Which Way to Go (2010-08-26)
"Would you tell me, please, which way I ought to go from
here?"
"That depends a good deal on where you want to get to," said the Cat.
"I don’t much care where--" said Alice.
"Then it doesn’t matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an explanation.
"Oh, you’re sure to do that," said the Cat, "if you only walk long
enough."
(Alice's Adventures in Wonderland, Chapter
6)
Paths for Mathematics & Science Education, a
related 2008-11-16 essay
Via the school of hard knocks, I have seen that student may prefer
rote learning and dislike explanations which start with the
supposition that explanations why are key to mathematics mastery.
Despite my aversion to rote learning, I see aspect of rote learning
in (a) Calculus instruction where students are told to learn to do
first and to worry about the theory second, and where technical
proofs in are inaccessible, or largely slowly, where the logical
structure is based on theorems stated with out proof, axioms or
assumed patterns given without motivations; and (b) before calculus
in the mastery of arithmetic, methods for addition and subtraction
may be explained in all or part while methods for multiplication
and long division are given without, and so learnt by rote.
When the mathematics teachers offers proofs in a classroom or
motivations or chains of reasons to support the use of method or to
reach the statement of a theorem, students may object on the
grounds that school authorities would not ask instructors to
present false formulas or false statements in class. Hence proofs
and explanation why are not needed. There-in lies a student
argument for rote learning and against the appearance of proofs in
mathematics.
Mathematics education in general may provide methods and
statements to met and master via (I) rote -that is by presentation
of methods and patterns to use; OR via (II) inductive learning
based examples that draw on and provide experience, hands-on or on
paper, to provide comprehension in all or part (hand waving); OR
via (III) deductive learning that draw methods and conclusions from
chains of reason which formally or informally depend on earlier
knowledge, that earlier practices or axioms (explicitly assumed
patterns).
While we may consider deductive education to be the highest form of
learning, that education is based on axioms or assumed patterns,
usually algebraically described, and which require mastery of the
algebraic shorthand way of writing and reasoning alongside the
ability and patience to follow chains of reason or implications.
But this edifice stands on axioms which are given, or drawn from
experience - that of the students or that of course designers and
their ancestors in course design and delivery. In essence there is
a bootstrap operation. Deductive codification begins with stated
axioms, typically but not necessarily given for rote learning.
That being said, site lessons via examples provide or indicate an
inductive development of high school mathematics in which axioms
(assumed patterns) - those needed for a more secure deductive
development of the discipline - are drawn or suggested by example
based experience.
Mathematic education besides off-paper measurements and perceptions
begins with on paper mastery of counting, arithmetic and drawing
methods, methods that lead to observable, repeatable, reproducible
and if need-be, correctable, results and thus feedback to students.
The ability to apply a method in a careful step-by-step manner to
obtain repeatable, reproducible results is the first source of
skill and confidence in mathematics. The combination of methods,
implications rules A IF B included, in sequence one at a time and
one after another, forms chains of reason or action that can be
followed to again provide repeatable, reproducible, observable and
correctable results on paper or off. Developing the ability to
combine rules and patterns to obtain further ones, theorems
included, is part of mathematics and key to its deductive
codification and foundation, modulo the limits of axiomatic
systems.
Within an axiomatic deductive framework, each of it statements is
subject to testing or the following question. Can we find a direct
or indirect chains of reason from the axioms - the explicit
assumptions of the framework or code - that imply the statement or
its negation. Those chains of reason provide a proof. In English
common law, the accused is assumed to be innocence and the
accusation or suspicion false until there is clear enough proof of
guilt - a convenience for the defendant or accused. In the French
civil law, the accusation or formally statement suspicion is assume
to be guilty until proof of innocence is available - a convenience
for the prosecutor. In contrast, in mathematics a conjecture or
formally stated suspicion is not presumed to be false or true.
In mathematics education, a rule or theorem that a student is asked
to meet and master is likely to be true, modulo the axioms and
practices of deductive logic in the discipline. The student
objection that following the chains of reason and verifying each
step there-in is not needed represents a plug-and-play approach to
mathematics and division of labour between the developers of
mathematics and those who might apply, including the students who
object to meeting and following proofs, or providing them. An
operational and empirical command of mathematics for the home and
for the workplace is possible in a repeatable, reproducible and
observable manner by rote learning, and without proofs. That
operational and empirical command of the necessary mathematical
methods should be an aim of mathematics education. In it, the key
to deductive reason, the ability to combine rules and patterns to
imply further one might be present in that operational command.
