Mathematical practice

Mathematical practice is used to distinguish the working practices of professional mathematicians (e.g. selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof can be formalised, and seeking peer review and publication) from the end result of proven and published theorems.

Quasi-empiricism

This distinction is considered especially important by adherents of quasi-empiricism in mathematics, which denies the possibility of foundations of mathematics and attempts to refocus attention on the ways in which mathematicians arrive at mathematical statements.

Folk mathematics

If modern mathematical practices are what distinguish modern professional mathematicians from older ideas of folk mathematics. Although such "folk" practices may well include useful formulae or algorithms, they are generally without the accompanying proof discipline.

Historical tradition

The evolution of mathematical practice was slow, and some contributors to modern mathematics did not follow even the practice of their time, e.g. Pierre de Fermat who was infamous for withholding his proofs, but nonetheless had a vast reputation for correct assertions of results. Likewise there is contrast between the practices of Pythagoras and Euclid. While Euclid was the originator of what we now understand as the published geometric proof, Pythagoras created a closed community and suppressed results; he is even said to have drowned a student in a barrel for revealing the existence of irrational numbers. Modern mathematicians admire Euclid's practices, and usually frown on those of both Fermat and Pythagoras. Nonetheless, all three are considered important contributors to mathematics, despite the variance in method.

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:

1. 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.
2. 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.
3. 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).

Assessment practice

Assessment practice overlaps with teaching practice in a sense (it is difficult to teach individuals to a certain level of mathematical competence without first having fore-knowledge of their current mathematical abilities).

These test practices sometimes require written exams to be sat (exams in which answers are in actuality written on exam scripts). However, given the usually lofty moral standards by which mathematical assessment has been tauted to have been conducted according to (together with the ease of statistical data interpretation that such test formats are associated with), multiple choice questions are often seen as useful in determining or verifying a given level of mathematical capability.

See also

• Informal mathematics
• Foundations of mathematics
• Philosophy of mathematics