Implementation Note
Make the Hard Easier
In mathematics course design and delivery serving the needs of
the many awareness of the logical or axiomatic structure possible for
the discipline is a plus, but that structure need not be based on a
minimal set of assumptions. The resulting level of rigour is enough for
an operational and empirical command of skills and methods of value in
calculus and/or of value in daily life. A larger redundant set of
axioms plus practices consistent with them delays development from a
minimal set of axioms, set theoretic or intermediate, to the study of
higher mathematics. At which point the set of assumptions may be
reduced in a logical manner.
The modern mathematics ideal of stating stating a minimal set of
assumptions or axioms is undermined when formulas, practices and theorems
have derivations or proofs too complicated to be given in full in class.
In the previous phase, the practices of counting, adding and multiplying
by forming and then adding or multiplying subcounts, subtotals and
subproducts was because of the take-home value of these practices for
daily or adult life should wait for algebra skill development. The
practices may be algebraically described using set and summation notation
combined, and then implied or derived from the axioms for whole and real
numbers. However, at this level, an operational command of algebra and
not its a derivation from a mimimal set of axioms is the goal. In the
next phase, discussion of operations on polymomials and functions or
computation rules, these practices may ease and accelerate the exposition
and operation mastery of the operations.
Assuming a larger set of axioms and practices makes the logical structure
of mathematics in arithmetic and algebra more accessible, and so may
encourage students to go further. Those who would like a closer adherence
to the modern mathematics curricula of the 1960s should remember its
axioms for real numbers, algebraically put, were assumed as starting
points for the introduction of modern mathematics. The manner in which
these intermediate axioms followed from the set theorectic, Zermelo
Fermelo axioms of set theory were neither mentioned nor explained in
secondary mathematics.
The recommendation here for an larger set of axioms, some algebraically
expressed, and some verbally introduced, gives and starting point for a
more relaxed and accessible form of modern mathematics.
Before this algebraic phase, assuming that products may be given
by forming and multiplying subproducts gives an assumption-based
justification for numerical practices with prime factorization and for
handling products of powers of ten, practices that would otherwise go
unsanctioned. After this phase, operations on polynomials tedious to
justify with the axioms for real numbers alone may like sanctions by
assuming the practice of adding and multiplying by forming sums and
products of subtotals and subproducts, respectively. Otherwise, the
sanction or operations of polynomials would be likely be omitted - be
too overwhelming to present in class, and so skipped.
In contrast to the modern mathematics high school programs of the 1960s,
the firt objective here is to sanction common skills and practices with
both pure and denominate numbers in mathematics, daily life and in
quantitative arts and disciplines at the high school and college level.
In that mathematics education is seen largely as service subject not only
the college level but also in secondary and primary schools. However,
site steps are designed to provide a full strength mastery of arithmetic,
algebra and logic etc needed by students, the minority, who choose to
study pure mathematics. There is no loss in rigour in that in practice
because university mathematics department in giving courses in pure
mathematics would review and refine students mastery of calculus high
school mathematics instead of taking it as is.
Clarity in Course Design
The introduction of modern mathematics curriculas in primary and
secondary mathematics in the 1960s was rather quick and urgent. The
East-West cold war led societies to want more and more engineers,
scientists and even mathematicians. The associated zeal for introducing
modern set-based mathematics as early as possible did not mention nor
explicit support common knowledge of decimal arithmetic. The modern
mathematics curricula talked about whole numbers, rational numbers and
real numbers but in axioms for them, employed decimals in notation for
them. And in calculus or real analysi at the university, a decimal free,
hard epsilon-delta view of limits, continuity, convergence.
As student of modern mathematics in secondary school and
college, I took the promise of a logical development of skills and
concepts very, very literally, perhaps more so than others of my age.
Because of that I was disappointed. The promise to explain everything
fully and consistently was not kept. Decimals and decimals methods were
employed in practice, but not mention and hence not sanction in my
secondary and college courses. Further, axioms for algebra and axioms
for geometry came from different domains, and did not quite sanction
common drawing and measuring practices with maps, plans and diagrams.
So unity and completeness was missing. But as a student, I sensed that
without having the station and courage to complain - I was still a
students struggling to master all the details and logic. Besides that I
had a strong sense that the algebraic way of writing and reasoning was
being employed and assumed in course delivery and materials, without
being fully rationalized. So I provided my own rationalizations, one I
would have shared with others, but for communication skills
insufficiently formed.
