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Chapter 8
Modern Math Instruction

A critique of modern mathematics curricula follows.

Oct 1, 2006, Postscript: Times have moved on. The modern mathematics curricula described below is part of the past of modern mathematics instruction. When I was writing this in 1995-96, I did not know to what extent modern mathematics curricula of the late 1950s to 1970s were being followed, or been supplanted.  Volume 1B in retrospect might be entittled, Rip Van Winkle reflects on mathematics education. What is modern changes with time.

Description and Analysis

An axiomatic codification of mathematics is provided by the Zermelo-Fermelo treatment of numbers and sets. Initially this treatment was just another way of looking at and organizing mathematics. From the 1920s to 1950s, recognition grew that it provided a more certain and penetrating framework for mathematical thought, or at least a rigorous codification.

The 1950s and early 1960s modern movements in mathematics instruction introduced into high schools and colleges simplified forms of the decimal-free axiomatic codification and thought-based foundation for mathematics [1].

The movements emphasized comprehension: the derivation of ideas from first principles, that is assumptions. Why mathematical assertions and formulas hold, that is, their derivation from given assumptions, was considered just as important as their use and statement. The movement represented and continued the hope in mathematics that advanced topics would be taught earlier and earlier in high schools and colleges.

Observations

As a high school student in the 1967-1969 period, I saw the claim that mathematics could be derived from a minimal set of assumptions. I found this near-certain derivation of conclusions from first principles, combined with the idea that mathematics would be useful in the mastery of other subjects, to be most appealing. My high school and early college physic courses also emphasized or echoed this appealing idea of derivation from first principles.

Within the codification, as indicated above, the decimal representation of numbers is not special and not required. It is absent. The simplification was faithful to original codification. It too remained decimal-free. Yet in schools, the exclusion of any link with the common decimal knowledge of arithmetic deprives the codification and students of a reasoning tool. This separates the explanation (and the formal comprehension) of mathematics from the common knowledge or use of decimal arithmetic. My experience in this matter follows.

First, primary school taught how to count, do arithmetic and use simple formulas. Then in high school came the codification, or axiomatic approach in a simplified form. The axiomatic approach (with its reliance on logic and the algebraic way of writing and thinking) offered rules for real numbers. But of decimals, the rules or axioms made no mention and offered no sanction. This was a source of logical distress. Given the emphasis on the logical derivation from axioms or assumptions, I wondered when my previous knowledge of decimal-based counting and arithmetic would be explicitly sanctioned. It was not[2].

Second, my high school science courses all employed decimal arithmetic and units of measurement as well, again without any formal sanction in math courses. The role of units in computations, a subject of interest in the business, geometric and physical computations, was ignored. Thus mathematics was further separated from the earlier acquired common knowledge and from the computational requirements of other disciplines.

Third, many college students, including myself, found the decimal-free discussion of limits, convergence and continuity non-intuitive. The underlying ideas appeared to be remote from comprehension, technically understandable perhaps for a brief moment, and at first not readily remembered [5]. But in retrospect, a decimal (significant digit) perspective of unlimited error control for computations or function evaluations gives a simple context for limits, convergence and continuity.

The simplified form of the codification met in modern math curriculums have been too faithful (due in the first instance to a rigorous adherence which later became an unquestioned tradition) to the decimal-free aspect of the codification at the expense of complicating the exposition of modern mathematics. The expense was incurred in both high schools and early college math courses. The decimal-free emphasis separated the codification in its original and simplified forms from the common decimal-based knowledge of arithmetic obtained in primary school.

Recommendations

For ease of exposition and a wider comprehension of mathematics and logic a departure may be warranted in the high school and college axiomatic development or codification of mathematics. In particular, assumptions about the decimal representation of real numbers, and assumptions about their convergence could be included. This would sanction decimal expansions and arithmetic along with the mature knowledge of convergence tacit in it. The initial explanation and description of decimals is a sufficient representation of real numbers for students not immediately specializing in mathematics. Further, in the exposition of mathematics, rules or axioms for the treatment of units in computation could be included (a) to link the exposition in mathematics with convenient practices in other disciplines and (b) to provide an explicit logical (thought-based) sanction for them.

