1 Chapter Guide

YOU are better than YOU think. Show yourself how:

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In mathematics, sooner or later you need to learn to read like a lawyer. For that read logic chapters 1 to 5 in Three Skills for Algebra. Sooner is better. Good luck.

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On the phone with a classmate or tutor, your studentscould
or twiddla to write & draw with each other on art, math & science etc.

Logic Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying, Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

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Caution: Site advice is approximately correct, for some circumstances, not all. Site How-TOs are logically developed, but not tried and tested. That leaves room for thought and refinement..

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After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;

For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Chapter 1.
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.

Analysis

The chapter 2, For and Against Mathematics, 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 5, 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.

The chapter 6, 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.

two unreconciled axiomatic perspectives of geometry

Linkage of synthetic and analytic approaches bring to the fore correspondences that need to be assumed and recognized in the discussion of geometry or the diagrams that are drawn in both analytic and synthetic geometry. These correspondences are not part of the formal exposition but they are part of the interpretation. Acknowledgement of these correspondences becomes part of the context for mathematics, a further codification linking set or arithmetic concepts to quantitative reasoning in geometry and other disciplines. Besides the inner arithmetic oriented codification and axiomization of mathematics, further outer codifications are present and needs to be taught.

The foregoing represents or suggests the systematic linkage of arithmetic concepts to quantitative considerations in other subjects. For instance, some physics courses and texts add to the pure development or codification of mathematical thought an axiomatic treatment of mechanics and also of units. Quantitative reason may have a layered codification in which the innermost one is the set theoretic framework of arithmetic based mathematics.

See Chapter 7 to learn more.

Synthesis

The chapter 8, Modern Mathematics Instruction, describes how this author met a modern mathematics curriculum in the late 1960s and makes observations about mathematics instruction which support the recommendations given in this work. Further support for the recommendations is given in the next chapter.

The chapter 9, The Two Ends, describes primary mathematics instruction and college level mathematics service courses. For most people entering college, this represents the start and finish, the two ends, of their math education. Observations here support earlier instructional thoughts.

A long chapter 10, The Transition, details how intermediate level courses may provide a smooth transition between the two ends. This chapter offers a program to develop algebraic and deductive thought apart from geometry. Again, teachers or curriculum committees may think of further topics to add or to refine the proposed core of this program. See the companion books or their table of contents.

The long chapter 11, Elementary Instruction, describes its subjects prealgebraic and predeductive, yet thought based, nature. This chapter describes how the common knowledge of counting, arithmetic and simple formulas might be cultivated or taught to a young child in predeductive fashion. Included at the end of this discussion is a recommendation. Complex numbers can be mastered via a simple operational approach. The approach is based on the addition and multiplication of points in the plane using rectangular and/or polar coordinates. There is a context here for the discussion of negative numbers and their square roots.

The chapter 12, Four Phases, describes a four stage development of skills, one suggested or implied by the previous chapters. The aim of the first three stages is to broaden the common knowledge of math and logic. In them, ease of exposition, preparation for the fourth phase, and preparation for quantitative reasoning in other subjects will be the guide. This work for the most part is dedicated to the first three phases: how to extend the common knowledge of TCPIT (the common person in the street). Implementation of the fourth phase is left to college level courses in mathematics.

Postscipt:: (October 1st, 2006): I have to re-read Chapters 8 to 12 above to see how my views may have changed. If so, I will include some further postscripts in the above chapters (online version only in the first instance).

Chapter 1. About The Next Chapters

Analysis

two unreconciled axiomatic perspectives of geometry

Synthesis

Chapter 1.
About The Next Chapters