Chapter 1. Introduction
Volume 1B, Mathematics Curriculum Notes
These pages describe the influence of rule and pattern based reason on
mathematics itself and on mathematics instruction. Images of what is done
and what has been tried before and why, provide a basis for further
discussion. In particular, recommendations for math and logic education
are first based on this analysis or description. They are justified again
by describing two views of mathematics, the initial view met in primary
school education and the last view typically met in college level service
courses – those courses taught to people not specializing in mathematics.
In mathematics and in rule-based reason (or logic) there are some ideas
which can be explained clearly and quickly, ideas with minimal
requirements for comprehension. Collecting and putting such ideas first
should make both disciplines easier and simpler to learn and teach. This
yields a scheme for instruction from primary school to college. By
putting the simplest ideas first, the common knowledge of both
disciplines may be strengthened. The strengthening may further offer a
context for the more formal deductive exposition and codification of
mathematics as well as quantitative reasoning in other disciplines.
Two Barriers
The first barrier opposing math education has been the lack of words to
introduce the algebraic or symbolic ways of writing and thinking.
One way of symbolic thinking is recorded and represented
in arithmetic. Further ways are recorded and represented in the
solution of a system of equation and in the comprehension of the
properties of real numbers.
Algebraic reasoning, more so than arithmetic figuring, is better seen and
written than expressed verbally. Many today find this visual and
nonverbal aspect of algebraic thought intimidating and mystifying. Words
have to explain this situation. The discussion and illustration at length
of the three skills for algebra, one of which is prealgebraic or
symbol-free can remove or lower this first barrier.
Alongside the first barrier, the second barrier opposing mathematics
instruction stems from the most secure and attractive features of formal
mathematics, its strict deductive hierarchy of concepts and assertions.
With the strict hierarchy, comprehension of later terms and ideas
requires a precise comprehension of previous ones. This has meant that
the concepts described at the end of a mathematics course are
incomprehensible to the student just starting that course, or even in the
middle of that course. The hierarchy implies a student, even one who
mastered all the concepts so far, has no inkling or vision of what comes
next. Thus for students past and present, including many teachers,
mathematical ideas beyond the last course passed or taken remain unseen
and unfathomable.
The strict hierarchical description of mathematics provides a hard and
rigorous route which only the most patient or the naturally adept follow
far. This strict route leaves the image of mathematics and its logic
incomplete and mysterious. Ease of exposition has not been the guide in
higher mathematics. Yet ease of exposition can be the guide in extending
the common knowledge of mathematics provided in elementary school before
any rigorous deductive exposition begins. Following a command of
arithmetic, counting and the use of simple formulas, there are wordy
lessons which offer a simple command of algebraic thought, and further
lessons which offer a math-free command of deductive reasoning: how to
use rules and patterns one at a time, or one after another to arrive at
conclusions or decisions. Such lessons offer strands of thought which are
easily described and repeated. Putting them first provides a context in
which more deductive accounts can be presented and discussed. In it,
previously mastered strands of thought can be threaded or rethreaded
together with more and more rigour.
Lowering or Removing Barriers
Mathematics courses from primary school to college consist of chains of
reason, some quite long, but not all deductive. The mastery of elementary
mathematical thought may be inductive, that is drawn from examples by the
primary school teacher. Primary school students of mathematics can be
given methods for computation and numbers to use in them, along with
suggestive reasoning to introduce and sometimes justify the computational
methods. All this may be contrasted with the algebraic and deductive
rigour of formal mathematics met in advanced instruction.
