Chapter 8
Modern Math Instruction (1983-1990 say)
A critique of modern mathematics curricula follows.
Oct 1, 2006, Postscript: Times have moved on. The
modern mathematics curricula described below is part of the past of
modern mathematics instruction. When I was writing this in 1995-96, I
did not know to what extent modern mathematics curricula of the late
1950s to 1970s were being followed, or been supplanted. Volume 1B
in retrospect might be entittled, Rip Van Winkle reflects on
mathematics education. What is modern changes with time.
Description and Analysis
An axiomatic codification of mathematics is provided by the
Zermelo-Fermelo treatment of numbers and sets. Initially this treatment
was just another way of looking at and organizing mathematics. From the
1920s to 1950s, recognition grew that it provided a more certain and
penetrating framework for mathematical thought, or at least a rigorous
codification.
The 1950s and early 1960s modern movements in mathematics instruction
introduced into high schools and colleges simplified forms of the
decimal-free axiomatic codification and thought-based foundation for
mathematics[1].
The movements emphasized comprehension: the derivation of ideas from
first principles, that is assumptions. Why mathematical assertions and
formulas hold, that is, their derivation from given assumptions, was
considered just as important as their use and statement. The movement
represented and continued the hope in mathematics that advanced topics
would be taught earlier and earlier in high schools and colleges.
Observations
As a high school student in the 1967-1969 period, I saw the claim that
mathematics could be derived from a minimal set of assumptions. I found
this near-certain derivation of conclusions from first principles,
combined with the idea that mathematics would be useful in the mastery of
other subjects, to be most appealing. My high school and early college
physic courses also emphasized or echoed this appealing idea of
derivation from first principles.
Within the codification, as indicated above, the decimal representation
of numbers is not special and not required. It is absent. The
simplification was faithful to original codification. It too remained
decimal-free. Yet in schools, the exclusion of any link with the common
decimal knowledge of arithmetic deprives the codification and students of
a reasoning tool. This separates the explanation (and the formal
comprehension) of mathematics from the common knowledge or use of decimal
arithmetic. My experience in this matter follows.
First, primary school taught how to count, do arithmetic and use simple
formulas. Then in high school came the codification, or axiomatic
approach in a simplified form. The axiomatic approach (with its reliance
on logic and the algebraic way of writing and thinking) offered rules for
real numbers. But of decimals, the rules or axioms made no mention and
offered no sanction. This was a source of logical distress. Given the
emphasis on the logical derivation from axioms or assumptions, I wondered
when my previous knowledge of decimal-based counting and arithmetic would
be explicitly sanctioned. It was not[2].
Second, my high school science courses all employed decimal arithmetic
and units of measurement as well, again without any formal
sanction in math courses. The role of units in computations, a subject of
interest in the business, geometric and physical computations, was
ignored. Thus mathematics was further separated from the earlier acquired
common knowledge and from the computational requirements of other
disciplines.
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Chemistry and physic teachers may show students how to carry units
through computations – at least one of mine did. The carrying
operation removes the need to convert all the quantities involved
to a single system of units. The conversion of units can be done
before, during or at the end of the calculations. Retention of
units in some form lessens the conceptual burden in performing the
calculations. By carrying units through a computation in an
algebraic or mechanical fashion, students do not have to think
immediately about what systems of units they are using nor do they
have to think about any unit-free formulation. Examples of units,
or their ratios, are given by the everyday use of terms like miles
or kilometers per hour, or dollars per pound or kilogram in
science, technology and business. Units of currency are related by
time-dependent constants.
Algebraic properties of arithmetic operations not only apply to
real numbers but also to real quantities – a real number times a
unit of measurement. The algebraic manipulation of units is
similar to that of monomial terms in the manipulations of
polynomials and rational functions with an exception or
restriction: addition requires that the monomials terms added
have the same degrees in the units present. [3] The present formal manipulation of polynomials
in high school algebra courses provides the necessary
background[4].
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Third, many college students, including myself, found the decimal-free
discussion of limits, convergence and continuity non-intuitive. The
underlying ideas appeared to be remote from comprehension, technically
understandable perhaps for a brief moment, and at first not readily
remembered[5]. But in retrospect, a decimal
(significant digit) perspective of unlimited error control for
computations or function evaluations gives a simple context for limits,
convergence and continuity.
The simplified form of the codification met in modern math curriculums
have been too faithful (due in the first instance to a rigorous adherence
which later became an unquestioned tradition) to the decimal-free aspect
of the codification at the expense of complicating the exposition of
modern mathematics. The expense was incurred in both high schools and
early college math courses. The decimal-free emphasis separated the
codification in its original and simplified forms from the common
decimal-based knowledge of arithmetic obtained in primary school.
