Mathematics and Logic - Skill and Concept Development

Chapter 1. Introduction

Volume 1B, Mathematics Curriculum Notes

These pages describe the influence of rule and pattern based reason on mathematics itself and on mathematics instruction. Images of what is done and what has been tried before and why, provide a basis for further discussion. In particular, recommendations for math and logic education are first based on this analysis or description. They are justified again by describing two views of mathematics, the initial view met in primary school education and the last view typically met in college level service courses – those courses taught to people not specializing in mathematics.

In mathematics and in rule-based reason (or logic) there are some ideas which can be explained clearly and quickly, ideas with minimal requirements for comprehension. Collecting and putting such ideas first should make both disciplines easier and simpler to learn and teach. This yields a scheme for instruction from primary school to college. By putting the simplest ideas first, the common knowledge of both disciplines may be strengthened. The strengthening may further offer a context for the more formal deductive exposition and codification of mathematics as well as quantitative reasoning in other disciplines.

Two Barriers

The first barrier opposing math education has been the lack of words to introduce the algebraic or symbolic ways of writing and thinking.

One way of symbolic thinking is recorded and represented in arithmetic. Further ways are recorded and represented in the solution of a system of equation and in the comprehension of the properties of real numbers.

Algebraic reasoning, more so than arithmetic figuring, is better seen and written than expressed verbally. Many today find this visual and nonverbal aspect of algebraic thought intimidating and mystifying. Words have to explain this situation. The discussion and illustration at length of the three skills for algebra, one of which is prealgebraic or symbol-free can remove or lower this first barrier.

Alongside the first barrier, the second barrier opposing mathematics instruction stems from the most secure and attractive features of formal mathematics, its strict deductive hierarchy of concepts and assertions. With the strict hierarchy, comprehension of later terms and ideas requires a precise comprehension of previous ones. This has meant that the concepts described at the end of a mathematics course are incomprehensible to the student just starting that course, or even in the middle of that course. The hierarchy implies a student, even one who mastered all the concepts so far, has no inkling or vision of what comes next. Thus for students past and present, including many teachers, mathematical ideas beyond the last course passed or taken remain unseen and unfathomable.

The strict hierarchical description of mathematics provides a hard and rigorous route which only the most patient or the naturally adept follow far. This strict route leaves the image of mathematics and its logic incomplete and mysterious. Ease of exposition has not been the guide in higher mathematics. Yet ease of exposition can be the guide in extending the common knowledge of mathematics provided in elementary school before any rigorous deductive exposition begins. Following a command of arithmetic, counting and the use of simple formulas, there are wordy lessons which offer a simple command of algebraic thought, and further lessons which offer a math-free command of deductive reasoning: how to use rules and patterns one at a time, or one after another to arrive at conclusions or decisions. Such lessons offer strands of thought which are easily described and repeated. Putting them first provides a context in which more deductive accounts can be presented and discussed. In it, previously mastered strands of thought can be threaded or rethreaded together with more and more rigour.

Lowering or Removing Barriers

Mathematics courses from primary school to college consist of chains of reason, some quite long, but not all deductive. The mastery of elementary mathematical thought may be inductive, that is drawn from examples by the primary school teacher. Primary school students of mathematics can be given methods for computation and numbers to use in them, along with suggestive reasoning to introduce and sometimes justify the computational methods. All this may be contrasted with the algebraic and deductive rigour of formal mathematics met in advanced instruction.

Primary Instruction

Primary or elementary instruction now defines the common knowledge of mathematics. This instruction introduces and explains arithmetic from whole numbers to fractions and decimals in a pre-algebraic and pre-deductive fashion. Physical reasoning, groupings and analogies are employed to explain and justify ideas and methods. The counting and computational methods taught in primary instruction are repeatable, reproducible and thus verifiable. Arithmetic is rule based. From it, student may learn that steps must be taken carefully. Two different people with the same arithmetic expression to compute are expected to obtain the same result, apart from small round-off errors. Primary instruction provides a secure, often satisfying, rule and pattern based command of mathematics. The security is based on methods which are repeatable and reproducible, and thus verifiable, apart from any deductive account.

Primary instruction in preceding the deductive exposition of mathematics, provides the first image of the discipline and more generally, of rule based figuring and reasoning. The latter touches many subjects. Advanced mathematics instruction presently ignores and thus discards this first image. It derives mathematics from decimal-free axioms or assumptions about points, numbers and sets without any mention or consideration of the student’s first command of the discipline. The knowledge of mathematics and its pattern based reason is thus not cumulative. It presently starts once in a pre-deductive fashion and then again separately in a deductive fashion. This represents a gap and discontinuity in the exposition of mathematics.

