POMME - Mathematics Course Design & Delivery Revisited
June 2010
Ends, Values and Methods for Instruction
"When I use a word," Humpty Dumpty said in rather
a scornful tone,
"it means just what I choose it to mean -- neither more nor
less."
"The question is," said Alice,
"whether you can make words mean so many different things."
"The question is," said Humpty Dumpty, "which is to be master - -
that's all."
(Through the Looking Glass, Chapter
6)
Foreword to Volume 1.
March 1996 (a POMME prequel)
The first part Pattern Based Reason of this volume
Elements of Reason (Volume 1) describes rule and pattern
based thought and processes in daily life, society, science and
technology. Reliable rules and patterns can be followed one at a
time or one after another to obtain conclusions or results. Not
solved is the problem of identifying reliable rules and patterns to
employ. Instead, the empirical method of coping with this problem
is discussed.
Rule and pattern based thought and processes touch many arts and
disciplines. Awareness of the difference between one- and two-way
implication rules will
improve reading, writing and argumentation skills. Students of
critical thinking, persuasion, philosophy, mathematics, science and
technology may find this first part worth reading.
In both arithmetic and logic, rules and patterns if followed
carefully lead to results which are repeatable and reproducible,
and thus verifiable and objective: two individuals following the
same rules and patterns with the same data or in similar
circumstances should obtain the same or similar results.
Arithmetic and deductive reason are but examples of
verifiable rule and pattern based thought or processes.
Verifiability, repeatability and reproducibility form a basis for
the appreciation of, if not reliance on, rule and pattern based
thought and processes. This appreciation should not be too firm.
The identification of reliable rules and patterns, or
reliable data to use with them is not certain. Further,
where rules and patterns do not apply mechanically, there is room
for thought. Still, verifiability, repeatability and
reproducibility may provide a basis for the common knowledge and
informal mastery of a subject.
The second part Mathematics Curriculum Notes is for
teachers and advanced students of mathematics or a quantitative
college discipline. This part describes simply yet precisely,
the role of rule-based reason, that is logic, in providing a
thought-based framework and codification for mathematical
thought. This second part further describes how an inductive
educational philosophy provides a context for math and logic
instruction from primary school to college. 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.
This two-part work and its the companion volumes Three Skills
for Algebra and Why Slopes and More Math stem
from a project to write a single book, namely Ideas that Might Count for Education, Reason and
Mathematics (1994). That single book (no longer available) was
written and distributed. It covered a vast number of topics. Some
of interest to one audience but not to another. With further
writing and rewriting, this first endeavor was divided into three
volumes, the first of which, the one before you, was divided into
two parts. Writing for some is an iterative affair.
The initial aim was to report some unique idea, innovations, for
math and logic instruction. These ideas or lessons had worked well
with college students, shy or curious about one or both
disciplines. But in writing and rewriting, the aim became wider.
The possibility of a consistent and coherent scheme for math and
logic instruction from primary school to college was seen and
explored. The scheme is comprehensive save for the treatment of
geometry. How to fit or emphasize Euclidean geometry in the
curriculum is not covered.
Formal mathematics can be difficult to follow for students who
fail to grasp deductive thought and the symbol-based algebraic
way of writing and reasoning. The latter like arithmetic is
better seen and written than spoken aloud. Symbols like pictures
can be worth a thousand words. Words have been missing to
explain the role of symbols in providing the shorthand notation
of mathematics or its algebraic way of writing and reasoning. The
latter consists of recording and developing thoughts on paper at
least for those among us afflicted with a short or too forgetful
memory.
The absence of a verbal culture to introduce and explain the
algebraic way of writing and thinking leaves its mastery to
immersion and osmosis. Comprehension depends on one's aptitude for
learning some basic ideas by immersion. I am in the radical
position of suggesting that a certain change is possible and
desirable. This work and its companions suggest how. They have
yet to be formally peer reviewed and so should be read with
caution. The discussion of math and logic instruction and the
discussion of reason and persuasion are both fraught with
controversy. Scrutiny or critical examination of this work may lead
to its refinement.