Where the theorectical development of mathematics was set-based and
decimal free, the common knowledge of arithmetic and computations,
including the discussion of error control in numerical methods, was
decimal-based. Thus mathematics has two faces, pure and common, with
common unaware of the pure, and the pure not mentioning the common. In
retrospect, at the advance level, the pure approach with its decimal-free
epsilon-delta dicussion of limits and continuity made calculus
algebraically hard than need-be - that compounded earlier silences or
gaps in the introduction of algebraic ways of writing and reasoning. The
decimal free nature of secondary and college mathematics from describing
sets of numbers to advance calculus or analysis stemmed from the decimal
free nature of ZF set-based codificaiton and axiomatic development of
pure mathematics, a set-based theory whose motivation was not
pedagogical.
The site remedy for the foregoing difficulties is put a practice
first, theory second approach to mathematics skill development in which
concrete decimal views of numbers is explicitly maintained in the
discusion of real numbers and axioms - the decimal view actual implies
some of the axioms; and in which limits and continuity are first and
fully introduced from a self-sufficient decimal perspective, one that
make the decimal-free perspective easier to digest. That is done in
Volume 3, Why Slopes and More Mathematics. It appendices push the
decimal perspective further to give proofs, using decimals, of theorems
usually stated in calculus without proof.
The need in pure mathematics for greater rigour in reasoning led
to and motivated its set theory foundations and codification in a
decimal-free manner. Ends, values and methods designed for the finer
aspects of mathematical research were not designed for common
use.
Modern mathematics curricula of the 1960s onward confirmed or
set the practice that mathematics course content in primary and secondary
school did not have be clear to parents or people with technical
backgrounds. However, the practice that mathematics course design
need not be understood by parents have be taken to the limit, so much so
that mathematicians themselves have difficulty in identifying and
undertanding what is being taught in the name pedagogical correctness
before college studies begin. Educational pyschologists of the
constructivist bent are calling for teachers, many half-trained in
mathematics and unable to describe its skills and concepts directly to
employ indirect methods for "mathematics mastery", methods that have yet
to be documented. That is besides attacks on the empirical premises of
science and technology, and mathematics itself, as part of a
justification for a spiritual approach to learning and teaching. It the
latter, true knowledge is a product of reflection, located in the mind,
and apart from masterial checks and balances. That irrational approach to
instruction in many school systems is one good reason for home-tutoring
in all or part.
My Local Problem. The Quebec secondary mathematics
program 1997-2006 was not only incomprehensible to parents, it was also
incomprehensible to your truly. To this day, despite having a 1983
doctorate in mathematics from the McGill University in Montreal,
Quebec, Canada, I have still cannot find nor follow the logic of the
provincial secondary mathematics program and the strange, strange,
texts it required and employed in grades 8 to 11 between 1997 and 2006.
The skill development program outline may provide remedy for that. More
recently, I met a mathematics 506 text, science and technical option,
whose statements appear to be mathematically correct, but there is no
explanation of the statements in the text. Self-instruction from the
textbook with comprehension of how and why the statements hold appears
impossible. The remarks here may see me first black-listed by the
Quebec Ministery of Education. However, I stand my remarks and say the
Ministery of Education may need to think out the box, and hire subject
experts from outside Quebec to rationally address the underlying
problems.
Clarity Here
The multiphase skill development framework here is outlined is in
language as simple and plain as possoble. Each phase requires more and
more technical knowledge to follow or refine. That being said, each phase
as described here should be clear to parents and teachers who have passed
through corresponding material in their calculus or secondary mathematics
education, except for the more technical asides written for people with a
strong undergraduate degree in mathematics.
Set Concepts and Notation in this framework.
More by accident than design, set notions provide a very technical
framework for the algebraic-deductive axiomatic approach and codification
of modern pure mathematics. Despite the limitation of this approach that
appeared around 1930s, the modern mathematics movement introduced set
notation and formality in to the development of primary and secondary
mathematics from counting to calculus. That may have represented the
latest advance in mathematics education, but the overzealous introduction
of modern mathematics in schools meant a new kind of mathematics which
parents, even those well-trained in mathematics and even those who done
well in their studies of mathematics, could not follow. In this 5 phase
approach, we want mathematics education with its end, values and steps to
be as clear as possible, as long as possible for parents and teachers
too. We want to step way from the situation where prepartion for the next
final examination is the most evident reason for mathematics studies in
secondary school and dare we hope, college too.