Between advanced and primary instruction, intermediate instruction is assigned the task of providing a smooth and continuous transition. The task includes directly but gradually describing and explaining deductive logic and the algebraic way of writing and thinking to mathematical novices.

Intermediate level instruction may place first those ideas easily grasped to extend the common knowledge, and put second the technical details. Here the ideas put first can be selected to ease the comprehension of those put second. Those put first can also chosen to give a command of mathematics and logic valuable in itself just in case the deductive exposition and codification is not reached. This design may give students immediate satisfaction in their studies and thus provide an invitation to continue. This perspective puts the student, and not the discipline first.

One way to introduce the deductive thought process is through math-free lessons on implication rules (two logic puzzles), chains of reason, longer change of reason, islands and divisions of knowledge given in companion works to this one. This introduction can be given in mathematics courses or in reading, writing and composition courses. The explanation of deductive thinking, while required in mathematics, should be across the curriculum alongside and in support of writing across the curriculum. [2]

[2] This author has only one comment about writing across the curriculum. Just as some people are today math phobic, say mathematically challenged, some are essay phobic or challenged despite good intentions.

The common knowledge, to be extended and developed further in intermediate instruction after primary instruction, can continue to be built first on rule-based methods that are repeatable, reproducible and therefore secure or verifiable. For ease of exposition or to create a more easily described image of mathematics, physical and geometrical arguments may be employed to support or suggest conclusions while deductive strands of reason are introduced as well. Example, see below, can be provided by discussing and illustrating three skills for algebra, and by presenting expositions of complex numbers and why slopes. All communicate essential, if not abstract, ideas in mathematics while requiring a minimal mathematical background.

Units and Decimals

two missing links

Mathematics courses, besides preparing students for the deductive exposition of advance courses, should also provide students with the deductive and quantitative reasoning skills required in other subjects. Quantitative and algebraic reasoning in science, technology and commerce involves units of measurement or quantity and the decimal, if not binary, representation of numbers. The set theoretic axiomization of pure mathematics is free of both units and decimals. Primary and intermediate level courses can be assigned the further task of sanctioning the use of both units and decimals in the quantitative or algebraic reasoning of other subjects. Presently, the presentation of axioms for real numbers makes no mention and offers no sanction for the use of decimals or the use of units.

Primary instruction provides, one hopes, a thought-based command and investment in the decimal representation of numbers. The latter representation is indeed adequate for those who will never see in full the decimal-free set theoretic codification of modern mathematics. Until the presentation of the latter, the decimal representation is also adequate for students of mathematics. For continuity between primary and further courses, added to the axioms or assumptions about real numbers in intermediate level courses may be two assumptions, first that real numbers have decimal expansions, and second, that infinite decimal expansions define a real number. This reflects the common knowledge or belief. Some sanction for it should be provided in mathematics so that the common knowledge and axiomatic perspective do not need to be reconciled.

Indeed, the decimal concept of convergence, with or without set theoretic wrapping, is sufficient for students not meeting the decimal free alternative. [3] Its discussion, see the chapter Error Control, Continuity and Limit in the companion work Why Slopes and More Math, can further provide a background and a context for the understanding of the decimal free approach – part of the motivation or explanation why. Abstraction by itself, without concrete examples, provides the student a vacuous knowledge. The vacuum is abhorred.

Primary instruction in quantitative reasoning introduces units of quantity or measurement. Their absence in the algebraically described axioms for real numbers, and in intermediate courses apart from trigonometry, separate mathematics courses from the quantitative reasoning required in other courses. While the need for units of measurement can be circumvented by the dimensionless (unit-free) development of formulas and equations, that circumvention is a complication not needed in elementary mathematics. In the first instance, the algebraic way of writing and thinking can be employed with calculations involving decimals and units.

Axioms for real numbers can be augmented with more axioms or assumptions about quantitative and algebraic reasoning with units, monetary or physical. Here for instance mathematics courses could present the innermost axioms, those for real numbers and for their decimal expansion, while other courses in quantitative reasoning, if not those in mathematics, could add further axioms, those involving units. For students, this would yield an axiomatic framework for numerical computations within a broader axiomatic framework for quantitative reasoning. A consistent discussion of units would be welcome in mathematics or adjunct disciplines. [4]

With the insertion of assumptions about decimals and units along with the discussion of implications rules and three skills for algebra, mathematics instruction could reach for or even achieve the following goals:

1. A continuous extension in high school and college of the common knowledge acquired in primary schools of decimal arithmetic, counting and the use of simple formulas.

