Foreword
Volume 1B, Mathematics Curriculum Notes
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.
Mathematics
Curriculum
Notes
understanding and explaining
reason and math
Volume 1
by
Alan M. Selby
Ph. D.
Printed in Canada
ISBN 0-9697564-6-1
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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.
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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.
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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 again 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, 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.
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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
Selby A, Volume 1B, Mathematics Curriculum Notes, 1996.
Postscript - February 2011
In two years of UK grammar school 1965-7, mathematics lessons consisted
of given rules and patterns in algebra, trig and geometry, all given in
what I presume was a mathematically correct manner. I was too young to
know otherwise. In then next three years of English Quebec secondary
schooling, rules and patterns were also given but in axiomatic
structure. That is, rule and pattern mastery started from axioms -
assumed patterns algebraically or geometrically put. The fine print in
my Quebec high school textbooks emphasized or valued starting from a
minimal set of axioms. The development was essentially logical. But
there were four flaws in the Quebec portion of my secondary school and
junior college education - nuances small and large.
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The arithmetic mastery of decimals and fractions met in my UK
primary and secondary school days was required, but not explicitly
sanctioned. That departed from the ideal in textbook fine print of
building mathematical knowledge on explicitly given axioms
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The algebraic way of writing and reason was required to understand
the shorthand role of letters and symbols in the axioms and in the
further development of algebra, geometry and science in my UK and
Quebec studies. But no course and the fine print in all of my
textbooks did not provide any sanction for this shorthand role.
While I found my own rationalization, self-constructed, I saw the
instruction of myself and fellow students slowed by the silent
assumption use of algebraic skill. Course design and delivery
assumed it without clearly or explicitly discussing it. Talking
about three skills for algebra, a lesson given in fall 1983,
represented my second effort to address this flaw. The first was in
a 1975 handout at a McGill University open house.
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The use of order pairs as coordinates in the plane ias in the
analytic approach to geometry was mixed with synthetic Euclidean
Geometry, the line, circle and triangle drawing approach. Having
two approaches unreconciled departed from the fine-print promise in
algebra of a minimal set of axioms.
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The use of drawings and diagrams, disowned in the algebraic course
view of mathematics, was present in both high school level
trigonometry and in later college level calculus. And geometric
drawings were employed along side mathematically and algebraically
deep utilization of epsilons and delta views of continuity and
convergence, with the underlying theory avoiding decimals, while
examples and illustrations employed or required decimals.
My strength and weakness as a student and a human being was and may
still be a reflex to take everything literally. So I was disappointed
with my high school and college education because the algebraic way of
writing and reasoning was not introduced in a clear step by step
manner. In all the courses I took and in all the textbooks I saw, this
shorthand way of writing and reasoning was required while the effort to
explain it was absent or, when present, insufficient. I was also
disappointed by the espousal of an ideal, the consistent and full
logical development of mathematics from axioms - assumed patterns.
Course design and delivery failed to deliver. In retrospect, following
graduate studies and doctoral degree in mathematics, and further
thought, the ideals espoused represented the hopes and motivation of
modern mathematics. But mathematicians if not mathematics educators
since Godel in the 1930s were aware that the hopes were not feasible.
None the less, the modern mathematics curricula echoed those hopes and
emphasized a rigour in ways that many tried to take literally.
In retrospect, mathematics is an empirical subject. Its development is
a mix of practice and theory. Given that, I first recommend K3-9
observable skills and practices with take home values be learnt and
taught in manner that emphasizes their value, with explanations to aid
mastery without overwhelming it, with the end of making students aware
of the domino effect of errors in calculations and reasoning, and with
the end of showing students how to do and record steps in manner they
and others can do or check. With skills that have take home value,
students may expect instructors to teach correct methods - methods that
can be learnt by rote if the student wishes, methods for which
explanations why are available for reading by students when or if they
want.
For skills that have high take home value for life in the street or
work, rigourous mastery is more important than comprehension. But in
the preparation of students for college programs, those that may employ
one variable calculus, or more. skill development needs to show
students how to use and combine rules and patterns in applications, and
in the development of further rules and patterns. The ability to apply
and the ability to reproduce in all or part what has been shown is in
itself an observable skill. Skills that can be seen can be described,
confirmed or corrected. But the thought-based development of skills and
concepts does not require a minimal set of axioms. Minimality here
represent a value of higher mathematics education. The development
requires a convenient and consistent set. This set may provides the
opportunity for students to see how rules and patterns may be applied
to obtain results and combined to obtain further rules and patterns.
The set develops and sanctions decimal, algebraic, geometric and logic
skills and practices, so that the flaws indicated above are avoided
while the stage is set for further studies in college programs. That
represents the current or last objective of site material.
The initial objective when writing began was not so large. Writing
began with the inductive criteria for course design and delivery, with
the question of how to motivate skill development, and with the hope of
making the modern mathematics curricula of the years 1965-90 more
accessible - less challenging to learn and teach. The initial aim was
to report the inductive principles and a few appetizers and starter
lessons to educational authorities, and then leave further work in this
matter to others. However, I was not formally qualified to present
ideas to educational authorities, or academic committees, more ideas
followed, More over, in 1989, before I started writing, education had
shifted from valuing skill development in reading, writing and
arithmetic, mathematics included, to saying true knowledge is a
personal affair, located in the mind, apart from observation and
correction of teachers, and not associated with perfomance, that is,
observable rule and pattern mastery. That subjective movement in
education, led in US and Canadian mathematic education by the NCTM
rtoday is the reverse of that espoused by the NCTM in the 1950s. Site
material provides a rational alternative.
A K1-9 emphasize of skills and practices with current or potential take
home value for work or life in the street represent student-centered
skill development. The K7-12 parallel or subsequent emphasis on skills
for one-variable calculus and college programs in technical fields
represents college oriented skill development for careers - not
guaranteed - that may benefit the student or society. Such instruction
has intellectual and/or take-home value for some, not all. That is not
ideal. But recognizing this situation represent a step forward from the
situation in which secondary mathematics instruction is clouded in
mystery, with the question why learn or teach this has the bureaucratic
answer: preparation for final examinations. Moreover, this college
oriented instruction can be offered, does not have to be taken nor
required, once most skills with take-home value have been covered. The
latter needs to be done first. It will be useful to all, including
those students aiming for college studies who might other wise miss it.
As a high school teacher, I once had to give a mathematics course
required for graduation to a group of students who have benefited from
a review and consolidation of skills with take-home value. Instead,
their time was wasted because of government standards for education
that forced the learning and teaching of topics with no academic nor
take home value for the students in question. That situation needs to
be addressed.
END OF POSTSCRIPT
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For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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