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www.whyslopes.com > Volume 1B, Mathematics Curriculum Notes, 1996 > Book Entrance | ||||
Book (Folder) ContentsThe Foreword provides inductive principles and standards for course design and delivery. Those inductive principles - support of them - guided book writing and site expansion. The first chapter introduces the main ideas in the rest of this work and describes gaps, flaws or inconsistencies in the exposition of the subject. The chapter 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 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. Skip on First Reading: The chapter Complex Numbers and Why Slopes 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 pre-algebraic 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. (Postscript: Where rote learning is the rule, introducing complex numbers as indicated in site pages would give an efficient ways to introduce trigonometry while gifted students may be pointed to the missing details). The chapter References identifies works which this author found useful and re-assuring 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 Rule-Based Reason in Mathematics describes the un-ruled and pre-codified 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 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 unreconciled axiomatic perspectives of geometry.
The chapter 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 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 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 Elementary Instruction describes its subjects pre-algebraic and pre-deductive, 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 pre-deductive 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 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. Next: Foreword with Inductive Principles for Instruction. |
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