More on Mathematics Education, Etc.
Covers: For a leaner curriculum, Education an empirical
art, More on testing, Constructivism versus Empirical Methods.
For a Leaner Curriculum
Where mathematics education reform is too bureaucratic or too rigid to
consider ideas that should count, more generations of students will
suffer from gaps in course design at the secondary & college level.
Education reform has led to more and more topics being
included in secondary school mathematics. while old shortcomings linger
and new ones born. A lean mathematics curriculum would focus on
fraction and algebra skills and sense, 2D geometry with and without
coordinates, trigonometric, and logic, all as preparation for calculus.
A lean mathematics curriculum might include some application to
demonstrate the usefulness of fractions, algebra and coordinates, and
so invite the further studies. Preparation for calculus is key to
college or university the thought-based as distinct from rote,
study and comprehension of accounting, science and mathematics. In
secondary school mathematics, statistics, 3d geometry, nets for 3D
polyhedra, and transformation geometry are digressions for learning
outside of mathematics, in say courses on social science, art or
technical drawing, if need-be. Including these digression in core
mathematics programs dilutes the preparation for calculus and
calculus-based studies in mathematics Inclusions leads to a loss of
focus in skill and knowledge development. Lean mathematics
instruction could and should focus on mastery of fractions, algebra, 2d
geometry with and without coordinates, logic, and trig.
Cut, cut, cut. Do the minimum well. Then enrich once the minimum is
well-taught. Further cuts or shortening are possible by dropping
artifacts in course design and delivery, topics not required for further
skill and concept development. That being said, teachers still have
to cover topics demanded by local school authorities. Site remedies may
be woven into lessons to support and enrich local curricula, lean or not.
Education, An Empirical Art
In empirical arts, practices with repeatable and
reproducible results come first, tested via trial and error, while
theories and principles come later to summarize, to codify, to refine
and even enlighten the practices. While practices or sequences of them
in some empirical or hands-on arts in science, technology and business,
assembly lines included, may comply with principles and
standards, even be connected and organized and designed around said
principles and standards, the forerunner to such organization
consists of experience where principles and standards in formation and
adaptation met reality - success and failure included.
Education is an empirical art. We may not read a student's mind, how a
student thinks or links together skills and patterns, yet we can
observe and test student performance, skill by skill, concept by concept,
and encourage, but not guarantee, mastery of standard calculations and
standard arguments or chains of reason in algebra, geometry and beyond.
In some disciplines, not all, there are right and wrong answers due to
methods that lead to repeatable and reproducible, and thus verifiable
results independent of whom-ever applies the method. Learning how to
apply and combine methods carefully to obtain reproducible and thus
verifiable results is an old sign of intelligence in many old arts and
disciplines in business, trades, science, engineering, technology
and bureaucracy. The latter is subject to the limitations of rule and
pattern based thought and practices, and the critical knowledge that not
all is certain in empirical based thought and practice.
Critical thinking in science and technology begins with an awareness that
what we hope for, dream of or construct in our minds remains speculation
or faith IF or WHILE it or its consequence cannot be observe or tested
directly, and thus corroborated if not confirmed. The foregoing is a
rebuttal to the constructivist theory of learning, the part which opposes
testing, the existence of questions with right or wrong answers, and
which says student knowledge, if individually constructed, should not be
contradicted. Empirically sound education must oppose wishful
thinking. That being said, constructivist methods for engaging,
authentic, genuine material and the development of critical thinking
could be incorporated into education as an empirical art.
More on Testing. Knowledge empirically found or
tested is relative and not absolute. Instruction which relies on
testing skills and concepts can only identify errors in the mastery of
the latter while correct responses only confirm, but do not guarantee
mastery. But the level of student competence in a discipline defined by
skills and concept mastery can be estimated from the degree of
difficulty, the unlikelihood of correct responses if skills and
concepts have not been mastered, and comprehensive of a test or
series of test. Here individualized testing may be informative that
mass testing. Empirical soundness of instruction and testing, the issue
of lessons and associated tests with repeatable and reproducible
results locally and beyond, should not be scrutinized in an absolute
manner. Cognitive theory should look at education as an empirical
art.
Constructivism versus Empirical Methods
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After all is said, I found myself advocating
an empirical approach to course design and delivery, an approach
which may be combined with constructivist educational methods,
those that work regardless of flaws in empirically unsound
constructivist principles or theories, - principles and
theories which imply subjectivity in mathematics and science, and
beyond, which emphasize the empirical weakness of testing in
education, if not in general, in place of the empirical merits.
Constructivism with its advocacy of critical thinking in
criticizing testing is contradicting the empirical basis of science
and technology, the readiness to test in order to eliminate errors
and so favor some success.
Managing or directing mathematics course
design and delivery by insisting that pedagogical methods will
work is a top-down approach to education reform. In the absence
of testing, of clearly explicitly defined steps or building
blocks which have worked, this top-down approach
becomes an empirical gamble, like marketing and
distributing a drug blindly in the hope that it work well and
have no side effects. Besides hope in education reform,
there needs to be verification - trust but verify.
Otherwise, great leap forwards may do more harm than good.
While we cannot read a student mind to see what has
been constructer or understood or not, or how, we can in good
empirical form observe, correct and mark what is written or
produced by students. Continuous testing, probing and observation
of student performance is part of a continuous educational
process. Through test feedback and/or direct
explanations, students learn to avoid or discount wishful
suppositions or constructs in contradiction with their environment
in and out of school. Thus schooling can shape students minds
rigidly. Or, schools can present rules and patterns of
various arts and disciplines, and indicate the origins, benefits
and limitations of rule and pattern based knowledge, the
presence of uncertainty, the open ended nature of many
situations or problems, a necessary disappointment for those of us
nostalgic for certainty.
Spelling in a language requires knowledge of all
the letters in its alphabet. We would oppose suggestions that
students have to learn only part of alphabet. Some
spellings are artificial. Students have to be given them.
Students cannot discover them. Likewise in mathematics, we should
oppose suggestions that students don't need fraction skills and
sense, the prerequisite to algebra, or suggestions
that pencils and paper calculation skills are not needed
because of calculators and technology, or suggestions that
students can discover mathematics by themselves. The structure of
mathematics is inherited, handed-down and varying over time.
Insistence on the discovery methods, insistence on cognitive
dissonance, in learning mathematics leads to a loss of clarity
and compounds existing confusions.
Putting constructivism subjective views of knowledge
in charge of mathematics and science education is akin to rejecting
the form of critical thinking in mathematics and science developed
since the 14th century A.D. The placement invites cognitive
dissonance (confusion) for all involved - students, teachers and
parents. Bon Appetite.
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