2. The formal and/or informal provision of a logical framework for computations in all mathematical disciplines with and without units.

3. The description and demonstration of a simple logical structure or codification for mathematics and quantitative reasoning in other disciplines.

The consideration and retention of both decimals and units in the exposition of mathematics would make it continuous and cumulative pedagogically from primary school to college. The common knowledge of calculation and logic, developed in primary and intermediate level courses, can be axiomatically codified in advanced courses, and provide a context for the discussion of the codification.

[3] The college book Calculus by Lipman Bers (Holt, Rinehart and Winston 1969) heads in this direction by talking about the decimal representation of numbers and other equivalent representations. The discussion of areas under a curve further begins with area approximation of a region based on its coverings by small squares. This approximation was taught in my primary school.

[4] Physicists and standards bureaus with their conventions on physical units may be of assistance here.

About The Next Chapters

The following chapters elaborate on the above ideas. Some repetition of the above ideas will be found. This introductory chapter and the forewords for this subwork, volume 1B, and complete work, volume 1, have previewed if not stolen the conclusions from the rest of this work. The rest of this work along with the companion works offers a simple frame for mathematics and logic instruction from primary school to college apart from the exposition of geometry. Curriculum committees will wish to add further topics to this frame.

The chapter 2, For and Against Mathematic s, indicates why people and not just mathematicians may interest themselves in the subject. No one reason can satisfy everyone. Reasons for student aversion to mathematics and scientific thought are noted.

The chapter 3, Algebraic Thought , describes the algebra barrier, its consequences in more detail, and offers words to lower or remove it. In brief, three skills, described with words and reinforced in examples, may introduce and explain the algebraic or symbolic way of writing and thinking clearly. Their discussion and illustration will further clarify as well two notions of a variable, one symbol-free. The mathematical adept are so accustomed to thinking in terms of symbols, that the pre-symbolic notion of a variable is often overlooked and taken for granted.

The chapter 4, Complex Numbers and Why Slopes (Calculus) offers two glimpses of mathematics. The first glimpse or example gives a simple exposition of complex numbers. Part of it motivates trigonometric reasoning and part of it, given say in early secondary or late primary instruction, defines multiplication so that the law of signs and the square root of (-1) both become clear and obvious to prealgebraic students – an immediate consequence of the product definition. The second glimpse previews the geometric interpretation of slopes in calculus. This example requires only a familiarity with the slope of straight line segment and the geometric significance of zero, positive and negative slopes. These two glimpses show how a minimal background is sufficient to understand significant strands of reason in mathematics.

The chapter 5, References , identifies four works which this author found useful and reassuring in the composition of this work. Given the scope of this work, I looked in the library for supporting and/or conflicting material. The ideas below are not in conflict with those I have seen in the literature. Further exploration of the math education literature is left to those employed in the field.

A chapter 6, Rule-Based Reason in Mathematics , describes the unruled and uncodified origins of mathematics apart from geometry. The algebraic and symbolic way of writing and reasoning was and still is, if done quickly, able to suggest more than can be proven. This chapter describes the advent of the deductive and axiomatic set theoretic foundation or codification for arithmetic based mathematics and the motivation for the advent. Geometry falls within the domain of this codification through coordinates. The next chapter says how.

Chapter 7, Two Treatments of Geometry , discusses and compares the older, ruler and compass oriented, synthetic treatment of Euclidean geometry, the synthetic treatment, with the newer analytic approach based on coordinates. Presence of two approaches, one older and one newer, gives at least two axiomatic developments of geometric knowledge – variants are possible. Both or all need to be recognized and reconciled in the exposition of geometry. That is, the correspondence between the two approaches should be discussed in class, else students are left with two un-reconciled axiomatic perspectives of geometry.

Postscript (2010): Chapters 1 to 6 describe barriers and remedies as seen in 1995-6. The remaining chapters explore ideas for remedies. However more effective remedies are available. See the ends and values for mathematics and logic in site pages. Clearer paths for skill and concept development imply a few changes or extras in the indicated remedies. But the foreword and chapters 1 to 6 provide background information.

Selby A, Volume 1B, Mathematics Curriculum Notes, 1996.

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