Alan Selby
Montreal 1996
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My initial aim in writing was to identify and fill technical gaps, and
may be motivational ones too, in the introduction and exposition of
modern mathematics, gaps I had glimpsed in my school and college days
first as a student and later as an instructor 1965-89. Writing was an
iterative affair. It explored and expressed different ideas for
instruction, whatever was easier, in the hope of forming or collecting
starter and further lessons to make modern mathematics instruction
stronger and easier to learn and teach. But in fall 2007, I started to
think about a path that diverged from the modern mathematics programs
1955-90 before converging to it at the senior high school level, with the
statement of axioms just before calculus.. Yet having seen and
addressed many technical gaps, there remained a question of motivation,
of clarifying ends, values and methods for instruction - why learn and
what to learn, and when.
In the forthcoming weeks and months, this site will outline and then give
in greater detail, a curriculum for POMME: Progressive, that is,
step by step, Observable, Motivated, Mathematics Education.
In POMME, the mid-level instruction for students aged 11 to 14
say (students could be slightly younger or older) provides skills and
abilities with take home value in the following areas:
- time and date matters
- money matters
- measurement calculations
- drawing, map and plan methods/usage
- decision making skills in risky and risk-free situation (a little
about the theory of chance)
with all the number and arithmetic skills these overlapping areas
require, with ends, values and methods for work and study - observable
and verifiable, and if need-be; and with a knowledge of the domino effect
in multi-step processes. That is, an error in one step of an numeric or
geometric process leads to errors in further steps. So attention to
detail or diligence is an end, a value and a must for observable and
verifiable skill development. That by itself will a source of skill and
confidence, self-esteem too, in all arts and disciplines in work and
study where mastery to be believed has to be observable.
In modern times, different societies may have different views of what is
important or needed in the mid-level application areas above. But naming
the application areas may imply a common core. The site to do is to
describe the application areas above in a form that may serve common
needs of the person in the street, the person who needs quantitative
skills and concepts sufficient for daily life. To the foregoing or a
late part of POMME, we may add logic mastery, and the explicit forward if
not backward use of implication rules written as IF A THEN B or
alternatively as B IF A.
Mid-level instruction with take home value and with the emphasis of work
and study skills provides a strong base for higher level instruction
from number theory to calculus. We will cover that below in some
detail. The mid-level instruction outlined above along the ends and
values for observable and verifiable skill development below provide a
destination and goals for the elementary instruction of younger
students, 4 to 10 years of age say.
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The mid-level applications above and what they require set
forth destination and values for elementary instruction. The
site section Helping Children &
Teens Learn identifies a collection of commercially
available work booklets for students aged 4 to 13 or 13 (that
is from pre-K to Grade 8) which altogether provide path for
elementary instruction, light enough for parents and
elementary teachers to understand and follow which giving
students a head start or preparation for mid-level
instruction. Doing more may be more trouble than it
worth. Elementary instruction or course design be lean and
simple enough for adult and teachers to provide.
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End and Values for Instruction, All Levels.
Logical and Quantitative skill development to be credible has to develop
observable and verifiable results. As a value and end for instruction,
what student think is less important here than what they can do. In
particular, we will show student how to do and record calculation or
reasoning steps in formats that aid figuring or reasoning, and do not
overwhelm students with too much formality. Lean is necessary and
sufficient. The aim of the formats is to allow the steps to be seen
and verified or corrected, one at a time, one after another.
Experience and teachers may explain and emphasize the aforementioned
domino effect of lack of care or mistakes in doing and recording
steps, including the data collection steps. In that domino effect, an
error in one step leads to errors in all further dependent steps, unless
by good fortune further errors intervene to cancel the effect of the
first. We may take as an end and value the ability to figure and reason
with arithmetic and geometric in observable and thus correctable or
repeatable and reproducible manner as a sign of practical diligence or
intelligence. The foregoing end and value will be an I can do it
source of skill and confidence, and thus self-esteem.
