In for a penny, in for a pound
Page Sections:
Writing began in 1990 to report ideas to local
educational authorities for review and refinement in a way that would
count. The bureaucratic impossibility of that, a reflection of the
extreme publish or perish competition in academic life, implied silence until
retirement (about 2020). Instead, self-publication followed offline in
1994 and online in 1995. Those ideas included inductive
principles (or standards) for instruction and three starter lessons (three
skills for algebra, two
logic puzzles, why
slopes - a calculus preview ), statistically if not universally effective
1983-89). Those three lessons by themselves could have changes the course of
mathematics instruction in the 1990's. Today Site how-TOs
go further. They supports the inductive
principles and standards essentially in full. Today, educational
authorities need to re-design secondary mathematics from arithmetic to
calculus with the aid of site innovations and its inductive principles and
standards, which Occam's Razor yet favour over
constructivism.
Common Needs and High School Mathematics
Secondary and primary school should aim
deliberately to define, reinforce and enlarge the common knowledge of
mathematics. What does TCPITS - The Common Person In The Street
- need?
With the high failure rate in calculus, there should be other ends
for high school mathematics.That should include mastery of
numerical, geometric and algebraic methods and concepts for solving
routine problems and for recognizing when those methods fails. In
particular, a textbook for learning a second language will not only
cover the spelling and grammatical rules and patterns (exceptions
included), it will may also introduce everyday activities to serve as a stage
for introducing or employing words and grammar. There-in a model for
mathematics education. Numbers are everywhere in street signs, on clocks
and in money matters. Geometry too appears in the use of maps and
designs, and in construction in the home or at work. And
algebra in form of the forward and backward use of the compound interest
formula (chapters 14)
and of geometric sums appears in banking, loan, mortgage and annuity matters (chapters
21-5) Apart from high school mathematics topics required only in
calculus (or in engineering and space travel), preparation for calculus
may serve these other ends while serving the other ends can be a platform for
re-enforcing skills and concepts required by calculus. The challenge for high
school course design is to include routine or common topics whose
potential or importance is clear or can be honestly and not artificially
emphasized, so that each course for the most part are relevant to common
needs. That would be besides i topics and skills explicitly
identified as present due to calculus only or mostly. In the
latter case remarks and digression mentioning routine calculations in
chemistry, physics, biology, accounting, and local construction or work trades
would provide further ends and value to skill and topic mastery. See
critical path analysis below.
Mathematics Teaching
Methods and Issues: Online Volume
Mathematics
Curriculum Notes begins with inductive principles for skill and concept development
and then points to olde flaws and inconsistencies in the exposition of
mathematics which predate and continue in course design today. Inductive
principles provide standards for course design and delivery, simple
subject-based, results oriented, and with site how-TOs,
provide in draft form at least a full set of values, ends and means for
instruction, direct or indirect, which address or avoid a multitude
of content difficulties - expositional gaps and inconsistencies in
modern mathematics curricula and their successors, diluted or not. Some gaps
were inherited from earlier times. Mathematics education, if was an
empirical art, would collect how-TOs easily understood and repeated in the
classroom, and document them for instruction and self-instruction. Site
pages explore remedies, logically designed but not fully tested in class - due to
qualification insufficient for employment in education. Clear and
precise skill and concept is a must - technical gaps in exposition
should but filled - but they need to be accompanied by discipline or
motivation, so that students will follow instruction.. Why bother to require
students to attend school, year after year, when motivation & discipline
for schooling are dilute or absent? Calculus is a motivation for
high
school mathematics. Yet we need to identify the
arithmetic, algebra and geometry in scenes from daily life and work to
say where is the math and how it helps, otherwise there are motivational
gaps.
