Phase 1: Basic Skills with Clear Take Home Value
In five or six years, this first phase essentially corresponds to
existing practices elementary or primary school level. It provides
basic skills in counting, measuring and figuring, most of clear value
for adult and daily life. Adult learners may master this material more
quickly.
Some small refinements to speed and enrich matters are implied
by site coverage of decimal subtraction, prime recognition and
factorization, and how to justify all operations on fractions by
raising terms. For that, the site algebraic account needs to
translated in into numerical examples which young students can
follow.
Children may accept primary school mathematics mastery as part of
growing up - preparing for adult hood. An ideal is outlined below.
Young children may see or be show how numbers are everywhere in daily
life in signs, in counting and in measurement. Counting, measuring and
figuring with numbers, measuring and drawing instruments, and even
maps-plans-diagrams drawn to scale can be presented as part of preparing
for the needs of adult or daily life.
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Whole numbers and small fraction appear in telling and
tracking time in years, months, weeks, days, hours, minutes and
seconds;
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Whole numbers, fractions and decimal appear in describing how
many or how much, exactly or approximately.
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Whole numbers, fractions and decimal appear in finding and
handling lengths, areas, volumes, capacity, weight or mass;
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Whole numbers, fractions and decimal appear in handling money:
counting, saving and spending in dollars, half-dollars, quarters,
dimes andand pennies, or the local equivalent in other decimal-based
currency systems
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Fractions, decimals and percentages may appear in describing
chance or likelihood, and choosing what chances to take or
avoid;
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Numbers may appear as coordinates or not in using maps, plans
and diagrams drawn to scale to describe angle, lengths, locations and
paths or routes;
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Decimals, fractions and decimals appear calculating with
distance, time and speed, or calculating with per unit costs.
Home-Tutoring - The Gift of Time
In practice, parent-friendly mathematics booklets are available for
home-tutoring before or beside school begins. Three hours of parental one
to one attention in home-tutoring could be equivalent to a full day or
week of mathematics in school. So if a students is struggline in
mathematic, a little investement at home may help them understand to
count, measure and figure with decimals, fractions, instruments and
maps-plans-drawn to scale. Just as toddlers or children start to walk and
talk at different ages and at their own pace, they can also learn to
figure, measure and count. Parents through the gift of time and attention
can help track and build skills, and do so deliberately, even a child
involvement slows down an activity a parent might do alone quickly and
more efficiently.
In school and at home, skill development may take many light to serious
forms. Schools and parents may encourage children to play games and do
activities involving numbers, measuring and drawing instruments, and
maps, plans & diagrams drawn to scale. Where parents can afford the
time, they may involve their children in play acting the buying and
selling of sevices and goods in their local communities and while
travelling for this or that reason. For home tutoring alone or to
complement in-school skill development, parents may buy mathematics
exercise booklets, see site suggestion, to cover or check basic skills
and concepts. A 80 to 90% mastery of the basic skills appearing should
provide a strong enough base for secondary school.
Educational pychologist correct advocate the use of manipulatives to
introduce numbers skills and sense, and perhaps geometric shapes as well.
The concept of manipulatives may also be extend to include drawing and
measuring instruments whose mechanism are visible, and not hidden in a
black box. As part of learning about counting, measuring and figuring,
children may learn how add, subtract, compare, divide and multiply
physically and geometrically. That may come before or besides the on
paper description of these operations on paper with whole numbers,
fractions, decimals and units.
Educational pyschologist incorrectly advocate the use of calculators to
avoid the drudery of computations. That may lead some teachers and some
schools to skip the teaching and verification of on-paper arithmetic
skills. But in my mind, the use of manipulatives in primary school should
include and imply mastery and full justification of
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addition and times tables including methods and patterns useful in
filling them, and using them backwards for subtraction and division
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decimal methods with justification and explanation of column methods
for addition, subtraction and multiplication, and how to do and check
long division.
