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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.
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.
Explanations of why these inter-related practices work - all or some can
assumed as axioms and those not assumed may explained in terms of the
others.
Oral Rules for Arithmetic
Each rule or pattern in below has value in its own right. But the earlier
ones lead to the last two. Moreover, these rules or patterns may
developed and understood verbally before any algebraic description of
properties of numbers, whole to real. The rules wordily given and
explained reflect and extend the common know-how in ways that may have
take-home value.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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