International Curriculum Reform
Site how-TOs
in accordance inductive
principles for education. offer the framework
for an applied
mathematics program. The program which may be woven into the
existing mixed or applied mathematics
curriculum in the UK and elsewhere. The program instead of repairing and
reinforcing the modern mathematics curricula in North America and Scandinavia
of the mid-1950s onward, as initially intended, gives a
logically-consistent successor . The modern mathematics curricula pretense of
following or supporting a context-free development of modern mathematics
from set theory stumble and ceased to be context-free with the mixed or
applied mathematics introduction of coordinates and drawings in the
development of Calculus, trig and analytic geometry and, if taught,
Euclidean Geometry. Secondary-level modern mathematics curricula was
inconsistent in that arithmetic with the decimal representation of
numbers was avoided in theory - not recognized in axioms
but required and used in practice. Site how-TOs
provide a consistent remedy that may be followed from arithmetic to calculus
to set the stage for applied disciplines and the optional study of modern,
context-free, mathematics at the undergraduate and university level.
Site how-TOs
are logically developed and consistent, modulo the applied math assumption of geometric
practices - those present in geometrical use of maps and plans with and without
the use of coordinates. Occam's Razor may favour them and inductive
principles for their development and refinement.
Goals and Methods for
Mathematics Education
Mathematics programs for primary and secondary schools need
to be based on tried and tested methods in accordance with inductive
principles. For instruction
with a lean inductive program to follow, student performance will
be a guide to remedial measures, what steps to retry as is or in expanded
form, and what is next in skill and concept development.
Mathematical induction
indicate how induction in general will fail if steps are
too large or not reachable. Leanness follows from foresight
based on Occam's Razor or critical
path analysis. In that analysis subprograms of study and effort,
will little or no benefit, those not required by later skill and concept
development, may be questioned if not omitted. Finally, there is a question of
motivation for programs of study. The key question that needs to be clearly
answered is as follows. What
are the ends to be met? There may be more than one: (i) an operational mastery
of numbers and mathematical methods needed in daily life with clear and
plentiful examples to span the needs and maintain mastery without too much
duplication; (ii) development of lawyer-like reading abilities for greater
skills and awareness in work, study and citizenship; (iii) calculus and
preparation for it. Site material supports the last two ends, at
least in part.
Sphagetti programming is a term that applies to code that is
bloated or needs replacement. In such code or any code, subprograms that
have input but no output may be removed. CPM is needed to identify that
sphagetti
Transparency is required in course
design. Calculations involving lengths, areas, volumes, weights, further
measures (direct or calculated) can be placed in a context. Drill and
practice with them may be cast in context or situations fictional or not, but
not too specialized. Course design could present a variety of activities
in daily life and identify the mathematics in them as part of skill and
concept development or as a motivation for the latter. When may before, during
or after. Mathematics instruction may emphasize the applications as ends
in themselves and as tools to develop, refine or consolidate skills and
comprehension that will reappear in further applications and/or the further
development of the subject alone and with others. Each topic in
mathematics should be accompanied by a clear statements in written and spoken
remarks of the one or more of the following: (i) where it will be needed
in daily life (local variety possible), and (ii) where it will be employed
in mathematics or science. Each technique should be named or have
an apt descriptive phrase, for example compound growth formula, rectangle
area calculation, completing the square, to permit oral and written
reference and discussion, and pointers to where it may be used.
Example: Quadratics are
employed in physics to describe projectile motions. Quadratics
are also required in the further mastery of polynomials,
geometry and calculus,
That motivation would be offered
besides skill and comprehension development and verification in accordance
with inductive principles. Students and teachers become pawns and cogs in a
bureaucratic machine when course design does not mention means, values and
ends. The latter may be short and long-term. The latter may also require
some compromises. The most direct route to the long-term ends may be too
dry and at the expense of early ends, means and values that could motivate and
drive learning and teaching. Preparation for (a) calculus and (b)
for engineering in secondary school is a long route.
Motivation should not be contrived nor too artificial.
Example: Compound
growth and decay formulas, and geometric sums too, appear in money matters
in the description of compound
interest and investment growth/decay, and in loans, mortgages and
pension plans calculations. The same or similar mathematics appears in
the description of population growth, animal husbandry, and
radioactive decay. The forward
and backward use of compound interest or compound growth and decay
formulas requires a knowledge of
logarithms, roots and exponentials.
