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YOU are better than YOU think. Show
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful,
Edifying, Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens
eyes. Leads to greater precision.
in reading and writing
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice is
approximately correct, for some circumstances, not all. Site How-TOs
are logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site
area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
For online automated help in senior
high school maths & calculus, visit quickmath.com
For Automatic Calculus and Algebra Help with derivatives, integrals,
graphs, linear equations, matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different
range of services, some free, some not, all based on webmathematica.
Good luck.
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Explore collaborative whiteboards from groupboard,
twiddla or
scriblink.
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for secondary mathematics and for calculus in secondary
schools or college
Innovations in site material in and outside of site books, that advances for
mathematics instruction, in providing clearer or alternate ways to learn and
teach for comparison, further identify or demonstrate past shortcoming - a
benefit of hindsight.
Current State of Site Lesson Plans
- First, site lesson plans and reforms for secondary I, II and for calculus
are well-put with supporting material online in full or almost so.
These lesson plans offer are innovations fresh or recycled likely to ease or
avoid difficulties in first time and remedial instruction.
- Site lessons plans for secondary III is a proposal. Online support may
come later through lessons here or links to lessons elsewhere. The study of
mathematics can seem endless. Here is a pause to provide examples and
more examples to give a context and hence motivation for fraction and
algebra skills and sense, seen or develop earlier,, and to give numerical
and algebraic experiences for mentioned or recall in further instruction.
- Online Lessons and links are online to support secondary IV and V
mathematics, say two pre-calculus years of studies in mathematics, in
areas on Euclidean and Analytic Geometry, and on Number Theory. Here the
Number theory sections provides more material than needed. Here the
secondary IV lesson plans are online in draft form, 75% done, but still
incomplete and subject to re-arrangement. The lean program to be written or
identified here will focus on the needs of a first course in calculus.
Just in time instruction is advocated for the sake of leanness and
effectiveness in course design.
A theoretical base for senior secondary IV and V mathematics and
calculus, a high school or college subject, follows below with details
sufficient for people with a mathematical background to provide what is missing,
and thus turn theory into practice.
Secondary I and II lesson plans cover the prerequisites. They develop the
fraction and algebraic skills and sense required for senior high school
mathematics.
Modern Mathematics Curricula Revisited
The set-based axiomatic, logic-based codification, of
modern mathematics was not designed for classroom, it was adapted to classroom
use in the modern mathematics curricula of the 1950's.
While modern mathematics may derives or codifies real numbers and the
properties from say ZF axioms (assumed patterns) about sets, the
presentation of the latter belong to advanced mathematics studies that few will
meet. In place of modern mathematics curricula and its present day echoes and
inconsistencies, after many reforms in the class-room, a deliberate mixed
mathematics approach is recommended.
Modern mathematics course designs of the mid-1950's onward, emphasized the
context-free view of real numbers while inconsistently (?) employing the
latter in a mixed mathematics manner as coordinates in 1, 2 and 3+ dimensions,
and while inconsistently drawing right triangles and taking advantage of
similarity properties from Euclidean Geometry, an older view of mathematics,
to define trig functions and after that in calculus, to use comparison of
geometric areas to evaluate the limit sin(x)/x as a x decreases to 0. At the
same time, despite foregoing inconsistencies with lip service to a context
free development of mathematics, the modern mathematics curriculum and its
echoes, knowingly or not, informally require place or decimal-value
representation of real numbers for calculations and coordinates, without
sanctioning them. Then calculus further uses decimal arithmetic in the
illustration of limits and continuity, while its theory depend on decimal-free
assumptions and viewpoints of limits, continuity and convergence which do not
sanction and which avoids all mention of decimal arithmetic. There-in
lies an inconsistency or lack of connection between practice and theory in the
development of mathematics alone. Finally, the use of geometric
diagrams, models and implications in trig and and in calculus of one, two and
three variables, the algebraic treatment of units, the concepts of
space, and figures in them, are all part of mixed or applied mathematics -
departures from pure mathematics necessary for the exposition or applications
outside of mathematics, and so necessary in the secondary and college
development of skills and concepts from arithmetic to advanced calculus.
