|
Skill in mathematics in arithmetic to calculus and in proofs
consists of the ability to do and record work in steps that can be
seen as done or later for confirmation or correction. The
ability requires adequate comprehension of what to do.
Each year of instruction covers only finitely skills
and practices. Skill development, one at a time, one
after another, should be a simple recursive task. The task is
feasible at the primary school level. Present-day
activity and exercise sheets and booklets show parents and teachers
how to build basic arithmetic and geometric abilities.
The general appearance of student weak in arithmetic in
college and secondary schools reflects a current lack of will in
primary (and secondary) instruction. There are no technical
barriers to overcome. In primary school, figuring well would imply
awareness of the domino effects of care and errors in
arithmetic. Awareness and avoidance of the domino effect of
errors represents an end, a value and a tool for building
skills and confidence in many arts and disciplines. The task is
also feasible or more so at the secondary and calculus level due to
my online steps. Just as
learning difficulties may compound, the effect of these steps to
ease or remove difficulties may also compound. The extent
remains to be seen.
Strong students in senior high school heading for business or
commerce activities may see some potential take home value or
utility in mastery of compound interest formulas and geometric
sums. In that, the backward use of the compound interest formula
offers a context for the introduction of logarithms and
exponentials. Strong students may also see the
secondary level, cross-curricular employment of mathematics.
- Chemistry and physics employ proportionality, arithmetic and
measures with units.
- Physics employs linear functions and quadratics
- Biology, physics and finance employ the forward and backward
analysis of growth and decay formulas in compound, half-life,
doubling time and continuous growth forms;
- Biology and the discussion of games employ combinatorics,
probability theory and fraction skills.
- Physics employs conic sections, vectors and periodic
trigonometric functions.
Yet the above cross-curricular role of mathematics will likely be
seen in all by at most one fifth of secondary school population,
and then only in senior high school years. The other
four fifth are skill development orphans in that they will not see
the foregoing cross-curricular connections. For them, the question
why learn this or that will be preparation for final examinations
for each of the five or more years spent in secondary school.
Without any further reason for learning, loss of interest and a
poor impression of mathematics follows. That loss of interest
will also touch students who could have been
strong. Apart from statistics and perhaps probability, most
topics in secondary mathematics in arithmetic, algebra, logic and
geometry appear to be present for the sake of college programs in
science, technology, engineer and mathematics. The latter
represent college programs which only a fifth may see. The
needs of the majority need to be considered. That also
suggests a change of direction in course design.
The placement of high school students in mathematics courses
covering topics needed for and by calculus-based college programs
may be done for the sake of inclusion and equality. It may be also
done to provide college programs with students. For
the majority of students, the four-fifths or more that
do not enter calculus-based programs, the coverage of
topics mostly required for calculus currently fails to give and
leave a good impression of mathematics and logic.
Students in many communities would benefit from treating each year
of mathematics instruction as if it was the last chance to give and
leave them with a good impression. That would raise the
question of what skills and practices would have the most benefit
for the students at hand. The foregoing approach would emphasize
skills and practices with actual or potential take-home value for
the sake of keeping students engaged. Such skills and
practices would be presented as soon as the level of student
know-how permitted.
Primary school mathematics in serving common or likely needs
usually has take-home value clear to students, parents and
teachers. Primary school and the
first years of secondary school can be explicitly dedicated to the
service or role of numbers, geometry and clear thinking
at home, at school and at work. Actual or likely needs
may be served by emphasizing time and date matters or practices;
money matters; measurement and figuring with decimals and
fractions; chance or probability comprehension for minimizing risk;
and logic or clear thinking. Maps, plans and diagrams use may
introduce geometry with applications to geography, navigation,
route planning and construction. In that, navigation and location
activities may emphasize actual or potential value of indirect
measurement and calculation of actual angles, lengths and areas
from maps, plans and diagrams drawn to scale. Navigation too
may employ arrows to denote movements alone and in sequence, head
to tail. In keeping accounts, signed numbers may represent assets
and debts. To further employ and introduce basic skills and
practices, exercises, scenes and role-playing may
illustrate traveling, use of maps and schedules, buying and selling
goods and services, book-keeping with positive and negative
amounts, cooking with weights and measures, paying taxes with
addition, subtractions and percentages, and money handling with
saving, checking and credit accounts; making clothes and objects
with plans and measures. The selection of skills
and their take-home may vary in accordance with local needs and
options for motivation and context.
