Standards and Principles for Mathematics Instruction as Skill Development
Rigour and Thought-Based Development
From arithmetic to calculus in mathematics, skill development depends on
showing and explaining what to do in an observable, repeatable and
reproducible manner. In that, mathematics programs for instruction may
set observable standards for the presentation and communication of work
in visible steps that can be corrected or confirmed. In mathematics as in
other skill-based arts and disciplines, some steps may performed by rote
or automatically while others steps may be done with full or partial
comprehension. In the latter case, reasons to justify steps may included
in the steps or not. Thus skill development is possible at many levels.
Which way to go depends on the ends, values and abilities of students and
teachers. The latter may vary between individuals and shift over time.
At the level of algebra and geometry, where one student has mastered
decimals and fractions by rote and another with more comprehension, there
may be no observable difference between their algebraic and geometric
works.
In my own case while studying what might be possible for the
thought-based development of mathematics from counting and arithmetic in
primary school to calculus in college or the end of high school, I
observed gaps in my comprehension of decimal arithmetic - my advance
studies in mathematic despite their algebraic-deductive bent had taken
arithmetic methods for granted.
Skill development in arithmetic, algebra, geometry, calculus and
statistics may involve varying mixes of explanation, comprehension and
practices between different classrooms in the same or different schools
while still leading to repeatable and reproducible results. In the
foregoing, the ability to do is a must while the ability to understand and
justify steps is optional, a question of depth and style.
Mathematics skill development is robust and flexible enough for practice
or learning to do to be accompanied by great variation in the depth and
extent of theory and comprehension. It appears that many aspects or steps
of arithmetic, algebra, geometry and calculus may be met and mastered
with varying degrees of compartementized logical or thought-based
development of skills and practices. In general, by showing students how
to do and record work in steps, steps that may be done automatically or
include a degree of self-validation - reasons for them, instruction sets
an observable and verifiable standards, standards that may vary between
modules or topics. Variation is to be expected because of variation
between different programs and within each program, because of variation
between different classrooms. The latter variation may be lessened by
course material in which common formats for doing and recording work in
visible steps are clearly put.
Domino Effect of Care and Mistakes
In the case of skills and practices with take-home value for work and
life in the street, rote mastery is likely to be more or as important has
learning to do with full comprehension of why. Explanation needs to provide
enough comprehension to do and record steps in a practice first, additional
theory optional manner. There explanation needs to be provided to make the
steps feasible without overwhelming students and their teachers with too
much detail.
In the case of skills and practices at the pre-college and college level,
student appreciation and need for learning to do alone or with partial or
fuller comprehension of why will depend on their hopes and estimation of
their work or academic destination. The extent and value of mastery of
the why may depend on and determine the career or academic destination of
students. Some career or academic destination will require greater
mastery of theory than others. The task of course design and delivery is
provide mathematical abilities with the greatest explanation of why that
does not overwhelm students and their teachers. The task of mathematics
instruction in the development of skills and practices is to provide
enough known-how for skill and practice of student work and academic
destinations. In primary school mathematics, mastery of arithmetic,
measurement practices and geometry can have a local take-home value clear
to parents, students and teachers. But in secondary mathematics, the
take-home value of most topics becomes less and less clear or more remote
from daily life.
The NCTM standards and principles for mathematics
education as part of a constructivist path for instructon emphasizes
true knowledge as a private matter, located in the mind and apart from
observable and reliable performance on tests. The constructivist may
say that performance on tests may vary from one to the next, and does
not truely capture what a student thinks.
do not yet specify in detail what
skills and concept should be mastered. They represents standard and
principles not for con
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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
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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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