A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox.
Real-world problems
Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems, they are used in mathematics education to teach students to connect real-world situations to the abstract language of mathematics.
In general, to use mathematics for solving a real-world problem, the first step is to construct a mathematical model of the problem. This involves abstraction from the details of the problem, and the modeller has to be careful not to lose essential aspects in translating the original problem into a mathematical one. After the problem has been solved in the world of mathematics, the solution must be translated back into the context of the original problem.
Abstract problems
Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been a rich source of inspiration.
Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically. Also provably unsolvable are so-called undecidable problems, such as the halting problem for Turing machines.
Some well-known difficult abstract problems that have been solved relatively recently are the four-colour theorem, Fermat's Last Theorem, and the Poincaré conjecture.
Computers do not need to have a sense of the motivations of mathematicians in order to do what they do. Formal definitions and computer-checkable deductions are absolutely central to mathematical science.
Degradation of problems to exercises
Mathematics educators using problem solving for evaluation have an issue phrased by Alan H. Schoenfeld:
- How can one compare test scores from year to year, when very different problems are used? (If similar problems are used year after year, teachers and students will learn what they are, students will practice them: problems become exercises, and the test no longer assesses problem solving).
The same issue was faced by Sylvestre Lacroix almost two centuries earlier:
- ... it is necessary to vary the questions that students might communicate with each other. Though they may fail the exam, they might pass later. Thus distribution of questions, the variety of topics, or the answers, risks losing the opportunity to compare, with precision, the candidates one-to-another.
Such degradation of problems into exercises is characteristic of mathematics in history. For example, describing the preparations for the Cambridge Mathematical Tripos in the 19th century, Andrew Warwick wrote:
- ... many families of the then standard problems had originally taxed the abilities of the greatest mathematicians of the 18th century.
See also
- List of unsolved problems in mathematics
- Problem solving
- Mathematical game
References
- (Newby & Newby 2008), "The second test is, that although such machines might execute many things with equal or perhaps greater perfection than any of us, they would, without doubt, fail in certain others from which it could be discovered that they did not act from knowledge, but solely from the disposition of their organs: for while reason is an universal instrument that is alike available on every occasion, these organs, on the contrary, need a particular arrangement for each particular action; whence it must be morally impossible that there should exist in any machine a diversity of organs sufficient to enable it to act in all the occurrences of life, in the way in which our reason enable us to act." translated from
(Descartes 1637), page =57, "Et le second est que, bien qu'elles fissent plusieurs choses aussy bien, ou peutestre mieux qu'aucun de nois, ells manqueroient infalliblement en quelques autres, par lesquelles on découuriroit quelles n'agiroient pas par connoissance, mais seulement par la disposition de leurs organs. Car, au lieu que la raison est un instrument univeersel, qui peut seruir en toutes sortes de rencontres, ces organs ont besoin de quelque particliere disposition pour chaque action particuliere; d'oǜ vient qu'il est moralement impossible qu'il y en ait assez de diuers en une machine, pour la faire agir en toutes les occurrences de la vie, de mesme façon que nostre raison nous fait agir." - Alan H. Schoenfeld (editor) (2007) Assessing mathematical proficiency, preface pages x, xi, Mathematical Sciences Research Institute, Cambridge University Press ISBN 978-0-521-87492-2
- S. F. Lacroix (1816) Essais sur l’enseignement en general, et sur celui des mathematiques en particulier, page 201
- Andrew Warwick (2003) Masters of Theory: Cambridge and the Rise of Mathematical Physics, page 145, University of Chicago Press ISBN 0-226-87375-7
- Newby, Ilana; Newby, Greg (2008-07-01). "Discourse on the Method of rightly conducting the reason, and seeking truth in the sciences by Rene Descartes". Project Gutenberg. Retrieved 2019-02-13., translated from
- Descartes, René (1637). Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les scienses, plus la dioptrique, les météores et la géométrie qui sont des essais de cette method (in French). Gallica - The BnF digital library.