For common occupations, those likely to be in the future of
many students, mathematics skill development could identify the
skills and practices in arithmetic, measurement, algebra and
geometry likely to be required. Nursing is mathematical in
the critical sense that measurement and dose calculation have to be
done accurately. Retailing and the aforementioned
buying and selling of goods and services give another occupation to
discuss fully to provide an operational command of basic
skills. Remember the aim is to serve the needs of the
majority, not the few. And in doing that, the needs of the
few will also be served – some may have to work in retail to
support their studies.
In secondary school mathematics and language courses, logic or
clear thinking may aided by writing one-way implications in the
form A IF B. This form will make the difference between saying A IF
B and saying A IF ONLY IF B clearer. Seeing the difference would
help students understand the terms and implication of agreements
and instructions. That has take home value for self-defense or
avoiding mistakes in work and studies. Explaining the difference
does not take much time. This explanation may be given year after
year to keep or develop awareness of the difference. Awareness of
the difference like awareness of the domino effect of mistakes may
lead to greater care and precision in mathematics and language at
home, at work and in school.
Over time, besides or after the coverage of skills and practices
with take-home value, instruction may weave a
consistent web of inter-related skills and practices in which some
imply others. That would introduce a deductive framework and
some intellectual value or pride in skill and concept mastery.
Among equivalent sets of axioms that might
be given for secondary mathematics, the choice that make skill and
concept development simplest would make instruction easiest for
students and teachers. In particular, rigor may
be based on axioms and definitions that provide the easiest entry
points for thought- or deductive-based development.
Operations with numbers can be defined in a manner that implies
most axioms, so the latter do not have to come out of the blue – a
flaw of the modern mathematics curricula in the 1960s.
Secondary mathematics skills and practices may be developed
in a manner that helps and sanctions the common skills with
take-home and the computational skills of value in science and
technology. By making early instruction learning and teaching
easier, more students may have the later choice of studying pure
mathematics or going in a different direction.
For students who choose to continue, the care and diligence needed
to follow and understand skills alone and in a deductive framework
provides a careful model for reason and planning.
Unlike skill development in physics, chemistry and biology
where rules and patterns must be accepted, and chemicals may be
taken off the shelf for use in a plug-and-play manner, skill and
concept development in mathematics permits a thought- or
logic-based organization and development. Strong students may
appreciate that.
In the debate between rote learn and explanation, note the
following. One secondary student witnessing my repeated inclusion
of reason why geometric formulas held informed me that the
explanations were not necessary. Teachers were hired to
give correct formulas. In saying that, I do not if she was
implying that explaining why formula held might be a
sign I was not sure of the formulas, and hence had to justify
them. Where an operational command of mathematics has more
take-home value that comprehension, explanation may be given to the
point that they do not overwhelm students and do not distract from
the operational command. Sometimes full is too much.
Critical Path Analysis
As assumed above, skill in mathematics may be seen in the ability
to do and record work in steps that can be seen as done or later
for confirmation or correction. The ability requires adequate
comprehension to know what to do. In serving
common needs and in preparing for calculus different paths through
different but overlapping sets of skills and concepts are
possible. Critical path analysis may
re-arrange and even select skill development steps in accordance
with constraints, ends and values. Some steps may be place in
parallel. Others may have to go in sequence.
With many unversed in mathematics assigned to teach it
in primary and secondary schools, there is a further constraint on
course design and materials, no matter which path is
follow. In them, steps need to be easily understood and
repeated in class. General direction on what skills ends and
values to emphasize are not enough. Steps, ends and values
need to be written or presently clearly in instructor
manuals alongside with advice and directions how to ease, avoid or
troubleshoot common difficulties. Such manuals should give a lower
bound for instruction. With all the topics area identified
above and with my online steps for algebra skill building, I do not
anticipate any technical difficulties in providing such manuals
|
Preparation for Calculus
Calculus is mathematics subject in college or late high school. College
programs in business, science, technology, engineering and mathematics
are calculus-based. High school mathematics mostly consist of topics
required by these programs, programs that less than a fifth of high
school students will enter. The one fifth or less that enter these
program are well-served by the current choice of topics in high school
mathematics.
Unfortunately, the topics usually do not have immediate value.
