In Math there´s ONLY ONE RIGHT WAY to get the answer
This is a convenient way to disguise the limitations of the instructor. It´s also a handy way to insist on the student jumping through the hoop set up by the curriculum designer. It is ridiculous.
-- With word problems, there are frequently legitimate differences in interpretation which lead to alternate formulations of the equation.
-- In arithmetic computations, there are standard algorithms for the operations, but there are also lots of creative shortcuts that are inspired by perceptive recognition of relationships. Multiplying by ninety nine is a snap done by subtracting one multiple from one hundred times the number.
-- The same problem can be used in calculus and in approximation theory with differing approaches to solution, one more widely applicable, one more elegant... none best.
-- Even in abstract areas like measure theory, there are synthetic and analytic proofs of theorems. For example the theorem that some entity must exist might be proved by disproving its nonexistence, analytically. Or, a stronger proof would be to whip an example of the entity into existence in some creative synthesis.
-- In fact whole subject areas have been developed working from different starting points and using different tools. The most famous example of this phenomenon was Newton´s differential calculus and Leibnitz´s integral calculus, two sides of the same coin, resulting in a nationalistic fracas over who invented calculus.
Math is cumulative and must be done in sequence
-- Algebra evolved over centuries, solving problems of interest from a variety of sources, and is laced with *facts* from other areas. One standard textbook just drops the little *piece* -- the slopes of perpendicular lines are the negative multiplicative inverses of each other -- on the learner without the respect to reveal the source of this tidbit.
-- The lovely creation, Probability Theory, was the abstraction of principles from a mammoth collage of information accumulated over years. Just as Euclidean geometry was. The result is the illusion that math is always just so.
-- Although a given course textbook will logically put the more fundamental material before the more elaborate, it´s still just one highway; but once built, later developers of textbooks are inclined to take the same route to make the life of the academics running the school programs simpler, not because the subject can only be approached this way or because the students are best sesrved this way.
-- The academic establishment is concerned over controlling the direction of masses of students and fears the chaos of diversity of viewpoints.
The sanctioned curriculum is the best way to learn
It likely wasn´t the way the material was discovered and the magic of the discovery process is lost without the need that inspired it. There certainly then is an alternate way to learn that could be more significant to some learners.
-- This control tactic reduces the diversity that the academic and publishing establishment has to deal with. Observe the pandemonium that ensues among the teaching staff when there´s a change like computers or set theory or back-to-basics.
-- It also provides a funnelling of prospective practitioners through a gateway whose purpose is to control numbers of practitioners. For economic reasons, scarcity is more porfitable to those inside the establishment. Calculus is frequently used for this purpose of "thinning the herd".
-- Even among professional mathematicians there are some who have little tolerance or aptitude for some math areas outside their specialty, who literally have nightmares from their exposure to those areas of the math curriculum. Had those areas been used as gateway courses, they could have been lost.
-- If math curriculum is sacrosanct, why does it go through fashions? The theme of the 60´s was Abstraction, the more abstract the better. The 70´s, after the collapse of the market for mathematicians when the space program went into hiatus, could only see the merit in Applied Math, except in those havens where the entrenched faculty were relatively immune from the job market.
To be empowered, you must derive the theorems.
Logically like bridge building, unless you want to learn bridge building, you only need to know where the bridges are and what the tolls are to use them. That´s why engineers seldom derive the theorems, actuaries pass their exams without needing to derive theorems, clinical scientists don´t either. Nor do they engage in this activity in their professional work.
Math problems as punishment
In grade school and middle schools, it´s not uncommon for teachers to use math problems as tasks to fill student time after hours. As a handy control tactic it´s convenient to use challenging problems the instructor has only to check the bottom line answer from some answer book. But the impact is devastating to the future attitudes and numeracy of students.
Only the devotees may enter the temple of the elect.
Every area has its arcane terminology and its insider customs, partly for the convenience, but partly to maintain an elite status. In math, during those periods when the space program made mathematicians scarce in the schools, this tendency took an ugly turn. In the hands of an instructor who had limited ability in math and less enthusiasm, the pernicious tactic of telling students that they "had no head for math" became a vile, damaging habit when confronted with students who didn´t manage to rise above the lack of teacher inspiration. It continues still, revealing instructors with limited understanding themselves and little skill at conveying ideas.
Some of these attitudes have been shored up by statistical studies linking currently defined math ability and music ability, with the suggestion that genetic factors were involved. But there are also research results suggesting that music is more fundamental than speech and that language arose from music. Something so fundamental is not going to be genetically precluded from anyone´s repetoire. And then there´s the dispute over the definition of math and the fact that there´s no evidence to preclude anyone from developing numeracy nor the quantitative skills to empower their chosen life´s work and lifestyle.
If it wasn´t painful, you didn´t learn anything worthwhile.
When you gravitate to a subject, it usually comes so naturally and painlessly that the ease dismays others. Obviously you´ve learned the same material as those who struggled. Whose learning was worthwhile?
This excuse of making dis-ease a virtue disguises inappropriate use of a course as a gateway to limit the number of practitioners entering a field. As an added benefit, it also glorifies the survivors and unifies them against the criticisms by outsiders on how things are done.
for Literature-based Mathematics
Curriculum & Support