The foregoing inconsistencies, the dependence on diagrams, depart from the
initial pure mathematics vocation of the modern mathematics curricula and
present-day echoes and delivers instead, an ad hoc or accidental,
inconsistent, mixed mathematics view of the discipline. The aversion to
decimals in axioms for real numbers and all consequences separated the modern
mathematics curricula from the common knowledge of arithmetic and real numbers,
without sanctioning nor supporting the latter.
Modern Mathematics Postponed
A Consistent Mixed Math Curriculum
Secondary IV and V mathematics after the informal consolidation of arithmetic
and algebraic or literal reasoning skills, may give a deliberate mixed
mathematics view and thought-based codification of the subject with the
following practices and axioms (assumed or suggested patterns).
In this approach, properties of real and complex numbers to
be implied by geometric- and decimal-based chains of reason . Then those
properties are codified - formally stated as axioms for real and complex
analysis with explicit mention on decimal representation or definition of real
numbers. The latter sanctions the use of decimals in calculations and in the
calculus-level development of limits, convergence and continuity.
Students of pure mathematics, a minority, will later also
meet the derivation of the same axioms and the decimal representation of
real numbers from ZF set theory and or another base for the current
form of modern mathematics. Here the earlier mixed mathematics
approach provides the numerical and algebraic experience and context to
appreciate the context-free development of algbera, and real and complex
analysis in pure mathematics.
Real Number, Geometric Development
- Numbers with or without signs as prefixes may be used as coordinates along
a line following the implicit or explicit choice of a unit length.
- Unsigned and then all real numbers may be represented as decimals.
- Unsigned and then all real numbers may be used as coordinates alone or in
ordered pairs and triplets in one, two and three dimensions following the
choice of a unit length.
- 1D vectors along a coordinate line exist, and can be added geometrically
in a head-to-tail manner.
- the coordinate description of the addition of vectors along the real
number line (following the choice of a unit length) geometrically
implies methods for adding and subtracting real numbers.
- The coordinate description of whole number multiples, and then proper and
improper positive fractions of vectors (following the choice of a unit
length) geometrically implies rules and methods for
multiplication of a real number (the coordinates) by whole numbers, proper
and improper fractions.
- The negative of a vector (-1 times it) can be defined geometrically.
The coordinate description of the latter, geometrically implies the
definition of multiplication of a coordinate or real number, by -1.
- The ability to changes the unit length (magnitude and then
direction) in the coordinate location of points along a line implies and
defines a multiplication of coordinates or real numbers.
- The geometric addition of vectors can be described using coordinates
following the choice of a unit length. That was assumed earlier. But
the result of this addition of a pair or several vectors in the line
is independent of the choice of unit length. The foregoing implies the
distributive laws for real numbers.
- The head-to-tail addition of vectors in the line is commutative. The
resultant of two vectors has a mid-point. Rotation of 180 degrees about it,
and reversal of the ordered of addition (if I remember correctly) implies
addition commutes geometrically. As a consequence, the addition of
coordinates (real numbers) is also commutative.
- The product of pair of unsigned whole numbers and fractions, proper or
not, mixed fractions included, may be defined or interpreted as the area
of a rectangle. Since the area is independent of the order of
multiplication of the sides of a rectangle, the product of unsigned
coordinates is commutative for coordinates with finite decimal expansions -
continuity implies for all unsigned coordinates. (Optional: If
multiplication of real numbers follows the rule, multiply the signs, and
multiply the magnitudes or unsigned part, independent, the commutatively of
products of real numbers follows from the study of 3 more cases, 4 in all).
- The product of unsigned numbers is zero when and only when one of the
factors is zero. The foregoing "follows" from the decimal method
of multiplication and from the area viewpoint of products.
- The product of triplet of unsigned whole numbers and fractions, proper or
not, mixed fractions included, may be defined or interpreted as the volume
of a box with square corners.. Since the volume is
independent of the order of multiplication of the sides of a rectangle, the
product of a triplet of unsigned coordinates is associative for coordinates
with finite decimal expansions - continuity implies for all coordinates.
- The resultant of a head-to-tail addition of three vectors in the
line in sequence implies the sum of the first two vectors with the third
equals the sum of the first with the sum of the last two. Hence
head-to-tail addition is associative geometrically. Hence, the addition of
coordinates is also associative.
- Points in the plane can be located using rectangular or polar coordinates.
