Formal or Informal Peer Review
Or, pne- and Two-Way Conversations with Society in the Individual
Construction of Knowledge - the scientific method as a form of social
(joint) constructivism
Educational theorists may enjoy the follow perspective on the
individual and social construction of mathematical skills and knowledge
For each of us mathematics is or should be a static and/or growing
collection of rules and patterns involving notation, geometry and logic
which can be used and combined in a repeatable, reproducible, recorded
(or described) and thus verifiable manner to arrive at numerical results
or further rules and patterns through calculation and/or some rules and
patterns logic. This collection may grow in a rigorous manner through the
addition of numbers, rules or patterns, explicitly assumed, for better or
worse, and through the introduction of further numbers, rules and
patterns that are tested in the following sense. The new numbers,
rules and patterns have to be implied by calculations or reasoning which
uses numbers, rules and patterns previously recognized as members of the
collection, all in a repeatable, reproducible, recorded (or described)
and therefore verifiable manner. In this growing individual collection of
assumed and derived numbers, rules and patterns, each of may recognized
certain sub- collections are more reliable than others, and certain
sub-collections are more agreeable with the present and past works of
colleagues through one-way or two-way social conversations with them.
Here authors, living or past, communicate with each of us, through their
written work. And over time, the social construction of mathematics has
become a social discourse with new adherence and new directions.
As students, not quiet ready to invent or re-invent rules and patterns of
arithmetic and algebra, we may be given rules and patterns to assume
along with drill and practice, so that their use leads to repeatable,
reproducible, recorded or well-described and hence verifiable results.
Social conversation with teachers physically present or manifested
through their spoken or written work may lead to the growth of a personal
collection of mathematical data, rules and patterns. Again, in this
growing individual collection of assumed and derived numbers, rules and
patterns, each of may recognized certain sub- collections are more
reliable than others, and certain sub-collections are more agreeable with
the present and past works of colleagues through one-way or two-way
social conversations with them.
There-in lies a common knowledge agreeable to others and hence
socially more authoritative, in which individual have become
like-minded due to the manner which they accept and grow their
collection or sub-collections of rules and patterns, in a repeatable,
reproducible, recorded and therefore verifiable manner.
There-in lies a standard which individual need to accept for their
hopes, dreams and speculation to be tested and accepted by others as
part of the common knowledge.
Thus each individual has a conscience or socially acquired rules and
patterns to guide and accept in the formation of his or her
personal collection and construction of knowledge. Individual departures
from those social rules and patterns leads to individual perspectives of
a subjective nature beyond the reach and sanction of social discourse and
beyond testing. Such subjective viewpoints may be challenged by standards
set in written work of others or be challenged in social discourse with
others in the neighborhoods, teachers, tutors and parents included.
Over time, the social discourse in mathematics has led to a courses that
present rules and patterns for students to meet and master in a
repeatable, reproducible and thus verifiable manner. Answers that are not
verifiable,] allow for the correction or challenge of student habits, and
the possibility of more prudent or careful answers in the future.
There-in lies a social discourse for the guidance and construction of a
student's growing collection of mathematical rules and patterns.
Student engagement so that they follow the guidance requires a context
and motivation that may very from culture to culture Where some
cultures produce students that are potentially active or too active
participants in their own education, other cultures, subcultures and
times produce students who are quieter, more passive and for whom
classroom procedures, even those of a constructivist nature, does not
work. The parent who does poorly in mathematics may inform his
son or daughter that mathematics after arithmetic, even before, is a
waste of effort. So the difficulties of one generation in mathematics,
the awkwardness or inappropriateness of instruction, may be seen or
ducked by the next. With students opposed to mathematics, a leaner
curriculum that covers and develops key skills and concepts, those
needed in practice or needed for father learning, with material that is
nice to know but not necessary or not mentioned later omitted,
may provide a shorter, less alienating program. Not all is certain.
Extreme constructivism may hold that the conclusions arrived at by an
individual should be respected and not challenged by an instructor. The
instructor should not be an authority. Less extreme constructivism may
hold that the conclusions arrived at by at a group of students should be
respected and not challenged by an instructor. Again, the instructor
should not be an authority. However, students in school and out learn
from their environment. The environment is authoritative. Child
learn to avoid extremes of heat and cold. For better or worse, the young
and aging individuals have non-verbal and then verbal interactions with
their environment, and in doing so may adopt habits and customs for
personal safety and survival. Nature takes care or provides the
growth - the increase in physical and mental capabilities. The
development of language skills adds an iterative verbal or word-based
communication to the abilities and knowledge of a child, and the customs
or rules the child may learn and follow.
The child's level of consciousness may vary between visual and verbal.
Each society in telling stories or providing histories provides the child
or teen or adult with a greater verbal awareness and image of the
surrounding environment, rules and customs included. With this
growing verbal knowledge of rules and customs, the knowledge may become
less hands-on The question of reliability appears for knowledge
that is more verbal than hands-on. There people, even a single
individual, may operate or function at different levels. See the three
signs of intelligence above.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
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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