Mathematics with Take-Home Value
Essay, Draft form.
Before a child goes to school, many mathematics, logic and language
skills will appear in home life. Which ones have the most take-home value
may vary. Different communities may have different needs. Some may serve
practical ends. Many skills are required in buying and selling goods,
property and services, in making informed decision in the presence or
absence of certainty. Practical applications done and recorded in an
observable and verifiable or correctable manner set an empirical standard
for rigour, one in which the domino effects of wishful thinking and
mistakes have immediate consequences.
We need to remember that the time in school of a child or teenager may be
limited because of family circumstances or because of a voluntary
termination of schooling. Thus Mathematics skill development should focus
in the first instance on what TCPITs, the common person in the street,
may need sooner or later in daily or adult life. There-in lies a place
for a broad development of mathematics and logic skill with take-home
value. That development should begin before the preparation for college
studies, but may overlap because the latter preparation in part may make
skills with take-home easier to learn and teach.
In daily life, many skills are mastered and used without a full
comprehension of how and why they work. In preparing a meal, very few if
any of us full understand the digestive system. Machines and mechanisms
may be also be employed without comprehension of how they work and why.
That being said, in counting, figuring and measuring, the ability to do
is required while comprehension is optional. In the case of methods with
take home value, the first task is to provide skill with methods without
overwhelming students with explanations of why, but explanations included
to the extent that they aid skill development, but do not overwhelm it.
Skill mastery is the first aim of instruction in methods with take-home
value.
Where primary instruction in methods with clear take-home value overlaps
secondary instruction preparing students for college and also prepares
students for further skills that have take-home value, Primary
instruction has skills with take home value to development explanations
of how and why rules and patterns give results is must to the point that
it aids skill development without overwhelming a student. Beyond that,
where secondary instruction preparation for college studies in business,
engineering, science, technology and mathematics education should
gradually emphasize comprehension of both how and why, with the
explanation that skill without comprehension becomes limited. That being
said, the theory or explanation of how and why should be lean. Where
topics do not have immediate take-home value, reasons for them should be
given. See site account of secondary ends and values - those to emphasize
in college oriented skill development.
A. Time and Date Skills
People have a sense of time, that before, after and simultaneous. That
sense of time and sequence holds not only in the right now, but also in
the telling of stories of past and future events, and also in fiction.
Modern family life may be arranged in accordance with the time of day,
day of the week, the month, the season and the year. Every one has a
birthday. Year round, daily life is governed by seasons and the day of
the week, what time to rise or eat, what time to sleep, when to work and
study and pray. School and work life are run or organized around time or
schedules. Events occur before, after or the same time. Civic and
religious calendars provide schedules, celebrations and holidays. The
teen or adult may have study and work schedule as well.
B. Money Skills
People buy and sell goods and services. People work for a living, There
are budgets to balance in private life and in business. Money is counted,
added, compared, subtracted, multiplied and divided. Life in families and
communities often involves money matters. Talking about ends, values and
methods for handling money at home, and in buying goods and services, or
balancing a personal or business budget would provide a context and
motivation for care and diligence in arithmetic with unsigned and even
signed numbers - the use of the latter is optional. Students may shown
how calculate net worth by adding subtotals. See corresponding remark
below.
In the case of money matters, students may master the forward and
backward use of compound interest/growth formulas and geometric summation
formulas by rote initially, simply because of the possible take-home
value. For example, numerical examples may be employed to empirically
suggest or confirm the geometric sum formula. To ease or avoid raising
the level of algebraic complexity, formulas may given and illustrated
instead of being derived independently or from each other. In the first
instance, the ability to apply methods with take-home value in a
repeatable and reproducible manner is still more important than know-why.
C. Spatial Skills
Child may learn to recognize the spatial positions of before, after,
behind, ahead, above, below and besides. That learning may come from
experience or from others at home and in class. The tutor or teacher may
check or develop this knowledge through simple questions and activities.