Explanations as to why topics are
covered should be brief, simple and non-fictional, or if fiction or nearly so,
presented as food for thought. While rates and proportions, probability
and averages are be useful in daily activities or pleasures,
the study of quadratics, further polynomials is motivated in all or
almost all by calculus. Economic
models and calculations involving quadratic functions in particular
provide food for thought, and
should be presented as such, rather than as great application and great
motivation for the study of quadratic and further polynomials.
Preparation for calculus provides the simplest explanation, even if that be
dry and boring. Offering motivation is fine, but artificial motivation
should be presented as such - food for thought and/or exercises to develop or
verify mastery.
Meeting the demands and requirements
of calculus would be a long term goal. In preparation for calculus,
instruction may emphasize the value of being able to apply arithmetic,
algebraic and geometrical methods in a show-work, repeatable,
reproducible and verifiable or correctable manner. The ability to apply
rules and patterns one at a time, one after another, alone or in combination,
with or without comprehension of why those rules and patterns work, is a
sign of care and diligence and a prerequisite to intelligence in such
applications.
The ability to follow rules and
patterns in one subject is a sign of specialized intelligence. With
regrets, it is not a guarantee of general intelligence. C'est la
vie.
The thought-based development of
mathematics is based on rules and patterns drawn from experience (or
not) and then assumed. An axiom in mathematics is simply an assumed
pattern.
Exercise for Parents, Students and Teachers -
Keep a record of all the out of school mathematics (arithmetic, geometric, algebraic) that you meet
That should give you
a list of uses and frequencies.
The student of calculus or before may
be advised to learn to do, and worry about the why later. That push
towards rote learning has merit. While a comprehension of the origins and
motivations for rules and patterns in mathematics is desirable, the
thought-based development of mathematics is based on the careful and even
mechanical application of rules and patterns, alone or in sequence and in
combination, to provide results or further rules and patterns to follow.
Comprehension in mathematics and its thought-based development is based on the
ability to see, follow and apply rules and patterns alone or in combination,
one at a time, one after another. Once that ability appears, the
student who has learnt by rules and patterns by rote, in a plug and play
manner, may return to their study to replace their rote mastery by a
thought-based mastery.
For instant in developing site
pages, I realized I had learnt decimal methods for arithmetic by rote and
not understood their origin. Site pages provide a remedy. Site pages endeavor
to provide a thought-based development and combination of the rules and
patterns required in calculus itself, and in what the latter demands from
arithmetic, algebra, geometry and logic. Those pages, quickly written,
are presently in need of polishing. But for that, they are
done.
The thinking part of an art or
discipline:
(repetition of preceeding theme)
The deductive thinking part of an art or discipline comes
after the assumption & careful mastery of some rules and patterns,
steps and methods, practices and conventions. Careful mastery means you
can use the rules and patterns etc to arrive at results in a repeatable, reproducible and,
if hence verifiably right or wrong manner. The thinking part of a subject
begins when you start to combine rules and patterns, steps and methods,
practices and conventions, to obtain new ones in a repeatable, reproducible
and hence verifiable manner.
Thinking or critical thinking within an art or
discipline continues through recognizing the benefits, origins and limitations
of rules and patterns, steps and methods, practices and conventions, so that
the approximations in the application of the latter are known or
avoided. The combination of rule and patterns, customs and practices,
steps and methods, one after another, may lead to short parallel strands of
reason and hence a thought-based development of an art or discipline besides
and even on the empirical mastery of rules and patterns etc with confidence building
results that should be repeatable, reproducible and hence
verifiable.
Once the ability to form or follow strands of reasons within an
art or discipline is present and respected or appreciated, fuller and fuller
thought-based developments can be offered, if not in class, then in print. The
first phase of education could be based on rote - here are the facts and
methods - learn to use them in a repeatable, reproducible and hence verifiable
right or wrong manner. Later phases may then build on that via a mix of
deductive and rote mastery of further rules and patterns. That is to say
deductive reason need not be explicitly axiomatic.
