Mathematical maturity

Last updated

In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right. [1] [2]

Contents

Definitions

Mathematical maturity has been defined in several different ways by various authors, and is often tied to other related concepts such as comfort and competence with mathematics, mathematical intuition and mathematical beliefs. [2]

One definition has been given as follows: [3]

... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas.

A broader list of characteristics of mathematical maturity has been given as follows: [4]

Finally, mathematical maturity has also been defined as an ability to do the following: [5]

It is sometimes said that the development of mathematical maturity requires a deep reflection on the subject matter for a prolonged period of time, along with a guiding spirit which encourages exploration. [5]

Progression

Mathematician Terence Tao has proposed a three-stage model of mathematics education that can be interpreted as a general framework of mathematical maturity progression. [6] The stages are summarized in the following table: [7] [8]

Overview of Terence Tao's stages of mathematical understanding
StageLevelPhaseDescriptionTypical durationNature of mistakesExample
1
Beginner
Pre-rigorous
This stage occurs before a mathematics student has learned mathematical formalism. Mathematics is learned in an informal manner that often incorporates examples, fuzzy notions and hand-waving. This stage is founded mostly in intuition and the emphasis is more on computation than theory. Natural talent is important at this level.
Until underclassman undergraduate years
Formal errors are made as a result of the application of formal rules or heuristics blindly due to the misunderstanding of rigorous mathematical formalism. Such errors can be difficult to correct even when pointed out.The introduction of calculus in terms of slopes, areas, rates of change etc.
2
Intermediate
Rigorous
Students learn mathematical formalism in this phase, which typically includes formal logic and some mathematical logic. This serves a foundation for learning mathematical proofs and various proof techniques. The onset of this phase typically occurs during a 'transition to higher mathematics' course, an 'introduction to proofs' course or an introductory course in mathematical analysis or abstract algebra. Students of computer science or applied mathematics may encounter this formalism during their underclassman undergraduate years via a course in discrete mathematics. Exposure to counterexamples is typical, and an ability to discern between sound mathematical argumentation and erroneous mathematical argumentation is developed.

This is an important phase for extinguishing misleading intuitions and elevating accurate intuitions, which the mathematical formalism aids with. The formality helps to deal with technical details, while intuition helps to deal with the bigger picture. However, sometimes a student can get stalled at this intermediary stage. This occurs when a student discards too much good intuition, rendering them able to only process mathematics at the formal level, instead of the more intuitive informal level. Getting stalled in this manner can impact the student's ability to read mathematical papers.

Upperclassman undergraduate years through early graduate years
Susceptibility to formal errors persists as a result of inapt formalism or an inability to perform "sanity checks" against intuition or other menial rules of thumb, such as a trivial sign error or the failure to verify an assumption made along the argumentation. However, unlike the previous stage such errors can usually be detected and often repaired once discovered.The reintroduction of calculus in the context of rigorous ε, δ notation
3
Advanced
Post-rigorous
At this phase, students return to a more intuitive mindset, similar to that of the initial pre-rigorous phase, except now, misleading intuition has been extinguished, enabling students to reliably reason at a more informal level. The assimilation of mathematical formalism equips students with an instinctive feel for mathematical soundness. This allows them to ascertain an intuitive sketch of a mathematical argument that can subsequently be converted into a formal proof. The emphasis is now on applications, intuition, and the “big picture". This optimizes problem-solving efficiency because now formal rigor is infused with a refined intuition. In this ideal state, every heuristic argument naturally suggests its rigorous counterpart, and vice versa. This readies the practitioner to tackle complex mathematical problems that otherwise would be inaccessible.

Students generally attain this phase towards their later graduate years, when they have typically begun reading mathematical research papers.

Later graduate years and beyond
Susceptibility to formal errors persists, albeit for a different reason; formalism is suspended for the sake of expedient intuitive reasoning and then later recalled for the translation into a rigorous argument, which can be sometimes be imprecise.The improvised and provisional use of infinitesimals or big-O notation; temporarily constructing informal, quasi-rigorous argumentation that can be later converted into a formal, rigorous argumentation

See also

Related Research Articles

Abstraction is a process wherein general rules and concepts are derived from the usage and classification of specific examples, literal signifiers, first principles, or other methods.

