Chapter 12. Four Phases

Education in mathematics and its logic or its rule and pattern based reason may be divided into four overlapping phases:

1. Elementary introduction: Pre-algebraic and pre-deductive with rule and pattern-based methods.
2. Starting the Transition: Algebraic and deductive thought introduced with more examples of rule and pattern based methods.
3. Continuing the Transition: Algebraic and deductive thought illustrated in many more examples or strands of reason.
4. Algebraic Codification: Algebraic and deductive derivation of mathematical knowledge from basic set theory axioms or more simply from assumptions about real numbers.

The aim of the first three phases is to broaden the common knowledge of math and logic. Here the curricula can take a path through easily described, repeated and mastered ideas. Ease of exposition and perhaps preparation for the fourth phase will be the guide. These phases are offered in support of an inductive philosophy for the communication of skills.

Elementary Instruction

The first phase is computational and rule-based. It ideally provides students with a mastery of arithmetic, counting and the use of simple formulas. It also provides them with the ability to recognize geometric shapes, employ or measure signed coordinates on a line and in the plane and employ or measure polar coordinates as well. The approach is inductive. The attention of students is drawn repeatedly to rules and patterns in many examples and situations. Computational and measurement skills are based on the mastery of methods with repeatable and reproducible results, exact or approximate. Mastery of such rules provides verifiable results and thus builds confidence – a secure knowledge of elementary mathematics.

Students may further learn about the approximation of linear measurements (temperature, distance, weight or masses) with decimal fractions, and the uncertainty in the last terms of an expansion (significant digits). They may also learn about infinite decimal expansion, repeating or not. Discussion of the latter provides a first sense of convergence. Numbers in the first instance are represented by finite or infinite decimal expansions. In this, discussion of the decimal number system provides the common thought-based understanding of this decimal representation. Powers of ten and their reciprocals can be introduced. The foregoing defines or introduces decimal notation for whole numbers, the denominators and numerators of fractions, and for decimal fractions.

The better and better approximation of the areas of regions by covering them with smaller and then smaller squares or rectangles, can be offered as a way to compute the areas. This covering process and the idea of a limiting value, the area, to provide a taste of calculus, albeit both students and teachers need not be aware that it is such a taste. Area estimation can be simply be presented as a measurement technique. From a technical perspective, it suggests to students that each region in the plane has an area, and this is the way to compute it. The thought that saying how to compute a quantity defines it can be expressed during this exposition of area estimation.

Simple formulas can be introduced for the calculation of perimeters, areas and volumes of planar or solid bodies and surfaces. Formulas can also be given for interest computations, simple or compound. Letters may appear here as shorthand for quantities that may be given, measured or computed. Calculations will involve units. The formulas may involve multiplication, addition and powers of both numbers and quantities.

Again, the first phase of mathematics is hands-on (manipulative). Both students and teachers may understand the applications and see how the repeatable and reproducible nature of arithmetic methods leads to verifiable results[1].

Set theoretic concepts (membership, union, intersection and complement) can be introduced here as well without too much emphasis on notation. Algebraic or symbolic shorthand has another role in the description of membership, inclusion, unions, intersections and complements.

The first phase is inductive – based on the recognition or identification of patterns to follow or watch for. The first phase provides students with a mastery of counting, arithmetic methods, and the use of simple formulas with or without units of measurement or quantity. Use of formulas begins the introduction of an algebraic skills – the symbolic description of calculations that might be done.

Starting the Transition

At the start of the second phase, students may expect to be given formulas or computational methods and data (number or quantities) to employ with them. Methods with repeatable, reproducible and therefore verifiable results, independent of whom obtains them, apart from approximations, are reassuring and confidence building. The confidence and secure knowledge thus attained can be retained and reinforced.

Cultivating Algebraic and Reasoning Skills

Once students have mastered counting, arithmetic and the use of simple formulas, they can be introduced (a) to the algebraic way of writing and thinking, and (b) to deductive logic. The average ages at which students are able to master the elements of (a) and (b), respectively, remain to be determined. But (a) and (b) together provide a foundation for the comprehension of the deductive exposition of mathematics.

