Mathematical practice comprises the working practices of professional

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

s: selecting

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

s to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking

peer review Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. Peer review ...

and

publication To publish is to make content available to the general public.Berne Conve ...

, as opposed to the end result of

proven Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants. The church and parish of Proven are named after Saint Victor. The Saint Victor Chur ...

and published

Philip Kitcher Philip Stuart Kitcher (born 20 February 1947) is a British philosopher who is John Dewey Professor Emeritus of philosophy at Columbia University. He specialises in the philosophy of science, the philosophy of biology, the philosophy of mathema ...

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 Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...

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 Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...

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 in everyday life relies on practicality (i.e., does it answer the question?) rather than formal proof.

Teaching practice

Mathematical teaching usually requires the use of several important teaching pedagogies or components. Most

GCSE The General Certificate of Secondary Education (GCSE) is an academic qualification in a particular subject, taken in England, Wales, and Northern Ireland. State schools in Scotland use the Scottish Qualifications Certificate instead. Private sc ...

A-Level The A-Level (Advanced Level) is a subject-based qualification conferred as part of the General Certificate of Education, as well as a school leaving qualification offered by the educational bodies in the United Kingdom and the educational aut ...

and

undergraduate Undergraduate education is education conducted after secondary education and before postgraduate education. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, in the United States, an entry-lev ...

mathematics require the following components: #

Textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...

s 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 Postgraduate or graduate education refers to academic or professional degrees, certificates, diplomas, or other qualifications pursued by post-secondary students who have earned an undergraduate ( bachelor's) degree. The organization and stru ...

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 The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except i ...

Historical tradition

Teaching practice

See also

Notes

Further reading