Logic mastery in all or part like a knowledge of the domino effect in
multi-step methods, is way to illuminate and so avoid study, work and
home-life difficulties stemming from the lack of precision reading and
writing skills. So logic mastery and knowledge of the domino effect has
clear value at home and for work and study, a value which parents,
teachers and students can see and appreciate.
Sooner or later, students should be able to understand as part of
language and mathematics reasoning skills,
- the forward, if not backward, use of rules IF A THEN B (that is, B
IF A),
- the use of the latter or implications IF B THEN A in chains of
reason; and
- the difference between saying A IF B and saying A IF and ONLY IF
B.
This logic mastery may come as part of mid-level or higher level
instruction. Earlier exposure is fine. But this logic mastery is hard
for the too young. What is hard for a typical 10 year old may be easier
for a 14 or 16 year old. But reading, writing and mathematics instructors
will have an easier time developing and verifying mastery of the above
logic skills and nuances in items A, B and C above for students who aware
of the domino effect of errors and who learnt to appreciate the benefits
of showing work, step by step, in given formats, for verification or
correction. The short and long chains reason written and recorded step
by step continue and extend the observable show work format and habits
present, we hope, in mid- and lower-level instruction.
A First Obligation for Course Design and Delivery
For students before the ages of 14, younger in some cultures, older in
others, the first objective and obligation of instruction - student
centered in a practical manner - is to give an operational command of
rules and patterns likely to be useful sooner or later in the work,
studies and home life of students and their present or future families.
The first objective represents an ethical duty for education. If a lack
of resources or poor environment does not encourage nor permit students
to continue, the first objective gives those students skills and
abilities with clear value for home and work, if not further education.
All the foregoing, includes good work habits for doing and recording the
arithmetic and geometric steps in an observable show work, manner
provides a good stopping point for mathematics instruction as well
a sound base for further instruction of a more technical kind for
intellectual development, or for trades, pre-college professions and
college studies.
Explanations and Connections
Explanations should be provide when and where they aid skill development
and do not overwhelm learning and teaching. Yet the take-home value of
rules and patterns for figuring and reasoning does not require all to be
fully explained. Rules and patterns may be learnt by rote if they is a
show work format for doing and recording the steps for observation and
verification or correction. In general, I expect students 14 years and
under want to be given methods without explanations of why they work in
the belief that their instructors are hired to present correct methods
only. So explanations or proof are not needed, and even resisted - seen
as a waste of time and effort.
In mid- and earlier instruction, above, students may see how to follow
rules and patterns, one at a time, one after another, in a recorded step
by step manner. As part of that experience, we hope students will see
some and then more examples of the combination of rules and patterns
leads to and hence justifies results including further rules and patterns
for figuring or reasoning. Thus the interdependence of rules and pattern
is seen and recognized, and even emphasized. In that, as said above,
explanations or linkages between rules and patterns should be given
where they aid skill development and where they are not overwhelming in
detail nor format for students and their teachers. The objective of
providing an operational command of arithmetic and geometric skills,
formula evaluation included, all in a show work style, has a greater take
home value. The informal if not formal combination of rules and patterns
by itself is a skill with take-home value for daily life and work, and
value to for higher level studies.
Higher Level Instruction
The Tasks. Higher level studies in mathematics for sake of
intellectual development or for the sake of careers in business, science,
engineering or technology can be delayed, and perhaps should be delayed
(there is room for local variation here) until most skills and concepts
with take home value, skill easily understood and repeated have
developed. Once the easiest skills and concepts with take home value
have been covered, higher level instruction has the more unfortunate or
harder tasks of (a) providing less obvious skills with take-home, skill
likely to depend on some technical skills and concepts, and (b) providing
technical skills and concept for intellectual development or for sake of
pre-college trades and careers, or for the sake of college studies -
those leading to business, science, engineering or technology.
Preparation and Selection. As far as I can see, without prejudice,
once most skills and concepts with take-home value clear to parents,
teachers and students have been met and checked, The further college- and
even trade/profession oriented mathematics instruction represents both
preparation and selection of students for further studies. Selection
may be based on performance - the written work and associated marks or
grades of students in one to several disciplines, mathematics included.
