Words For Teachers, Mathematicians and
Mathematics Education Faculty
Instructors: (i) Calling this a T3 (Teach the
Teacher) site is not a claim to perfection. It is a
claim that site ends, values and methods should be taken as a
lower bound and a lower standard for skill development - do
better - in your instruction and in course design.
With regrets, the mathematics curricula put forth by English
educational authorities in North America and the UK, those I have
seen, are unsatisfactory. Technical gaps persist, are
accepted or go unnoticed. The underlying premise
in secondary course design of preparing for advance mathematics
(calculus) does not serve common needs.
(ii) Encourage strong readers in your classes to explore site
skill and concept development pathways. - that may help them catch-up,
leap ahead or help others in your classes. Assigning
students to evaluate site skill development methods for class
projects and presentation would help teacher training and tutor
training programs. (iii) It may be
improper to say that reforms of any kind should be accompanied by
well-documented how-TOs, and preceded by critical path analysis of what
is to done, for whom, and why. However, in business, government and
education, reform in haste, repent at leisure appears to be a
rule. Changes too good to challenge, too good to wait for how-TOs
to be collected, studied and documented, are reforms in haste. (iv)
Just as computer have to boot-up by loading elementary
instructions, quantitative skill development have to
boot-up in the
education of learners 4 to 14 years of age.
The 21st
yearbook of the NCTM (1951) ends with statements to the effect
that mathematics instructors need to be master technicians,
learning engineers. Its focus was on content matters or options.
The aim then appeared to be the preparation of students for
university level studies in mathematics. The underlying premise was
that these students would be prepared for advanced mathematics in the
form of calculus etc to do the work in business, engineering, science and
technology that society or governments wanted. The needs of students not
in the university track , for a general education in mathematics etc was
acknowledged but not analyzed. Leading Mathematician of the
1940s to the 1960s were involved in the discussion of possible
of material (content options) for pre-university students.
Student Centered Skill Development: In education which
values skill development, calls for education to be student centered
may be answered by putting first in mathematics or quantitative skill
development, the common or likely needs of students and their
families. The common person in the street does not need
preparation for calculus, he or she first needs a command of
quantitative skills and concepts to the greatest extent possible,
before preparation, if any, for advanced mathematics begins. Options
here could be fine-tuned to give a base for the preparation for
advanced mathematics, but more importantly, it would also provide a
safety net and some appreciation for what should be the common
knowledge of mathematics by the many, many students who begin the
preparation but do not complete it.
Skills and Concepts with Take Home Value: To provide take
home value, mathematics education should focus on the progressive
development of skills and concepts likely to have immediate or
long-term take home value for students. Teach geometry in
the form of map and plan usage. Mastery of similarity would be
implicit in that. Teach arithmetic (the first R) and formula
evaluation for working with lengths, measures and all money
matters. Knowledge of error propagation, an apolitical
domino effect, in multi-step methods would follow from
that. Teach logic (the difference between saying
A if B and saying A if and only if B, and chains of reason) for
the sake of precision in the 3 more Rs reading, writing and
reasons. More precisions and less confusion in work and study
would follow from that. Teach about chance, averages and
probability along with logic as a base for making decisions not only in
playing games, but also in making decision in daily life. All the
foregoing should provide ends, values and methods for careful attention
to details in work and study.
Future course design to be student oriented in a practical
manner needs to put firsts the common and likely
needs of daily life in the streets before preparation for advanced
mathematics (calculus) begins in earnest.
Where is the discussion of content options in the education reforms 1990
onwards? Should not content possibilities, what to teach and why be
the main force in reform? This site present options for filling
gaps and repairing inconsistency in skill (and concept)
development. This site introduction of fractions,
algebra and complex numbers, and calculus is especially strong.
- Raising terms in fractions, is shown as method for going from simple
to the general case not only in the comparison, addition and
subtraction but also for multiplication and division. That
makes the justification clearer, more accessible;
- The algebraic way of writing and reasoning is introduced at length
with words and geometry. The unifying concept that all rules and
patterns will be employed forward and backwards appears.
- The geometric development of complex numbers and their
properties may stand on the properties of real numbers and the
observation that a midpoint of two points (a line segment) is moved by a
rotation in to the midpoint of the images of those two points (or the
line segment). That provides an option for mastering complex
numbers geometrically before the geometric introduction and mastery of
periodic trig functions. Why that was not done before is a question
to resolve - what is new in the route here?