Then a theoretical command of mathematics in addition requires
mastery of the proofs, or the ability to produce proofs within a
given set of axiomatic and logical practices. Full comprehension of
the axiomatic method may be based, as indicated above, on drawing
the axioms themselves from experience of students - that provided
by a careful selection and discussion of examples, along side a
deliberate and earlier development of algebraic, shorthand, ways of
writing and reasoning. Those could be prerequisites to the
axiomatic development. For classrooms full of students (a)
wanting to learn how to to do and nothing more, and (b) wanting to
understand as well, course design and materials may cater to (a) in
class, and provide full or fuller support for (b) in further
material for reading or viewing. There-in lies the site
objective.
Mathematics is art or discipline whose methods do have to be plug
and play. Thought-based development is possible. In contrast, the
methods of science and technology can be described in the
classroom, and supported in part by basic instruments -
mechanical preferred - and simple experiment that do not involve
plug-and-play components. The question for science education is how
to minimize the role of plug-and-play components in the science lab
to maximize the hands-on experience and evidence for scientific
process in the high school and/or college lab. Unlike mathematics,
where plug and play can be avoided, science education and labs
must rely on plug and play elements - elements that are beyond the
reach of the classroom lab. That irks.
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The big question which way to go, how and why hangs over
mathematics learning and teaching.
-
People not in school (and older students) will say
they were good in mathematics until ... and then say they wished they
had gone further. Most study it until they lose interest or until
it becomes too hard.
-
In many high schools and colleges, asking why this or
that is in a mathematics course has the answer: Preparation
for final examinations. In that environment, explanations of why
a formula or method works may be met by the view: mathematics
teachers are hired only to give correct methods, and so explanations of
how or why are not needed.
-
In algebra, the placeholder roles of letters and further
symbols in formula for areas, perimeters and volumes may be clear and
obvious for most. But in algebra, less clear and less obvious are the
further role of letters and symbols in dealing with polynomials, in
solving equations, and in rules for algebra - also called properties
of numbers or axioms for numbers. And in college or senior high school
mathematics, the role of letters and symbols in epsilonics (the
technical discussion of limits in calculus or beyond) is hard even for
the best students and teachers.
The site answer for which way to go, how and why follows.
The why itself follows from a reading of the history of
mathematics education in North America and Europe. The how follows
reflections on what has been done and why, plus the assumption that
showing how to do in an observable and reliable manner may be a source
of skill and confidence. Course designs or curricula published online
by educational authorities and organizations vary between (i) a focus
on standard for delivery style, content loosely or grudging given, and
(ii), a focus on content with delivery style barely mentioned.
Perhaps, putting content first and deciding what skills and concept
develop should come first. Otherwise, what to teach is not
certain.
The site answer for which way to go, how and why reflects and
reacts to what has been done or not done before.
Three Paths & Ends for Mathematics Studies (one iffy)
- Serve the common or likely needs of daily life and livelihood (work
included), and beyond that, for specialized trades that require a little
more mathematics, not much.
Six application areas that serve
common or likely needs provide a focus and destination for primary and
junior high school mathematics that teachers and parents without a
strong background in mathematics may easily understand and
support.
- Prepare for college programs in non-scientific fields - the social
arts and education apart from mathematics and science.
Here consolidation and extension of six application
areas which serve common or likely needs would maintain or extend
skills and know-how. The further study of mathematics and preparation
for calculus would be optional - might have intellectual value, or keep
options for entry into technical fields open.
- Prepare for college programs in education engineering, science,
technology and business where mathematics in the form of calculus and
beyond is a must. Most students heading for business might be served by
following Path III, but in a manner that skips material they will not be
needed.
Many, many small technical innovations in site material
imply simpler paths for the thought-based introduction development of
fraction skills; number theory skills (prime number factorization
& use included); the introduction and development of algebra from
solving equations and backward use of formulas algebra to its
full-strength roles in calculus; and the introduction and development
of complex numbers and trigonometry. The innovations in question are
sufficiently close to present day instructional practices to be of
immediate service, but altogether they may be woven into alternative
curriculum for senior high school mathematics and calculus, one that
includes a thought-based development of skills and concepts, and axioms
for mathematics that better serve the needs of college programs outside
of mathematics, while being consistent with those previously employed
in courses design and delivery. While big changes in course design and
delivery are not to your liking, the smaller ones - ideas to make the
hard easier for students - may be.
The preparation for college programs in scientific fields would build
on the careful development and mastery between ages 5 and 14 of skills
that serve common or likely needs.
Mathematics instruction may follow the first path only in primary and
secondary school. Or Learners from ages 4 and 14 say may follow Path I
and then continue with Path II or III.
Path I: Serving Common Needs
The common or likely needs of daily and livelihoods appears in most
places. In most societies, counting, measurement and arithmetic skills
and know-how are employed in several areas: (i) time & date
matters, (ii) money usage, (iii) mass, weight and measure usage, (iv) in
map and plan usage - geometry, (iv) avoiding or controlling risks -
matters of chance; and (vi) in understanding the logic of instructions
and contracts.