The spatial sense of up and down, or above and below, may be easier to
learn and remember than the sense of left and right.
In my child-hood window on life, reading of words and decimals from
left to right was no problem. But I still had difficulty identifying
left and right. I have no experience with literal and numeric
dislectsia, I will ask a question. Would it be easier for some students
with dislectsia in reading and writing English, or in reading and
writing decimals horizontally have less difficulties in reading and
writing words and decimals if the latter were written vertically.
Column place value methods for counting and figuring with decimals
could easily be transposed to provide row place methods. Calculators
displays could also be transposed.
D. Geometry, Navigation and Construction Skills with maps and plans
Maps and plans drawn to scale (or not) are everywhere. In going to
school and in traveling children may see maps and plans. Larger
schools, colleges and workplaces may use building plans or maps to
provide directions.
Geometry in part consists of direct measurements with units of length, on
rulers and tape measures, with angle measures in terms of degrees and
fractions of a circle or revolution. Students may identify a right angle
with a quarter of circle before thinking about quarter revolutions.
Scaling of Direct Measurement of lengths and angles on maps and plans
drawn to scale then gives another method, not direct, for finding and
using lengths and angles not measured directly. Triangle construction
algorithms ASA, SAS, SSS and even AA(A) will be included among methods
for drawing triangles, rectangles and the circles to scale or full size.
In travel and in construction, we may use maps drawn to scale to
estimate or calculate lengths and even areas, and thus solve problems
without or before any knowledge of higher mathematics in the form of
trigonometry. Solving problems with maps drawn to scale has take-home
value, and may be emphasized and illustrated in geography and science
lessons. Reading maps and plans, understanding contour levels, are all
parts of quantitative skill development.
For geometry the foregoing students may learn on paper to evaluate a
formula for an area or volume given the necessary lengths and measures.
They should also be able determine those lengths or measures from
hands-on experience with the actual figure or scale drawings of it.
Instruction should point out applications robustly and fully.
E. Counting, Measuring and Figuring Skill
Besides counts and measures of time, dollars and length, master measures
of area and volume, and of mass and weight. In buying and selling goods
and services, and in making things - that includes cooking and
construction - people use and combine measures alone and in proportions.
Examples include speed and the cost of a unit (a prequel to the
discussion of per unit rates) of a good or service.
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Express Measure & Counts in terms of given and alternate units of
each, respectively.
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Add, Compare and Subtract Counts and Measures - when possible.
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Multiply: Form the product of both with a number.
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Divide Measures & Counts by another measure or number.
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Each rate has a reciprocal. For example, the rate of 5 apples per 10
pennies is equivalent to the reciprocal rate of 10 pennies per 5
applies. The transformation of one rate into reciprocal may be taught
along side the use of rates in arithmetic. Mastery here may be
verbal. Skill here can be extended with the arithmetic with multiples
of units and fractions with such mutliples in numerators and
denominators.
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Example of the chain rule for rates is given by observation that 5
apples for each 10 oranges, and 10 oranges per 20 pears implies 5
applies per 20 pears as well. With the mastery of arithmetic with
fractions, the chain rules can associated with a product of fractions
with units.
Exposure to measurement matters in the home and in shopping will depend
on the family and community life of students. Due to the variation, skill
development in school will have provide the missing experience and a
context for it. But all has take-home value.
F. Chance and Risk Skills
In decision making, not all is certain. Risks are present even for people
who avoid games. Learning about chance and probability may help avoid
situations or decision where the risk are high, or help in making
decisions that lower risk and make the chances of success greater. Again,
due to the variation, skill development will have provide the missing
experience and a context for it.
G. Logic and Decision Making Skills
Logic mastery in mathematics or apart - say in a reading and writing
course - may lead to greater care or precision in reading and writing,
and so avoid or lessen difficulties in studies and in work. Precision in
reading and writing may follow here from study of logic and from an
awareness of the domino effect of errors.
