Mathematical practice
Encyclopedia
Mathematical practice is used to distinguish the working practices of professional mathematicians (e.g. selecting theorem
s 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 theorem
s.
, which denies the possibility of foundations of mathematics
and attempts to refocus attention on the ways in which mathematicians arrive at mathematical statements.
. Although such "folk" practices may well include useful formulae or algorithms, they are generally without the accompanying proof discipline.
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 number
s. 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?)
or components. Most GCSE
, A-Level and undergraduate mathematics require the following components:
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.
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s to prove, using informal notations to persuade themselves and others that various steps in the final proof can be formalised
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
, and seeking peer review
Peer review
Peer review is a process of self-regulation by a profession or a process of evaluation involving qualified individuals within the relevant field. Peer review methods are employed to maintain standards, improve performance and provide credibility...
and publication) from the end result of proven
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
and published theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s.
Quasi-empiricism
This distinction is considered especially important by adherents of quasi-empiricism in mathematicsQuasi-empiricism in mathematics
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics...
, which denies the possibility of foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
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 mathematicsFolk mathematics
As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that are found by investigation and may circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or...
. 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 FermatPierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
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
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
and Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
. 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 number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s. 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
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 pedagogiesPedagogy
Pedagogy is the study of being a teacher or the process of teaching. The term generally refers to strategies of instruction, or a style of instruction....
or components. Most GCSE
General Certificate of Secondary Education
The General Certificate of Secondary Education is an academic qualification awarded in a specified subject, generally taken in a number of subjects by students aged 14–16 in secondary education in England, Wales and Northern Ireland and is equivalent to a Level 2 and Level 1 in Key Skills...
, A-Level and undergraduate mathematics require the following components:
- TextbookTextbookA textbook or coursebook is a manual of instruction in any branch of study. Textbooks are produced according to the demands of educational institutions...
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 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 BMOBritish Mathematical OlympiadThe British Mathematical Olympiad forms part of the selection process for the UK International Mathematical Olympiad team. There are two rounds, the BMO1 and the BMO2.-BMO Round 1:...
(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 OlympiadInternational Mathematical OlympiadThe International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate 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 in 1980...
).
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.