Rethinking Where is the Logic in Mathematics
Should mathematics be
based on (i) logic and formally stated patterns (axioms) or on
(ii) that appear to provide repeatable and reproducible, so observable
and verifiable, results. Option (ii) with hints of (i) may be best for the
most accessible form of a mathematics curriculum, while inclusion of all
logic, formal or informal, that explains why the patterns hold would provide
the most complete or comprehensive form. Each instance of a curriculum
or its delivery might vary between the inclusive and comprehensive
forms.
Page Content
- Pre-deductive Reason with comprehension
- Deductive Reason in Arithmetic with or without comprehension
- Deductive Reason in mathematics after arithmetic and before the end of
calculus.
- Contrast between the thought-based development of mathematics and science.
- Essays on College Level Logic (Food for thought)
- Options for College Instruction in logic.
In primary school, students may learn to do arithmetic with whole numbers,
fractions and decimals with the aid of practice and learning by rote or learning
with explanations. One prerequisite for this is a mastery of decimal notation
for whole numbers and decimal fractions, and a mastery of addition and
times tables for all pairs of whole number from 0 to 10 (or higher) in the first
and further year of primary school, if not before.
A fraction is said to be decimal if it equivalent to a
fraction whose denominator is a power of 10, or to a fraction whose
denominator is given by a product of the primes 2 and 5 to integral
powers.
For example of learning with comprehension, students who have mastered place
value in decimal notation for numbers may also understand the carries or
conversion which take place in place-value based, column methods for addition
of whole numbers using decimals. There in counting or adding the ones, if
the result is ten or more ones then there will a carry of one etc in the ten
column. Similar considerations apply to the tens and hundred columns in
the column methods for addition of multi-digit numbers. A thought-based
development of column methods for addition occurs when conversion or
carrying process is explained and understood by example.
Some implication rules IF A THEN B may be implicit in this, but the formal
or algebraic statement of such rules might be too complex for primary school
level instruction.
Thought-based understanding of why a method works is one source of skill,
competent and confidence in arithmetic.
The Arithmetic Reference Page provides
or indicates a thought-based development of place-value or column methods for
addition, subtraction, comparison and multiplication and long-division of
decimals. But the underlying explanations in full are too complex for primary
school instruction. A thought based development of addition, comparison
and subtraction methods for decimals, without and then a decimal point,
may aid skills and concept development when students are meeting and mastering
arithmetic. See the Arithmetic Reference Page
for options. However, the thought-based development of methods for
multiplication and long division is too complex. That development is best
left as options for enriching senior high school mathematics during or besides
the geometric explanation of distributive laws and/or geometric and algebraic
development of operations (addition, subtraction, multiplication and division)
on polynomials. That lack of explanation implies multiplication and long
division methods have to be mastered by rote through drill and practice.
That points to further elements of pre-deductive reason in mathematics.
Primary students may master decimal methods with explanation or indication
of why they work in the case of addition, comparison and subtraction, but only
partially or most likely, not at all in the case of multiplication and
division.
Primary school students may be shown how to do and write decimal arithmetic,
that is addition, subtraction, multiplication and variants of long division, in
well-formatted manner that writes and so record the method, step-by-step, in an
observable, legible and verifiable or correctable manner. While teachers cannot
read student minds, teachers can observe the steps students write and draw on
paper in developing and recording ideas for others to see, verify and correct in
an objective (right or wrong) manner. Drill and practice with arithmetic methods
with single, double, triple and finally multiple digit decimal representation of
whole numbers and mixed number may and should make students aware that an
mistake in one step makes all that follows and/or the result, wrong. Here
we say and/or since further mistakes may remedy the effect of the earlier ones.
None the less, with or without a thought-based understanding of why arithmetic
methods work, the mechanical and written practice of doing arithmetic, one
step at a time, one step after another, to arrive at a final result communicates
the steps in a verifiable or correctable manner.
In arithmetic writing and then showing and verifying the steps in
operation provides a test or proof of correctness of the work independent of any
student comprehension of why the underlying methods. In other words, in
arithmetic, agreement on how to add, subtract, multiply or divide in a written
and well-formatted manner allows the student or a parent or a teachers to judge
results and the extent to which a student has mastered an arithmetic method in a
repeatable and reproducible manner. In daily life before the common use of
calculators, the derivation of arithmetic values for sums, difference, products
and quotients were derived from the application of arithmetic methods with
details and proof depending on comprehension of how, if not why, arithmetic
steps are to be done. There-in lies a kind of deductive reason based on the
careful use of arithmetic rules and practices, one at a time at a time and one
after another, to arrive at results through we hope well-recorded or
reproducible and verifiable steps. Implicit here is the assumption that
arithmetic methods are reliable (lead to repeatable and reproducible
results) and are also correct. As indicate above, the correctness of arithmetic
methods, their thought-based explanation could be a subject or an aside in later
mathematics studies at say the current, senior secondary school level.
