Chapter 2. Communication of Ideas
This chapter describes the inductive educational principles or pedagogy
which guided the organization and content of this book on reason and
its mathematical companions. The principles are echoed in Volume 1B,
Mathematics Curriculum Notes. They reflect the pedagogical
principles of the NCTM in 1950-89, but not those of the same
organization 1990-2010. Masters of mathematical induction in knowing
how the latter may fail will easily recognize how a like failure may
hit step by step skill development.
Initial Assumption
No area of knowledge is properly mastered until it can be readily
explained to others. Each subject needs paths (or curricula) passing
through easily described and easily repeated ideas and skills. Each such
path permits those who have traveled along it to tell others what to
expect and hopefully why. The existence of such paths may show that an
area is well-understood.
Differences
Differences in our alertness, how awake we are, imply that a
lecture, a book, a picture, or a film is seen and understood differently
by each of us, depending on the time of day, etc. All of us are
witnesses. Some witnesses see more than others. In describing and
explaining ideas to several people, we need to speak in a way that each
listener (or witness) will understand as much as possible. When we speak
to several people, our words will be understood by each differently. In
communication, especially in teaching, these differences need to be
considered.
Principles for Instruction
For learning and teaching in all disciplines, the following overlapping
principles appear self-evident, at least once stated.
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Each discipline needs to be taught or presented, so that students
understand what they are learning and why. Without a knowledge or an
opinion of why, students may lose interest and not go further. The
why could be approximate a little uncertainty leaves room for
thought.
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Pathways through easily described and repeated ideas may extend
knowledge of any discipline, area of thought or belief. One or more
paths through easily described and easily repeated topics may allow
those who travel further to tell others willing to listen, what to
expect and possibly why. Of course, differences of opinion exist
on which disciplines should be taught or what pathways in them should
be followed.
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Awkwardness with an idea or skill often signals difficulty with
previous ones. It may indicate at least one earlier skill has been
missed or forgotten. When an awkwardness is felt or seen, learners
should go or be taken back to practice the missing skills, and
possibly the ones just before them. This retreat aims to restore
confidence and build skills, so that the learner can go further. This
requires a diagnostic skill and a knowledge of or opinion on how the
topics in question can be organized and taught. Here again,
opinions may differ.
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Each collection of mental and physical skills could be organized into
a ladder-like sequences of steps with the basic ones first and the
more advanced ones second. Learning in any subject stumbles when a
first or succeeding step is not easily reachable from those before
them. To climb a ladder, the initial steps must be reachable, and
each further step must be reachable from the one or ones before it.
Explanations should follow chains of reason or persuasion which
begin at the level of the student before advancing further.
Remark 1. An alternative to ladder-like structure is a tree-like
structure. Here skills and ideas are represented by the branches of a
tree or bush. The tree can be climbed when those branches closest to the
ground are accessible while the higher ones are accessible from branches
below. For ease of exposition and comprehension, the organization of
ideas into a flat tree-like structures where each branch can be reached
via a few lower branches or directly from the ground is to be preferred
to the case in which most branches are high and can be reached only from
the one just below it. This is to simply observe that short chains of
reasoning are better for explanation and comprehension than longer ones.
Remark 2. These words or thoughts on communication of skills echo
a course on how to be a cross-country ski instructor. The course was
taught one weekend early in 1981, by an instructor-trainer from CANSKI,
the CANadian association for Nordic, that is cross-country, SKIing. The
course gave a piece by piece approach to instruction. The objective was
to build both the confidence and ability of students. The course
emphasized that difficulty with a skill signaled the need for a retreat
to, or even before, previously mastered skills. The detailed structure
provided turned cross-country ski instruction into an art. Arts of this
kind are required in other areas of instruction.
Plots and Subplots
A writer normally offers a main plot along with a few or several
subplots. The main plot and ideas should be obvious to everyone the first
time. After the main plot is noticed, the subplots themselves and links
between them may become apparent. On reading for the first time a book or
an article, we grasp and master some of its ideas. The rest remains to be
found. Others ideas and messages just pass us by. A second or third
reading may help us see them.
In the Classroom
A teacher has to explain ideas to students with different backgrounds and
knowledge. One approach to this is to divide students into groups, and to
speak to each group separately. A second approach is to speak to all in a
way that speaks to each group at its own level without being too
imprecise. A teacher may try to explain and broadcast ideas or knowledge
at several levels at once. The intent here is to allow each listener to
tune to the level most suited to him or her with reinforcing echoes from
the other levels.
Echoes can be provided by the repetition of words and phrases with
similar, like or related meanings. They can be used one after another in
a single sentence. Familiarity with one word or phrase leads to an
understanding of the others. The latter in turn favors variety instead of
monotony in speech. Multiple themes and multiple levels of meanings may
challenge the listener or student and take some beyond what was expected
of them. Some redundancy and repetition in communication is fine. Too
much may be boring.
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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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