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Evaluation of the North American Mathematics Curriculum - Hook, Line
and Sinkers
The modern mathematics curricula, say 1955-80 inconsistently
introduced ideas from higher level mathematics but provided a
nearly expert, discipline-based, discipline-centered approach to course
design and delivery, with a few awkward elements. The olde problem of too
many symbols and not enough words in the introduction of algebra was not
recognized and so persisted. The decimal-free nature of modern
mathematics - its lack of dependence on the decimal representation of
real numbers - meant the common use of decimals, required in high school
arithmetic, was not sanctioned and implied the decimal viewpoint of error
control and continuity, a view that lingers with the study of scientific
notation a * 10k for measurements (with 0.1 < a < 1),
was otherwise avoided. The discussion of ratios a :b and multiple a:b:c
also continued in an awkward manner. The sprit of the modern mathematics
curricula was not child-centered. It was discipline centered. It focused
on the elements of mathematics which appeared, which would be needed for
comprehension of high level mathematics in a context-independent
matter. That focus provided a hard route to follow due to the
lack of a clear introduction of algebraic concepts and due to the
avoidance of decimals - the sanction of their use in daily life (weights,
measures and calculations) and the absence of any dependence in the high
school & college development of mathematics. That focus
made learning and teaching harder than need-be. The new fashioned
(context-free) description of functions as sets of ordered pairs
that satisfy a vertical line property appeared too suddenly and too
absolutely. The companion concepts of - how one number
depended on others - and function notation y = f(x) should be emphasized
first. The modern mathematics curricula selection and introduction
of skills and concepts was not optimal. Its introduction was nearly
expert, but not expert enough - too much enthusiam, not enough thought.
In recent decades, factors outside of the discipline
led to curriculum reforms 1989 onward that have ignored and compounded
the earlier difficulties in course design and delivery
First, the end of streaming in course design and delivery, the
merging of course content for enriched instruction into general
instruction added topics not essential into the high school education of
every student. Second, the rich treatment of Euclidean Geometry was
judged too hard, too intimidating for the general student, so it was
dropped - site pages indicate a leaner, minimal treatment of Euclidean
Geometry, one that depends on direct use of logic. Third, arithmetic
drill, practice and correction was considered not important and so
de-emphasized in North America and UK(?) schools in favour of calculator
use and spreadsheet use. But students need to have an automatics,
efficient command of exact arithmetic with whole numbers and fractions,
one that does not require them to reach for a calculator for every simple
calculation, if they are to master algebra, trig, functions and
calculus That is a discipline-based view. Anything less delays or
dilutes high school and college level mathematics - changes the
discipline in a way that earlier masters would not understand - and so
undermines any reason for the study of mathematics, year after year in
high school. So course content needs to be maintained and protected
by discipline experts.
Mastery of the skills and concepts through their ability to do
calculations and follow rules and patterns in a repeatable, reproducible
and hence verifiable manner. That requires care and precision. It
can be a struggle to understand precisely the chains of reason, verbal
and symbolic, in a mathematics text due to steps to large,
Not every one has the patience for it. The high school and college
exposition of mathematics from algebra to calculus may make that
mastery harder than need-be with algebraic skills and concepts introduced
awkwardly. Site pages point to a remedy for that.
In the past, mastery of arithmetic, figuring skills, was regarded
as a sign of intelligence. In brief, it meant a student or a
worker had the wits or ability to follow rules and patterns in a
repeatable, reproducible, verifiable and reliable manner.
But factors not expert in mathematics, the soft science in the form
of psychology and theories of learning may call for critical thinking and
independent judgment but oppose the mastery of rules and patterns, alone
and in sequence, for the sake of repeatable, reproducible, verifiable and
reproducible results. That points to a conflict or inconsistency
between expert views of mathematics and hard sciences - how university
professors in the hard sciences and mathematics might value and define
their disciplines - and the anti-rule, anti-bureaucratic but still
bureaucratic applied and developed theories and practices for education
reform. Factors who are not expert in mathematics may influence and
control course design and delivery
Mathematics course design and delivery should identify what skills and
concepts are essential to provide a curricula which is learn but
effective. The advance in site pages for the exposition of the
mathematics suggest how. Those advances and the question
how to select topics to interest students - can we design a sequence of
courses so each one if it was the last taken by a student, would leave
a satisfying image of the discipline and with or through that an
invitation to further studies?
provides mathematics education committees in schools and colleges with
opportunities to make learning and teaching simpler and more
effective. Course design and delivery with some variations may be
built on the collection and development of appetizers and lessons easily
understood and repeated by teachers and effective in the classroom.
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