- The site geometric and algebraic preview of calculus may ease or
avoid some algebra difficulties by providing a context for the
calculation of derivatives and by putting first those elements of
algebraic reasoning easiest to understand.
Stepping Back in Time
The advent of writing and drawing on parchment or paper (or
whatever) allowed geometric reasoning to done and recorded in books at
least since the time Euclid, 300 B. C. Since 1500 A.D.,
writing on paper and slates has permitted or led to the development
of arithmetic notation and written mechanical methods for
arithmetic to be developed and become part of the common knowledge. The
technological advance (olde and low-tech) of writing and drawing ideas on
paper set the stage for the development of mathematical thought,
done and recorded step by step. Reasoning steps present in
proofs and problem solving being drawn and written on paper extended the
power of the mind and became a medium for recording or
communication of observable verifiable, correctable and if verified,
thoughts independent of any scribe. To say the development of skills and
knowledge is a private matter for the individual, beyond observation and
verification is to ignore the common bond and interaction made possible
by communication.
Diagrams and symbols drawn, and word spoken or written provide a
means for individuals to share their ideas (or lies) in manner that
makes private thoughts and ideas part of the common discourse and
knowledge between a few or the many. A common knowledge
appears in the power to draw and write thoughts on paper to
record and imply rules and patterns for the original scribe and others
to follow or adapt.
Education in an established art or discipline, call this training if you
like, requires an observable mastery of current tools and practices. Both
research and education in mathematics, science and everyday technology
requires the development and mastery of methods in an observable,
repeatable and hence verifiable manner for credibility, that
is, potential peer review by fellow researchers or students, or
teachers. Mastery of knowledge in mathematics and science is not
private affair. It is a public affair. Skill in these disciplines has to
been seen to be credible. And in mathematics, mastery of
the tools and practices of arithmetic, algebra, geometry,
trigonometry, probability, calculus and so must be expressed on paper in
order to be observed and checked or double-checked and corrected as
need-be. There are rules and patterns to learn and follow with care to
avoid errors, errors of the observable kind.
Progressive Skill Development
Skills which can be observed directly may be judged and
corrected by observers. Skills with observable results can be
judged by those results. Mistakes can be identified and advice or
directions for skill improvement can be given. In many arts and
disciplines, skill development is progressive: later skills may follow
from earlier ones via self-instruction or deliberate
instruction.
While mathematics may be logically and progressively developed, skills
and concepts, one at a time, one after another, once students have
grasped logic and the algebraic way of writing and reasoning, the
prerequisite to that progressive development is a mastery of logic and of
the shorthand roles of letters and symbols in algebraic reasoning.
To make matters simpler, the mastery of logic can be learnt apart from
mathematics. See Volume 2.
What is obvious to some, is not obvious to all. The algebraic
shorthand way of writing and reasoning with letters and symbols, beyond
say the evaluation of formulas, is not obvious for all. The
difficulty in algebra may stem for the fact that picture is worth a
thousand words. Pictures are better seen in glance than described with
words. When arithmetic and algebraic expressions are written
or seen on paper, they are often best seen and grasped silently
than expressed in words. In general, there has been a lack of
words, a lack of rationalization, for the use of letters and symbols in
mathematical reason. That nonverbal aspect of art and mathematics
is offset by the use of names for pictures (the Mona Lisa) and the use of
names or identifying phrases for formulas: the compound interest
formula, this rectangle area formula.
A General Plan: To aid skill and conceptual mastery
of algebra, steps for that may begin with giving and evaluating
formulas for length, area, volume and speed, distance or time; may
continue with the the use of letters to denote the unknown measure, say
length or area in a geometric figure, which is visible before
allowing letters to denote an unknown count or number; may
continue with the algebraic description of methods or patterns for
describing arithmetic with fractions; may continue with the use of
letters to describe arithmetic identities; and may continue in the
explanation of the forward and backward use of formulas and
proportionality relations. Algebraic descriptions of
arithmetic methods for fractions appears as part of site lessons on
fractions. Those lessons need to read twice - once for the mechanical
mastery of fraction skills and concepts apart from the algebraic
description, and a second time to introduce the role of algebra as a
means to describe fraction arithmetic patterns or how-TOs.
