Phase 2:More Basic Skills with Take-home Value in 1 to 2 years
Number, Geometry and Formula Evaluation Skills
This phase may emphasize more counting, measuring and figuring skills and
practices with numbers, with measuring and drawing instruments, with
maps-plans-diagram drawn to scale. Now we include more and more formula
evaluation in that. In daily or adult life, there are many potential
activities in common practices with maps-plans-diagrams drawn to scale
before mastery of trigonometry may assist planning or making or maintaing
objects and structures with fabric, wood, paint, bricks, metal etc. We
also include the applications of map-plans-diagrams drawn to scale in
travel, surveying and navigation for the calculation of missing lengths
and measures, and for the description of contour lines.
Besides formula evaluation, we will not insist students be able to solve
equations numerically or algebraically. That will come later mainly
because I am not certain that site smaller and extra steps for algebra
skill development will work with all students. Where students are
streamed in accordance with ability, the algebra skill development steps
in the next could appear in this phase. Then instead of memorizing
equilavent formulas, only one need to be mastered because the rest are
easily derived.
The Oral Dimension
The oral dimension of mathematics has been missing or weak as arithmetic
expressions and formulas are often better seen and understood in silence,
the expressions and formulas being too difficult to read aloud, term by
term. Moreover, calculations may be described with words or formulas,
whichever is the easiest. The example the perimeter of a polygon and the
calculation of averages are both more easily introduced and described
with words and examples than they are with algebraic expressions. Some
verbal slogans to describe a calculation may be as easy or easier to
understand than the algebraic description of a calculation:
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The area of a rectangle is the product of the lengths of its sides.
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The volume of a box is the product of the lengths of its sides.
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The area of a triangle is one half the product of its base and its
height.
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The area of a parallelograms is the product of its base and its
height.
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The perimeter of a triangle, quadrilateral and further polygons is
the sum of the lengths of the sides.
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The surface area of a polyhedron is the sum of the areas of its
faces.
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The lateral surface area of a polyhedron is the sum of the areas of
its faces, apart from its base.
All depends on the examples at hand.
Talking about the Mona Lisa names a picture and may invoke a common image
in our minds. Likewise, many calculations or formula have names. Names
and descriptive phrases or slogans have power. At the elementary level,
we may speak of the circle perimeter calculation formula, a trapezoid
area computation formula; the distance computation for constant speed,
given time in conversation where writing these formulas is less
convenient. Students of higher mathematics will hear and see simple
interest, compound interest and the quadratic formula. The latter are
more easily mentioned that read aloud term by term. Naming well-known
formulas allows them to dicuss over the phone or in preliminary
discussions of what calculation to do. Naming formulas or rules and
patterns for use in communicaton is part of the oral dimension.
The oral dimension may be extended further. By verbally introducing
students to counting, adding and multiplying by adding and/or multiplying
subcounts, subtotals and subtotals, we may verbally describe and extend
mastery of arithmetic and number theory. Mastery of decimals, fraction
operations and prime factorization will included here - the prime
factorization being included for the sake of efficient and exact
arithmetic with whole numbers and fractions.
Without waiting for mastery of algebra, students may be shown how to
count, add and multiply by forming and adding or multiplying subcounts,
subtotals and subproducts. With words, we may emphasize count
computation by forming and adding subcounts, sum or total computation
by by forming and adding subsums or subtotals with unsigned and signed
amounts; and the calculation of products by forming and multiplingly
subproducts. The latter will be useful in prime factorization, a must
or plus for efficient arithmetic with small denominator fractions.
The orally introduced arithmetic skills or practices here steal some of
the thunder from the later algebraic statement of axioms for real
numbers but are consistent with the latter. Courses in higher
mathematics may show how our orally introduced arithmetic practices are
consequences of the axioms. However, our first aim in this framework
for mathematics education is to provide skills and practices with
take-home value.
Redundant Formulas - Equivalent Formulas
For student for whom algebra mastery may be difficult, we may give
formulas that later studies will show or imply be equivalent because each
formula has a take-home. In this phase, we may be satisfied with
numerical consistency of equivalent formulas - each may serve as a check
on its partners. Equivalent and consistent arise in several routine
situation with actual or likely value for adult and daily life.
In the study of say distance, time and speed, formulas for each quantity
will be given for the sake of take-home value. In the study of money
matters, equivalent or redundant formulas may be given for compound
interest and growth, and for loan, mortgage and even pension plans. Then
in the later development of algebra skills, how each of the formulas
leads to the others will covered. The aim is this part is to provide
mastery of computation. That may include giving equivalent formulas -
formulas that imply each other algebraically. That being said, teachers
and tutors may experiment with students individually or group to see if
they can understand the algebra needed to show equivalence. If so, the
presentation of equivalent formulas may go after that mastery. With
skills which serves a common needs, a sound empirical command in a
repeatable and reproducible way is more important than comprehension of
why practices work.
In multi-step written work, doing and recording steps will make the
domino effects of care and mistake clear. With that being carefull and
avoiding mistakes will become an end, value and methods for skill use and
development in many arts and disciplines at home, at school and in
work.> Lean and effective formats for doing and recording written work
on paper, formula evaluation included, will make skill mastery evident in
steps that can be seen and checked as done or later. That provides a
trail of work if not reason to see and check in primary and early
secondary mathematics.
Higher mathematics will widen the steps by showing students how to
include reasons for the steps. The result will be derivations or proofs
that may seen and checked as well.
Arithmetic with Units - Denominate Numbers
A quantity denominate number is a multiple of a unit of measure.
The site section on arithemtic or fraction with units serves the common
needs for the pre-algebraic mastery of arithmetic with units, that is
denominate numbers, with rates or unit costs or speed.
In the later algebraic, forward and backward analysis of proportionality
relations, mastery of arithmetic with units helps with the calculation
and use of proportionality constants. While pure mathematics in college
tends to avoid working with units in calculation, common skills for daily
life and common skills in the later study of chemistry and physics
require it. Thus an operational command of arithmetic and fractions with
units needs to be developed if not for pure mathematics, then for for
adult and daily life, and cross-curricula value in science.
Algebra skill development as seen today or recently, has students
multiply and divide monomials in one to several variables, even before
the concept of what is a variable is understood. Similar skills is
required in arithmetic involving with units. The latter has take-home for
mathematics mastery sans and with algebra as indicated above.
Implementation Notes
In this phase or before, students should see and master decimal methods,
five operations with fractions - efficiently; and the use of primes and
prime factorization to aid the latter.
In modern times, counting-measuring-and-figuring skills and practices
with numbers, maps-plans-diagrams drawn to scale and even formulas are
very much present in time and date matters, money matters, likelihood
matters, and so on. Numbers are also present in signs and addresses. In
daily and adult life, parents and teachers may identify common or routine
activities and problems where numbers or geometric know-how is required.
Those common or routine activities and problems need to be seen and
discuss in skill development. So the skill development advocated in
primary school and early secondary school phases are open ended. As a
rule of thumb, skill development is free to include any and all
mathematical and logical routines with value for adult or daily life or
value for motivating skill development. Playing games online and off may
serve to build skills and confidence in light to challenging, but
entertaining ways.
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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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