The good work habits for doing and recording arithmetic and geometric
steps coupled with logic mastery emphasized in mid-level instruction
provides a good base, a must, for the demands of higher level
mathematics.
The competition may be unavoidable. None the less,
instruction may still aim to make skills and concepts development
simpler and more accessible - as inclusive as possible. To that end,
site sections and pages so far written provide innovations small to
large for the introduction of skills and concepts in number theory,
algebra, geometry, complex numbers, trigonometry, functions and
calculus starter lessons.
How mid-level instruction may serve high-level instruction
Before the age of 14, the introduction of algebra may be limited to the
evaluation of geometry, monetary and distance-speed-time related formulas
in a step by step manner, with equal signs present and aligned
vertically, in a format chosen to standardize the evaluation process - to
make observable and verifiable.
Starting before the age of 14, a do-this, do that approach to the
identification of primes and the prime factorization of whole whole
numbers may be included in the operational mastery of whole numbers and
fractions. That mastery has take home value because its illustrates the
domino effect and because algebra skill development requires an efficient
mastery of exact arithmetic (no decimal approximations here) with whole
numbers and fractions. That being said, the discussion of primes and
prime factorization might be delayed to high level instruction if
priority is given to other skills and concepts with more immediate
take-home value.
The observation that methods carefully follow give repeatable and
reproducible results may also support the initial student view that an
explanation or logical derivation of the method is not necessary. That
represents a practice first, theory second, view of knowledge.
But mid and earlier level instruction in focusing on observable, that is
show work formats for figuring and reasoning represent a prequel to the
inclusion of proofs and demonstrations in higher level mathematics. One
further task of mid- level instruction is to introduce the combination
of rules and patterns to obtain further rules and patterns, all in a show
work style, without insisting that all be explained and connected. Yet
the letters set the stage for students to see and even appreciate (or
at least reproduce) the combination of rules and patterns to derive some
from earlier ones. So reasoning and chains of reason informally may and
(?) should appear in mid-level instruction to set the stage for higher
level instruction. The inclusion should be compatible with the first
obligation above - it should overwhelm students.
Algebra Skill Development - Step by Step
The use of formulas to describe calculations that may be done is a first
use of algebra. In it, the shorthand role of letters and symbols is
clear. Beyond that, the algebraic shorthand roles of letters and symbols
in mathematics needs to be slowly and carefully expanded. Secondary and
college mathematics programs in general (the modern mathematics programs
of the 1950s in particular) are to quick to employ algebraic notation in
the statement of axioms and methods for algebra and logic. The algebraic
and deductive ways of writing and reasoning obvious to some, are not
obvious to all. The introduction and rationalization of the latter,
especially the algebraic aspect, represents an olde gap in present-day
course design and delivery. Steps A to J include a remedy.
In steps A to J, besides ideas for the senior if not mid-level
gradual development of algebra skills from the forward and backward use of
formulas to calculus, you will find a treatment of complex numbers that
may be placed before the study of periodic trigonometric functions. That
treatment provides a base for well-known, complex-number shortcuts in the
study of trig identities and for the expression of dot- and cross-products
in the plane in terms of sines and cosines. Online Volume 3 includes ideas
for easing and avoiding algebra shocks in calculus and beyond. Before
calculus, the expression of roots and powers in terms of logarithms and the
exponential function, saying how to calculate the former with the latter,
provide a systematic development, a clearer alternative to other fuzzier
treatments that prevail. The treatment is present in site pages.
Theorems and Proofs
Practice first and theory second or not at all may be good policy for
elementary and mid-level instruction where explanations are given where
they aid skill development and do not overwhelm it, where students learn
about the domino effect of errors in numerical and geometric figuring,
and where observable and verifiable format for work, written or drawn,
is required for the sake of observation and hence verification or
correction of performance. Students may not appreciate the value of
formally combining rules and patterns together to obtain results or
further rules and patterns. The formal combination as in the modern
mathematics curricula of the period 1955-1990s may require an operational
command of algebra and logic. Instruction has to wait for that
operational command to be developed. Instruction also to gradually shift
the values of students from their expecting to be given methods or
formulas, and circumstances in which to apply them, to the question of
the logical development and structure of skills and concepts in
mathematics. The shift in values has to be promoted. Reasons are
required, other wise instruction becomes a ritual.