This description is dependent on the choice of a unit length, and the
direction of the x-axis. Coordinates may involve degrees in the first
instance.
- Points in the plane can be added using rectangular coordinates. This
addition is commutative and associative due to the properties of
coordinates, a.k.a. real numbers.
Complex Numbers and Trig, Geometric Development.
For details start with the first site lesson on
complex numbers for details - it is outside the complex number site area.
Then visit the complex number site area.
- Points in the plane can be multiplied using polar coordinates via the
rule: add the angles, multiple the lengths. The properties of
coordinates (real numbers) implies this multiplication is commutative and
associative.
- The identification of a horizontal axes with a real number line, the
introduction of real and imaginary parts (rectangular coordinate viewpoint
and a change of notation) implies real numbers can be multiplied using the
rule: add the angles and multiple the lengths. That rule is consistent with
the earlier rules for multiplying real numbers. It could obviate the need
for an earlier definition of multiplication for signed numbers.
- The distributive law for complex numbers is a consequence of a change of
unit length. (Rectangular and polar coordinates are dependent on the choice
of a unit length and orientation in the plane. The assumption that
the head-to-tail addition of vectors in the plane is independent of an
selected coordinate system - in other words, independent of the
length and direction of the "horizontal" unit vector, for the
latter determines the "vertical" unit vector - implies
multiplication of complex numbers distributes over addition.)
- With the aid of rectangular and polar coordinates, (periodic) trig
functions can be defined for all (real) angles - obtuse or acute included -
with the aid of a unit circle. Similarity of right triangles implies (?)
this unit circle definition is independent of choice of unit length.
Similarity of right triangles also implies that trig functions for acute
angles may be calculated using the ratios of sides in a right triangle.
Courses have the option of introducing trig functions with the unit circle
before introducing right-triangle based or related trig
calculations. The properties of trig functions are easy consequences of
the field properties of complex numbers. The latter can be from geometry and
properties of real numbers (decimal arithmetic).
Calculus
Cognitive Dissonance: In my earlier and literal
adherence to modern mathematics curricula, the use of diagrams and decimal
calculations, and other hand-waving devices not sanctioned by the axioms in
the modern mathematics curricula was a source of discomfort - a departure from
the rigorous development ideas and concepts which I was suppose to support or
encourage. The discomfort began in trigonometry with the use of right
triangles and ratios of sides to say how to compute and thus define trig
functions.
Calculus is the subject which requires algebraic ways of writing and
reasoning, and arithmetic skills and sense at full strength. Calculus
courses tend to use diagrams and decimals to develop or illustrate concepts
along side the statement of theorems and rules which may or may not be proven.
Here adherence to pure mathematics and the decimal-free viewpoint of real
numbers makes the exposition harder to follow - brings about more algebraic
shocks than need-be. Exposition demands some hand-waving, some departures from
pure mathematics. The diagram-free pure mathematics representation and
definition of trig function is not for begginners. That the exposition or
introduction of trigonometry and calculus requires a mixed mathematics approach.
The latter can be presented or developed in a thought-based or logical fashion
in a manner, self-contained, sufficient for the needs of other disciplines, and
sufficient for the development of the algebraic-deductive and computational
ability prerequisite to the study by a few of modern mathematics.
Modern Mathematics Postponed
A Consistent Mixed Math Curriculum
In the foregoing development, properties of real and complex numbers are
geometrically implied. Elements of this mixed mathematics development can
be seen in the geometric or vectorial illustrations of properties of real
numbers when they are introduced earlier in high school or primary school, at
least where real numbers and there properties are not learnt fully by
rote.
Once the properties of real and complex numbers have been geometrically
implied, and once the decimal representation of coordinates, that is real
numbers, assumed, a reformed or modified modern mathematics program, echo of the
late 1950's, can be begin again with the set-based statement of the axioms -
here arithmetic patterns algebraically described for both real and complex
numbers - plus explicit assumptions about the decimal representation of
real numbers. The latter provide continuity with the common knowledge of
arithmetic with decimals, alone or in the numerators and denominators of
fractions. The latter provides a mixed mathematics framework for the further
development of trigonometry and calculus.
Then limits, continuity and convergence in elementary or advance calculus
explicitly exploit the decimal representation of real numbers.