The site J-method for subtraction could unify and simplify
conversion or borrowing needed in decimal subtraction method. Here
J, the tenth letter of more than one alphabet, Aramic included, is
shorthand for ten.
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fractions sense and methods with full mastery of the latter implied
by raising terms - translate the site algebraic account of that into
numerical examples suitable for late primary school students or if
need-be early secondary school students.
Not all is perfect. There is a great variation is what elementary school
cover and how. Parents need to ensure above 80 or 90 percent is covered.
The leading years of high school have a chance to include the missing
skills and concepts. Where not, parents may hire a tutor. Good luck.
Manipulatives and Hands-On Experience
Manipulatives may start with handling of square, circular, triangular
blocks, and the discover that a square block does not fit into a round
hole and vice versa. Manipulatives may continue with children holding and
drawing shapes and letters. Manipulatives actual or drawn may further be
used to show how to count objects direct or after physical operations
that correspond to addition, subtraction, replication or multiplication,
and division. So number sense and geometric sense may be linked together.
While higher mathematics involves manipulation of letters, digits,
symbols and drawings on paper, the latter may begin and be introduced
off-paper with physical actions on spatial and planar objects, objects
that can be seen, touched, counted, replicated or divided. Arithmetic may
be introduced as but an on paper representation and extension of physical
operation of addition, subtraction, multiplication and division. Fingers
too in counting, represent hands on derivatives.
The use of calculators introduces a black box into arithmetic. Done too
early, the use undermines the hands-on mastery and comprehension of
operations with decimals, fractions and mixed numbers. Hands-on
experience with ruler, compass and protractors, with measuring cups and
graduated cylinders; with scales and balances and with thermometers whose
mechanism can be seen and probeld all extend the role of manipulatives in
physically introducing quantitative skills and concepts. The use of
automatic instruments or black boxes for measurement without any internal
mechanism be visible deprives students of hands-on experience with
measurment and observation. Children should be shown how to find lengths,
areas and volume via counting units directly and then with arithmetic
methods. Children should also see how to find volumes of small objects by
immersion - the measurement of displaced of water in a graduated
cylinder.
Children should also be shown how to measure the capacity of a container
by filling it and then emptying its contents to fill other containers of
known volume and/or to fill graduated cylinders. Seeing that a pyramidal
cone needs to be filled and empty thrice to fill a box with the same
hieght and base area sets the stage for later mastery and comprehension
of volume formulas. Seeing that a circular cone needs to be filled and
empty thrice to fill a can with the same hieght and diameter also sets
the stage for later mastery and comprehension of volume formulas.
Through out primary school, division of lengths, areas, volumes, liquid
and granular quantities, and servings of food into amounts of equal value
all provide a context first for unit fractions like one half, one-third
and one-quarter, and then multiples there-of. In the division of a single
object, all fraction must be proper. But in measuring how may unit
lengths, areas, volumes, weights and so on go into a giving quantity,
improper fractions will appear.
Caution and Hopes for Schooling
Some school promote students year after year for the sake of self-esteem.
Eventually, that kind of promotion undermines skill development and hence
the confidence of students. Such promotion is penny wise, pound
foolish.
For skill development in reading, writing and mathematics, parents need
to ask for report cards that essentially chart clearly and honestly what
skills have been seen and fully mastered, which have been observed as
partially mastered, and which are missing. School systems which devise
such skill checklist may help their instructor and parents monitor
individual skill level in a way that would allow individual help and even
homework. For example, when a teacher hands-out an assignment, some or
all exercise there-in could be identified with which skill they check,
master or develop. Then students might be directed only to do those
exercises or activities where skill mastery has yet to be fully seen and
reported in report cards or earlier assignments. Well done that might
imply more work in tracking skills, but less work in marking for
teachers. It might also allow students to learn at their own pace or, if
they shown mastery to a required, to work on another subject.
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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:
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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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