Preparation for Calculus
Context and Motivation
Calculus is mathematics subject in college or late high school. College
programs in business, science, technology, engineering and mathematics
employ calculus skills and concepts as language for developing practice
and theory. The practice and theory may be approximately correct, for
some circumstances, not all. To learn about that, calculus needs to be
mastered.
Most high school mathematics and logic sets the stage for mastery of
calculus and pure mathematics. Many students have difficulty because many
skills and practices in high school mathematics do not have immediate
value. On arrival in college,
many students have to take remedial mathematics courses before starting
calculus because of low skill development standards in their high school
mathematics courses. Forewarned is forearmed.
Raising skill development standards.
Many students have difficulty in algebra because of weak arithmetic
skills and because of old gaps in course design and delivery.
- site coverage of decimals, fractions, signed numbers and primes
offer a firm base for algebra. Algebra at full strength
requires an exact and efficient mastery of
arithmetic with integers and fractions. Anything less is substandard.
- Site coverage of algebra skills and practices before and in calculus
included many innovations to ease, address and avoid common difficulties.
This claim is fully-supported by steps in site material developed since
my fall 1983 presentation of two lesson on three skills for
algebra and why study slopes In site algebra steps, all
the technical troubles in learning and teaching algebra skills and practices that
I have seen are essentially addressed. Courses for gifted students may start
with site steps as appetizers before a more rigourous development.
- Site online logic chapters are mathematics-free, informative and
enterntaining. They sharpen logic and reasoning skills in mathematics and
language courses.
- Site coverage of geometry is not as complete as site coverage of
algebra. Geometry here is based on skills and practices associated with
maps, plans and diagrams drawn to scale. Future, if not present course
design, may show how maps, plans and diagrams may be used to measure or
calculate missing angles, lengths and areas to provide students with
skills and practices that have value in planning and making things -
clothes or buildings; and have value in planning or plotting routes for
navigation or orienteering or treasure hunting. The associated activities
may range from playful to serious. The foregoing would not have take-home
value, it would also allow the use of trigonometry to be introduced as an
numerical alternative to drawing maps, plans and diagrams to scale for
the sake of finding missing angles and lengths. The site geometric
introduction of complex numbers follows in the footsteps of Wallis and
Gauss 1840 and earlier but does so in a deductive manner that makes unit
trigonomtry easier to learn and teach. Easy consequence include algebraic
methods for justifying trigonometric identities and for deriving
trigonometric formulas for dot- and cross-products. The approach could
make future high school mathematics before calculus and present-day
college mathematics courses serving STEM easier to learn and teach.
Senior high school students who do well in junior high school may see
some mathematical skills and concepts employed in advanced biology, chemistry,
science and personal-business finance courses. The steps outlined above
may help more students do well. That being said, with most post-primary logic and
mathematics topics being present as preparation for calculus-based studies in college of
a fifth or less of high school students, the mathematics instruction does not
serve well the other four-fifths. While site innovations for making the hard
easier will help more succeed, the question remains of how to provide student-centered
skill development, a development that maintains and extends common skills and knowledge
to serve actual or potential everyday needs at home, at work, in business and
on the street. While cross-curricula activities may engage about one fifth
of high school students, I would like to see secondary mathematics, logic
and language skill and practice development give and leave a good last impression
in each high school year. What mathematics instruction would a homeschooling parent seek
for an adolescent not able or not interested in calculus-based college studies?
Presently, secondary mathematics instruction serves the needs of one-fifth without
giving or leaving a good, useful and favourable impression of the subject to the other four-fifths.
not
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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.
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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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