Mastery of method which work, which appear to give
repeatable, reproducible and thus verifiable or correctable results is a
second source of skill, competent and confidence in mathematics.
There-in an empirical knowledge. Some students when offered a
thought-based explanation of why the methods work will decline, not seeing the
need for such explanation. There-in a source of opposition to the
inclusion of explanations in mathematics - for some students that inclusion
with a must for their intellectual comfort while for others, the same
inclusion will be a source of stress, and proof that mathematics is too fussy,
or includes too many uninteresting topics.
Mastery of written methods for arithmetic or figuring skills develops the
ability to learn multi-step methods in and outside of mathematics and
develops the knowledge that lack of care, lack of attention to detail in one
step, will generally lead to incorrect results. Because of that figuring skills
in the past have been a preparation, a sign and an indication of intelligence
for the common person in the street and for employers and institution of higher
learning who want people able to follow and apply rules and patterns, one at a
time, one after another, with care and precision. There-in lies value to be
promoted in mathematics in arithmetic and beyond for the sake of future studies
and work both and outside of mathematics.
When mathematicians think of logic, they think if deductive reason in which
one and two-way implications rules B IF A and B IF AND ONLY IF
A are used in sequence, forwards and backwards, as part of definitions,
assumptions and chains of reason to arrive at conclusions and further
implication rules. The following chapters indicate the role of implication rules
in generating and describing or codifying islands and bodies of rule and pattern
based knowledge in mathematics.
- One versus two Way Implications.
Seeing the difference is a must for precision reading, writing and
reasoning.
- Chains of Reason. See how
implications rules B IF A can be used directly, alone and in sequence
to arrive at conclusions and further implications. The
thought-based development of arithmetic, algebra and geometry at this site
contains many examples of short and direct chains of reason without mention
of implication rules B IF A.
- Longer Chains of Reason or Mathematical
Induction. Here is a ladder climbing metaphor which introduces the main
ideas in mathematical induction
- Islands and Division of Knowledge -
This chapters points to the existence of different ways to codify and
explain deductive bodies of knowledge. Different starting points are
possible. For the description and development of skills and concepts,
mathematics course designers may look for the starting
easiest for students and teachers to follow.
In site steps for development of skills and concepts in arithmetic, algebra,
geometry, trig and calculus, implication rules are applied directly and not
indirectly with two exceptions.
First Possible Exception Before Calculus: One may use decimal
arithmetic (or area arguments) to imply
If two numbers or two length are nonzero then their product is also
nonzero.
The contrapositive form of this implication is as follows:
If a product of two numbers or lengths is zero then at least one of the
factors must be zero.
Here the contrapositive form of an implication must hold for the sake of
consistency. Otherwise, the original implication would not always hold.
Second Possible Exception in Calculus: If a series
converges then its terms tend to zero. The contrapositive of this say if the
terms of a series do not converge to zero then the series does not converge -
it diverges.
In calculus and in preparation for calculus (as indicated in site
pages), there appears to be no further use of indirect logic. That being
said, methods of indirect reason and the exclusion of possibilities via
consistency requirements, that is, proofs by absurdity or contradiction,
may be a subject of discussion beyond calculus in mathematics and a
subject of discussion of evidence, proof and rebuttals, in the training
of lawyers and detectives in law enforcement, and in scientific
researchers.
In mathematics education from arithmetic to calculus
& beyond, there is the option of a full thought-based, and not just a plug
and play, development skills and concepts. See site pages. The thought-based
option separates applied mathematics education from education in science and
technology.
Science, technology, industry and even farming today rely on
routines, methods and well-labeled products for use in a plug and play spirit
or manner to obtain repeatable and reproducible results on a short In science and
technology, the full, verifiable development of methods, routines and
materials is beyond the reach of instruction in schools, universities and
industry. The great, great division of labour in science, technology and so
on, yields too many routines, methods and labels to check in primitive school
labs and modern research facilities. All is plug and play. The creation
of new plug and play objects is based on combination of earlier routines,
methods and materials, and a verification that the combination leads to
repeatable and reproducible results in areas harmful or helpful to daily life.