Counting principles as is or disguised in the form of area calculation
may be employed to imply the commutate, associative and distributive
properties of arithmetic.
Two Gaps in Skill Development
Or why mathematics is harder than need-be.
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The olde Algebra Gap: The shorthand roles of letters
and symbols are not fully explained or rationalized from
solving equations to the very challenging use of algebra in
advanced mathematics (calculus). Solving linear
equations starting with
fractional operations on stick diagrams gives an entry level,
geometric introduction to algebra with letters referring to
visible lengths. Chapters
8 to 12 in Volume 2 and the essay
What is a Variable put more words into the explanation and
comprehension of algebra. Chapter
14 in the same Volume 2 with its detailed discussion of
the direct and indirect use a formulas identifies a unifying
theme for algebra and logic - all rules and patterns may
and will be used forward and backwards in mathematics, science,
technology and logic or reason. The very challenging use of
algebra in calculus is made easier by (i) this why
slopes,
geometric preview of calculus, by (ii) this factored
polynomial, algebraic preview in Chapters 2
to 6 in Volume 3, and by (iii) the further
discussion of slopes, limits, derivatives and integration
in Chapters
11 to 18 of Volume 3. Mathematical Fact:
Calculus requires earlier high school mathematics and logic at
full strength: (i) This long complex numbers lesson
on shows how to simplify the development of periodic trig
functions, the derivation of their properties, and the
derivation of trig identities and formulas in the plane for
vectors dot and cross-products. For further
algebra skill development, see the site coverage of
fraction with units,
proportionality,
polynomials,
quadratics
functions and
straight line slopes and equations. And for logic
mastery, start with the math-free chapters
1 to 5 in Volume 2 as early as possible for the sake of
precision or greater precision in reading, writing, reason.
See too the site coverage of Number Theory.
The New Arithmetic
Gap: An exact and efficient mastery of arithmetic with
decimals and fractions is needed for proper, full strength,
high level study of mathematics alone and in science,
technology and business. In site material,
webpages with html, and real player and flash format
webvidoes on arithmetic with
decimals and integers, on fractions
and solving linear
equations with
fractional operations on stick diagrams may help fill this
gap. Calculators and computers (cash registers too) can be
and should be employed to do arithmetic.
Note: An exact and efficient command of arithmetic should
be obtained in the last years of primary school and the first
years of secondary school, partly to serve these
ends, values & methods for work & study -
learning to avoid mistakes in multistep methods
via the early mastery of exact arithmetic with
decimals; and partly to set the stage for an exact
and careful mastery of algebra. The division of
polynomials (a requirement for calculus) will be easier for
students well-practiced long division with whole numbers
(decimals).
Mastery of arithmetic with decimal and fractions may be taught to
give student experience with the domino effect: an error in the
data or one step of a methods leads to results, intermediate to
last, most likely in error. Whence the old view that figuring
well is a sign of intelligence (intelligence of the practical
kind) stays alive as end and value for primary and junior high
school mathematics.
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Quantitative skill development should reflect a critical path
analysis and knowledge of the ends, values and methods of instruction
which have been chosen. After arithmetic mathematics
education may continue with a lean but adequate coverage of the
algebra, geometry and trig required for common trades or for further
study - preparation for calculus included. That
being said, with regrets, I am not exactly sure what preparation
common trades would require, but I do know what calculus requires.
Skill Development Pathways
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Decimal Arith. Lessons
(Flash) ] Students
ages 9+ should master decimal methods for exact and only
later approximate arithmetic with whole numbers and decimal
fractions (proper or not). Practical intelligence comes
from learning how to do decimal operations carefully and via
that becoming aware of the domino effect: an mistake in the
data or one step of a multistep method leads to incorrect
results, intermediate to last. Furthermore mastery of decimal
methods and explanations of why they work provide a greater command
of decimal place value. The video lessons here provide a full
explanations - too much for most students, but just right for
tutors and teachers
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Fraction
Operations (comparison, multiplication & division
included). developed by raising terms to transform into easy
cases. (A must read for the instruction of students 11 to
adult and a must read for students 14+ ).
