Mathematics and Logic - Skill and Concept Development

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.

1. 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.
   

2. 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.

3. 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.
   

4. 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).

5. 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.
   

6. 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.

7. 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.

8. 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.
   

9. 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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