Foreword to Volume 1 (an intro to all site books)

Links to Book Forewords: 1. Elements of Reason 1A. Pattern Based Reason 1B. Mathematics Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes and More Math

Elements of Reason understanding and explaining  reason and math Volume 1 by Alan M. Selby Ph. D. Printed in Canada ISBN 0-9697564-1-0

Mathematics teachers hould see the second part  - the first part provides a context for the second part and further site volumes. Bon Appetit.

The first part  Pattern Based Reason of this volume  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 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  and 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.

     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
 
Reprinting may lead to new ISBN numbers.

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