Mathematics Education - Leaving Students with a Good Impression
Presently, mathematics education ends in failure and alienation for many
students in many communities. The algebraic way of writing and reasoning
is unclear to many. How can these troubles be mitgated? A solution may lie
in a shift of ends and values and in elementary steps to make algebra more accessible.
Three Rip Van
Winkle quotes from The 21st year book The Learning and Teaching
of Mathematics, Its Theory and Practice of the National Council
of Teachers of Mathematics, Washington D. C. 1953, provide directions for
mathematics instruction
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page 349. .. the teacher must be a master technician. He must know
how to build any known kind of learning. .. must weigh, balance, and
appraise the possible learning. ... know their relative worth both
for the individual and for society.
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page 348. a teacher is a learning engineer, a builder of minds that
will solve problems. As such, he must first know the total
mathematics he will teach, that is, the framework.
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page 248. There are some persons who say one who knows cannot teach
for he cannot fathom the difficulties of his students. These persons
say that as a teacher work with his students through a problematic
situation which is new to both teacher and student, real learning
takes place and then only. We believe this assumption to be entirely
erroneous and assert that a teacher is a learning engineer ...
Primary school mathematics appear to give and leave a better impression
than secondary and college studies. That may be due to development of
common skills with time and dates; money counting and use; measurement
with decimals and fractions; maps and plans drawn to scale; arithmetic
with numbers and amounts; taking or avoiding risks; and solving some
logic puzzles. In exercises or activities adults and eventually students
may see the cumulative take-home value of skill development. Many
students graduate from secondary school without the basic or ordinary
abilities likely to be needed in daily or adult life. In eating in
restaurants, in travelling, in buying and selling goods and services, in
paying taxes, in using saving, chequing and credit accounts; and finally
in distant mortgages and annuities, quantitative abilities are needed. In
textbooks for learning a second language, scenes of an invididual or
family eating in a restaurant provide an opportunity to introduce
vocabulary. Similar scenes or stories in early secondary textbook might
span common activities and in the process emphasize needs - more common
for some more than others. Pirmary school and the first year of secondary
school might be explicitly dedicated giving students with favourable and
largest possible impression of the role of numbers and geometry in
society, first locally and then more globally. There-in lies a first end
and value for instruction. That impression may also provide a foundation
for further studies. Primary schools should emphasize avoidance of the
domino effect effect of errors in arithmetic and more generally in
multistep methods as an end, value and tool for further skill mastery at
home, at work and in school.
The first years and level of mathematics education should serve the
mathematics education needs of communities where students are unlikely to
attend college programs in STEM or are unlikely to complete high school.
The first years should give and leave a favorable impression of
mathematics before any preparation for college studies begin. The first
duty of mathematics education to TCPITS is not to make advance
mathematics visible, but to provide a first practical command of
mathematics and logic of service in daily life at home.
In secondary school, students with dim prospects or none of going to
college are included in mathematics courses largely serving the needs of
calculus-based programs in business and STEM. For these students, the
answer why learn or teach this or that is preparation for the next final
examination. Year after year that is a source of alienation. A better
route for such students might be focus on the algorithmic development of
mathematics and logic skills with actual or potiential take-home value,
from a be prepared for adult hood perspective. Preparation for college
studies which cannot be done well or which alienates should be put aside.
It may be better to do less and leave a good impression and some respect
and desire for further learning than to alienate students with skills and
concepts in an overwhelming manner. The motivational difficulties here
may be eased by advances in skill engineering that makes the technical
requirements of calculus-based programs easier to satisfy. In serving
that preparation, putting first skills and practices with some take-home
value clear to students and teachers might help further instruction leave
a good impression.
For stronger students, secondary mathematics instruction may continue
with the explicit goals of preparation for calculus-based college
programs in business or science, technology, engineering and mathematical
disciplines. For some honesty, this preparation would mention three
facts. First, that entrance into college programs is selective and
success in those programs in not certain. Second, calculus requires
arithmetic from times tables, decimal and frction skills and prime
factorations plus algebra and geometry at ful strength, with success in
calculus being uncertain. Third and last, calculus may provide a language
for understanding the practices and theory, as well as the limitations in
college programs requiring it.