The leading chapters of Volume 2, Three Skills for Algebra, develop
deductive reason (logic mastery) in a math-free way. Altogether, those
chapters hint at the partial Euclidean organization and codification of
rule and pattern based arts and disciplines.
Awareness of the difference between say A if B and saying A if and
only if B (or equivalent expressions) will sharpen reading and writing.
And seeing how to chain implication rules together will help reasoning in
general. These two elements of logic and further elements may be
introduced when students are ready for them - the age level for that may
depend on the student. The net result should fewer difficulties in work
and study, and better results in general.
Self-Defense
People in health care and instruction may sense an obligation to help
others. But in buying and selling goods, properties and services, and in
working for a living, decision may require money skills and logic or
language skills to negotiate, to recognize the good and bad aspects of
deals and contracts, and to recognize agreements that should not be made.
H. Proof
The aim of instruction in the above areas and in the parts of arithmetic
they require is to develop observable and verifiable skills and know-how.
As part of that, we will show learners how to do and record numerical and
geometric steps in learn but sufficient show work formats that allow
steps and results, from start to end, to seen and confirmed or corrected.
Emphasizing that care and patience, and precision too, are needed to
avoid the domino effect of errors and approximations in short and long
chains of reason will be emphasized. The ability to figure well, and to
read and write with precision, is an observable sign of intelligence of
the practical kind for work and studies in general.
The word proof is usually associated with higher mathematics. But
figuring with decimals, fractions and geometric reason on paper is
based on doing and recording written and drawn steps, one at a time,
one after another, in an observable and thus confirmable or correctable
manner.
Errors are possible in arithmetic and geometry. When a student or
another checks the students work, that work is serving as visible proof
of the correct application of a skill or method, a proof that can be
checked for fullness and correctness. The work done and recorded should
show and demonstrate that figuring steps have been used properly.
Results should be repeatable and reproducible in routine applications
of mathematics methods, even before any acquaintanance with implication
rules A if B. The call to show work in school or in a business,
account keeping included, represents a common form of proof.
The students who does and records the steps of a calculation, one at
a time, one after another, carefully, without errors, is not only
proving skill but also providing a proof of the result. The recorded,
error-free work is proof if it provided by correct method even if the
composer of proof does not have a full comprehension of why the
method is correct. That is situation is likely to arise in arithmetic
where skill with arithmetic methods has more take home value than
full comprehension of why the methods work. Rigour and proof in skill
development is based on the observable mastery of skills, with steps
done and recorded in a way that gives and implies a repeatable,
reproducible and verifiable presentation and results. In that
comprehension of why methods give results is optional.
I. Routing and Non-Routine Problem Solving
Rigour in observable skill development and problem solving does not
refer to deductive logic, see the next item, it refers to observable
and thus hard results, done and recorded for the doer and others to
confirm or correct. In this the record or trace of figuring and drawing
steps implies correctness, modulo that of the underlying methodsd.
Steps done and recorded mechanically represent the underling rule and
pattern based reasoning - for better or worse.
Rigourous skill development begins with an awareness of the domino
effect of mistakes in following steps or instructions. That should
imply more careful work and study effort. Proof and problem solving
skill further includes approaching problems or puzzles with a
systematic or deliberate trial and error exploration of what might fit
or work. The combinatorial trial and error solution of jigsaws puzzles
with edges first provides an example.
Routine problem solving is largely mechanical in that one requires or
looks for existing rules and patterns directly and carefully. When a
problem does not appear to be routine, or when routine solution methods
are not satisfactory, the habit of always looking for pieces that fit
or methods that may work represents a creative, combinatorial and
opportunistic approach. Problem solving may ends with an awareness of
what is is not yet routine or solutions not found. That leaves room for
further thought. But problem solving may also end the careful or
diligence application of rules and patterns, the accomplishment and
record of steps that the solver or others may follow now to later to
check and confirm results. With work done and recorded in an observable
or repeatable and reproducible manner, solutions becomes rigourous.
Moreover, the paths they follow may make further problems of a like
kind, routine.
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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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