The logic chapters common to the books Pattern Based Reason and Three Skills for Algebra introduce the main elements of deductive, that is, rule and pattern-based thought, with examples that are math-free. These examples can be gradually understood by most students from the ages of 11 to 16 say. They can be employed in any subject in which chains of reason or deductive thought is important.

The introduction of the algebraic way of writing and reasoning, based on the presentation and illustration of the three skills, was discussed in earlier chapters. The algebraic or symbolic way of writing and thinking is to be introduced and illustrated before and not while the arithmetic properties of real numbers etc are described in an algebraic fashion.

Arithmetic properties (axioms) indicate or say when two different calculations or formulas yield the same result. Deductive algebraic reasoning is based on the replacement of such formulas (descriptions of calculations) by one another or by a shorthand symbol that represents their common value or result.

Continuing the Transition

A purely deductive approach would not use the arithmetic methods met in primary school without deriving them from first principles or axioms. Of course, that derivation is too complicated for secondary school students, and should be reserved to math students in college – those interested in the full story. The immediate justification, via long chains of reasons, for operations already mastered may be of little immediate interest to secondary school students. The operations in question work – they give repeatable and reproducible results. The operations of decimal arithmetic fall in this category – justified, introduced or explained via examples and description in primary school. So they are not justified again in high school nor college courses. The justification of decimal arithmetic (based on mathematical induction) is a forgotten subject, of little interest today. The justification however of arithmetic operations could be an illustration of algebraic and deductive thought, and it would give experience with polynomial like manipulations of expansions in powers of 10 or some other base. It would further reinforce the command of arithmetic. _But the omission of any justification represents the first departure from the ideal of deriving conclusions from axioms in math classes. This is a precedent. And in view of it, other departures may be tolerated.

Secondary school mathematics after the second phase can be devoted to illustrating chains of algebraic and deductive thought in ways easily understood and repeatable by both students and teachers, especially teachers seconded from other subjects to present mathematics. Solutions of math problems consists of one or more chains of reasoning based on formal deduction, the drawing of diagrams and computation. The proof of a statement or theorem represents another chain of reason. The objective of the higher level math in secondary school can be limited to demonstrating to students how to follow or create chains of reasons, and thus justify a conclusion. The conclusion can be a numerical result or the correctness of a proposition. Cultivating in many the ability to follow chains of reason, here deductive thought, is more than important in the first instance than presenting a strict and rigorous perspective accessible only to the few. The few can see and study the more rigorous approach later[2].

Examples

The justification of previously mastered operations is not enough – many students may lose interest and the concern for it may appear to be legalistic. Deductive chains of reason should be employed in the derivation and justification of operations not previously met. The issue then is to show the value of long chains of reason through new examples, not old, albeit some students will be curious. They can be offered an enriched program, or be informed that later courses should satisfy their curiosity. Examples to explore follow.

1. In algebra, the exploration and justification of money computations (growth, geometric sums, mortgage and annuity computations – present or future value, finite math, combinatorics & probability computations) may provide further examples of practical chains of reason. The justification of some formulas, summation formulas for geometric and arithmetic sums for example, is based here on mathematical induction.
2. Nonanalytic/synthetic geometry in the plane and/or the theory of linear algebra (as distinct from the mastery of matrix computations) provide bodies (islands) of rule and pattern based thought, each connected internally by long and short paths or chains of one and two-way implication rules.
3. A preview of calculus, a discussion of why slopes, offers an informal and very physical chains of reason. This preview may be accompanied by an indication that the chains of reason are not strictly acceptable in pure mathematics or that physical arguments, while suggestive, are not reliable enough for use in pure mathematics. The preview offered here can provide motivation for the study of slopes in algebra courses.
4. Trigonometry is required by students wishing to retain the option of studying science, engineering or mathematics. And if its exposition is made simple enough [3], students heading in other directions may master some trigonometry as well. The complex number chapters in the companion book Why Slopes and More Math (or the earlier discussion) show or indicate how to add and multiply points or arrows in the plane, and thus introduce or define the complex numbers. The trigonometric derivation of formulas for real and imaginary parts of a product, in terms of those of the factors, gives an application of the cosine and sine addition formulas. But the multiplication idea of adding angles or rotating is also present in one unit circle triangle-rotation proof of these addition formulas. So after the introduction of the complex numbers via the addition and multiplication of points or arrows in the plane, the triangle rotation proof of the cosine addition formula can be given. A prior knowledge of the multiplication rule add the angles, multiplying the lengths makes the triangle rotation proof less unexpected. The combined explanation of trigonometry and complex numbers provides another example of a chain or chains of reason in mathematics.