A mathematical problem is a problem that can be represented analyzed and possibly solved with the methods of mathematics This can be a real world problem such as computing the orbits of the planets in the solar system or a problem of a more abstract nature such as Hilbert s problems It can also be a problem referring to the nature of mathematics itself such as Russell s Paradox Real world problemsInformal real world mathematical problems are questions related to a concrete setting such as Adam has five apples and gives John three How many has he left Such questions are usually more difficult to solve than regular mathematical exercises like 5 3 even if one knows the mathematics required to solve the problem Known as word problems they are used in mathematics education to teach students to connect real world situations to the abstract language of mathematics In general to use mathematics for solving a real world problem the first step is to construct a mathematical model of the problem This involves abstraction from the details of the problem and the modeller has to be careful not to lose essential aspects in translating the original problem into a mathematical one After the problem has been solved in the world of mathematics the solution must be translated back into the context of the original problem Abstract problemsAbstract mathematical problems arise in all fields of mathematics While mathematicians usually study them for their own sake by doing so results may be obtained that find application outside the realm of mathematics Theoretical physics has historically been a rich source of inspiration Some abstract problems have been rigorously proved to be unsolvable such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry and solving the general quintic equation algebraically Also provably unsolvable are so called undecidable problems such as the halting problem for Turing machines Some well known difficult abstract problems that have been solved relatively recently are the four colour theorem Fermat s Last Theorem and the Poincare conjecture Computers do not need to have a sense of the motivations of mathematicians in order to do what they do Formal definitions and computer checkable deductions are absolutely central to mathematical science Degradation of problems to exercisesMathematics educators using problem solving for evaluation have an issue phrased by Alan H Schoenfeld How can one compare test scores from year to year when very different problems are used If similar problems are used year after year teachers and students will learn what they are students will practice them problems become exercises and the test no longer assesses problem solving The same issue was faced by Sylvestre Lacroix almost two centuries earlier it is necessary to vary the questions that students might communicate with each other Though they may fail the exam they might pass later Thus distribution of questions the variety of topics or the answers risks losing the opportunity to compare with precision the candidates one to another Such degradation of problems into exercises is characteristic of mathematics in history For example describing the preparations for the Cambridge Mathematical Tripos in the 19th century Andrew Warwick wrote many families of the then standard problems had originally taxed the abilities of the greatest mathematicians of the 18th century See alsoList of unsolved problems in mathematics Problem solving Mathematical gameReferences Newby amp Newby 2008 The second test is that although such machines might execute many things with equal or perhaps greater perfection than any of us they would without doubt fail in certain others from which it could be discovered that they did not act from knowledge but solely from the disposition of their organs for while reason is an universal instrument that is alike available on every occasion these organs on the contrary need a particular arrangement for each particular action whence it must be morally impossible that there should exist in any machine a diversity of organs sufficient to enable it to act in all the occurrences of life in the way in which our reason enable us to act translated from Descartes 1637 page 57 Et le second est que bien qu elles fissent plusieurs choses aussy bien ou peutestre mieux qu aucun de nois ells manqueroient infalliblement en quelques autres par lesquelles on decouuriroit quelles n agiroient pas par connoissance mais seulement par la disposition de leurs organs Car au lieu que la raison est un instrument univeersel qui peut seruir en toutes sortes de rencontres ces organs ont besoin de quelque particliere disposition pour chaque action particuliere d oǜ vient qu il est moralement impossible qu il y en ait assez de diuers en une machine pour la faire agir en toutes les occurrences de la vie de mesme facon que nostre raison nous fait agir Alan H Schoenfeld editor 2007 Assessing mathematical proficiency preface pages x xi Mathematical Sciences Research Institute Cambridge University Press ISBN 978 0 521 87492 2 S F Lacroix 1816 Essais sur l enseignement en general et sur celui des mathematiques en particulier page 201 Andrew Warwick 2003 Masters of Theory Cambridge and the Rise of Mathematical Physics page 145 University of Chicago Press ISBN 0 226 87375 7 Newby Ilana Newby Greg 2008 07 01 Discourse on the Method of rightly conducting the reason and seeking truth in the sciences by Rene Descartes Project Gutenberg Retrieved 2019 02 13 translated from Descartes Rene 1637 Discours de la methode pour bien conduire sa raison et chercher la verite dans les scienses plus la dioptrique les meteores et la geometrie qui sont des essais de cette method in French Gallica The BnF digital library Wikimedia Commons has media related to Mathematical problems
