Yes, something indeed is, as everyone knows. Of the many regrettable consequences that come to mind there are two I believe to be the most pernicious. With overemphasis on drills and standardized multiple-choice tests students (and teachers) develop a perception that
To Err is Wrong
This is despite the overwhelming evidence that mistakes often serve as stepping stones for
correct answers, while failures of expectation form a most basic element of the learning process.
When test after test after test, we are conditioned to provide (or select) just one possible answer, the idea that
Free vs. Pedantic Thinking
becomes deeply ingrained in our process of thinking. It's almost obvious that, in Mathematics, if the sought
answer is to prove something then in most cases it is impossible to expect a single correct one.
Thus with the current education system Mathematics is being emptied of its most basic and characteristic element - deduction.
The system violates two fundamental principles of creativity and learning methodology:
- Do not be afraid of failure.
- Do not be satisfied with a single answer.
As a complement to the above I began collecting quotations from various sources:
- Hugo Steinhaus, One Hundred Problems in Elementary Mathematics, Dover Pubns, 1979
This small book is a response to the deplorable state of mathematical education of high school students sensed by college educators after
- Edmund Landau, Foundations of Analysis, Chelsea Pub Co, 1960
Please forget whatever you've been studying at school; for you have not learned it.
- Martin Gardner, Mathematical Carnival, Vintage, 1965.
A teacher of mathematics, no matter how much he loves
his subject and how strong his desire to communicate, is perpetually faced with one overwhelming
difficulty: How can he keep his students awake?
- G. Polya, How To Solve It?, Princeton University Press, 1973
Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling
his students in routine operations he kills their interest, hampers their intellectual development,
and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems
proportionate to their knowledge, and helps them to solve their problems with stimulating questions,
he may give them a taste for, and some means of, independent thinking.
- A. K. Dewdney, 200% of Nothing, John Wiley & Sons, 1993
A faltering education system is colliding head-on with the world-wide emergence of a competitive
technological economy in which mathematics, more than an single science, will be the main determinant
- National Research Council. Mathematical Sciences Education Board. Everybody Counts. Washington, D.C., National Academy
(The traditional mathematics curriculum is described as) "a long dimly lit
journey through a mountain of meaningless manipulations, with the reward of power and understanding available only to those who
complete the journey."
- J. A. Paulos, A Mathematician Reads The Newspaper, Anchor Books, 1995
Furthermore, because of the mind-numbing way in which mathematics is generally taught, many
people have serious misconceptions about the subject and fail to appreciate its wide applicability.
- G.-C. Rota, Indiscrete Thoughts, Birkhauser, Boston, 1977.
Failure to conclude has been an outstanding characteristic of philosophy throughout its history.
Philosophers of the past have repeatedly stressed the essential role of failure in philosophy.
Jose Ortega y Gasset used to describe philosophy as "a constant shipwreck." However, fear of failure
did not stop him or any other philosopher from doing philosophy.
- M.Guillen, Five Equations that Changed the World, Hyperion, 1995.
A. Einstein: "It is, in fact nothing short of miracle that the modern methods of instruction
have not yet entirely strangled the holy curiosity of inquiry; for this delicate little plant...
stands mainly in need of freedom; without this it goes to wrack and ruin without fail."
- S. K. Stein, Strength in Numbers, John Wiley & Sonse, 1996
If you browse through The Mathematics Teacher, the main journal devoted to instruction in mathematics you will find constant lamentation, going back to its first volume in 1908, where one teacher wrote, "One of the most obvious facts about mathematics in our schools is a general dissatisfaction." The tone in 1911 was even less cheery, "Our conference is charged with gloom. I have attended funerals, but I do not remember a more mournful occasion than this. We are failures and our students are not getting anything worthwhile."
Year after year, the complaints in The Mathematics Teacher persist. I will skip ahead to 1958, when we read, "The traditional curriculum is meaningless, and by heading for abstract mathematics the modernists are moving further from reality." This was an early warning about the group developing what came to be called "the New Math." More about that reform later. Still, in 1994, the University of Chicago School Mathematics Project complained, "The student today still encounters a variant of the elementary school curriculum designed for the pupil of a hundred years ago."