Problem Solving Skill Development
Routine to Non-Routine
Quantitative skills and methods (mathematics) represents
a growing body of rules and patterns that can be carefully, in other
words intelligently, used one at a time and one after another, alone
and in sequence, to arrive at repeatable, reproducible, observable and
hence verifiable results.
Mathematics in the first instance, is an art form, a discipline, with
simple and then more complicated rules, patterns and methods to master.
Once students have sufficient drill and practice, sufficient exposure,
the use of some skills and concepts should become familiar, automatic,
and their use no longer an adventure. In daily life for school and later
for work, students need to learn to address and solve routine problems
with ways that lead to repeatable and reproducible results - reliable
results. Creativity is not over-rated when and where reproducible and
reliable results are wanted. Figuring skills for problem of repeated or likely
value in daily life should be mastered, by rote if need-be, with deeper comprehension
if possible, so that the routine numerical and geometric
problems of daily, present or future, are addressed automatically.
With that end, the first several of mathematics learning and teaching
will have take home value clear to students, their parents and their
teachers. To avoid re-invention of the wheel,
students need to meet and master problems and situations in which mathematical
skills and concepts can be applied in routine or predictable manner. People have to learn to walk become they can run.
The first aim of mathematics instruction is to give students skills and
concepts - previously found or hard-won by previous generations - for
addressing the routine or likely problems and puzzles of daily
life in a straightforward manner - creativity not required.
For that, knowledge of and avoidance of the domino effects
of errors would be a must. For mastery of chains of reason and of
the difference between one and two way implication rules would
encourage precision in reading and writing, and so help avoid
the domino effect of errors.
Skill development may begin with
learning or showing how to do and record steps in problem solving
in formats readable and clear to by peers, teachers and one-self, so that
figuring and reasoning steps may be seen and confirmed or corrected. In this, work
that does not exist or cannot be read is an error to be addressed. In this, the doer
and others may see the domino of effects of being careful and the domino
effects of mistakes, small and large. The foregoing shows how
skill and competence may seen and judged. With that observable and tangible
standards can be set.
Challenging as it may be to some, routine problem solving represents a
first step in developing the critical thinking and problem solving
skills. Routine problem solving provides a standard. Seeing what kinds
of problems have been met and/or solve before, and how, provides a
model and a base for further problem solving.
Problem solving from a state of ignorance is over-rated.
While creativity in the combination of previously mastered skills and
concepts and in the invention of new ones is possible with any level of
knowledge, the ability to be creative and in that produce methods to
solve problems in a verifiable manner - a manner that peers can follow
or reproduce - increases with the level of knowledge and level of skill
and competence. Students need to learn when creativity is required and
when previous methods give satisfactory results. Problem solving
exercises with incomplete information of what has been done before may
be given to students to show how a greater knowledge of previous
solution reduces problem solving challenges, or to set the stage for
further learning. With a combinatorial or creative mind, problem
solving which systematically
stands on prior knowledge of what has worked before or not, is
better.
Routines and methods in society for
"solving" problems may also lead to harmful results in
a repeatable, reproducible manner. The ability to follow instructions carefully and precisely is
a plus for getting results but not a guarantee that the results will be
ethnical or that practices will be sustainable. So people should not
be trained to follow methods or instructions without reflection on the
benefits and limitations of the methods. Routine solution methods may
be challenged and should be for the everyone's sake. But routine
methods cannot be challenge, cannot be considered and examine carefully
if their study is avoided.
Practice in solving problems which have become routine
may prepare students for open problems. Practice in solving routine
problems and puzzles in a straightforward or combinatorial or
opportunistic manner when solution methods are not given provides a model
for tackling non-routine problems, a model that stands on and then looks
beyond previous methods. In the case of non-routine problems,
problem solving aims to find solution with repeatable
and reproducible, and reliable results. With that open problems may become
routine. That being said, the origins and limitations of solutions methods need
to be seen, attention to detail helps with that, to avoid the misapplication
of solution methods. Not all problems are routine.
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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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