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Ends,
Values and Format for Work and Study. Communication,
showing reason and problem solving step by step, with proper
format, provide ends, values and base for observable and verifiable
(or correctable) skill development. Vertical alignment of
equal signs in the evaluation or simplification of arithmetic and
algebra expression sets the stage for proper use of equal signs by
students 11 to adult. Here communication, problem solving and
reason are taken as part of skill development.
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Early Geometry in the form of Plan
and Map usage provide a practical, self-contained, early math
(trig-free) approach to solving similarity problems by drawing
diagrams to scale and then measuring on the diagram. Here is a
self-standing prequel to trig. (Ages 10 to 13, not as is, Ages 14+
as is.
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Solving
Linear Equations via fractional operations on sticks
(line segments) makes use of letters and stick-free solution
understandable. Proper format for use of equal signs is
followed here. Ages 13+
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Easily Solved
Systems of equations in essentially one unknown gives a model
for formulating word problems, as a way to ease or avoid
mental gymnastics with algebra. Ages 13+.
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Integers
introduced geometrically in a way that gives a model for the
geometric introduction of most operations on signed
numbers. See Arithmetic
How-TOs for more.
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The Forward
and Backward use of formulas (also rules and patterns)
is a unifying theme for senior high school and college
mathematics and science. The theme appears here with
the compound interest formula. In retrospect ( a site or teacher to
do), simpler formulas introduce the theme. In
proportionality problems this theme appears with backward
use (finding the proportionality constant) put
first.
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Wordy
Introductions to Logic may develop precision reading and
writing needed in maths, all further studies, home life and work
for better performance or self-protection - Romeo and Juliet
make mathematical
induction easier to understand and explain - Chains
of reason provide a model for reason in Euclidean Geometry
outside mathematics.
(Ages 15+ but earlier for avid readers, gifted
students).
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Three
Skills for Algebra (Talking about Numbers, Describing
Calculations, Describing when calculations are equal, what is a
variable) may ease or avoid fears & difficulties and
clarify concepts that obvious to some, but not ALL. The
algebraic way of writing and reasoning needs to be introduced with
words - rationalized. Ages 14+
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Essay
What is a variable puts words before and beside symbols
at a level the calculus or precalculus student will
understand. Fellow Mathematicians: Arguments against past
verbal descriptions of variables do not apply here. Ages 14+
and before calculus.
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Complex numbers &
properties introduced geometrically & rigorously before
the development of periodic trig functions will simplify
simplify the high school level 2D geometric development
of trig and vectors. The simple geometric
proof here of the distributive law is the key. The advantages of
using complex numbers in the exposition of trig was well-known in
the 1940s or earlier.
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Why complex numbers were not geometrically developed before
trig in the course designs of the 1950s or 60s is a bit of
mystery. Some inquiry or research may explain
why. Since 1976, this site author looked for a simple
proof of the distributive law, found or re-invented several,
only to learn in February 2010 that giving a geometric proof
was an exercise in Secondary Mathematics, A Functional
Approach for Teachers, H. F. Fehr, D. C Heath and
Company Boston 1951.
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multiplication, addition, subtraction and long division of
polynomials geometrically,.Quickly. Justification is
another matter. The
aim is to make the algebraic processes more
accessible. Ages 15+
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Quadratics
- The algebraic way of writing and
reasoning is employed at full strength in calculus. The
aim again is to make the algebraic process more
accessible.
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Function
Theory for Senior High School and Calculus Students -
Multiple Viewpoints explained and reconciled. Ages 15+.
May begin before and finish in calculus.
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Geometric and Algebraic (Chapters
2 to 6 in Volume 3) Calculus Previews:
These offer an end earlier studies or a start for calculus in a
manner that strengthens algebra skills and eases or postpones
calculus difficulties.
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The decimal representation of real numbers with limits and
convergence related to the possibility of unlimited error control
(decimally described) in the evaluation of functions and
limits might make epsilonics easy for
undergraduates specializing in mathematics or advanced students of
calculus/real analysis.
Preparation for Calculus etc - Steps not covered or not covered
fully in site material.
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Logarithms and
Exponentials. (i) The
natural log may be quickly introduced as the area
under a curve (proofs skipped) or simply assumed. (ii)
Calculators may use to illustrate or confirm properties. (iii) All
other logs may be expressed in terms of the natural log. (iv)
The inverse exp(x) to the natural log may be introduced by using
the graph of the latter backwards - introduce the horizontal line
method for calculating a function from a set or graph in the plane.
