Chapter 9
The Two Ends
Secondary or intermediate mathematics instruction should provide a smooth transition or bridge between the start and finish of this instruction. For most who attend colleges, the start is met in elementary or primary school and represented and the finish is met in college service courses.
College Service Courses
College service courses refer to those courses taught to students not specializing in mathematics, usually the majority of students in a college. Calculus, often the lowest level of mathematics taught for credit in a college exposes students to algebraic thought at its full strength. Miscomprehension of the algebraic way of writing and thinking usually leads to failure or intellectual hardship if not in a first, then in a subsequent calculus course. Calculus instructors would be most content if their students had previously mastered logical reasoning and the symbolic or algebraic ways of writing and thinking along with some trigonometry and optionally geometry before entering calculus [1]. College faculties today in their exposition of mathematics start almost from scratch [2].
College level instruction in math, commerce, engineering, science and technology may assume or build on a knowledge of logic, of basic trigonometry, and of the symbolic and algebraic way of writing and thinking. These skills are required not only for the exposition and mastery of calculus but also for computational methods in various disciplines and more advanced mathematics courses (analysis, abstract algebra, differential equations, numerical methods, differential geometry).
In college math courses presently taught in North America, there is presently little emphasis on set theory or derivation of mathematics from first principles. Algebraic and computational skills are emphasized and graded instead. Science, engineering and commerce departments want mathematics courses to prepare their students for computation and not the set theoretic codification of modern mathematics.
Only students specializing in mathematical disciplines [3] require or employ the set notions of membership, unions, intersections, ordered pairs and complements. But for students in disciplines not concerned with the set theoretic codification, wrapping concepts in set theoretic terms may be a distraction. The wrapping should only employed when it clarifies the exposition, or provides a second perspective.
Elementary Courses
Primary or early secondary school instruction has say the role of providing a knowledge of decimal arithmetic, counting and the use of simple formulas. This instruction leads to a mastery of some rule and pattern reasoning or figuring and a familiarity with repeatable, reproducible processes and thus their verifiable results. This rule and pattern reasoning appears before the use of implication rules that also establish conclusions in a repeatable, reproducible and thus verifiable manner.
The demands on primary instruction are described next. Note the innovation, the discussion of rectangular and polar coordinates, and then associated discussion of navigation and complex numbers. More comments or details will be given later.
1. Counting, weights and measures.
2. The conversion units of measurement.
3. Formulas for perimeters, areas, volumes and interest (simple or compound) along with illustrations. These formulas show how shorthand notation describes calculations that may or might be done or postponed.4. Decimal positional notation for whole numbers and then denominators and numerators of fractions. Arithmetic computations should be done by hand and explained in such a way that students understand or see from examples, their meaning and justification. The explanation of powers of 10 and perhaps scientific notation for numbers is part of the comprehension of decimal positional notation.
5. Repeating and non-repeating decimal expansions for fractions (rational numbers) and for irrational numbers such as Ö2 and p.. The presumed convergence of these decimal expansions. The physical notion that each decimal place serves to better locate a point on the real number scale. The presumed correspondence with numbers and a coordinate axes is exploited here.6. The need for care in arithmetic and the objective nature of arithmetic. Arithmetic in primary school should make students aware that a single false step in a calculation cast doubts on the results of all the following steps. Arithmetic in primary school should also lead to the expectation of objectivity. Results obtained exactly should be independent of the computer, here a student with a pencil and some paper.
7. Round-off problems in calculations, inaccuracy in measurements, and the number of significant digits in decimal expansion. There should be an uncertainty of at most half a unit in the last retained decimal place, if the accuracy range is not otherwise indicated.8. Measurement of regular and irregular areas by covering them with triangles, squares or rectangles, and then summing. This approximation of area, sometimes exact, represents a practical skill. When a region is covered by small squares, the convergence of inner and outer approximations as the squares are made smaller could be illustrated. (The outer approximation is the sum of the areas of squares in the covering which intersect the region. The inner approximation is the sum of the areas of the squares in the covering which are included fully in the region.) The approximation discussed here is also a foretaste of approximation, convergence and summation in mathematics after arithmetic – an example which can be later recalled in the explanation of integral calculus, so that the later explanation of mathematics coheres with and is not disjoint from the earlier description.
9. Coordinates on the line. Students only familiar with unsigned decimal numbers can be introduced to signed numbers as a means to signal a position on one side or another of the origin of a line. The height of water above and below a zero-level mark provides a first example. Temperature measurement in Fahrenheit and Celsius provide two further examples and show again that the choice of the origin may be arbitrary. Moreover, addition and subtraction of positive numbers can be identified with displacements in the positive or negative direction respectively. Similarly, addition and subtraction of negative numbers can be identified with displacements in the negative or positive direction.10. Cartesian and Polar Coordinates, their use in locating points and their measurement on maps. Here elements of navigation could be introduced. Students could be given a map, a starting location for a boat or airplane, and then asked to track the location of the latter through a sequence of displacements. The latter could be described with numbers by coordinates shifts or angles and lengths. They can also be described by arrows or vectors drawn on a map to represent a sequence of motions. This leads to the (map) addition of arrows or vectors. Beyond this, the multiplication of arrows or points in the plane can be easily defined using the add the angles, multiply the lengths rule for complex numbers. The latter in a pre-algebraic fashion will justify the law of signs and allow square roots of negative numbers to be identified. Ease of exposition and visualization is the justification for this last innovation. It demystifies negative and complex numbers even before algebra is studied.
At the end of primary school, or at the start of secondary school, students could be shown the mechanics of buying and selling. They could play games which in the simplest case involve transactions between a customer and retailer, and in the more complicated case between retailer and wholesaler, and wholesaler and suppler. Keeping track of the discounts, accounts and methods of payments would be an exercise in a flexible rule-based reasoning process. Here students could be challenged and required to do arithmetic without a calculator — tell them to imagine a power failure. Questions of how to verify results could be done. Those students required to master the most complicated computations and transactions would be favoured or better prepared than students shown only the simplest material[4]. Mastery of decimal arithmetic first became popular with merchants as a means of doing and recording transactions and balances. This provides another setting for the discussion or introduction of positive and negative numbers or balances. The above topics and others are discussed in the chapter Elementary Instruction.
[2] Many colleges have remedial programs for arithmetic and algebra skills alongside more advanced, but none the less secondary school level, pre-calculus courses in algebra or trigonometry.
[3] For instance: probability and statistics, mathematics itself, electrical engineering, theoretical physics computer programming or database organization, ... .
[4] In retrospect, the chapter should be callled Symbolically or Algebraically Described Rules for Arithmetic.
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