Ends, Values and Methods for work and studies
Some work and study values to emphasize when needed, follow.
Pay attention to details
- no one else can do that for you.
Mathematics becomes simpler if you take care to learn and apply steps
carefully - the child in primary school who insists that the alphabet is
too long to learn will have difficulty - the teenager who insists that
details are not important in mathematics will likewise have difficulty.
An error in one step of a method leads to incorrect results - not knowing
this domino effect of errors is reason for poor performance in many arts
and disciplines. In work and studies, people who refuses to pay attention
to detail will have difficulties or troubles, sooner or later.
Calculators and Black Boxes versus Hands-on Manipulatives
These have many practical uses outside and inside mathematics. But
learning to do arithmetic well with decimals reinforces place values
comprehension and provides the first opportunity for students to learn
that an error in one step of a method has a domino effect - all further
steps are likely to be in errors as well. Secondary, if not primary
mathematics, requires the ability to understand and do arithmetic exactly
and efficiently without a calculator. In some aspects that requirement
may be an a hang-over from pre-calculator days. But present day practices
in higher level maths assume and require and early mastery of exact and
efficient arithmetic with whole numbers and fractions.
To tell primary and secondary math instructors that their students need
not master arithmetic with whole numbers and fractions, exactly and
efficiently is inconsistent with the pedagogical advice that students
needs to learn skills and concepts with manipulatives. Beyond the
elementary use of counters and blocks to introduce counting and geometric
shapes, mastery of manipulatives may also be extended to include
- Counting with fingers - a temporary measure.
- Filling and using the addition and times tables forwards and
backwards. The observation of the pattern with hard or drawn
manipulatives that two objects and another three objects gives five
objects explains why we put 2 + 3 = 5 in the addition table, and then say
5 less 2 is three. The observation that 3 sets or rows of 4 objects give
12 when counted explains why we put 3 × 4 = 12 in multiplicaton tables,
and then say 4 goes into 12, exactly 3 times. Many other arithmetic
patterns can be seen in and used to fill addition and times tables with
say 12 by 12 or more entries.
- Measuring may be done with rulers, tape measure, protractors,
graduated cups or cylinders and mass or weight balances - all the
foregoing involve instruments with no hidden inner mechanisms.
- Addition, subtraction, comparison, multiplication and division may be
first introduced using physical operation on integral and fractional
fractional multiples of objects and unit measures before being described
with whole numbers and fractions on paper
To tell primary and secondary math instructors that their students need
not master arithmetic with whole numbers and fractions, exactly and
efficiently is also to undermine the content and quality of further
instruction in algebra, trigonometry and calculus. A balance is required.
Calculators may serve as an aid in prime number factorization or the
recognition of primes. Calculators are often, but not always needed in
trigonometry and in the evaluation of logs and exponential functions.
Students need to be given pride in using calculators when necessary, but
also in a minimal manner, as part of skill development and enrichment. In
the current technical paths for preparing students for the practical use
of mathematics and beyond that, perhaps, for calculus-based college
programs, hands-on skills with pencil and paper are required - to
ignorantly minimize them is to undermine mathematics education. The
educational pyschology oversteps its bound when in ingorance of the
technical or practical application of an art or discipline, it calls for
some and concept not to be taught in schools and colleges. Mathematics
today includes a bundle of skills and concepts with dependencies that
need to be known and respected in pyschological advice and directions for
course design and delivery.
Calculator use should not distract from the mastery of exact arithmetic
needed for doing algebra and deriving formulas there-in exactly.
Hands on Work Ethic for parents, tutors and teachers
The young child may complain that the alphabet is too long to learn. We
know better. We know that being able to name and write all the letters
in an alphabet is needed for reading and writing, and in that for
spelling carefully and recognizing words. So the alphabet mastery is
required. No letter can be skipped. Children in Grade 1 likewise need
to given and be told to master mathematical or quantitative skills
fully and properly.
Even though calculators may be used later to aid arithmetic, the full
mastery of high school mathematics requires efficient and exact skills
in arithmetic methods with decimals and fractions. Anything less will
undermine the coverage of algebra, geometry, trig and calculus in high
school and college. While we want mental agility in arithmetic - the
mental ability to do some arithmetic in the head, mathematics training
of students age 5 to adult should aim for development and mastery of
observable skills on paper for the most part, with connections to
off-paper geometric or physical measurement and construction skills.