Primary Instruction
Primary or elementary instruction now defines the common knowledge of
mathematics. This instruction introduces and explains arithmetic from
whole numbers to fractions and decimals in a pre-algebraic and
pre-deductive fashion. Physical reasoning, groupings and analogies are
employed to explain and justify ideas and methods. The counting and
computational methods taught in primary instruction are repeatable,
reproducible and thus verifiable. Arithmetic is rule based. From it,
student may learn that steps must be taken carefully. Two different
people with the same arithmetic expression to compute are expected to
obtain the same result, apart from small round-off errors. Primary
instruction provides a secure, often satisfying, rule and pattern based
command of mathematics. The security is based on methods which are
repeatable and reproducible, and thus verifiable, apart from any
deductive account.
Primary instruction in preceding the deductive exposition of mathematics,
provides the first image of the discipline and more generally, of rule
based figuring and reasoning. The latter touches many subjects. Advanced
mathematics instruction presently ignores and thus discards this first
image. It derives mathematics from decimal-free axioms or assumptions
about points, numbers and sets without any mention or consideration of
the student’s first command of the discipline. The knowledge of
mathematics and its pattern based reason is thus not cumulative. It
presently starts once in a pre-deductive fashion and then again
separately in a deductive fashion. This represents a gap and
discontinuity in the exposition of mathematics.
Between advanced and primary instruction, intermediate instruction is
assigned the task of providing a smooth and continuous transition. The
task includes directly but gradually describing and explaining deductive
logic and the algebraic way of writing and thinking to mathematical
novices.
Intermediate level instruction may place first those ideas easily grasped
to extend the common knowledge, and put second the technical details.
Here the ideas put first can be selected to ease the comprehension of
those put second. Those put first can also chosen to give a command of
mathematics and logic valuable in itself just in case the deductive
exposition and codification is not reached. This design may give students
immediate satisfaction in their studies and thus provide an invitation to
continue. This perspective puts the student, and not the discipline
first.
One way to introduce the deductive thought process is through math-free
lessons on implication rules (two logic puzzles), chains of reason,
longer change of reason, islands and divisions of knowledge given in
companion works to this one. This introduction can be given in
mathematics courses or in reading, writing and composition courses. The
explanation of deductive thinking, while required in mathematics, should
be across the curriculum alongside and in support of writing across the
curriculum. [2]
[2] This author has only
one comment about writing across the curriculum. Just as some people
are today math phobic, say mathematically challenged, some are essay
phobic or challenged despite good intentions.
The common knowledge, to be extended and developed further in
intermediate instruction after primary instruction, can continue to be
built first on rule-based methods that are repeatable, reproducible and
therefore secure or verifiable. For ease of exposition or to create a
more easily described image of mathematics, physical and geometrical
arguments may be employed to support or suggest conclusions while
deductive strands of reason are introduced as well. Example, see below,
can be provided by discussing and illustrating three skills for algebra,
and by presenting expositions of complex numbers and why slopes. All
communicate essential, if not abstract, ideas in mathematics while
requiring a minimal mathematical background.
Units and Decimals
two missing links
Mathematics courses, besides preparing students for the deductive
exposition of advance courses, should also provide students with the
deductive and quantitative reasoning skills required in other subjects.
Quantitative and algebraic reasoning in science, technology and commerce
involves units of measurement or quantity and the decimal, if not binary,
representation of numbers. The set theoretic axiomization of pure
mathematics is free of both units and decimals. Primary and intermediate
level courses can be assigned the further task of sanctioning the use of
both units and decimals in the quantitative or algebraic reasoning of
other subjects. Presently, the presentation of axioms for real numbers
makes no mention and offers no sanction for the use of decimals or the
use of units.
Primary instruction provides, one hopes, a thought-based command and
investment in the decimal representation of numbers. The latter
representation is indeed adequate for those who will never see in full
the decimal-free set theoretic codification of modern mathematics. Until
the presentation of the latter, the decimal representation is also
adequate for students of mathematics. For continuity between primary and
further courses, added to the axioms or assumptions about real numbers in
intermediate level courses may be two assumptions, first that real
numbers have decimal expansions, and second, that infinite decimal
expansions define a real number. This reflects the common knowledge or
belief. Some sanction for it should be provided in mathematics so that
the common knowledge and axiomatic perspective do not need to be
reconciled.