The foregoing includes doing and recording figuring
and reasoning data and steps for the sake of accounting - remembering
and checking what was done. Skill development standards also require
people to know about and avoid the domino effect of errors and
approximations in counting and measurement, in figuring and drawing,
and in logic itself. To serve the common or likely needs, skill
development may focus on the how and not the why. The why can be
provided for students who will not learn without it and where it does
not overwhelm students or teachers.
Many teachers not trained in a mathematical fields are
told their schools need a mathematics teachers and, if you want to keep
your job, you are one of them. So skill and know-how development in
the service of common or likely needs, has to made as a simple as
possible. As a step in that direction, this site includes a list of
topics for
Learning to do in an observable, verifiable and reliable manner, while
avoiding the domino effect, provide a standard for students and teachers
to emphasize and respect. The underlying care and work habits in learning
to do in a reliable manner is likely to be a source of skill and
confidence for many students for their present and future work and
studies.
Path II: Preparing for College Programs in Non-Scientific Fields
Preparation for college programs in non-scientific fields - the social
arts and education apart from mathematics and science. This preparation
may continue and conserve the mastery of mathematical and logic skills
needed for common or likely needs. Logic mastery here may develop the
ability to read agreements and contracts before signing them - that could
be a matter of self-defense for students. Students may learn about the
use if not origins of compound interest formulas and geometric sums for
the sake of later skill in handling savings, mortgages, investments and
annuities/pension plans. The same material plus origins may be included
in path III.
Path II students may be further exposed to statistics.
However, intellectually, college programs that require statistics
forwards and backwards are usually programs in the social sciences data
and graphical analysis will be done with the aid of statistical
conventions and methods, the benefits and limitations of which require
an advance mastery of mathematics, one that will not be seen or had by
students and instructors in the social sciences. Exposing Path I and
Path II students to path III concepts without the explanation that the
latter are required for high school and college programs in scientific
fields, may leave those students with these sense that mathematics
after arithmetic has no end nor value.
Path III: Preparing for College Programs in Scientific
Fields
Preparation for mathematical college programs in science and business
which employ mathematics strongly in practice or theory, or require some
level of mathematics mastery as a means of student selection. While
earlier instruction may be inclusive and non-competitive in the attempt
to prepare students for daily life etc, preparation for college programs
has to respect subject matter. In particular, mastery of mathematics in
the form of calculus, where not a filter, is the key to understanding
the practice and theory of college programs in scientific and technical
fields. But calculus is very demanding, It is the main reason for number
theory, algebra, geometry, trig, functions, set, logic and probability in
senior, if not junior high school mathematics.
The foregoing preparation for college programs has
intellectual value in the logic and thought-based development of skills
and concepts. While children and young teens may think a teacher is
hired to give correct mathematical methods in class, so that
explanations in full or part of why in skill and know-how development
is not important, the higher level preparation for college programs
requires and values comprehension not only of method use, but also
their origins or why they work. In that, knowing more is better than
knowing less. But the why is often limited to seeing how examples may
confirm or imply "initial" assumptions (rules, patterns, practices
included) and how the latter in turn may be combined to imply more
rules and patterns in a cumulative manner. That may be the best that
can be done in most technical and scientific fields.
Path III or college mathematics, may cover how roots,
powers trigonometric functions and logarithmic are defined, and how the
definitions imply the properties. That all part of the thought based
development of mathematics. But most numerical values of these
functions are full provided by electronic calculators. In the past,
they were found from tables or slides rules. But most students and
teachers of mathematics, will not study how tables and calculators
provide or got those numerical values. So while people may understand
the definition of calculations - what is or should be calculated, the
how of the calculation remains a mystery for almost all, professional
mathematicians included, because the how is not studied - and because
the how may vary between various calculators and tables. Meanwhile in
biology, chemistry, physics and computer technology, there are great
limitations on what experiments and what theories can be done or
confirmed in school and college labs. So the high school and college
level mastery of these subjects is limited to learning theory and its
empirical origins, to learning about lab materials, equipment and
processes in a plug and play manner - material comes in labeled
containers filled and labeled by others, equipment for observations and
measurement (electronic calculators included) provide understandable
results, but with inner working often unclear to students and teachers.
That being said, in biology dissections (or images thereof) and in
biology, chemistry and physics, the use of mechanical methods for
observation and measurement where possible may bring students and
teachers closer the roots of the art or discipline in question.
For those who are interested, mathematics in which the
explanation of why, the thought-based development of skills and
concepts, is the fullest apart from the use of table and calculators
for obtaining numerical results.
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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.
-
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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