The young mathematic student for
instance may benefit from a thought-based mastery of decimal methods for
addition, comparison and subtraction, a comprehension built inductively from
examples without or before the formal deductive IF-THEN use of implication
rules. But the young student in learning decimal methods for
multiplication and long division would not immediately benefit from an
immediate in-class thought-based explanations of why the methods work.
None the less, skill and confidence in their results might and should follow
from drill and practice, and methods for verifying or checking
results. Later on, arithmetic methods may be employed as part of
deductive arguments with the implicit assumption that they work - give correct
results. That being said, site pages include and point to the
thought-based development or mastery of decimal methods for multiplication in
full and long division at least in part.
Teachers: Site how-TOs,
inductive
principles, and values also give and define a necessary
alternative to constructivism and like Alice in Wonderland, subjective theories of
education. Ideas that cannot be expressed on paper with diagrams, words and
symbols are not part of observable skill & comprehension. Compare
and contrast that view with the Allegory
of the Cave in Plato's work The Republic where knowledge
is based on shadow interpretation. Compare and contrast that view with the
dominant constructivist theory of skill and concept learning, in which
mastery is a subjective affair, not for observation nor correction in an
objective manner; and in which changes in delivery style in a shadowy manner
was suppose to lead to a subjective (anarchistic) view of knowledge, one that
in retrospect resembles the state of knowledge before striving for objectivity
was the norm in science and technology, if not law.
Ends, Values and Means:
- The will and ability to read
notes and textbooks like a lawyer, so that no nuance, no subtlety and no
clause escapes attention is an end, values and means for
mathematics education in the training of students and in course design and
delivery.
- Skill and confidence in mathematics, a written art
or discipline, may follow from care and precision in following and
recording the steps in methods and routines in a well-formatted, readable,
reproducible and repeatable manner for verification or correction. Here care
and precision is another end, value and means for skill and concept
development.
- Mastery of thought-based paths for understanding and
developing skills, concepts and comprehension is a further end, value and
means for mathematics education, and may be considered an extension of the
care and precision, required so that no subtlety, no clause and no detail
escape attention..
Writing began to support and strengthen the thought-based
development of mathematics along the lines of the modern mathematics curricula
of the mid-1950s onward. Identification of gaps in exposition and logic
in Volume 1B, Mathematics
Curriculum Notes, did not change that plan. It indicated more work to do. As
a late 1960's student of the modern mathematics curricula, I was engaged or
hooked by its purported thought- and logic-based development, Euclidean style of
mathematical knowledge. Thus I was
against rote learning.
In retrospect rote learning co-exists with
thought-based development of skills and concepts in mathematics.
(1) At the high level, in the
introduction of calculus, students are given key theorems (rules and patterns)
to apply and told the proofs are too complicated for immediate mastery - site
pages may lessen the level of complication, but the advice to mastery the key
theorems and learn how to apply, but to skip the proofs still applies. As a
student, I did not like the gap that represented in my skill and concept
development.
(2) Precalculus students will
learn formulas for areas and volumes via mix of rote learning and derivation.
The formulas that cannot be derived before calculus can be derived or
explained as applications of calculus in the form of slope calculation
reversal (integration).
(3) Site pages point to the inductive (drawn from
experience) and thought-based development of primary and secondary school
mathematics. But in decimal arithmetic, the easy inductive or thought-based
mastery of place-value addition, comparison and subtraction aids skill
development. But multiplication and long division methods, must for late
primary and early secondary school, are easily mastered by rote but may be too
complicated for a thought-based development. The latter may come, if it comes
at all, at the senior high school level.
Note: Calculus requires and extends the
earlier full strength use of algebra in secondary mathematics. The latter in
turn requires a full strength mastery of exact and approximate arithmetic with
the decimal format of numbers alone or in fractions. That mastery begins
in primary school.