<span class="mw-page-title-main">David Hilbert</span> German mathematician (1862–1943)

David Hilbert was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics.

<span class="mw-page-title-main">History of mathematics</span>

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy to record time and formulate calendars.

Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics that include number theory, algebra, geometry, analysis, and set theory.

Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.

Foundations of mathematics is the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality.

In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and consequential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."

<i>La Géométrie</i> Mathematical appendix to Descartes Discourse on Method, published in 1637

La Géométrie was published in 1637 as an appendix to Discours de la méthode, written by René Descartes. In the Discourse, Descartes presents his method for obtaining clarity on any subject. La Géométrie and two other appendices, also by Descartes, La Dioptrique (Optics) and Les Météores (Meteorology), were published with the Discourse to give examples of the kinds of successes he had achieved following his method.

<i>Critique of Pure Reason</i> 1781 book by Immanuel Kant

The Critique of Pure Reason is a book by the German philosopher Immanuel Kant, in which the author seeks to determine the limits and scope of metaphysics. Also referred to as Kant's "First Critique", it was followed by his Critique of Practical Reason (1788) and Critique of Judgment (1790). In the preface to the first edition, Kant explains that by a "critique of pure reason" he means a critique "of the faculty of reason in general, in respect of all knowledge after which it may strive independently of all experience" and that he aims to decide on "the possibility or impossibility of metaphysics". The term "critique" is understood to mean a systematic analysis in this context, rather than the colloquial sense of the term.

Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. In other words, to be abstract is to remove context and application. Two of the most highly abstract areas of modern mathematics are category theory and model theory.

<span class="mw-page-title-main">Pure mathematics</span> Mathematics independent of applications

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.

<i>Prolegomena to Any Future Metaphysics</i> 1783 book by Immanuel Kant

Prolegomena to Any Future Metaphysics That Will Be Able to Present Itself as a Science is a book by the German philosopher Immanuel Kant, published in 1783, two years after the first edition of his Critique of Pure Reason. One of Kant's shorter works, it contains a summary of the Critique‘s main conclusions, sometimes by arguments Kant had not used in the Critique. Kant characterizes his more accessible approach here as an "analytic" one, as opposed to the Critique‘s "synthetic" examination of successive faculties of the mind and their principles.

<span class="mw-page-title-main">Outline of thought</span> Overview of and topical guide to thought

The following outline is provided as an overview of and topical guide to thought (thinking):

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups.

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive logical or mathematical truth—and the ability to solve mathematical challenges efficiently. Humans apply logical intuition in proving mathematical theorems, validating logical arguments, developing algorithms and heuristics, and in related contexts where mathematical challenges are involved. The ability to recognize logical or mathematical truth and identify viable methods may vary from person to person, and may even be a result of knowledge and experience, which are subject to cultivation. The ability may not be realizable in a computer program by means other than genetic programming or evolutionary programming.

References

  1. ↑ Lynn Arthur Steen (1983) "Developing Mathematical Maturity" pages 99 to 110 in The Future of College Mathematics: Proceedings of a Conference/Workshop on the First Two Years of College Mathematics, Anthony Ralston editor, Springer ISBN   1-4612-5510-4
  2. 1 2 Lew, Kristen. "How Do Mathematicians Describe Mathematical Maturity?" (PDF). Special Interest Groups of the Mathematical Association of America (SIGMAA). Retrieved 2019-12-07.
  3. ↑ Math 22 Lecture A, Larry Denenberg
  4. ↑ LBS 119 Calculus II Course Goals, Lyman Briggs School of Science
  5. 1 2 A Set of Mathematical Equivoques, Ken Suman, Department of Mathematics and Statistics, Winona State University
  6. ↑ Lew, K. (2019). How do mathematicians describe mathematical maturity? Cognition and Instruction, 37(2), 121-142.
  7. ↑ There’s more to mathematics than rigour and proofs. (2022, November 26). What’s New. https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
  8. ↑ Numberphile2. (2017, March 18). Terry Tao and “Cheating Strategically” (extra footage) - Numberphile [Video]. YouTube. https://www.youtube.com/watch?v=48Hr3CT5Tpk
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.