Indeed, the decimal concept of convergence, with or without set theoretic
wrapping, is sufficient for students not meeting the decimal free
alternative. [3] Its discussion, see the chapter
Error Control, Continuity and Limit in the companion work Why Slopes and
More Math, can further provide a background and a context for the
understanding of the decimal free approach – part of the motivation or
explanation why. Abstraction by itself, without concrete examples,
provides the student a vacuous knowledge. The vacuum is abhorred.
Primary instruction in quantitative reasoning introduces units of
quantity or measurement. Their absence in the algebraically described
axioms for real numbers, and in intermediate courses apart from
trigonometry, separate mathematics courses from the quantitative
reasoning required in other courses. While the need for units of
measurement can be circumvented by the dimensionless (unit-free)
development of formulas and equations, that circumvention is a
complication not needed in elementary mathematics. In the first instance,
the algebraic way of writing and thinking can be employed with
calculations involving decimals and units.
Axioms for real numbers can be augmented with more axioms or assumptions
about quantitative and algebraic reasoning with units, monetary or
physical. Here for instance mathematics courses could present the
innermost axioms, those for real numbers and for their decimal expansion,
while other courses in quantitative reasoning, if not those in
mathematics, could add further axioms, those involving units. For
students, this would yield an axiomatic framework for numerical
computations within a broader axiomatic framework for quantitative
reasoning. A consistent discussion of units would be welcome in
mathematics or adjunct disciplines. [4]
With the insertion of assumptions about decimals and units along with the
discussion of implications rules and three skills for algebra,
mathematics instruction could reach for or even achieve the following
goals:
1. A continuous extension in high school and college of the common
knowledge acquired in primary schools of decimal arithmetic, counting and
the use of simple formulas.
2. The formal and/or informal provision of a logical framework for
computations in all mathematical disciplines with and without units.
3. The description and demonstration of a simple logical structure or
codification for mathematics and quantitative reasoning in other
disciplines.
The consideration and retention of both decimals and units in the
exposition of mathematics would make it continuous and cumulative
pedagogically from primary school to college. The common knowledge of
calculation and logic, developed in primary and intermediate level
courses, can be axiomatically codified in advanced courses, and provide a
context for the discussion of the codification.
[3] The college book Calculus by Lipman Bers (Holt, Rinehart and Winston
1969) heads in this direction by talking about the decimal representation
of numbers and other equivalent representations. The discussion of areas
under a curve further begins with area approximation of a region based on
its coverings by small squares. This approximation was taught in my
primary school.
[4] Physicists and standards bureaus with their conventions on physical
units may be of assistance here.
About The Next Chapters
The following chapters elaborate on the above ideas. Some repetition of
the above ideas will be found. This introductory chapter and the
forewords for this subwork, volume 1B, and complete work, volume 1, have
previewed if not stolen the conclusions from the rest of this work. The
rest of this work along with the companion works offers a simple frame
for mathematics and logic instruction from primary school to college
apart from the exposition of geometry. Curriculum committees will wish to
add further topics to this frame.
The chapter 2, For and Against Mathematic s, 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 3, 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.
The chapter 4, Complex Numbers and Why Slopes (Calculus) 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 prealgebraic
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.
The chapter 5, References , identifies four works which this
author found useful and reassuring 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 6, Rule-Based Reason in Mathematics , describes the
unruled and uncodified 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.
Chapter 7, 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 un-reconciled axiomatic perspectives of geometry.
Postscript (2010): Chapters 1 to 6 describe barriers and
remedies as seen in 1995-6. The remaining chapters explore ideas for
remedies. However more effective remedies are available. See the ends and
values for mathematics and logic in site pages. Clearer paths for skill
and concept development imply a few changes or extras in the indicated
remedies. But the foreword and chapters 1 to 6 provide background
information.
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