Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
For students of reason in society, science and technology:
Pattern Based Reason describes
origins, benefits and limits of rule- and pattern-based thought and
actions. Not all is certain. We may strive for objectivity, but not
reach it. Postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theories and practices.
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics.
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
Mathematical Induction - a light romantic view that becomes serious.
Responsibility Arguments - his, hers or no one's
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design. Site Theme: Different entry
points may be easier or harder for knowledge mastery.
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals.
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this.
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign.
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function?
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms.
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
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www.whyslopes.com >> - Volume 1B Mathematics Curriculum Notes >> Foreword Next: [Chapter 1 Introduction.]               
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.
understanding and explaining
reason and math
Alan M. Selby
Printed in Canada
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
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.
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.
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.
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.  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
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.
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.
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
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.
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.
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
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
END OF POSTSCRIPT
www.whyslopes.com >> - Volume 1B Mathematics Curriculum Notes >> Foreword Next: [Chapter 1 Introduction.]               
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
1996 - Magellan, the McKinley
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology
-- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The
NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4,
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. May buildings in
space are similar by design.
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.
Why factor polynomials - this 1995-96 lesson introduces calculus
skills and concepts. It may also may be given to introduce further function maxima
and minima both inside and at the ends of closed intervals.
Check Arith. Skills - too many calculus and precalculus
students do not have strong arithmetic and computation skills. The
exercises here check them while numerically hinting at
equivalent computation rules.
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