In steps A to J below include ideas for the senior if not mid-level
gradual development of algebra skills from the forward and backward use
of formulas to calculus. That being said, chapters 1 to5 in Volume 2,
Three
Skills for Algebra, introduce logic, Euclidean style, apart from
mathematics. Those chapters are for students who are not too young. This
naive introduction Euclidean-Geometry leanly employs
logic with a minimum amount of algebra, and it includes theorems and
proofs, with the proofs being based on direct logic only. The backward
use of implication rules is avoided. But in general, the site approach to
the thought-based development of rules, patterns or practices of
mathematics from arithmetic to calculus starter lessons resides in their
informal statement and combination.
Just as students were expected to follow and apply steps in drawing and
figuring in mid- and lower-level instruction, students will be expected
to follow and reproduce or recreate steps in the thought-based
development of higher level skills and concepts, all in an observable
show work style. Where young students object to theorems and proofs, it
may be possible to give a problem that say show that instead of prove
that, so the solution of the problem is essentially a proof of the the
corresponding theorem. In other words, a problem that say show that is a
theorem in disguise. In senior instruction as in mid- and lower
instruction, the form and content of steps written or drawn on paper can
be verified and corrected. An operational command has to be seen and
verified to be credible. Modern notations and concepts will be present
when and where they aid the operational command of arithmetic, algebra,
geometry, trig, functions, probability and calculus, and do not distract
from that command. The notation and concepts should be more trouble
than they are worth at senior and mid-level instruction of youth or
adults in mathematics and logic. Yet the foregoing as it help context
for modern notations and concepts implies their presence sufficient for
the needs of university courses in modern mathematics at the post
calculus level.
All the foregoing represents a critical path analysis of
what should be covered and when. In that skills, concepts and notation
or format will be introduced as needed or just ahead of that in order
to avoid overwhelming students and teachers with formalities. The aim
to keep instruction as simple as possible. An informal approach is
warranted because the algebraic-deductive styles and formats for
reasoning and concept development and codification in modern pure
mathematics has to be introduced slowly. Too early is too much. And an
informal approach is easier to understand and repeat in the classroom.
An informal approach to the practices of mathematics and logic is
broader. Within it, ends, values and methods that serve common needs of
the person in the street, or serve the calculations with decimals,
coordinates and units needs of science, technology and business can be
woven into mathematics instruction. The more stringent and narrower
domain of pure mathematics can be left to university level instruction
at the calculus level and beyond, or it can be included at that
pre-university level instruction of interested or exceptional
students. Finally, an informal approach allows the axioms for real and
complex numbers - algebraic stated properties of arithmetic with real
and complex numbers - given in the modern mathematics curricula for
secondary mathematics to be implied by arithmetic and geometric rules,
patterns or practices met in the informal approach before calculus.
Thus the informal thought-based development converges to the modern
mathematics secondary curricula, and does so after a development of
algebraic and deductive reasoning skills sufficient for the
latter.
POMME reflects a critical path analysis with some preference for a just
in time skill development, the latter to avoid giving students
information they will not appreciate.
Going Further
If I was to stop work the outline above and site content would be
sufficient for others to complete it. Most pieces are online. The essay
above and the end notes below reflect and point to the end of the
intellectual development of program for logic and quantitative skill
development that ranges from calculus for college or senior high school
students backwards to the question of what students 4+ should see and do
in mathematics. Almost surely, the presentation may be refined and
improved. However, I am quite satisfied to see or believe that no great
challenges remain in the design and implementation of a new mathematics
education program. The objective in the next months, this year and if
need-be next, is to provide a route and manual for the step by step
development of skills and concepts at the secondary level from the
mid-level application areas onward in an observable and hence verifiable
and correctable or improvable manner, with local variations in
implementation expected.
