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
|
|