(v) Properties of exp(x) may be derived or assumed.
(vi) How to compute rational powers and rational roots for
non-negative numbers may be shown to build algebraic skills and to
define bx as exp( x ln(b)) (vii). How
to calculate rational powers 1/n and m/n of real numbers x
when n is odd may be shown. That is x 1/n =
sign(x) (|x|)1/n and x m/n = sign(x)
(|x|)m/n when n is odd and n is real.
(viii). Logarithms and Exponentials may be used in the forward and
backward use and equivalence of (i) compound
growth formulas A = P(1+r)n , (ii) doubling time
formulas A = 2t/TP (ii)
half-life formulas A = 2-t/TP, (iii) continuous
growth/decay formulas A = P exp(t ln B). The
equivalence of these models and their forward and backward
use in biological, chemical, physical and financial growth
and decay formulas may be implied or
explored. Reference: the site coverage of See Exponents, Radicals
& logs. and chapter
14 in Volume 2.
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Probability.
The Study of Probability provides a
stage to practice and maintain exact arithmetic skills with whole
numbers and fractions. The study also employs function
notation P(E) say and set concepts/operations to codify and
calculate chances or probabilities. Operation
with sets may be linked to logical operations, mathematical
induction and function notation may serve as a deliberate prequel
to function theory, and then employ functions to codify and clarify
concepts.
Motivation: Chance and probability concepts and
considerations are met in daily to estimate the odds of success in
uncertain situation from playing games (optional) to running a
business, a school or a civil service department. The exposition of
probability concepts may employ sets, functions, counting
principles and logic as part of a deliberate development of
algebraic thinking skills before calculus, time
permitting.
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Sets: Sets and their elements may be specified via
lists (rosters), indicated or sketched using diagrams. After some
mastery of algebra, subsets may be specified with with a mix
of set builder notation and algebraic conditions - equalities and
inequalities included. Whence sets may appear as the
"solution" of algebraic conditions. Counting in groups
(non-overlapping sets) is implicit in primary school maths.
That style of counting may be made described algebraically and
extended to cover the case of overlapping sets. Counting
skills may be codified and extended with the use of injective and
surjective functions. Motivation: Sets have immediate
value in high school probability theory, counting methods
included, in illustrating logical ideas and in
simplifying the description of functions - their graphs &
domains. Talking about sets of numbers further
lightly sets the stage for modern mathematics.
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Conic Sections: (i) How the intersection of planes and
cones imply the loci description of conic section in the plane
should be available. The 1942 text What
is Mathematics by Courant and Robbins gives this for an ellipse
and references for other cases. (ii) How completing the
square for quadratics in two variables x and y in the
simplest cases (no xy terms) leads to standard forms for
equations of ellipses, parabolas and hyperbolas. (iii) Application
of the standard forms to graphing conics sections in the plane and
recognizing their form. In physics conic sections
appear in planetary and projectile motion. Other
serious application appear to be scarce.
Note: The study of conic sections or their
equations is not needed to begin calculus. The study in
advance calculus of functions of two or more variables
requires completing the square or equivalent linear algebra
concepts to classify "critical points" as maxima, minima and
"saddle-points" and to graph level sets around such points.
That classification is of service in engineer (mechanics). Thus the
study of conics could begin before calculus and be finished besides
or after calculus in courses in analytic geometry, in linear
algebra or in mechanics. The study of conic sections
before calculus is a possibility to build algebra skills. To
what extent and level it should be done as part of the preparation
for calculus is not answered here.
Changing Course
Design
Writing switched in 2007 from returning the modern maths curricula of
the 1960s to favor, to developing an successor to it and the
constructivist reform movement of the mid-1980s to the
present.
Calls for competence, reason, communication, reason and student
centered instruction may all phrases to support and call
for observable and verifiable skill development in pure and
applied disciplines. Future course design (the
preparation for calculation part) in secondary mathematics should not
begin with axioms for real numbers, it should end with that. Before
that end, in primary and secondary school, explanations of why methods
work should be included where it aids comprehension and does not
overwhelm students and their tutors or instructors. That explanation of
why should always be available as a reference. But common practices in
arithmetic (the assumption that counts do not depend on how counted)
and in map usage (geometric assumptions) may be employed to imply the
algebraic statement of properties of both real and complex numbers, a
possibility only after the slow and careful introduction of the
algebraic way of writing and reasoning. Once those properties are
implied by common means, sufficient for the common person in the street
to follow, the axiomatic development and codification of real numbers
may begins. But the axioms envisioned here would include the decimal
representation of real numbers with limits and convergence defined in
terms of decimal concepts in order to make calculus more accessible.