The primary or junior high school description of the decimal arithmetic and the decimal representation of numbers, with positions hundreds, tens, units, tenths, one-hundredths, etc., is an inductive thought-based process. The decimal-based perspective is ample for the common knowledge of mathematics and for most college students and people not specializing in mathematics – all those who will not see the more rigorous perspective presently favoured.

Elementary courses, even though they be remote from the axioms or properties of real numbers and the modern set theoretic codification of mathematics, provide a thought-based framework for counting and decimal arithmetic. Here a student should understand the positional decimal-based representation of whole numbers by themselves or in numerators and denominators of fractions and in the decimal expansion of rationals and irrationals. The concepts of a > b, a < b or a = b for whole numbers are initially understood in primary instruction from comparison of decimal expansions, and not how the real line is ordered. (When the latter is introduced, comparison by magnitude and by the linear ordering of the real line need to be compared and contrasted. See below.)

There is an intellectual investment in the decimal representation of numbers. The notions in it should be respected and strengthened, and no concept be discarded or replaced until a student is positioned to understand the alternative, why it is introduced and, for the sake of rigour, its equivalence to the original perspective.

The decimal-free perspective of real numbers should be postponed to advanced lessons where it studied by a few and linked to the previously taught decimal-base perspective. There the transition from the decimal perspective to the decimal-free can be designed to give a context and motivation for the decimal-free perspectives of calculus and modern math.

Axioms for real numbers that do not mention the decimal representation or provide an alternative accessible to students leave a vacuum. Axioms for real numbers which do not mention or are not reconciled with the primary school perspective, provide a second separate perspective. Modern math instruction has split the discipline into two, and not reconciled the parts. Here the parts are the decimal based common knowledge and the deductive exposition or derivation from decimal-free axioms.

Assumptions about decimals together with assumptions about sets, along with the emphasis of some restrictions on their formation, could provide a simpler, more accessible, but still precise decimal-based image of the codification. From this precise image, the decimal-free codification is but a small step away. Again, the latter codification, in more detail or in full, could be inserted in university courses to students concentrating in mathematics and learning about the set theoretic foundations for real numbers and arithmetic. Colleges students who make a finer study of mathematics also can be shown or be given a reference for the derivation of decimal arithmetic from basic assumptions about sets. This would represent extra work for the few students who specialize in mathematics, but it would ease the earlier exposition of mathematics for most others, teachers and students included. In the high school classroom, mention that other bases could have been used would imply the more general viewpoint to receptive ears that the decimal perspective is convenient, but not special.

[2] The strict set-theoretic derivation of the decimal representation of numbers is too long or complicated for a beginning student to appreciate or follow – and it would not add much to the development of his or her mathematical knowledge and reasoning power.

[3] Exception: adding units of different kinds is of interest in some special situations. For instance in taking a census, the population and property of a small village could be represented by the expression 176 persons + 10 dogs + 30 bicycles + 40 houses + 50000 square meters of floor space. The plus sign here represents the word and. Further exceptions may be found in the programming language C and possibly in some mathematical software which allow algebraic structures and not just numbers in their computations. Also note that the requirement that terms added together have the same dimensions or degree in units provide a error check for students of physics and chemistry, if not mathematics, in the formulation and manipulation of formulas and equations. As a student prone to nodding off in class, I remember following in part a long derivation in calculus and then realizing that the dimensions or units in two adjacent terms did not agree, and thus the calculation was false. When I reported this flaw, this old concept of checking the units was new to many in the math class and possibly the instructor RR. None the less, a joint effort found the error, a typo.

[4] From the strict mathematical perspective, units are just indeterminates or formal polynomial variables with inverses over a number field.

[5] Even Prof. E. McSquared’s original, fantastic & highly edifying, Calculus Primer, by Swann and Johnson, ISBN 9-913232-17-3, William Kaufman Inc, 1975, with its comic strip explanation of the decimal-free approach attempted to lessen this difficulty.

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