Reasons and Motivations for High School Mathematics
Many topics in mathematics and logic appear in primary and secondary
schools. Some have
take home value clear from daily life in society, or easily described
by parents and teachers. What may obvious to older folks from experience
may not be obvious to younger folks, especially those brought in a too
sheltered environment. But at the high school level, given parental
pressure to include students in pre-university studies, most high school
mathematics consists of technical skills and topics needed in college
programs. Unfortunately, skills and topics with take home value are not
emphasized, except in the education of students placed in courses not
required for college studies. I think that is a mistake.
take home value can covered once or twice for the take-home value as
part of preparation for university or college studies.
Different programs required different topics. But high school courses
have to prepare for all. The theorectical parts of science, engineering
and business programs are very mathematical. So in the end what may seem
to be over-preparation does not hurt. Developing the ability and finding
the patience to master skills in full is just important as skill mastery.
In between a social life [recommended for the sake of energy or to avoid
lack there-of], attention to detail is needed. See other work and study
tips for further advice.
There are two or three kinds of skill development in high school and college mathematics. Student
oriented skill development - not enough of it - provides know-how with actual or possible
value for work or home life. Over-preparation does not hurt. Subject or program oriented
instruction focuses on the skills and topics the subject might require in practice or theory.
That may be dry and technical. Over-preparation does not hurt. Intellectual value
may be found in the habit of doing and record reasoning or work steps in a way that shows
the doer or others what was done, how and even why. Proof and reason of the public,
share or objective kind, is based on that habit.
Modern fashion in education [constructivism and cognitive theories of learning] calls for teachers to engage students, to provide
real-life motivations for course topics and to provide situations in which
students may learn by themselves, what is important or not. Modern fashion
say direct instruction where teachers give students skills and concepts to learn,
and then test skills and concepts mastery is an improper form of education.
But that was how I was taught. I view mathematics and science as skill based
subjects. The skills in them were developed and recorded over years, decades
and centuries through the collective efforts of many. The skills not developed
by any one individual. Instead, the most useful were shared. In academic and work
subjects built on skill mastery, skill has to be seen to be confirmed or corrected.
Anything less leads to unreliable or absent skill development.
Site material aims for know-how development. In that development, the identification
of skills having take-home, being preparation for college programs, or having intellectual
value, offers ends and values for learning and teaching. All students should and many may
appreciate the take-home value of some skills and concepts. Some students may also
be engaged by college-oriented skill development, and its ends and values. Beyond that
where modern fashion in education provides documented examples of real-life problems
to engage students, modern fashion is very welcome. However modern fashion aversion
to direct skill development and testing is inconsistent with the needs of daily life
and college programs. Modern fashion
in education is incorrect where its aversion to observable skill development and its claim
that true knowledge is located in the mind apart from confirmable skills leads to
high school instruction that does not provide nor aim for know-how development. That
modern aversion also sabotages the formation of primary and secondary teachers for
mathematics skill development.
Arithmetic and Number Theory Skills
If you need a calculator to do arithmetic, your mathematical preparation for
college studies has been sabotaged by your earlier instruction. You have been
misled. In entering or finishing first year high school mathematics, you should be
able to fill in the 12 times and sum tables quickly, do arithmetic with decimals
on paper, and do arithmetic with fractions efficiently. Site material explains that.
All the foregoing has take home value for counting, measuring and figuring in daily
life. Good arithmetic skill is a sign of intelligence. It implies a knowledge of the domino
effect of mistakes and it implies the will and patience to learn step-by-step methods
carefully not in mathematics but also work and studies. In applying for jobs, emphasize
that while saying your arithmetic skills are very, very good.
Experience with
counting, measuring and figuring with whole numbers, fractions and decimals first without
calculators and then with calculators is plus for jobs in
buying and selling goods and services, in earning wages and in saving or investing. In daily
life and business, others including people you trust may make mistakes in counting, measuring and figuring what they
owe you. So daily life and business, it is trust but verify in all matters involving arithmetic and measures. Try
to develop that skill before you leave school.
Primes and prime factorization are number theory skills. With experience in adding,
subtracting, multiplying and dividing fractions with small whole numbers in numerators and denominators,
skill with recognizing primes and obtaining prime factorization of other whole numbers may not
be needed. That skill may be over-preparation for mathematics with take-home value. Yet it
does not hurt. It may help fractions skills and sense when you go to work, or if you have
children to educate yourself. But in preparation for college programs, recognizing primes and
obtaining prime factorization of other
is must for exact arithmetic without a calculator - or if you like aided by one.
Algebra may begin with formula evaluation. You can do that with a calculator. But
practice in the evaluation of formulas and the exact manipulation of algebraic and
arithmetic without using decimal approximations - those provided by a calculator -
sets the stage for backward use of formulas in an exact and precise manner.
Early skill in mathematics may come from teachers providing correct methods
for students to master. But preparation for college mathematics requires
not only mastery of methods, but also comprehension of why those methods
work. The comprehension is relative. In a dictionary
the meaning word is explained by other words. Those other
words in turn is given or explained by still further words. In this
chain of looking up meaning of words, you hope to find a word you understand for
a clearer or better comprehension. You hope not to go around in cycles. In mathematics,
later skills and concepts may be also come from other skills and concepts - earlier
ones. Seeing how represents comprehension. Unlike the study of skills with take home
value, where skill mastery provides the value, preparation for college programs
finds values not only skill mastery but also in the mastery of their explanations.
Slid preparation for college level studies requires rules and pattern mastery and comprehension of
how some follow from earlier ones. That represents a shift in values. Exact arithmetic
with whole numbers, fractions, primes, square roots and further radicals and powers in needed
in the explanation and comprehension of the origins of rules and patterns, formulas included,
in college oriented studies. In the foregoing, over-preparation does no harm.
