Extending Arithmetic Skills Orally
The rules wordily given
reflect and extend common knowledge and common
practices in counting, figuring
and measuring. Technically, mathematics education needs to recognize these rules
to sanction them and to build on them.
Counts are independent of the order in which elements of a
(finite) set are counted. In practice, people may
count twice in the hope of detecting mistakes. Recounting is
required when earlier counts disagree. This pattern is met
and assumed in learning to count before high school.
Counts may be obtained by forming and adding non-overlapping
subcounts in any order. In practice, addition of
whole numbers may be introduced as a counting shortcut. People may add
twice in the hope of detecting mistakes. Disagreement between
additions will require a recount - here another addition.
Counting by addition is the practiced in elections where polling
stations report their totals for inclusion in larger totals.
Measures of lengths, areas and volumes may be obtained by
forming and adding non-overlapping sub-measurements -
measurement (counting how many units and then how many fractions
of units) is a form of counting. Counting by addition justifies
measurement by addition.
In accounting, money may be counted may be obtained by
forming and adding non-overlapping subtotals. In
metric or decimal based, money is counted in terms of whole units
(dollars, pounds, francs, Yen and so on) and in terms of one hundredths
of those units (cents and pennies).
In arithmetic, sums of whole numbers, fractions and decimals
may also be obtained by forming and adding non-overlapping subtotals.
If one looks carefully enough, this practice or principles
is another consequence of the ability to count wholes by
addition.
In accounting, sums of revenue (positive amounts) and costs
(negative amounts) may be obtained by forming and adding
non-overlapping subtotals. The subtotals themselves may
be zero or signed. Here adding twice is a process
to check for mistakes. Students may be told that the
uniqueness of the total means in daily live that the sum of assets and
debts can be made more nor less by subtotaling in different ways. That
knowledge has take home value. This property may be
presented as special case of the next, or it may be employed to
introduced the next.
In arithmetic with signed numbers (integers and then rationals,
sums integers and rationals may be obtained by forming and adding
non-overlapping subtotals. Note when students use both
signed coordinates and arrows to represent movements on a map or plan,
the observation that the head to tail addition of arrows in sequence
may be done by subtotalling and that addition is commutative informally
implies most of this property and the previous one with signed
numbers.
In Arithmetic, Products of numbers may also be obtained forming
and multiplying non-overlapping subproducts. This rule
find application when students are shown how to group like primes in
the the prime factorization of whole numbers. It has also has
application in the discussion of decimals - the optional explanation of
how or why decimal methods for multiplication work.
Extension: This rule also applies to
division as division by a number is replaced by or identified with
multiplication by its reciprocal in the case of fractions and a
multiplicative inverse.
In arithmetic, products of nonzero numbers are nonzero, but if
one or more factors is zero, the associated product is zero.
This rule may answer questions about whether or not a
product is zero. This rules may be used to speed the calculation
of a product in which one of the factors, one given by the value of an
expression, happens to have the value zero. The
rule may be implied by observations about how and why the product of
two nonzero counts or two nonzero lengths cannot be zero.
The first two of the last three rules or patterns imply that sums and
products of terms and factors may calculated and grouped (carefully) in
different ways even before algebra begins.
The distributive property of arithmetic with real numbers is introduced
elsewhere with the aid of geometry and coupled with a column methods for
calculating products of sums. See using geometry in
algebra. Mastery of the algebraic form of properties of
real numbers etc may be left to courses in pure mathematics. The
modern mathematics course designs seen in my student days
emphasized the algebraic form of arithmetic properties, but presented
as axioms for real numbers etc. The use of algebra in that manner
raise the level of complexity beyond the level of many
students and teachers. Furthermore, in the senior high school
development of mathematics, the necessary extensions to
aid if not justify polynomial addition, subtraction, multiplication
were indicated orally and not written algebraically. Thus the
use of oral rules to justify if not explain has been part
of secondary mathematics previously, that being for ease
of exposition.
With the expansion of the role of words before and
in algebra to expand and enrich the common know-how or knowledge in
mathematics, Upper high school mathematics and calculus instruction
may have balance the verbal and algebraic description of arithmetic
properties of numbers, whole to real or complex. Course design
and delivery will have to adjust.
Words before Symbols: In the first instance, the
use of letters in formulas to denote lengths or amounts stems from their
shorthand role in providing a more compact description of a calculation.
But that shorthand role of letters and symbols has limitations. For
example, calculation of the perimeters of a triangle, quadrilateral and
polygons in general may be simply given by the instruction: add the
lengths of the sides. Before the introduction of algebra, that
instruction can be understood and followed. In contrast, the algebraic
description of the calculation of these perimeters introduces many
letters and symbols, alone or with subscripts, and in doing so raises the
level of complexity. That introduction of letters and symbols is has a
role in the introduction of algebra but when the aim to show how to
compute perimeters, algebraic expressions for perimeters are not needed.
Notes
Before algebra begins, words may be also used to say when different
counting or arithmetic methods lead to the same result. Here again the
words may be simpler to understand and follow than the corresponding and
far more complicated algebraic descriptions of the same mathematical
rules or patterns. In particular, the following sequence of phrases
describe common practices in primary and secondary mathematics more
easily explained and understood with words. In mathematics lessons
given by teachers not fully versed in algebra, the use of words in place
of symbols makes instruction simpler with little or no loss of content
and rigour.
Formulas may be given for the
perimeters of regular polygons from equilateral triangles, squares
and pentagons to equalateral n-gons. Formulas may be given for
perimeters of triangles, quadrilaterals, and multi-sided polygons in
general. But the level of complexity in learning and teaching how to
calculate perimeters may be lowered by describing perimeter
calculations with words. In general, the instruction Add the
lengths of the sides or more briefly Add the side lengths
has lower complexity in learning and teaching than denoting each side
by a letter alone or with adornments (subscripts say), and giving a
formula. In the case of equilateral triangles and equilateral
polygons, the perimeter p =n × L is given by multiplying the number
of sides n by the common side length L. That is not very complex. But
in general, words may be included with the common knowledge of
arithmetic and its applications. Without raising the level of
complexity, common and useful properties, patterns and calculations
may be understood and mastered with words and simple phrases or
slogans in place of letters and symbols. There are arithmetic
practices with take home value, practices that are easily described
with words. Adding words to arithmetic sets the stage for adding
words to algebra. Both additions make the hard easier, proof being in
the details.
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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.
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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
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justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
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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.
-
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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