To be part of the fifth or less that enter calculus-based college,
students have to do well in high school mathematics and science. And in
mathematics, that requires students to find or bring their own
motivation, because during high school studies, most skills and
practices are covered because they appear on the next test or final
examination. Many students, teachers and adults, have no idea how or
why skills and practices need to mastered - theirs is but to learn or
teach without understanding why and with knowing what standards to
maintain or seek in skill development. The fifth that do well, a fifth
not fully known in advance, may see applications of mathematics in
biology, chemistry, physics and even money calculations in senior high
school, but the other four fifths or more will not see any
applications. With present high schoo course design, four-fifth or more
of students who graduate will not understand why they studied
mathematics.
What we are missing in secondary mathematics is a student-oriented
course design. Primary school mathematics may give and leave a good
impression in developing skills and concepts with actual or potential
value at home, at work and in the street. Four fifths or more of
students, secondary mathematics instruction too should try to give and
leave a good impression before preparation for calculus begins in
earnest. May be topics only present because of calculus should be skipped
or postponed for long time in the education of students unlikely to
attend calculus-based college programs. The alternative secondary program
present ideas for providing students with skills and practices from
everyday life and money-matters for home and business. Covering less
might best to give and leave a good impression - to avoid overwhelming
students with skills and concepts unneeded or very, very
confusing.
Making Preparation for Calculus Easier
The introduction of algebraic ways of writing and reasoning is
harder than need-be in mathematics. That is due to old gaps. Talking
about that may lead old instructors to say site material is nonsense. As
a student who would not use the quadratic formula until I understood it
origin - a be true to myself moment, I was forced to justify by myself,
for myself, the algebraic shorthand way of writing and reasoning employed
in its justification. The textbook I had was of no use. Since then, while
shyly thinking that the shorthand role of letters and symbol in
mathematics was used but not properly introduced, I have been looking for
remedies in course material and textbooks. Not founding them, I slowly
developed my own.
The shorthand roles of letters and symbols beyond the use of
formulas is meaningless formality for many students and adults. There is
a more to mathematics than being given a method or formula, and numbers
to use in it. In my days as student and then teacher of algebra and
calculus students, I have thought steps were missing or too large in the
development of algebra before and in calculus. Those missing or too large
steps explain common difficulties. While natural abilities or talent is
required to walk, talk and argue, too much natural talent has be required
in algebra skill development. Site material provides an informal but
effective remedies.
Informal here refers to the current lack of set notation and
formalism in site material. I favour leanest possibly emphasize of set
operations in combinatorics, probability theory, logic with Venn
Diagrams, function representation and calculus to helps students with
an operational command of skills and concepts without overwhelming
students and teachers with formalism. The latter can be left to
university level courses in mathematical subjects.
As strongly as possible, let me emphasize that site steps for
easing difficulties and building technical skills and confidence in
algebra and calculus are second to none - they address the technical
troubles I have seen in course design and delivery. In too many schools
and colleges, mathematics education fails to develop a good mastery of
arithmetic sabotages the full strength mastery of algebra needed in
calculus and preparation for it.
-
site coverage of decimals, fractions, signed numbers and primes
provide a firm base for algebra. The full-strength mastery of algebra
before and in calculus requires this firm base - an exact and efficient
mastery of arithmetic with integers and fractions.
-
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 to address and
circumvent technical difficulties. Each step consists of one to a dozen
online lessons presented in webvideos, online chapters and further
webpages. The algebra and calculus teps in stem from years and decades
of in-class experience and out-of class writing provide lower bound for
skill development. The claim is not existential - a pointer to lessons
or ways for circumventing difficulties that should exist. The claim is
more concretely based on lessons posted online and ready for testing.
Site review and personal experience imply some work in easing or
avoiding common difficulties. In them, all the technical troubles in
learning and teaching algebra skills and practices that I have seen are
essentially addressed.
-
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.
-
Site online logic chapters are mathematics-free, informative and
enterntaining. They sharpen logic and reasoning skills in mathematics
and language courses.
In the high school mathematics path leading to calculus, students
need to acquire a full-strength mastery of logic, arithemtic, algebra and
geometry. Presently only a fifth or less succeed. The trouble with
calculus-oriented mathematics courses in junior and senior high school
mathematics stems from the immediate lack of concrete value for life at
home, at work and on the street. The courses introduce skills and
practices which do not have immediate application. Less than one fifth of
high school students will see applicatons of algebra and geometry in
science and technology courses.
|
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.
|
|