The vision of mathematics here departs from full strength employment of
sets as in the modern mathematics curricula of the 1960s. A more balance
approach is required. In it, the notation and formality of modern
mathematics can be introduced as needed to enhance an operational command
of mathematics by students unlikely to enter pure mathematics. So
mathematics is regarded as a service subject not only at the college
level, but also at the precollege level.
Where Sets Concepts and Notions are Useful: Talking about
disjoint and intersecting sets is useful in counting, adding and
multiplying. In particular, it useful in describing counting, adding
and multiplying by forming disjoint subsets to obtain counts, totals
and products from adding or multiplying subcounts, subtotals and
subproducts. Higher mathematics may describe the latter with set and
summation notation. Showing how to find the union, intersection and
complements of sets given by lists - the roster methods, or depicted in
Venn Diagrams may aid skill development in counting, logic and
probability in ways easily understood and repeated in school. The list
or roster method of describing sets can also be used to denote or
indicate collections whole and natural numbers, integers, fractions and
further rational numbers. Line segments on the real number line may
also be described with endpoint included/excluded interval notation.
After students mastered algebra, set builder notation may be employed
to form subsets of a given set of numbers by being used to expression
conditions required for subset membership.
In general, there is no harm keeping set notation in pre-university
mathematics education where it aids an operation command, does not
overwhelm, and improves precision in the description and definition
of skills and practices. That being said, we may deliberately depart from
modern mathematics twice, once in dealing with what is a variable and
second with the concept of a function.
First, the site explanation of what is a variable is verbal, and more
closely tied to the informal notion of changeable than it is to
formal concepts such as function. The site explanation is clearer to
the common person in the street. Whereas the technical identification
of a variable with say an identify function on a domain is not for
beginners.
Because modern mathematics codifies and represents function as sets
of ordered pairs, the modern mathematics curricula of the 1960 to
1980s, did the same in the high school development of functions and
in the calculus of one variable. However, in calculus of two or
more variables taught as a service subject, the perspective of
functions as set of ordered pairs I have not seen, albeit the
notion does arise in pure courses on advanced calculus and beyond
in mathematics. Talking about equivalent computation rules and
giving them in various forms eliminates the overhead and
double-talk present when and where high school mathmatics courses
identify functions and relations with sets of ordered pairs.
Critical path analysis and calls for greater accessibility,
suggests we keep concepts clear and simple.
Second, instead of identifying functions with sets of ordered pairs
that satisfy a vertical line rule, we take an operational veiwpoint
closer to the needs of most students, those who will not being
studying pure mathematics. That is functions may be identified with
and represented by equivalent computation rules. The latter may be
given by formulas and also by sets of order pairs that satisfy the
vertical line rule. For such sets, a vertical line methods may be
used to compute function values, and thus provide a computation rule.
The tongue-twisting task or overhead of explaining that functions in
mathematics are "really" set of order pairs is thus avoided.
Mathematics Education Committees
Participation in a mathematics education committee is likely to
result in a clash of wills and compromises regarding what should be
in or out. Imagine a mad hatter's tea party. The compromises will please no one.
As a mathematician, published
and perished, but a still a mathematician, I have looked at past and present
curricula and textbooks to look for best approaches and practices, and to
look ways to fill the gaps and remedy the flaws I have seen in
course design ends, values and methods. Indeed, when writing began I looked
into the current efforts of the 1990s in the hope that my ideas would not be too
different. Then I did not expect to spend the next decades trying
to understand and address technical and then context difficulties. In that, I have been thinking
out of tbe box in the well-trodden field of mathematics skill development
because I saw issues and problems that routine methods did not address.
I am prepared to discuss and debate what should be done and how
in the context of addressing those issues and problems in an empirical,
manner. The manner in which pure mathematics codifies itself for the
sake rigour is not necessarily manner that elementary mathematics should
be taught to people who need quantitative and logical skills and practices
for life on the street or life in a technical subject.
Apart from self-doubts, this Committee of One reached
a nearly unaminous decision. In developing this framework for instruction,
I think is rational and coherent, and generally correct, the best I can
do without constructive feedback. However, given its scope, I wonder if my
exposition is correct in its use of terms and notation, and if my possible
abuse of the latter is justified. That remains to be seen if mathematics education committees
with more than one member try to review and refine this refine this framework.
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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
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
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
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
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
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
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.
-
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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