Presentation and Communication Skills
Teachers cannot read your mind. The proof of that is the fact that
most students are not detained after school for all the stray thoughts that
crossed their minds during the day. I know what I am talking about. I was
a distracted school boy or teen a long, long time ago.
Your skill in arithmetic, in algebra, in geometry and outside of mathematics in writing needs
to be seen to be observed or corrected. That is because teachers cannot read your mind. Outside
of school, you may have a favourite meal, you may like a movie, or you may admire what others
have done in deed, or to their appearance. Remember not to stare.
Or, you may criticizes what others show. In all the foregoing, you are seeing
judging the skill of others. School may provide observable skills, judge them, and tell you
how how they will be judged or marked. In school, you need to watch for clues from your teachers
on how they will judge your skills. That will lead to better marks and performance. Skill has to be
seen to be appreciated.
In mathematics, skill may be seen in how arithmetic and algebra are done, and how diagrams
are drawn. Site material includes lessons on how to evaluate arithmetic expressions and how
evaluate formulas. Those lessons show how to do and record steps in ways that you and others
may follow as done or later. The objective is not to do the work quickly - any fool can do that - but to
the work skilfully. Once you have learnt to do and show work skilfully, your marks will improve,
doing the work will become routine, and your teachers may say that they seen what you can
do, further practice is not needed.
If you can show your teachers that you know how to present your work in a way
that shows skill and full comprehension, you may be allowed to skip work or do more advanced
work for the same course or a different one to make better use of your time. Or, if you
are in class where your instruction is being slowed by the speed of others not as careful as yourself,
you can ask to serve as in-class tutor. Helping others do the work in class may lead a teacher
the option of having your marks being based on test results instead of homework. If enough students become
in class tutors [I am dreaming here], the instruction of yourself and others may become quicker
and more efficient. See what is possible. Good luck.
Algebra Skills
Algebra may begin with formula evaluation. In formulas, letters may denote
lengths, areas and volumes. Letters may denote distance, speed and time.
All these quantities may be seen or sensed. Letters act like names, placeholders
or pronouns for them. The letter $\pi$ may denote a distance. In formula evaluation,
the values of the numbers and quantities in them are given.
Algebra become
mysterious when we say IT is a number without giving the value of IT. The pronoun IT
is easier to undersand, if that pronoun IT is the measure or length of a physical
quantity, one that we may see or draw.
There is more to algebra that formula evaluation. Formulas instead of being
given, may be explained. The explanation point to the comprehension of how or
why mathematical methods work that preparation for college programs require.
But formulas can also be used backwards. The gradual task of mathematics instruction
in high school and college is to show how more and more rules and patterns - formulas and
equations included - can be used forwards and backwards. In algebra, that requires
the ability to solve equations for numbers [one or many ITs] whose value is not given.
The algebraic way of reasoning is difficult for many.Site algebra starter lessons begin with formula evaluation, and continue with a four step
program for solving linear equations. All steps emphasize formats for doing and recording work in a way
that you and others can follow as it is being done, or later. Remember skill and know-how has to be
seen to be confirmed or corrected.
In the first step, linear equations have to be solved for a number x [an IT] that stands
for the length of a line segment or stick. In a three column format, fractional operations on stick diagrams
drawn in the first column lead to the solution geometrically. The equivalent algebraic operations
on equations give the solution algebraically. Through many, many examples, the aim is to improve
your fraction skills and sense, and to move you away from the easily understood geometric
solultion - one that is drawn - to a mastery of the algebraic. In the process, the
intent is also to move you away from solving linear equation in which unknown is the length
of a visible stick to being comfortable with solving equations in which the unknown is the value
of a letter x, a value that has no geometric meaning. That is the subject of the second step.
The third step intoduces easy systems of linear equations - triangle systems and systems
in essentially one unknown. Both kinds of easy systems aim to solidify and build substitution
skills. Some examples or exercises require skill with the order of operations in arithmetic.
The fourth steps for more general systems introduces three ways to do Gaussian eleminaton. The presentation
here is simple. Students 14 to adult may be able to follow it after the earlier steps.
Once upon time, a class I taught included three ways to solve quadratic equations - a topic
in junior or senior high school mathematics - depending on your location or school system. The student
on failing explained to me that he decided only to master one way, three being more than he needed.
He did not understand that besides solving problems, the objective was to develop algebraic
skill. Each different way represent a different skill. Here, solving linear equations or solving
quadratic equations has little or no immediate take-home.
Modern mathematics offers a logical structure for itself - a codification in which
comprehension and justification stems from assume rules and patterns - axioms. The axioms
are expressed algebraically. Site material provides a computation rule context and meaning
for most of the assumed patterns. That should make them easier to understand and explain.
The observation that formulas, rules and patterns may be used forwards and backwards
represents a unifying theme in high school and college programs of the mathematical kind.
The algebra starter lessons provides many examples very carefully and slowly to introduce
numerical and algebraic forms of the theme.
Arithmetic and algebra are too often
done and learnt in silence. Site pages and chapters on three skills for algebra, what is a variable,
extending arithmetic with words, and descirbing this unifying theme with words, altogether set the
stage for a greater and clearer role of words in mathematics learning and teaching. The verbal
extension of arithmetic with words has take home value.
Closing Remarks
One of my university Professors CR had the habit of saying where the ideas
and methods of his course were used. Some of the application areas were
over my head - beyond my ability or available time and energy to explore
and understand in full. But at least there were mentioned. I would like
to see a similar habit or effort in textbooks and course design.
Above we have seen that skill development may have take-home value
may have value for college programs, or may have intellectual value. To learn more,
see site books and further site areas on geometry and more algebra.
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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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