in primary school.
The coverage of arithmetic, algebra, geometry and even calculus in
site pages was intended to eliminate rote learning in mathematics
education, and provide a thought-based development instead in
accordance with inductive principles and
standards for instruction in rule and pattern based arts and
disciplines - those advocating or striving for
objectivity. Yet as indicated some rote learning is needed in
the development of arithmetic.Mathematics education, yours or mine,
may have involved more rote learning than need-be. Some pages
were written to provide myself a thought-based understanding of
skills and concept met earlier by rote.
Mathematics education may become a bureaucratic set of rituals in
rules and patterns are mastered one a time and one after
another because that is required for a forthcoming end of
year final examination, without any further foresight into the
aims, ends and even values that might be part of mathematics
education. Preparation for calculus provide one end, albeit the
high failure rate in calculus makes one wonder if the route to
calculus is too hard for students - does it represent a pyramid
scheme in which many start and few benefit. The remedy for
that is to include along side and even as part of preparation for
calculus, an identification and development of the
arithmetic, algebraic, geometry and even logic routines and methods
that are present and useful in daily life. That should be besides
the development of skills and topics explicitly required by
calculus. Then each course would provide useful and hence
motivated skills and concepts.
Is it possible for students to end their studies in mathematics
before calculus is manner that provides skills and confidence along
side the decision not to prepare further for calculus?
Students need to master routine methods for the routine
mathematical questions that appear in daily life. The question what
would not work in society if there were no numbers, no
compound interest or no growth formulas, no geometric sums, no
geometry (maps and plans include) might motivated consumer or
people-oriented mathematics studies along side calculus
preparation. I suspect most students and teachers, and guidance
counselors, are unaware that preparation for calculus is or
was the reason - the only possible reason - for many or most
topics in secondary and primary school mathematics.
Preparation for calculus should set a standard for and not a hidden
agenda in earlier skill and concept development.
Each course mathematics represents an opportunity to see and
recognize the though-based development of skills and
concepts. Students may enter with a mastery of rules and
patterns (we hope), some mastered by rote and some with a
thought-based development. In the course, further
rules and patterns may built on earlier ones or built separately by
emphasizing the origins and derivations from immediate observations
or previously met if not mastered rules and methods, regardless of
how those earlier rules and methods were met and (?) mastered. The
latter emphasis represent the introduction or continuation of the
though-based development of skills and comprehension. Courses
are generally too short for a full review of earlier rules and
patterns with a thought-based development. Modulo that, or
except for that, each course may stand on a mix of rote and
thought-based learning to provide a thought-based development via
the careful use of rules and patterns alone and in combination to
get results or to imply further rules and patterns.
Students may object to the thought-based development of
skills. That development may not be tested and hence not
required by forthcoming final examinations. Students may be
in learn only what is required for the final. They may be no
long-term goals. Further more, students may not appreciate
explanations of rules whose earlier rote mastery gives results,
repeatable and reproducible, or not - It works, why do we need
to see more, may represent the underling attitude or question,
to which there is no reply. And students may object to the
inclusion of a development of a rule or method, and ask instead
rote mastery of the rule and method with minimum explanation and
many examples. Most activities outside of mathematics are
plug and play. There is no need to ask why. That sets a
standard, not optimal, for mathematics education. It represents the
view, give us the formula and numbers to plug in it. Such
student needs to be in courses which emphasis routine problem
solving skills for the multiple ways in which mathematics touches
daily life. Students aiming for calculus would be benefit
from a solid grounding the precalculus application of mathematics,
every one is likely to meet.
The thought-based development of mathematics is not for all. There
are some dilemmas and gaps in the exposition of high school
mathematics that will not be fully addressed here. Course
design and promotion needs to reflect a critical path analysis of
what is to be done and why. Site pages identify what might be
included in secondary mathematics as preparation for calculus and
as aid (?) to routine problem solving likely to be needed in the
lives of students and families - problem solving with geometry and
money matters say. The challenge is to provide a mathematics
education with aims, methods and values that students and teachers
can appreciate.
|
The Rote, Plug and Play, Descriptive, Aspect of Education in
Science and Technology
Question: How can the latter be reduced? Can it?