Recommendations
For ease of exposition and a wider comprehension of mathematics and logic
a departure may be warranted in the high school and college axiomatic
development or codification of mathematics. In particular, assumptions
about the decimal representation of real numbers, and assumptions about
their convergence could be included. This would sanction decimal
expansions and arithmetic along with the mature knowledge of convergence
tacit in it. The initial explanation and description of decimals is a
sufficient representation of real numbers for students not immediately
specializing in mathematics. Further, in the exposition of mathematics,
rules or axioms for the treatment of units in computation could be
included (a) to link the exposition in mathematics with convenient
practices in other disciplines and (b) to provide an explicit logical
(thought-based) sanction for them.
The primary or junior high school description of the decimal arithmetic
and the decimal representation of numbers, with positions hundreds, tens,
units, tenths, one-hundredths, etc., is an inductive thought-based
process. The decimal-based perspective is ample for the common knowledge
of mathematics and for most college students and people not specializing
in mathematics – all those who will not see the more rigorous perspective
presently favoured.
Elementary courses, even though they be remote from the axioms or
properties of real numbers and the modern set theoretic codification of
mathematics, provide a thought-based framework for counting and decimal
arithmetic. Here a student should understand the positional decimal-based
representation of whole numbers by themselves or in numerators and
denominators of fractions and in the decimal expansion of rationals and
irrationals. The concepts of a > b, a < b or a = b for whole
numbers are initially understood in primary instruction from comparison
of decimal expansions, and not how the real line is ordered. (When the
latter is introduced, comparison by magnitude and by the linear ordering
of the real line need to be compared and contrasted. See below.)
There is an intellectual investment in the decimal representation of
numbers. The notions in it should be respected and strengthened, and no
concept be discarded or replaced until a student is positioned to
understand the alternative, why it is introduced and, for the sake of
rigour, its equivalence to the original perspective.
The decimal-free perspective of real numbers should be postponed to
advanced lessons where it studied by a few and linked to the previously
taught decimal-base perspective. There the transition from the decimal
perspective to the decimal-free can be designed to give a context and
motivation for the decimal-free perspectives of calculus and modern math.
Axioms for real numbers that do not mention the decimal representation or
provide an alternative accessible to students leave a vacuum. Axioms for
real numbers which do not mention or are not reconciled with the primary
school perspective, provide a second separate perspective. Modern math
instruction has split the discipline into two, and not reconciled the
parts. Here the parts are the decimal based common knowledge and the
deductive exposition or derivation from decimal-free axioms.
Assumptions about decimals together with assumptions about sets, along
with the emphasis of some restrictions on their formation, could provide
a simpler, more accessible, but still precise decimal-based image of the
codification. From this precise image, the decimal-free codification is
but a small step away. Again, the latter codification, in more detail or
in full, could be inserted in university courses to students
concentrating in mathematics and learning about the set theoretic
foundations for real numbers and arithmetic. Colleges students who make a
finer study of mathematics also can be shown or be given a reference for
the derivation of decimal arithmetic from basic assumptions about sets.
This would represent extra work for the few students who specialize in
mathematics, but it would ease the earlier exposition of mathematics for
most others, teachers and students included. In the high school
classroom, mention that other bases could have been used would imply the
more general viewpoint to receptive ears that the decimal perspective is
convenient, but not special.
[1] For an overview of the modern
mathematics movements, its origin and motivation, present and would-be
teachers should see the still-excellent and previously mentioned, 1965 work
Secondary School Mathematics by J. J. Kinsella, published by The
Center for Applied Research in Education, Inc., New York. This
reference also includes a description of options or different axiomatic
frameworks for the exposition of Euclidean geometry.
[2] The strict set-theoretic derivation of
the decimal representation of numbers is too long or complicated for a
beginning student to appreciate or follow – and it would not add much to
the development of his or her mathematical knowledge and reasoning
power.
[3] Exception: adding units of different kinds is of interest in some
special situations. For instance in taking a census, the population and
property of a small village could be represented by the expression 176
persons + 10 dogs + 30 bicycles + 40 houses + 50000 square meters of
floor space. The plus sign here represents the word and. Further exceptions may be found in the programming
language C and possibly in some mathematical software which allow
algebraic structures and not just numbers in their computations. Also
note that the requirement that terms added together have the same
dimensions or degree in units provide a error check for students of
physics and chemistry, if not mathematics, in the formulation and
manipulation of formulas and equations. As a student prone to nodding off
in class, I remember following in part a long derivation in calculus and
then realizing that the dimensions or units in two adjacent terms did not
agree, and thus the calculation was false. When I reported this flaw,
this old concept of checking the units was new to many in the math class
and possibly the instructor RR. None the less, a joint effort found the
error, a typo.
[4] From the strict mathematical
perspective, units are just indeterminates or formal polynomial variables
with inverses over a number field.
[5] Even Prof. E. McSquared’s original,
fantastic & highly edifying, Calculus Primer, by Swann and
Johnson, ISBN 9-913232-17-3, William Kaufman Inc, 1975, with its comic
strip explanation of the decimal-free approach attempted to lessen this
difficulty.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
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See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
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of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
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Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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work with maps and plans, and measure length, weight-mass and
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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