Students heading for business or commerce activities while avoiding
science may see some potential utility in the forward and backward
mastery of compound interest formulas and geometric sums. The forward and
backward use may provide a context and motivation for the study of
logarithms and their inverses. Students heading for STEM as well may see
in chemistry and physics, proportionality and fractions with units in
chemistry, in physic linear functions and quadratics; in biology, physics
and finance the forward and backward analysis of growth and decay
formulas in compound, half-life, doubling time and continuous growth
forms; and in biology and the discussion of games, some probability
theory. The applications here are largely cross-curricular and point to
the repeated appearance of mathematical patterns in different forms.
There are topics in secondary mathematics present to serve the technical
needs of calculus and beyond, with little or no application in other
subjects. Operations with polynomials in algebra belong to this category.
Calls to provide cross-curricular application or geniune applications of
mathematics in real-life may distract- be more trouble than they are
worth - in some parts of skill mastery. Sometimes the shortest path is
the most efficient.
Each year of mathematics instruction spans finitely skills and practices.
Many operations sans and with clear take home value at the primary and
early secondary school level may shared and taught algorithmically.
Instructors may show students how to do and record work in steps that can
be seen and checked as done or later. In this process, avoiding the
domino effect of errors and enjoying the domino effect of diligence
becomes an end, a value and tool for mastery of skills and practices.
Showing how to do and record work provide observable standards for
gradual and cumulative skill development.
In each level of skill building, some skills may be reasonably assumed as
mastered, with further mastered in a recursive manner. That is known
outside of mathematics as progressive instruction. In mathematics today,
the recursion fails in arithmetic and in algebra. The failure in
arithmetic represent a soft problem dues to shifts in pedagogy. In the
case of algebra, the problem is hard. Skill development steps have been
too large for many to follow immediately or after long exposure to the
shorthand roles of letters and symbols. That role is gibberish to many
mathphobics. The website www.whyslopes.com offer steps to ease or avoid
difficulties in high school algebra and to ease or avoid algebra shocks
in calculus. The precalculus steps each consists of a dozen lessons
easily understood and repeated in class alongside an expanded role for
words to introduce and describe arithmetic and algebra practices and
concepts. In that talking about the forward and backward use of formulas
and rules vocalized a theme common to algebra, logic and calculus. An
essay on what is a variable provides another example. The following
elementary steps may also change college, secondary or late primary
instruction:
- Quick Prime factorization algorithm
- Fraction multiplication and division justified by raising terms
- Arithmetic and algebra practices efficiently described with words.
- Complex Numbers developed apart from trigonometry from elementary
ideas efficiently
- A verbal introduction to what is a variable
The above steps or the lessons that illustrated them can be woven into
present day course design to lessen difficulties or speed instruction.
The site 15 step treatment of geometry begins with common practices with
maps and plans to set the stage for an alternative development of
analytic geometry and a very simplified account of Euclidean Geometry.
The site coverage of functions includes a few expositional inventions -
not too damaging we hope.
Systematics algebra skill development steps fill an old void present in
the exposition of our discipline from it origins. With these extra steps
course design and course materials may document and illustrate skills by
showing how to do and record work in ways that can be seen and checked as
done or later. Completeness here means all instructional steps for skill
development are described and statistically effective. Completeness
appears to be possible. With it plans and paths for skill engineering may
be written and illustrated, and subject to critical path analysis, just
in time design principles and pareto optimality with respect to competing
ends and values. One path may emphasize inclusion to make skill
development simpler to learn and teach. Another may emphasize the logical
or thought-based for the instructon of gifted students. I suspect
inclusive paths may be documented and supported by illustrations,
multiple video demonstrations, in a manner that adults and teachers
unversed in mathematics may use to guide the studies of their charges.
That in one extreme could provide an clear alternative to present-day
efforts in curriculum design beyond the comprehension of mathematicians
and people well-trained in calculus.
The 1953 quotes from the NCTM are provocative. They do not reflect NCTM
adherence to constructivism since 1989. Contructivism in representing a
subjective view of learning may appear in different forms to each
adherence - the lack of any need for consistency is the advantage of
subjectivity. In all applied arts and disciplines, the form of
constructivism which says knowledge is a private matter, beyond
observation and not reliable connected to any testing that may done is
ill-suited to instruction aiming for verifiable skill development.
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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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