Advice to repeat to your son or daughters
Communicate - Show Work
Teachers cannot read what is in your mind, but they see what you write
and draw - so for your own sake, try to record the steps and data in your
solutions in a way that others can see, follow, duplicate or correct your
reasoning. Showing work is required in the written response part of final
examinations. Look at course notes and texts, look at the written work of
your friends, ask everyone how to do and show work. That detail will help
you a lot. In arithmetic and beyond in mathematics, follow this
written work formats for developing and showing skill. It shows how
to record work in steps that can be seen as done or later for
confirmation or correction.
In mathematics training, we cannot read the minds of students, but we can
see what and correct what they do, and through drill and practice,
encourage steps to become familiar with methods and to follow and record
their steps in an repeatable, reproducible, observable and hence
correctable or verifiable manner. As a parent or teacher, you want and
should require your charges to keep binders full of their writings and
drawings to record, monitor, correct and report progress. That provides a
hands-on, tangible approach to education in which neatness and quality
should be encouraged over haste - that advice is given with some doubt,
since from school-boy days, I do not remember being neat, so the word
encouraged is employed here in place of required or forced. When students
meet arithmetic methods for addition, subtraction, multiplication and
long division with decimals, they will learn that an error in one step
makes all that follow doubtful or wrong. Students need to learn how to
follow and combine multi-steps methods carefully. That ability and work
ethic may begin in arithmetic.
Problem Solving:
If you have the skill and patience to solve jigsaw puzzles by fitting its
pieces together by trial and error, while looking at the desired picture
and pieces themselves for hints of how to do so, then you may have the
skill and patience to fit or apply the rules and patterns of any
discipline to arrive at results and to understand explanations, in
detail. Show the reasoning or work that leads to a solution is part of
problem solving. See the previous advice.
For better problem solving skills, besides worked examples, study the
proofs or explanations in your notes and textbooks. Both problem solving
and theory combine rules, patterns and methods to arrive at results. The
problem that says show that is actually asking for a proof. See
the Volume 2 chapter on painless theorem proving to
learn more. Amusement is not required.
Seeing how explanations combine rules and patterns to arrive at
conclusion may also help you combine rules and patterns to solve
problems. But there are two kinds of problems: Routine and non-routine.
A problem is routine if a method for it solution is known and given. In
using such methods, knowing about their limitations - exactly what
problems they solve, how and why - will lead to more precise or exactly
reasoning solutions. Understanding of a solution method, the deeper the
better, lessens the chance of making an mistake in deciding to use it -
Does it truly give an solution?
A problem is non-routine for you can find no standard or given path to
solve it in a repeatable and reproducible manner. Then you will have to
think out of the box, and try to combine or extend you previous know-how
to obtain a solution. In that, success is not guaranteed. Not all is
certain. Practice in non-routine problem solving may be given by tutors
or instructors providing a problem close or distant from the previous
experience of a student in order for the latter to find and communicate a
solution step by step. Solutions have to be seen and recorded step by
step, with reason for each step given or well-known, in order to be
checked and believed.
More on Problem Solving and Proofs:
If you learn how to apply methods to obtain correct results and if you
learn how combine rules, patterns and methods carefully and precisely in
problem solving then explanations here and elsewhere will become easier.
Indeed, there is a convergence between checking and writing solutions to
problems and checking and writing proofs or explanations. Both types of
checking and writing involve the careful use or application of rules,
patterns and methods one at a time and one after another to arrive at
results - numbers or more rules, patterns and methods. Thus problem
solving and deductive explanations (proofs) involve similar skills and
concepts. So problem solving skill helps with proof writing and following
skills, and vice-versa. The latter remains true even when the underlying
rules, patterns and methods are learnt by rote.
Still More Advice
Reflection at home and in school may be done in any way you please. But
skill and concept mastery in mathematics and logic-English is different.
There are rules and patterns to master. Mastery of logic and awareness of
the domino effect of care and errors will help avoid confusion and
difficulties in work and studies. Logic mastery in particular will help
you follow and write recipes, instruction, explanations and stories with
greater precision and more comprehension. Good luck.
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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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