• 1A Pattern Based Reason
• 1B Mathematics Curriculum Notes

Foreword

Volume 1, Elements of Reason

The first part Pattern Based Reason (Volume 1A) of this work Elements of Reason describes rule and pattern based thought and processes in daily life, society, science and technology. Reliable rules and patterns can be followed one at a time or one after another to obtain conclusions or results. Not solved is the problem of identifying reliable rules and patterns to employ. Instead, the empirical method of coping with this problem is discussed.

Rule and pattern based thought and processes touch many arts and disciplines. Awareness of the difference between one- and two-way implication rules will improve reading, writing and argumentation skills. Students of critical thinking, persuasion, philosophy, mathematics, science and technology may find this first part worth reading.

In both arithmetic and logic, rules and patterns if followed carefully lead to results which are repeatable and reproducible, and thus verifiable and objective: two individuals following the same rules and patterns with the same data or in similar circumstances should obtain the same or similar results. Arithmetic and deductive reason are but examples of verifiable rule and pattern based thought or processes.

Verifiability, repeatability and reproducibility form a basis for the appreciation of, if not reliance on, rule and pattern based thought and processes. This appreciation should not be too firm. The identification of reliable rules and patterns, or reliable data to use with them is not certain. Further, where rules and patterns do not apply mechanically, there is room for thought. Still, verifiability, repeatability and reproducibility may provide a basis for the common knowledge and informal mastery of a subject.

The second part Mathematics Curriculum Notes (Volume 1B) is for teachers and advanced students of mathematics or a quantitative college discipline. This part describes simply yet precisely, the role of rule-based reason, that is logic, in providing a thought-based framework and codification for mathematical thought. This second part further describes how an inductive educational philosophy provides a context for math and logic instruction from primary school to college. Ideas which are easily repeated and understood may provide a common knowledge of mathematics and the rule-based reason sufficient for a more formal and rigorous comprehension.

This two-part work and its the companion volumes Three Skills for Algebra Why Slopes and More Math stem from a project to write a single book, namely Ideas that Might Count for Education, Reason and Mathematics (1994). That single book (no longer available ) was written and distributed. It covered a vast number of topics. Some of interest to one audience but not to another. With further writing and rewriting, this first endeavor was divided into three volumes, the first of which, the one before you, was divided into two parts. Writing for some is an iterative affair.

The initial aim was to report some unique idea, innovations, for math and logic instruction. These ideas or lessons had worked well with college students, shy or curious about one or both disciplines. But in writing and rewriting, the aim became wider. The possibility of a consistent and coherent scheme for math and logic instruction from primary school to college was seen and explored. The scheme is comprehensive save for the treatment of geometry. How to fit or emphasize Euclidean geometry in the curriculum is not covered.

Formal mathematics can be difficult to follow for students who fail to grasp deductive thought and the symbol-based algebraic way of writing and reasoning. The latter like arithmetic is better seen and written than spoken aloud. Symbols like pictures can be worth a thousand words. Words have been missing to explain the role of symbols in providing the shorthand notation of mathematics or its algebraic way of writing and reasoning. The latter consists of recording and developing thoughts on paper at least for those among us afflicted with a short or too forgetful memory.

The absence of a verbal culture to introduce and explain the algebraic way of writing and thinking leaves its mastery to immersion and osmosis. Comprehension depends on one's aptitude for learning some basic ideas by immersion. I am in the radical position of suggesting that a certain change is possible and desirable. This work and its companions suggest how. They have yet to be formally peer reviewed and so should be read with caution. The discussion of math and logic instruction and the discussion of reason and persuasion are both fraught with controversy. Scrutiny or critical examination of this work may lead to its refinement.

Alan Selby
Montreal 1996.

December 2011 Postscript

Site chapters and steps now stands at the sharp edge of mathematics education reform. Site material stems from olde and continuing gaps and inconsistencies in ends and methods - there was no pleasing all. The essay which way to go "lightly" introduces a more detailed, five phase framework and nearly plain-language remedy. Phases 1 to 3 focus on skills of value for adult or daily life - precision in reading-writing-figuring included. Phases 4 & 5 focus on calculus and preparation for it. Many university programs demand calculus. Preparing for it has value in senior high school science too, and some value for trades-professions not taught in university. The framework addresses and remedies all the difficulties identified above, and implement most of the ideas in the subvolume 1B, Mathematics Curriculum Notes. The framework being done sets the stage for yet another consolidation of site material.

 Canadian Cataloguing in Publication Data
 Selby, Alan M,
 Understanding and Explaining reason and math


 Contents: v. 1. Elements of Reason - v. 2. Three Skills
 for algebra - v.3. Why Slopes and more math.
  ISBN 0-9697564-4-5 (set) -
  ISBN 0-9697564-1-0 (v. 1) -
  ISBN 0-9697564-2-9 (v. 2) -
  ISBN 0-9697564-3-7 (v. 3) -
   1. Mathematics--Philosophy.  2. Reason.
   3. Algebra. 4. Calculus. I. Title. II. Title: Elements of
    reason. III. Three Skills for algebra. IV. Title: Why
    Slopes and more math.

 QA8.4.S44    1995 510'.1  C95-900945-0

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