Mathematical practice comprises the working
practices of professional mathematicians:
selecting theorems to prove, using informal
notations to persuade themselves and others
that various steps in the final proof are
convincing, and seeking peer review and publication,
as opposed to the end result of proven and
published theorems.
Philip Kitcher has proposed a more formal
definition of a mathematical practice, as
a quintuple. His intention was primarily to
document mathematical practice through its
historical changes.
== Historical tradition ==
The evolution of mathematical practice was
slow, and some contributors to modern mathematics
did not follow even the practice of their
time. For example, Pierre de Fermat was infamous
for withholding his proofs, but nonetheless
had a vast reputation for correct assertions
of results.
One motivation to study mathematical practice
is that, despite much work in the 20th century,
some still feel that the foundations of mathematics
remain unclear and ambiguous. One proposed
remedy is to shift focus to some degree onto
'what is meant by a proof', and other such
questions of method.
If mathematics has been informally used throughout
history, in numerous cultures and continents,
then it could be argued that "mathematical
practice" is the practice, or use, of mathematics
in everyday life. One definition of mathematical
practice, as described above, is the "working
practices of professional mathematicians."
However, another definition, more in keeping
with the predominant usage of mathematics,
is that mathematical practice is the everyday
practice, or use, of math. Whether one is
estimating the total cost of their groceries,
calculating miles per gallon, or figuring
out how many minutes on the treadmill that
chocolate éclair will require, math as used
by most people relies less on proof than on
practicality (i. e., does it answer the question?).
== Teaching practice ==
Mathematical teaching usually requires the
use of several important teaching pedagogies
or components. Most GCSE, A-Level and undergraduate
mathematics require the following components:
Textbooks or lecture notes which display the
mathematical material to be covered/taught
within the context of the teaching of mathematics.
This requires that the mathematical content
being taught at the (say) undergraduate level
is of a well documented and widely accepted
nature that has been unanimously verified
as being correct and meaningful within a mathematical
context.
Workbooks. Usually, in order to ensure that
students have an opportunity to learn and
test the material that they have learnt, workbooks
or question papers enable mathematical understanding
to be tested. It is not unknown for exam papers
to draw upon questions from such test papers,
or to require prerequisite knowledge of such
test papers for mathematical progression.
Exam papers and standardised (and preferably
apolitical) testing methods. Often, within
countries such as the US, the UK (and, in
all likelihood, China) there are standardised
qualifications, examinations and workbooks
that form the concrete teaching materials
needed for secondary-school and pre-university
courses (for example, within the UK, all students
are required to sit or take Scottish Highers/Advanced
Highers, A-levels or their equivalent in order
to ensure that a certain minimal level of
mathematical competence in a wide variety
of topics has been obtained). Note, however,
that at the undergraduate, post-graduate and
doctoral levels within these countries, there
need not be any standardised process via which
mathematicians of differing ability levels
can be tested or examined. Other common test
formats within the UK and beyond include the
BMO (which is a multiple-choice test competition
paper used in order to determine the best
candidates that are to represent countries
within the International Mathematical Olympiad).
== See also ==
Common Core State Standards Initiative: Mathematical
practice
Foundations of mathematics
Informal mathematics
Philosophy of mathematics
== Notes ==
== Further reading ==
Mancosu, P. (2008). The Philosophy of Mathematical
Practice. OUP Oxford. ISBN 978-0-19-929645-3.
Retrieved September 19, 2018. 447 pages.
