A discipline (a
organized, formal field of study) such as mathematics tends to be defined by
the types of problems it addresses, the methods it uses to address these
problems, and the results it has achieved. One way to organize this set of
information is to divide it into the following three categories (of course,
they overlap each other):
1.
Mathematics as a human endeavor. For
example, consider the math of measurement of time such as years, seasons,
months, weeks, days, and so on. Or, consider the measurement of distance, and
the different systems of distance measurement that developed throughout the
world. Or, think about math in art, dance, and music. There is a rich history
of human development of mathematics and mathematical uses in our modern
society.
2.
Mathematics as a discipline. You are familiar
with lots of academic disciplines such as archeology, biology, chemistry,
economics, history, psychology, sociology, and so on. Mathematics is a broad
and deep discipline that is continuing to grow in breadth and depth. Nowadays,
a Ph.D. research dissertation in mathematics is typically narrowly focused on
definitions, theorems, and proofs related to a single problem in a narrow
subfield in mathematics.
3.
Mathematics as an interdisciplinary language and
tool. Like reading and writing, math is an important component of learning and
"doing" (using one's knowledge) in each academic discipline.
Mathematics is such a useful language and tool that it is considered one of the
"basics" in our formal educational system.
To a large extent,
students and many of their teachers tend to define mathematics in terms of what
they learn in math courses, and these courses tend to focus on #3. The
instructional and assessment focus tends to be on basic skills and on solving
relatively simple problems using these basic skills. As the three-component
discussion given above indicates, this is only part of mathematics.
Even within the third
component, it is not clear what should be emphasized in curriculum,
instruction, and assessment. The issue of basic skills versus higher-order
skills is particularly important in math education. How much of the math
education time should be spent in helping students gain a high level of
accuracy and automaticity in basic computational and procedural skills? How
much time should be spent on higher-order skills such as problem posing, problem
representation, solving complex problems, and transferring math knowledge and
skills to problems in non-math disciplines?