The coverage of arithmetic, algebra, geometry and even
calculus in site pages was intended to eliminate rote learning in mathematics
education, and provide a thought-based development instead in accordance with inductive
principles and standards for instruction in rule and pattern based arts
and disciplines - those advocating or striving for objectivity. Yet as indicated some
rote learning is needed in the development of arithmetic.Mathematics
education, yours or mine, may have involved more rote learning than
need-be. Some pages were written to provide myself a thought-based
understanding of skills and concept met earlier by rote.
Mathematics education
may become a bureaucratic set of rituals in rules and patterns are mastered
one a time and one after another because that is required for a
forthcoming end of year final examination, without any further foresight into
the aims, ends and even values that might be part of mathematics education.
Preparation for calculus provide one end, albeit the high failure rate in
calculus makes one wonder if the route to calculus is too hard for students -
does it represent a pyramid scheme in which many start and few benefit.
The remedy for that is to include along side and even as part of preparation
for calculus, an identification and development of the arithmetic,
algebraic, geometry and even logic routines and methods that are present and
useful in daily life. That should be besides the development of skills and
topics explicitly required by calculus. Then each course would provide
useful and hence motivated skills and concepts.
Is it possible for students to end their studies in
mathematics before calculus is manner that provides skills and confidence
along side the decision not to prepare further for calculus?
Students need to master routine methods for the
routine mathematical questions that appear in daily life. The question what
would not work in society if there were no numbers, no compound interest
or no growth formulas, no geometric sums, no geometry (maps and plans
include) might motivated consumer or people-oriented mathematics studies along
side calculus preparation. I suspect most students and teachers, and guidance
counselors, are unaware that preparation for calculus is or was the reason
- the only possible reason - for
many or most topics in secondary and primary school mathematics.
Preparation for calculus should set a standard for and not a hidden agenda in
earlier skill and concept development.
Each course mathematics represents an opportunity to see and
recognize the though-based development of skills and concepts. Students
may enter with a mastery of rules and patterns (we hope), some mastered by
rote and some with a thought-based development. In the
course, further rules and patterns may built on earlier ones or built
separately by emphasizing the origins and derivations from immediate
observations or previously met if not mastered rules and methods, regardless
of how those earlier rules and methods were met and (?) mastered. The latter
emphasis represent the introduction or continuation of the though-based
development of skills and comprehension. Courses are generally too short
for a full review of earlier rules and patterns with a thought-based
development. Modulo that, or except for that, each course may
stand on a mix of rote and thought-based learning to provide a thought-based
development via the careful use of rules and patterns alone and in combination
to get results or to imply further rules and patterns.
Students may object to the thought-based development of
skills. That development may not be tested and hence not required by
forthcoming final examinations. Students may be in learn only what is
required for the final. They may be no long-term goals. Further more, students
may not appreciate explanations of rules whose earlier rote mastery gives
results, repeatable and reproducible, or not - It works, why do we need to
see more, may represent the underling attitude or question, to which there
is no reply. And students may object to the inclusion of a development
of a rule or method, and ask instead rote mastery of the rule and method with
minimum explanation and many examples. Most activities outside of
mathematics are plug and play. There is no need to ask why. That sets a
standard, not optimal, for mathematics education. It represents the view, give
us the formula and numbers to plug in it. Such student needs to be in
courses which emphasis routine problem solving skills for the multiple ways in
which mathematics touches daily life. Students aiming for calculus would
be benefit from a solid grounding the precalculus application of mathematics,
every one is likely to meet.
The thought-based development of mathematics is not for all.
There are some dilemmas and gaps in the exposition of high school mathematics
that will not be fully addressed here. Course design and promotion needs
to reflect a critical path analysis of what is to be done and why. Site
pages identify what might be included in secondary mathematics as preparation
for calculus and as aid (?) to routine problem solving likely to be needed in
the lives of students and families - problem solving with geometry and money
matters say. The challenge is to provide a mathematics education with
aims, methods and values that students and teachers can appreciate.