Remark.The definition of trigonometric functions is dependent, in the secondary school exposition at least, on drawn or imagined triangles and on assumptions about the ratios of the lengths of sides of similar right triangles. Secondary school level trigonometry, and the trigonometry met in the typical first course on calculus, are not derived purely from assumptions (or axioms) about real numbers (the decimals say). There are additional assumptions that points, lines, circles and triangles we locate or draw, can all be represented in analytic geometry. These assumptions represent correspondences that need to be acknowledged.

Algebraic Codification

The operational command of mathematics provided by the first three phases just described may be sufficient for students of art, engineering, science and technology in their further studies of mathematics, if any, and other subjects. Comprehension of mathematics may initially stem from an exposition of informal or mixed chains of reason along with a cultivated and growing appreciation for rigour. The first three phases have the aim of illustrating and giving a command of arithmetic, counting, algebraic thought and deductive logic through a vast number of examples. Such examples may also provide the mathematical maturity for the fourth phase: understanding rigorous derivations of modern mathematics from axioms about real numbers and sets, if not geometric objects.

The logical (thought-based) codification of mathematical ideas and results within a set theoretic foundation is a technical endeavour. But the endeavour provides a single framework for the discussion and rule-based development of the arithmetic-oriented parts of pure and applied mathematics. Analytic geometry is included in this development by means of arithmetic based coordinates. The endeavour follows many long chains of reasoning from basic assumptions about sets to the set-theoretic (decimal-free) representation of real numbers. Further chains of reasoning yield complex numbers, analytic geometry, trigonometry and calculus from the real numbers, all in a diagram-free fashion. Diagrams can be employed to illustrate concepts or communicate ideas. While conclusions from physical concepts cannot be employed to obtain results in the codification, they do offer motivation.

Within the codification, geometric and physical concepts can be modeled. This requires some further assumptions of a physical nature to establish a correspondence within the analytic framework provided by the codification and reality. Such correspondences are present in the geometric illustrations of the codification (and even in its algebraic expression). Whatever is being illustrated is being modeled within the codification via some correspondence. The codification represents a mathematical universe, but there is more to mathematics than this. There are the correspondences that we assume in linking the codification with illustration or applications. Beyond this, there are the practical and sometimes uncodified algebra of the applied sciences. The rigorous communication of mathematics describes not only the codification but also acknowledges the extra correspondences – those needed to relate the symbolically or algebraically described concepts to geometric or physical interpretations.

The fourth phase is not for all. It should not be emphasized before students have an appreciation of deductive reason. It should not and cannot be emphasized before the completion of first courses in trigonometry and calculus that mix the assumptions made in both algebra and geometry – a reliance on diagrams.

More than one line of thought may be followed in math instruction. The first line of say the first three phases aims to extend the common knowledge of mathematics through the informal description of ideas and methods with repeatable and reproducible results and through the offering of short and longer chains of reason. The second and further lines of thought in the fourth phase, college level, could be the more and more deductive and rigorous derivation of mathematical results from axioms about real numbers or sets. Deductive strands of reason presented earlier could be linked together.

With the axioms about real numbers, for the benefit of those not majoring or specializing in mathematics, the explicit assumption that an infinite decimal expansion defines a real number should be included. An initial emphasis on the first three phases may allow more students and people to appreciate mathematics and logic in general and possibly the fourth phase, the modern or present-day axiomatic organization of mathematics, than linear and more direct exposition of the latter. That is the hope, thesis and conclusion. Criticism and refinement are welcome.

[2] Given the non-uniform comprehension of students graduating from diverse high school programs, instruction at the college level mathematics in North America simply hopes for a mastery of some logic and the algebraic way of writing and thinking. The latter may or may not follow from immersion in calculus.

[3] Arithmetic may fall in this category. In previous centuries mastery of arithmetic is regarded as a sign of intelligence. Now the mastery is common place.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science