End Notes: This recent This New Zealand curriculum
page describes the mathematics and statistics strands favored in
present day course design in clear content oriented manner. Older
Mathematics Ed. References (1940-1970) provide a content
viewpoint and reflect older ends, values and methods for stronger,
university oriented students. The needs of other students are put
aside. The 21st Handbook of the NCTM old fashioned view that
teachers (I include course designers in that) should be learning
engineer is to my liking. Post 1990 calls for education to be
inclusive and to engage students are reflected in POMME the ends,
values, content and methods for mid-level instruction for ages 10 to
14 say. But higher level POMME provide ends, values and methods for
instruction which have clear and appealing pre-1990 merit. Learning
to do in a clear show work manner for sake of observable and
correctable skill and concept development is back. Ends and values
that are feasible required methods to achieve them.
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More End Notes
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Missing Elements: In site pages, there is not a systematic
inclusion of sets and set related concepts. But sets and set operations
may assist in the development of counting methods and in the
illustration of logic, and in the precise development of probability
theory and practices. While the set viewpoint of real functions of two
variable is online in complete and full manner (with one innovation -
the presentation of both horizontal and vertical line methods for
calculating function values from a set of order pairs in the plane), I
have to decide how much of that treatment is must for skill and concept
development of students not specializing in mathematics. In calculus
texts for functions of two or three variables, the set of order pairs
view of functions appears to take a holiday.
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Long-Term Value - providing a context: I once attended a
university mathematics courses in which the professor for each method
or topic covered indicated clearly where it was applied. There-in
lies a model for senior high school mathematics course design and
delivery, a model requires the indication of the long-term if not
immediate reasons for each topic or skill covered. And in the case of
polynomials, we might say their study is required for calculus and
beyond - there are no immediate applications - and that mastery of
operations with polynomials is part of algebra skill development. The
study of polynomials provides one example of a required topic with no
immediate real world or genuine application accessible at the secondary
level. Any counterexample might be more trouble than it is
worth.
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First Comparison with Modern Mathematics Secondary Curricula: .
The initial steps in this route depart from the modern mathematics
programs of the the period 1955-1990, programs whose content still
lingers in North American schools and elsewhere, but there will be
convergence. The algebraically encoded axioms for real numbers will be
provided or emphasized after a progressive development of algebraic
shorthand roles of letters and symbols. Where the modern and current
post-modern mathematics present the axioms as assumptions (algebraic
codified and mysterious for many), the senior mathematics part of
POMME will expand algebraic role of letters and symbols slowly and with
that imply the axioms using numerical and geometric practices which are
a necessary for operational command of pre-university mathematics. That
provides the option or base for a later axiomatic development of
mathematics in or after calculus.
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Second One of the pitfalls of the modern mathematics curricula
was it lack of formal support for the use of or common practices with
coordinates and for calculations with decimals and units, two practices
met in applications outside of pure mathematics. To address those
pitfalls, POMME as a matter of practice employs and refers numerical
and geometric methods, decimal arithmetic and coordinates included, in
its pre-axiomatic development of skills and concepts. The
introduction of axioms in POMME points to the axiomatic codification of
the arithmetic properties of real and complex numbers while allowing
and encouraging the use of decimals, units and coordinates. That could
be sufficient for most students at the secondary and university level.
The pure development of mathematics is left to undergraduate courses in
mathematics. The foregoing is recommended in a half-hearted manner due
to nostalgia or affection on my part for set theoretic development or
codification of my subject. The key question is to provide secondary
and undergraduate an operational command of mathematics while providing
not necessarily to all, but to some or a few, a knowledge of pure
mathematics.
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Practice First, Theory Second: In mathematics instruction, the
presence or availability of a path or innovation for skill and concept
development does not imply its employment. Inclusion may be more
trouble than it is worth. Mathematics education needs to be kept
simple in terms of ends, values and methods to be feasible.
If I cannot have peace my way, what kind of peace are you offering?
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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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