Courses on analysis, real or complex, may switch to decimal free viewpoint and
even included the context, coordinate-free, development or derivation of
real and complex numbers, and then functions of real and complex variables, from
ZF decimal-free assumptions about sets. See site volumes 2 and 3, and site
areas on number theory and complex numbers to learn more.
The foregoing program I suspect is generally solid. Most, if not all,
elements are online. That being, the program is understood when and only when
readers see possibilities for improvement.
Remark 1, Why Sets: Before pure
mathematics courses on real and complex analysis, set formality in the
development and description of of real numbers and complex numbers and
functions provides a precise framework for this development, for counting
methods in combinatorics and probability. So set-based language and
properties of sets can be woven into a mixed mathematics curriculum without
harm and with some benefit.
Remark 2, abrupt introduction of a concept: Between
the presentation of functions as calculation, mapping or assignment rules, and
the identification of functions with their graphs, there should not be an
abrupt transition. The identification of functions with their
graphs, a set of ordered pairs which satisfies a vertical line property,
is a feature of modern mathematics. We might avoid the transition
altogether by identifying the graph of a function with a set of ordered pairs,
and explaining how a set satisfying the the vertical line test yields a
function (a computation or assignment rule) via a vertical line based
calculation or assignment method. In secondary and college level mathematics,
I would recommend talking about functions as rules and not identify functions
with their graphs, even though there is a one to one correspondence between
functions and sets of ordered pairs which pass the vertical line test.
The identification is a technical complication but left for later, a
technicality introduced the thoughts of progress in education extended to the
inclusion of more and more college level material in high school courses. See
the site area on analytic geometry coverage of functions for an effort to
avoid the abrupt transition.
Remark 3, the problem of units: In applications, in
the physical sciences and in economics, quantities of length, time, mass and
money appear alone or in ratios. Axioms for calculations for quantities
need to be devised to sanction the calculation, numerical or algebraic, that
involve units and changes of scale in units. While students may first obtain a
pre-axiomatic, thought-based knowledge of mathematics through calculation
practices with and without units that yield repeatable, reproducible and hence
verifiable results, the statement and use of axioms for real and/or complex
numbers without mention of units may be sufficient to introduce the logical
organization and codification of pure mathematics, but is insufficient for the
requirement of applied in the domain of numerical and algebraic calculations
with units. While quantities and operations on them can be mapped into numbers
and operations on them, and so into the domain of pure mathematics, via
an explicit choice of a system of units which eliminates or factors the units,
axioms for real and complex quantities would be useful if stated explicit or
verbally described and sanctioned along side the statement of axioms for real
and complex numbers.
Remark 4, solution (?) for the problem of units: The
associate law for addition and multiplication of three real or complex numbers
can be stated algebraically. However in practice, the sum and products of
terms can be computed in many ways. While advance mathematics can inductively
define and thus formally describe and imply how the order of addition
and multiplication in sums and products of terms and factors does not affect
their values, in early classes we may state the associative law or axiom for
sums and products of three terms or factors, and then verbally imply the more
general law or consequence. A similar approach may extend axioms for
real and complex numbers to real and complex quantities, and so permit units
to be carried through calculations. Axioms for real and complex numbers
sufficient for the codification or formalization of pure mathematics then
represent a partial codification or formalization of mixed mathematics.
Remark 5. In modern mathematics, the context-free
development of real numbers from axioms (assumed) patterns involving sets is
not for novices. The connection of context-free modern mathematics to the
concrete and hands-on use of coordinates in diagrams or physical diagrams
requires or implies mixed mathematics assumptions about geometry with or
without coordinates, assumptions that may have predated the context-free
development of mathematics. Since those assumptions or equivalent ones have to
made in secondary school mathematics, the mixed mathematics program above
exploits such assumptions or equivalent one to geometrically or physically
imply the properties of real numbers
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Mathematics Education Essays etc
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Theory of Knowledge
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The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
In for a Penny
Constructivism and Cognitive Theory
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
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words before symbols
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videos on primes, lcm, gcm,lcd, square roots etc
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preview, algebraic
preview,
3 study guides,
much more
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consequences for trig & vectors in the plane
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algebra
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hindsight
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construction, duplication & Isometry - Failure
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sum in triangles -//
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Similarity
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and of proofs
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& (?) derivatives
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& proportions - slopes & rates included
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dot & cross - cosine
law
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