Principles and Standards for Skill-Oriented Mathematics Education
In reading, writing, reason and mathematics, learners 4 to 16 need skills and practices which
might be useful for life at home, at work, and on the street. Some will also need skills and practices
for calculus-based college programs.
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In and after arithmetic, students need to be shown how to do
and record work in steps that can be seen and checked. Students
also need to be given enough information for the work. Before
algebra begins in ernest, skills with decimals, primes, fractions
and signs are required. In that quality and accuracy is more
important than speed. In doing and writing work in clear steps, the
domino effect of errors will be seen. Avoiding this domino effect
is an end, value and tool for building skills and confidence at
home and on the street.
The child who says there are too many letters in the
alphabet is mistaken. The child will eventually learn that all have
to be learnt in order to read and write. Likewise in mathematics,
students wanting to learn algebra, geometry and calculus without a
proper command of arithmetic are mistaken. Later skills depend on
earlier ones. If skills are skipped or are not properly mastered,
difficulties will follow.
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Learning to think and act carefully is one reason for
arithmetic mastery. But careful thinking can also be learnt in
logic. It is a language skill. Seeing the difference between
saying A IF B and saying A IF AND ONLY IF B is a key step along
the path to careful thinking, reading and writing. The student
who learn to figure well and think, read and write with precision
may avoid some bad decisions - avoid actions or agreements in
which the numbers or words are not favourable. Learning to figure
and think careful is not just a matter of mastery useful skills,
it is a matter of self-defense.
Skills and practices may be mastered with full, partial or
no comprehension of why. In my pure mathematics education, the
axiomatic method provided a logical home for understanding and
eventually explaining university and late secondary mathematics.
In this home, the statement of axioms for sets, algebra or
geometry gave a base and framework for a deductive comprehension
for a large part of pure mathematics at the high school and
college mathematics. In some school districts, the axiomatic
approach may still be strong. In others, only a ghost may remain.
In retrospect, the axiomatic method I met - swallowed hook, line
and sinker - did not fully sanction all skills met in practice in
mathematics itself, in science and on the street. There were some
logical practical and instructional gaps. See Volume 1 and
1B.
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Mathematics skill development is modular. In primary and early
secondary school, counting and calculating with decimals, fractions and
signs may be mastered through a mix of rote learning and comprehension.
For learning to do and record work in steps that can be seen, students
need to be shown how with enough explanation to understand how. In each
module, rules and methods for doing and recording work in visible steps
for immediate or later checking may be given and/or explaining.
Mechanical rigour in logic and of arithmetic, algebra, geometry
and calculus may be seen in the first instance as the ability to do,
record and present work in steps that can be seen as done or later for
confirmation correction. This mechanical rigour requires adequate
explanation and comprehension of the work for the steps to be done.
From pure mathematics, we have the notion that mastery of skills and
practices should be part of an axiomatic framework, with reasons to
justify each step fully understood if not written, in practice steps
may be learnt by rote or done automatically in ways that require
avoidance of the domino effect of mistakes. Rigourous mastery of
arithmetic, counting and measuring practices only requires the ability
to do in an observable manner for the sake of confirmation and
correction. In this, rigour consists of knowing how to do in an
observable, repeatable and reproducible manner. Higher level rigour as
in a deeper comprehension of the origin and a thought-based
justification of methods may come later in an optional manner. Rigour
in algebra and geometry may vary from learning to do by rote or
automatically to understanding the nuances as to why each step and
substep is justified. In the latter, learning to record the nuances or
reason for each steps and substep could be part of skill development.
When the latter is emphasized, theory too becomes within the reach of
observable and verifiable skill devleopment.
Mathematics can be learnt and taught in a modular manner with a
focus on providing and fine-tuning mechanical rigour in a practice
first, theory second or optional manner. Given a students who can do
arithmetic with decimals and fractions efficiently, the secondary
school algebra or geometry teacher will accept that and have no reason
to review arithmetic that has been mastered in practice. When
solutions, derivations or proofs appear in further mathematics, the
main concern is not what a students thinks, but what a student can do
in an observable and correctable manner in accordance with the rules
and methods. The latter provides observable and verifiable performance
standards which parents, students and teachers may see as right or
wrong.
The NCTM principles and standards for school mathematics, the
right way to teach, begin with the assumption that true-knowledge is a
private matter, located and built in the mind of each, apart from any
reliable form of testing, and apart from correction. At the school
level, that implies instructors do not have the authority to correct
students. At the research level in mathematics, science and technology
that implies the peer review process is a pedagogically incorrect
process. The NCTM maintains instructors should provide students with
circumstances and food for thought for students to discover and build
their true-knowledge. In the principles and standards for school
mathematics skill development advocated here instructors have an
obligition to show students how to do and record work in visible steps
for the sake of checking, and in that try to offer enough explanations
for students to be able to do in a repeatable and reproducible
manner.
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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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