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The Rote, Plug and Play, Descriptive,
Aspect of Education in Science and
Technology
Question: How can the latter be reduced? Can it?
Mathematics is an art and discipline
in which a thought-based development is possible on first meeting or on
review, revision or consolidation of its skills and concepts. Most can
be developed with drawing and writing instrument (pencil and paper or modern
successors included). In contrast, science, and technology in theory and in
practice has become plug and play. The cook in the kitchen often
and the chemist rely on bought ingredients, properly labeled. The high school
and college science or technology lab may include instruments that are
used in a plug and play manner. The use of electronic balances may provide
greater accuracy and convenience, but they also represent that plug and play
aspect of a school or college lab. The use of balances where physical
weights are employed is closer to the origins of science and technology, more
primitive, but less plug and play. That is to suggest simple measurement
of mass and volume may be done in the school lab. The use of batteries,
ammeters and voltmeters represent a further plug and play element of the
school or college science lab. Is it possible to verify Kirchoff's current and
voltage laws with them? I tried and failed. The foregoing raises the
question of how to illustrate chemistry, physics and biology in lab in a way
that will confirm or corroborate theories alongside their in class description
in a manner that is, minimally plug and play. Dissection in biology does not
count as artificial plug and play. In the classroom, the periodic table
represent a cumulative effort which can be described as is with its
development, and its modern variations. But it cannot be derived.
And in chemistry, students may be surprised by parallel theories which
disagree on the identification of acids, bases and salt. The term
acid-salt refers to a compound that when dissolved in water is acidic
(turns litmus red) while the chemical formula theory suggested it should be a
salt. Students may have to view science and technology as a plug and
play process in which methods are subject to verification - do they give repeatable
and reproducible results in practice, and in which theories, hypothesizes or
hopes are subject to refutation (it does not work) or partial success,
small or wide ranging, instead of absolute confirmation.
Do you insist on putting a round peg into a square hole
Does it make sense for subjective
complicated, incompletely defined theories of knowledge to be applied to
mathematics or science - these being arts and disciplines striving for
objectivity? The answer is no. The attempt to do so lends an Alice
in Wonderland experience for students and teachers in precalculus
mathematics in North America and less so in the UK. The majority is not always
right.
The 1990 NCTM standards and principles fail to answers to two
concrete questions: What mathematics is to be taught, why and how.
The 1989 standards as is and reformulates in year 2000 focuses on delivery, constructivist style and so represent a subjective view of knowledge in
which teacher or discipline centered view instruction (my discipline has many
steps,
let me cover them) is replaced by calls individuals to construct
their own comprehension from authentic, realistic, genuine activities.
Yet most primary and secondary school
instructor do not have a calculus background - the motivation for much of
secondary school mathematic program, its content, and many are seconded from
other disciplines, impressed into mathematics instruction. Pre-1990 expositional difficulties in mathematics
in the form of steps too large or missing and in form of expositional
inconsistencies are also not recognized.
The NCTM
standards ignore the instructor content mastery problem and
expositional difficulties, while calling for and a specifying a
subjective delivery style, and depreciating rule and pattern mastery as a form
of rote learning. Contrast that with the old fashioned view good
figuring skills were necessary to demonstrate intelligence. Contrast that with
the Euclidean model for rule and pattern based reason prized in the work of
Euclid and in modern mathematics. Contrast that as well with the
content of Volume 1A, Pattern Based Reason,
and its description of the origin, benefits and limitations of rule and
pattern-based processes in thought and deed in mathematics and science.
Modern society, for better or worse, with great variation, strives for
objectivity and not subjectivity in the development and application of rules
and principles in law, mathematics, science and technology. Site
how-TOs and inductive principles and standards (or lower bounds) for
instruction support the latter. The NCTM needs a change of
course.
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Mathematics Education Essays etc
Area Intro
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
In for a Penny
Constructivism and Cognitive Theory
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links
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