Mathematics is an art and discipline in
which a thought-based development is possible on first meeting or on
review, revision or consolidation of its skills and concepts. Most
can be developed with drawing and writing instrument (pencil and paper or
modern successors included). In contrast, science, and technology in
theory and in practice has become plug and play. The cook in
the kitchen often and the chemist rely on bought ingredients, properly
labeled. The high school and college science or technology lab may
include instruments that are used in a plug and play manner. The use of
electronic balances may provide greater accuracy and convenience, but
they also represent that plug and play aspect of a school or college
lab. The use of balances where physical weights are employed is
closer to the origins of science and technology, more primitive, but less
plug and play. That is to suggest simple measurement of mass and
volume may be done in the school lab. The use of batteries,
ammeters and voltmeters represent a further plug and play element of the
school or college science lab. Is it possible to verify Kirchoff's
current and voltage laws with them? I tried and failed. The
foregoing raises the question of how to illustrate chemistry, physics and
biology in lab in a way that will confirm or corroborate theories
alongside their in class description in a manner that is, minimally plug
and play. Dissection in biology does not count as artificial plug
and play. In the classroom, the periodic table represent a cumulative
effort which can be described as is with its development, and its modern
variations. But it cannot be derived. And in chemistry,
students may be surprised by parallel theories which disagree on the
identification of acids, bases and salt. The term acid-salt
refers to a compound that when dissolved in water is acidic (turns
litmus red) while the chemical formula theory suggested it should be a
salt. Students may have to view science and technology as a plug
and play process in which methods are subject to verification - do they
give repeatable and reproducible results in practice, and in which
theories, hypothesizes or hopes are subject to refutation (it does
not work) or partial success, small or wide ranging, instead of
absolute confirmation.
Do you insist on putting a round peg into a square hole Does it
make sense for subjective complicated,
incompletely defined theories of knowledge to be applied to
mathematics or science - these being arts and disciplines striving for
objectivity? The answer is no. The attempt to do so lends an
Alice in Wonderland experience for students and teachers in
precalculus mathematics in North America and less so in the UK. The
majority is not always right.
The 1990 NCTM standards and
principles fail to answers to two concrete questions:
What mathematics is to be taught, why and how. The 1989
standards as is and reformulates in year 2000 focuses on delivery,
constructivist style and so represent a subjective view of knowledge in
which teacher or discipline centered view instruction (my discipline has
many steps, let me cover them) is replaced by calls
individuals to construct their own comprehension from authentic,
realistic, genuine activities. Yet most primary
and secondary school instructor do not have a calculus background - the
motivation for much of secondary school mathematic program, its content,
and many are seconded from other disciplines, impressed into mathematics
instruction. Pre-1990 expositional difficulties in
mathematics in the form of steps too large or missing and in form of
expositional inconsistencies are also not recognized.
The NCTM standards ignore the instructor content mastery
problem and expositional difficulties, while calling for and
a specifying a subjective delivery style, and depreciating rule and
pattern mastery as a form of rote learning. Contrast that
with the old fashioned view good figuring skills were necessary to
demonstrate intelligence. Contrast that with the Euclidean model for rule
and pattern based reason prized in the work of Euclid and in modern
mathematics. Contrast that as well with the content of Volume
1A, Pattern Based Reason, and its
description of the origin, benefits and limitations of rule and
pattern-based processes in thought and deed in mathematics and
science. Modern society, for better or worse, with great
variation, strives for objectivity and not subjectivity in the
development and application of rules and principles in law,
mathematics, science and technology. Site how-TOs and inductive
principles and standards (or lower bounds) for instruction support
the latter. The NCTM needs a change of course.
|
|
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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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