Chapter 9 The Two Ends
Secondary or intermediate mathematics instruction should provide a smooth
transition or bridge between the start and finish of this instruction.
For most who attend colleges, the start is met in elementary or primary
school and represented and the finish is met in college service courses.
College Service Courses
College service courses refer to those courses taught to students not
specializing in mathematics, usually the majority of students in a
college. Calculus, often the lowest level of mathematics taught for
credit in a college exposes students to algebraic thought at its full
strength. Miscomprehension of the algebraic way of writing and thinking
usually leads to failure or intellectual hardship if not in a first, then
in a subsequent calculus course. Calculus instructors would be most
content if their students had previously mastered logical reasoning and
the symbolic or algebraic ways of writing and thinking along with some
trigonometry and optionally geometry before entering calculus [1]. College faculties today in their exposition of
mathematics start almost from scratch [2].
College level instruction in math, commerce, engineering, science and
technology may assume or build on a knowledge of logic, of basic
trigonometry, and of the symbolic and algebraic way of writing and
thinking. These skills are required not only for the exposition and
mastery of calculus but also for computational methods in various
disciplines and more advanced mathematics courses (analysis, abstract
algebra, differential equations, numerical methods, differential
geometry).
In college math courses presently taught in North America, there is
presently little emphasis on set theory or derivation of mathematics from
first principles. Algebraic and computational skills are emphasized and
graded instead. Science, engineering and commerce departments want
mathematics courses to prepare their students for computation and not the
set theoretic codification of modern mathematics.
Only students specializing in mathematical disciplines [3] require or employ the set notions of membership,
unions, intersections, ordered pairs and complements. But for students in
disciplines not concerned with the set theoretic codification, wrapping
concepts in set theoretic terms may be a distraction. The wrapping should
only employed when it clarifies the exposition, or provides a second
perspective.
Elementary Courses
Primary or early secondary school instruction has say the role of
providing a knowledge of decimal arithmetic, counting and the use of
simple formulas. This instruction leads to a mastery of some rule and
pattern reasoning or figuring and a familiarity with repeatable,
reproducible processes and thus their verifiable results. This rule and
pattern reasoning appears before the use of implication rules that also
establish conclusions in a repeatable, reproducible and thus verifiable
manner.
The demands on primary instruction are described next. Note the
innovation, the discussion of rectangular and polar coordinates, and then
associated discussion of navigation and complex numbers. More comments or
details will be given later.
1. Counting, weights and measures.
2. The conversion units of measurement.
3. Formulas for perimeters, areas, volumes and interest (simple or
compound) along with illustrations. These formulas show how shorthand
notation describes calculations that may or might be done or postponed.
4. Decimal positional notation for whole numbers and then denominators
and numerators of fractions. Arithmetic computations should be done
by hand and explained in such a way that students understand or see from
examples, their meaning and justification. The explanation of powers of
10 and perhaps scientific notation for numbers is part of the
comprehension of decimal positional notation.
5. Repeating and non-repeating decimal expansions for fractions
(rational numbers) and for irrational numbers such as Ö2 and p.. The presumed
convergence of these decimal expansions. The physical notion that each
decimal place serves to better locate a point on the real number scale. The
presumed correspondence with numbers and a coordinate axes is exploited
here.
6. The need for care in arithmetic and the objective nature of
arithmetic. Arithmetic in primary school should make students aware
that a single false step in a calculation cast doubts on the results of
all the following steps. Arithmetic in primary school should also lead to
the expectation of objectivity. Results obtained exactly should be
independent of the computer, here a student with a pencil and some paper.
7. Round-off problems in calculations, inaccuracy in measurements,
and the number of significant digits in decimal expansion. There should
be an uncertainty of at most half a unit in the last retained decimal
place, if the accuracy range is not otherwise indicated.
8. Measurement of regular and irregular areas by covering them with
triangles, squares or rectangles, and then summing. This
approximation of area, sometimes exact, represents a practical skill.
When a region is covered by small squares, the convergence of inner and
outer approximations as the squares are made smaller could be
illustrated. (The outer approximation is the sum of the areas of squares
in the covering which intersect the region. The inner approximation is
the sum of the areas of the squares in the covering which are included
fully in the region.) The approximation discussed here is also a
foretaste of approximation, convergence and summation in mathematics
after arithmetic – an example which can be later recalled in the
explanation of integral calculus, so that the later explanation of
mathematics coheres with and is not disjoint from the earlier
description.
9. Coordinates on the line. Students only familiar with unsigned
decimal numbers can be introduced to signed numbers as a means to signal a
position on one side or another of the origin of a line. The height
of water above and below a zero-level mark provides a first example.
Temperature measurement in Fahrenheit and Celsius provide two further
examples and show again that the choice of the origin may be arbitrary.
Moreover, addition and subtraction of positive numbers can be identified
with displacements in the positive or negative direction respectively.
Similarly, addition and subtraction of negative numbers can be identified
with displacements in the negative or positive direction.
10. Cartesian and Polar Coordinates, their use in locating points and
their measurement on maps. Here elements of navigation could be
introduced. Students could be given a map, a starting location for a boat
or airplane, and then asked to track the location of the latter through a
sequence of displacements. The latter could be described with numbers by
coordinates shifts or angles and lengths. They can also be described by
arrows or vectors drawn on a map to represent a sequence of motions. This
leads to the (map) addition of arrows or vectors. Beyond this, the
multiplication of arrows or points in the plane can be easily defined
using the add the angles, multiply the lengths rule for complex
numbers. The latter in a pre-algebraic fashion will justify the law of
signs and allow square roots of negative numbers to be identified. Ease
of exposition and visualization is the justification for this last
innovation. It demystifies negative and complex numbers even before
algebra is studied.
At the end of primary school, or at the start of secondary school, students
could be shown the mechanics of buying and selling. They could play games
which in the simplest case involve transactions between a customer and
retailer, and in the more complicated case between retailer and wholesaler,
and wholesaler and suppler. Keeping track of the discounts, accounts and
methods of payments would be an exercise in a flexible rule-based reasoning
process. Here students could be challenged and required to do arithmetic
without a calculator — tell them to imagine a power failure. Questions of
how to verify results could be done. Those students required to master the
most complicated computations and transactions would be favoured or better
prepared than students shown only the simplest material[4]. Mastery of decimal arithmetic first became popular with
merchants as a means of doing and recording transactions and balances. This
provides another setting for the discussion or introduction of positive and
negative numbers or balances. The above topics and others are discussed in
chapter 11 on Elementary
Instruction.
[1] The introductory chapters in the companion book Why Slopes and
More Math however assumes a fragile command of the algebraic way of
writing and reasoning, and tries to reinforce it. Instruction in calculus
can be regarded as another or last opportunity to demonstrate and explain
the algebraic thought process.
[2] Many colleges have remedial
programs for arithmetic and algebra skills alongside more advanced, but
none the less secondary school level, pre-calculus courses in algebra
or trigonometry.
[3] For instance: probability and
statistics, mathematics itself, electrical engineering, theoretical
physics computer programming or database organization, ... .
[4] In retrospect, the chapter
should be callled Symbolically or Algebraically Described
Rules for Arithmetic.
|
|
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.
|
|