Chapter 10: Explaining Logic

For logic and pattern based thought, there are algebra-free lessons on implication rules, deception, chains of reason, longer chains of reason – mathematical induction, and islands and division of knowledge. One shows students how to apply rules and patterns one at a time or one after another. Another shows students the need to read statements and definition in a precise fashion – every word counts. And one, the essay Islands and Divisions of Knowledge offers a model for rule and pattern based thought that can be easily understood before the study of Euclidean Geometry, the original model. These lessons in full should be understandable to the typical fifteen year old but only to a precocious ten year old.

The discussion of logic, the use of the terms and, or, not etc., can be further illustrated with Venn Diagrams and with the symbolic or notational description of sets. The discussion of truth tables for implications and the logical interpretations of the operation NOT and the connectives AND and OR provide a further symbolic or algebraic perspective of logic.

The symbolic perspectives should be presented in full after and not before the algebraic way of writing and thinking is introduced. See the next section. To keep these comments on how to explain logic in one place, truth tables or a variation of them are discussed in the next paragraphs.

Truth or Obeyance Tables. Entries in the truth table for a material implication if A then B has left many instructors, yours truly included, at a lost for words. With this, students have been told to accept the entries as is, without question. But the three notions of an implication rule being obeyed, disobeyed or not disobeyed provide justification for the entries. In particular, an implication rule A implies B or if A then B is said to be false in situations where it is disobeyed and it is said to hold (or be true) in those situations where it is obeyed or at least not disobeyed. Finally, the implication rule is said to be always true in the circumstances of interest, provided it is never disobeyed in those circumstances[1].

There is a distinction to be made between describing instances where a rule or conditional statement IF A THEN B is obeyed, not disobeyed or disobeyed and identifying the respective circumstances in which the rule or statement is never disobeyed. See the logic chapters common to the companion books Three Skills for Algebra and Pattern Based Reason. They describes how the three notions of a rule being obeyed, not disobeyed or disobeyed can be used to describe and explain or justify the entries in truth tables for material implications.

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Volume 1B, Mathematics Curriculum Notes,

 Foreword + Chapters 1 to 10 + 12

Area Map
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5  Books & Articles to Read
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links

Chapter 11: Primary School Mathematics

11 Primary Math
11 Cue Cards
11 Counting
11 Decimals - Addition
11 Decimals -Times
11 Decimals & Subtraction
11 Fractions and Division
11 Notational Conflict
11 Reciprocals Etc
11 Decimals - Ratios
11 Size Comparison
11 Numbers, +ve or -ve
11 Rename < Sign
11 Complex Numbers

Will provide an alternative to Chapter 11 later, most likely in the Parent's Area: Help Your Child or Teen Learn 

Most students in high school are not heading for calculus, but most topics in high school mathematics are present due to calculus.  Preparation for calculus demands their coverage at  full strength.

See too, this site 55+,  Math Education Essays. Site areas and pages provide pieces of the a Mathematics Education, Jigsaw Puzzle, in formation.

-Inductive principles for course design & delivery  require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for  what was missed or forgotten in earlier studies. 

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Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.

Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.

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Modern mathematics curricula introduced an inconsistency into course design and delivery. They did not sanction the use of decimals nor the use of diagrams in skill and concept development but decimal arithmetic and diagrams are needed for student comprehension and for an operational mastery of quantitative skills. That implies the need for an mixed-math curricula based on a systematic development of operational skills, sufficient for applications and sufficient to provide a base & context  for  the very optional study of pure mathematis.

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