Education in science & technology can only describe key theories and
practices, and try to give students the skills and concepts necessary to
participate in the plug and play development and maintenance of routines,
methods and materials.
While labs for the training of people to apply technology
may rely on plug and play instruments and methods, labs for future science may
need more primitive instruments, construction or workings self-evident, to
provide a fuller thought-based development of methods and concepts. It appears
to me that science labs in secondary school and early college are more to familiarize
students with lab methods (out of date or not) than they are to formulate and
test hypotheses. An aim of the science lab could be to illustrate and verify
basic facts assertions or implications given in class. Labs could have that
purpose hypotheses (the implication to be tested) given by the instructor.
There is a question of what can be tested. For example, To what extent are
Kirchoff's laws, DC and AC, verifiable in school labs using voltmeters and ammeters?
The test or experiment is easily set-up, safe for the most part, and with
instruments that were at my disposal, disappointing. I had wanted to show my
students that Kirchoff's law could be verified. That raises the question of
what skill and confidence building tests are feasible in a primitive high
school lab. Appendix: Further Thoughts about Logic
Running Out of Steam: When this site author is sufficiently
energized, the following readings as is or revised may be employed to
continue the above thoughts on rethinking logic at the primary and
secondary level to the college level.
Question: Running out of Steam refers to what means of
locomotion?
Related (overlapping) Readings in the site area: Math
Education Essays
- 18.
Problem Solving Hints Introduction to Problem Solving Stragetires and
Methods: Problem solving is like putting together a jigsaw puzzle. In the
case of textbook problems,
- Problem
Solving Methods - Trial and Error, Logic, Problem solving is like
putting together a jigsaw puzzle. In the case of textbook problems, all the
pieces are present and just need to be fitted together
Related Reading in Volume 1A, Pattern Based Reason,
online in full with postscripts
These chapters and postscripts explore but do not prescribe options
for the discussion and development of logic at the college or very senior
secondary level.
Chapter 20, Shorthand
or Pronouns in Logic, introduces the use of letters A and B, and
possibly others first to represent situations that can occur or not, and second
to represent phrases or statements that may be true or false (or neither).
Talking about pronouns, the pronoun metaphor, and talking about shorthand,
represent one or two ways to introduce the the shorthand role of letters in
logic and more generally in mathematics.
The online Volume 2, Three Skills For Algebra, in Chapters 8
and 9, and in the online postscript, What is a Variable, go further in Elucidating
or clarifying the shorthand role of letters and symbols in logic and algebra,
or symbol based, shorthand paths, for arriving at conclusions with implication
rules and formulas (or numbers)
Chapter 21 coins or introduces Occurrence
Tables. for three
phrases A AND B; A OR B; and NOT A; for one
way implications B IF A, and for two-way
implications B IF and ONLY IF A. The last section of Chapter 21 defines Converses
to One Way Implications and so digresses from the earlier content of the
chapter.
The occurrence (or obedience) tables invented and introduced
in Chapter 21, Occurrence
Tables, identify those situations in which implication rules are obeyed,
disobeyed or not disobeyed. The latter notions are intended to simplify or
justify the explanation of truth tables for the implication B IF A, or if you
prefer, the implication, IF A THEN B.
Chapter 22, The
Contrapositive shows the equivalence of an implication rule with its
contrapositive formulation - meaning B IF A holds when and only when NOT A
IF NOT B holds. The analysis is based on the three notions of a rule being
(i) obeyed, (ii) disobeyed or v(iii) not disobeyed. An implication rule B
IF A or IF A THEN B is Vacuously
True when and only when it never applies - that is when situation A never
occurs. In the latter case B or NOT B implies NOT A is a tautology.
Chapter 24, Direct
and Indirect Reason describes and explains direct and indirect methods for
reaching or proving conclusions. Among the indirect methods, this chapter
describes in particular, how an implication rule can be shown to always hold by
(a) showing its contrapositive form always hold, or by (b) looking for
absurdities that would occur if the implication rule did not hold. The second
method (b) is more indirect than the first method (a).
Online Postscripts: While we may not know that a
theory is consistent, we use the requirement for consistency as part of
the reasoning process without loss of generality or harm we hope. See
Proof
by Absurdity alias proof by contradiction and see How
the demand for consistency supports the law of the excluded middle
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