Mathematics
Curriculum
Notes
Printed in Canada
ISBN 0-9697564-6-1
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This folder has a tree like structure. The
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Every master of mathematical
induction knows how induction may fail in and by analogy in education.
Foreword
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Postscript - December 2008
Mathematics Training Revisited: Mathematics is an art and
discipline with observable tools and practices, and tangible rules and
practices to master via a mix of rote learning with explanation of why to
accompany the how an option. In the first instance, training students to do
arithmetic and to plug in numbers into formulas carefully may build
skills and confidence in individual abilities and in the ability of
mathematics to give repeatable, reproducible, reliable results. At higher
level, learning how to
combine rules and practices to arrive at further ones provides the base for an
awareness and then further understanding of how mathematical tools and
practices depend on each other, are built from each other. Learning how
to combine a few rules and practices to arrive at further ones provides the
starting point for the further and more detailed understanding of
of a logical or thought-based
development of mathematics and its logic from arithmetic to calculus.
Teachers, you are not using enough words: Writing
and drawing on paper in a 2D manner is often but not always worth a thousand
words - occasionally, using letters and symbols to describe and calculation
can be complicated than a verbal description. That 2D writing and drawing
provides tools and practices with conventions to follow for recording,
developing, reasoning and problem solving in an observable and verifiable
manner. That writing and drawing in providing a format for recording and
developing thoughts on paper extends our minds and memories in a very
helpful visual manner - expressions and formulas and drawing may be
grasped in a glance even while their verbal and linear description is absent
or poor. Words have been missing in the development of algebra.. The
tools and practices have been met in a silence like a silent movie with too
few captions for all to follow the plot and connections. Talking
about three skills for
algebra and what
is a variables, the forwards
and backwards of rules and formulas, and using geometry to introduce
students to the shorthand roles of letters or symbols would enlarge and
clarity the use of words and lessen the natural talent required of
students in discussing and developing mathematics mathematics.
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Four principles offer an inductive philosophy for the explanation and
comprehension of math and reasoning skills. Three of the principles were met in
a course on how to teach Nordic, that is cross-country skiing. The course was
taught one weekend early in 1981, by an instructor-trainer from CANSKI, the
CANadian association for Nordic SKIing in Flin Flon, Manitoba. Nordic ski
instruction may begin with a lesson on how to put on the boots and attach them
to the ski and also how to hold the ski poles – to be precise one holds not
the poles, but their straps in way that will guide the poles.
There is a technique here, one that is not obvious. The course gave minute
attention to the details which novice and even experienced skiers might not
know. In this course on ski instruction, the more complicated movements or
skills were deliberately preceded by simpler motions. Each of which was easy to
describe, master and/or review separately. This course turned Nordic ski
instruction into an art. The four principles follow.
1. Each discipline needs to be 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.
2. 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 again possibly
why. Of course, differences of opinion exist on which disciplines should be
taught or what pathways in them should be followed.
3. 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, more precisely 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 – a knowledge of or opinion on
how the topics in question can be organized and taught. Here again opinions
may differ.
4. Each collection of mental and physical skills should 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. [1] To
climb a ladder, the initial steps must be reachable, and each further step
must be reachable from the one or ones before it, else failure occurs.
Explanations should follow chains of reasons or persuasion which begin at the
level of the student.
In mathematics education there are two barriers to comprehension to be
lowered or removed. First, the algebraic or symbolic way of writing and thinking
is better seen and read silently than read aloud or spoken. This has been an
obstacle to the comprehension and communication of mathematical thought. Second,
the deductive nature of formal mathematics exposition with its long chains of
reason and preparation implies that concepts appearing at the end of a course
are not comprehensible to students in the middle of the course nor at its
beginning. Mathematics beyond the last concept mastered may seem impenetrable
and mysterious.
To lower both barriers, students may be given lessons, easily described and
repeated, which require a minimal formal comprehension of mathematics and logic
while presenting ideas essential to deductive and to algebraic or symbolic
thought. Recognizing, collecting and offering first such lessons may extend the
common knowledge of mathematics beyond the mastery of arithmetic, counting and
simple formulas that should be obtained in elementary school. This work
identifies such lessons and indicates ideas for math and logic instruction from
primary school to the start of college. Some of the ideas may be worth reading,
repeating or refining – the three Rs that this author hopes for.
Alan Selby
Montreal 1996
Note: The jumpmath
program unconnected to whyslopes.com
says the following: One feature distinguishes
our workbooks from regular math textbooks, however: in the JUMP workbooks,
teachers are consistently shown how to help students who are having trouble
moving forward by breaking mathematical concepts and operations into the most
basic elements of understanding and perception on page 2 of the Jump Teacher
Manual - Fractions, one of three plus pdf files in the Jump publication
page. The quote italized here complies with inductive principles given
above.
Next: Three Remarks (Postscripts)
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