Problem Solving Tips and Methods
Solving problems in mathematics and other subjects sounds practical.
There are two kinds of problems, routine and not. The solution of routine
problems may be given in class for students to apply. When a problem is
routine, routine solutions should be employed. So routine solution
methods for routine problems should be met and memorized to avoid the
extra work required for solving problems whose solution is not given. The
rule of thumb is a follows.
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For Routine Problems, learn and use Routine Methods.
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For non-routine problems, be combinatorial, follow strageties, and
try whatever may work, near or far.
There is a need to master or at least identify what problems are
routine. Otherwise, you will spend time in looking for and inventing
solutions for problems whose solutions should routine or automatic.
The Jigsaw Puzzle Approach
Problem solving may be like putting together a jigsaw puzzle. In solving
a jigsaw puzzle, we may begin with the sides as pieces with straight
edeges are fewer in number and must be aligned, after that the more
difficult to place inside pieces may be fitted together. Jigsaw puzzles
may be made more challenging by hiding the picture they are suppose to
form, or by assembling the pieces upside down. That being said, with the
pieces picture side up, we may put try to put them together with trial
and error as needed, but with continuity and drawn shape limiting the
trial and error. This trial and error combination of pieces that go
together may be ad hoc, opportunistic and in general combinatorial. The
trial and error requires persistence. With that, over time, more and more
of the puzzle will be solved until, if all the pieces are present, the
problem is fully solved. Unfortunately, jigsaw puzzle pieces may walk
away over time, so there no guarantee that all the effort made will lead
to complete picture to solve the puzzle. More generally, when we are
tackling a nonrouting problem or puzzle, the existence of solutions is
not always certain.
Text Book Problems and Exercises
For most textbook problems and exercises, all the pieces or elements
needed to solve the problem are likely to be present in the current or
previous chapters. They may just need to be fitted together in ways
similar to the worked problems or examples in the text or course notes.
The similarity will be close for the easiest problems and further for the
more complicated ones. Skill in following textbook patterns may become
routine or almost so with practice, with careful reading of the text or
notes, with care not to forget earlier skills and methods. In senior high
school and college mathematics and mathematical subjects, problem solving
may remain routine and feasible with time and effort to see and master
all the problem solving methods present in notes or a textbook.
Thinking Outside the Box when need-be
What is routine for one is not for another. Experience counts. Where a
student may have to think hard to solve a problem, an older student or an
instructor may tackle a problem based on past experience. Problem solving
may think out of the box or the confines of earlier problem solving
practices, when none of the latter practices apply, Thinking out of the
box means look for new angles or different perspectives for tackling or
addressing the problem. Or, it may involve tackling what appears to be a
related, similar or easier problem in the hope that experience with the
latter will make the original problem addressable. Not all certain. And
for problems from real life, solutions may be routine, solutions may be
difficult to find, or the existence of solutions may be not be known.
Some trial and error may be required with success not always certain for
the original formulation of a problem.
Real World Problems
In real world problems and questions unlike most problems and exercises
in a book, there may be no given pattern to follow. Not all is certain.
Here may be missing pieces or extra pieces, and no guarantee that the
solution can be done.
Preparing to Solve Problems
Master Logic: Again, poblem solving is like putting together a jigsaw
puzzle. In the case of textbook problems, all the pieces are present
and just need to be fitted together following the clues, and an
possible a picture showing the desired result. In the case of real
world problems, there may be missing pieces or extra pieces, and no
guarantee that the solution can be done.
Problem solving besides thinking out of the box and being opportunistic
an combinatorial in looking for clues to use alone or with others
requires precision in reading, writing and figuring. Imprecise logic and
language abilities will lead to difficulties. Precision reading and
writing, and opportunistic trial and error skills for problem solving may
be refined and developed (we hope) by reading the following chapters in
site Volumes 1A and 2.
- Implication Rules (Volume 1, Part I, Pattern Based Reason)
- chains of reason (Volume 1, Part I, Pattern Based Reason)
- longer chains of reason (Volume 1, Part I, Pattern Based Reason)
- islands and divisions of knowledge (Volume 1, Part I, Pattern Based
Reason)
- painless theorem proving (Volume 2, Three Skills for Algebra)
These appetizers and lessons show how rules and patterns may fit together
to arrive at conclusions or solve SOME problems. Other problems are just
too hard. We can't prevent that.
Master Fractions
Many applied mathematics problems involving chopping and combining
lengths, areas and volumes. So you need to know how to take a
proper or improper fraction of a length, area or volume. You need to
understand that one length may be 2.5 times or 2½ times or
(5/2) times another. Any if you do calculation, you need to do it with
care or at least do it with the knowledge that an error in one step makes
all that follows wrong. The ability to figure well and precisely, so that
you answer is correct, shows or suggests the ability to follow methods,
one step at a time and one step after another in any subject, and in
problem solving as well.
Algebra Word Problems
If your interest is in solving algebra word problems at the high school
level, I would recommend learning how to solve linear equations in
several unknowns in an effortless fashion. High school students who can
solve linear equations in one unknown are often given word problems where
extra variables have to be eliminated to formulate a single equation in
one unknown quantity to solve. The trick here is to draw or extract a
single equation from the given information. But in most such words
problems, it is easier to extract or draw from the given information
several linear equations in several unknowns to solve. Each sentence in
the word problem gives an equation in one or more unknowns or quantities.
Now the algebraic way of writing and thinking can be used to eliminate
variables and to solve for the one or more quantities of interest in an
effortless fashion.
The algebraic solution of linear equations involves the elimination of
variables to obtain say one equation in one unknown. This elimination
process may be better done and recorded with algebraic notation. Going
directly to one equation in one unknown to solve a problem requires more
work to be done with words.
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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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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