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Chapter 8
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students how to carry units through computations – at least one of
mine did. The carrying operation removes the need to convert all the
quantities involved to a single system of units. The conversion of units
can be done before, during or at the end of the calculations. Retention
of units in some form lessens the conceptual burden in performing the
calculations. By carrying units through a computation in an algebraic or
mechanical fashion, students do not have to think immediately about what
systems of units they are using nor do they have to think about any
unit-free formulation. Examples of units, or their ratios, are given by
the everyday use of terms like miles or kilometers per hour, or dollars
per pound or kilogram in science, technology and business. Units of
currency are related by time-dependent constants.
Algebraic properties of arithmetic operations not only apply to real numbers but also to real quantities – a real number times a unit of measurement. The algebraic manipulation of units is similar to that of monomial terms in the manipulations of polynomials and rational functions with an exception or restriction: addition requires that the monomials terms added have the same degrees in the units present. [3] The present formal manipulation of polynomials in high school algebra courses provides the necessary background[4].
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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.
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.
[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.
www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,Foreword + Chapters 1 to 10 + 12
Area Map Inductive Principles 1 Introduction 2 For & Against Math 3 Algebraic Thought 4 Why Slopes & SQRT of -1 5 Books & Articles to Read 6 Unruly Origins of Algebra 6. Axiomatic Civilization 7 Geometry, 2 Ways 8 Modern Instruction 9 The Two Ends 10 The Transition 10 Explaining Logic 10 Explaining Algebra 10 Why Sets in Math. 12 Four Phases Links
Chapter 11: Primary School Mathematics
11 Primary Math 11 Cue Cards 11 Counting 11 Decimals - Addition 11 Decimals -Times 11 Decimals & Subtraction 11 Fractions and Division 11 Notational Conflict 11 Reciprocals Etc 11 Decimals - Ratios 11 Size Comparison 11 Numbers, +ve or -ve 11 Rename < Sign 11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's Area: Help Your Child or Teen LearnMost students in high school are not heading for calculus, but most topics in high school mathematics are present due to calculus. Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math Education Essays. Site areas and pages provide pieces of the a Mathematics Education, Jigsaw Puzzle, in formation.
-Inductive principles for course design & delivery require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for what was missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.
Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.
Modern mathematics curricula introduced an inconsistency into course design and delivery. They did not sanction the use of decimals nor the use of diagrams in skill and concept development but decimal arithmetic and diagrams are needed for student comprehension and for an operational mastery of quantitative skills. That implies the need for an mixed-math curricula based on a systematic development of operational skills, sufficient for applications and sufficient to provide a base & context for the very optional study of pure mathematis.
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