The introduction to modern maths in secondary school should not begin
with axioms for real numbers, it should end with them or better yet
elevate to axioms, properties of numbers, real and
complex, that serve common and advanced skill development in daily
life and in university mathematics, science and engineering. That
elevation should be done after the properties have been
implied. See below.
The rigour of modern mathematics may not be possible in quantitative
skill development from counting and measuring to calculus but it should
be possible to include as a reference or option, a thought-based
development of skills and concepts. In that sense, the exposition and
introduction of mathematics may self-contained. In contrast, all other
applied disciplines (chemistry, physics) beyond hands-on methods for
lengths and measures employ tools and concepts, rules and
patterns, in an off-the-shelf, plug and play manner because
their insufficient time and capacity in high school and college
laboratories to produce or fully explain all the substances and
tools used in practice. Mathematics is simpler. Finally,
mathematics education, its extent and feasibility, will depend on ends,
values and methods for instruction, and the number of years available
for it. To keep interest alive as long as possible, skills and
concepts with take-home value should be put first.
The first task of quantitative
education is to foster an practical command of skills and concepts.
After mastering practices, mathematical and logical, students may see how
to combine them and compound them, and eventually through that see how
some practices imply the others. Algebraic and deductive practices in
mathematics cannot be applied until they have been introduced and master
inductively or progressively. That boot-up might begin with skills
and concepts that have take-home value before preparation for college
mathematics (calculus) begins in earnest. In other words
primary school and early secondary school instruction in quantitative
skills with numbers, measures and maps may progressively develop
skill in calculation and in handling concepts in a do-this, do that
manner, with notation chosen to optimize that development, not overwhelm,
and with explanation given only when it helps skill mastery and does not
distract from it. In that mathematical and logic methods may be trusted
if they provide and require results in an observable and hence verifiable
or correctable manner. Again the method for presenting work may be
chosen to favour that. Methods with take home value are emphasized
for the sake of context and motivation. Along side, the algebraic methods
of writing and reasoning and logical methods for arriving at conclusions
will be introduced in a direct, do-this, do that manner to serve
quantitative skill development. The should be introduced with take home
value, to foster sharper wits for reasoning & writing in work and
study, and to provide a wide base and practical motivation/context for a
fuller but optional Euclidean approach to "derive" or codify or
frame mathematics practices, or a large subset of them, in terms of
logical consequences of axioms - assumed patterns.
The boot-up at the pre-K and primary level may cover (not all in
order) counting and further operations with decimals - including
exact arithmetic with units of money to two decimal places and site
shortcuts for prime factorization of whole numbers <
169; arithmetic with units and mixed units of measure for
lengths and time; arithmetic with fractions where the mixed units
are now whole, halves, thirds, quarters and so on; and fraction
arithmetic with units of measure, to enable calculations with
time-speed-distance and more general rates describing proportionality
of quantities with different units of measure.
The boot-up process and the logic
development both provide stories to follow and repeat. The early
ability to remember sensations (sights, sounds, taste, touch, ...), to
tell or communicates past events and future possibilities, and to master
skills alone and one after another is a base for skill development
and knowledge, fictional or not.
While my secondary mathematics education emphasized logic and the
derivation of theorems from axioms, there is a possibility of
recognizing common practices in counting (the assumption that counts
have to be independent of counting methods have to give the same
result) and in the practical use of maps and plans (tacit assumptions
about similarity and coordinates) may be explicitly employed to develop
and imply what mathematicians would call the field, order and
completeness properties of real and complex numbers. Those
properties may serve as a base for the modern mathematics that follows
those properties. The derivation of those properties from set
theory may be left to advanced undergraduate course in
mathematics. That said, before such courses are
taken, the formalism of modern mathematics in earlier mathematics
should be introduced when and only when it aids the development of
skills and ideas - the precalculus treatment of probability
theory and functions provide examples of that. But beyond that
formalism should be employed where it speed learning or does no
harm.
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