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Mathematical maturity
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<h2 id="definitions">Definitions</h2> <p>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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>One definition has been given as follows:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <blockquote><p>... fearlessness in the face of symbols: the ability to read and understand <a href="/facts/Mathematical_notation/vHgEE8cE">notation</a>, 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.</p></blockquote> <p>A broader list of characteristics of mathematical maturity has been given as follows:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <blockquote> <ul><li>The capacity to <a href="/facts/Generalization/v0b1pwUx">generalize</a> from a specific example to a broad concept</li> <li>The capacity to handle increasingly <a href="/facts/Abstraction_(mathematics)/JCYl47fF">abstract</a> ideas</li> <li>The ability to communicate mathematically by learning standard notation and acceptable style</li> <li>A significant shift from learning by memorization to learning through understanding</li> <li>The capacity to separate the key ideas from the less significant</li> <li>The ability to link a <a href="/facts/Geometry/wTQf5ffQ">geometrical</a> representation with an analytic representation</li> <li>The ability to translate verbal problems into mathematical problems</li> <li>The ability to recognize a valid <a href="/facts/Mathematical_proof/7ECDrU80">proof</a> and detect 'sloppy' thinking</li> <li>The ability to recognize mathematical patterns</li> <li>The ability to move back and forth between the geometrical (graph) and the analytical (equation)</li> <li>Improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude</li></ul></blockquote> <p>Finally, mathematical maturity has also been defined as an ability to do the following:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <blockquote> <ul><li>Make and use connections with other problems and other disciplines</li> <li>Fill in missing details</li> <li>Spot, correct and learn from mistakes</li> <li>Winnow the chaff from the wheat, get to the crux, identify intent</li> <li>Recognize and appreciate <a href="/facts/Mathematical_beauty/HYwRap01">elegance</a></li> <li>Think abstractly</li> <li>Read, write and critique formal proofs</li> <li>Draw a line between what you know and what you don’t know</li> <li>Recognize patterns, themes, currents and eddies</li> <li>Apply what you know in creative ways</li> <li>Approximate appropriately</li> <li>Teach yourself</li> <li>Generalize</li> <li>Remain focused</li> <li>Bring instinct and intuition to bear when needed</li></ul></blockquote><p>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.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><h2 id="progression">Progression</h2> <p>Mathematician <a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a> has proposed a three-stage model of <a href="/facts/Mathematics_education/6hXltmmF">mathematics education</a> that can be interpreted as a general framework of mathematical maturity progression.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The stages are summarized in the following table:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> Overview of Terence Tao's stages of mathematical understanding<table><tbody><tr><th scope="col">Stage</th><th scope="col">Level</th><th scope="col">Phase</th><th scope="col">Description</th><th scope="col">Typical duration</th><th scope="col">Nature of mistakes</th><th scope="col">Example</th></tr><tr><td>1</td><th scope="row">Beginner</th><td>Pre-rigorous</td><td>This stage occurs before a mathematics student has learned mathematical formalism. Mathematics is learned in an informal manner that often incorporates examples, <a href="/facts/Fuzzy_concept/UXc3n4nN">fuzzy notions</a> and <a href="/facts/Hand-waving/SsMMAaHY">hand-waving</a>. This stage is founded mostly in intuition and the emphasis is more on computation than theory. Natural talent is important at this level.</td><td>Until underclassman <a href="/facts/Undergraduate/GVvPMLFN">undergraduate</a> years</td><td>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.</td><td>The introduction of <a href="/facts/Calculus/MEmUTYoz">calculus</a> in terms of <a href="/facts/Slopes/a2L3k0j5">slopes</a>, <a href="/facts/Areas/KsNG1G9t">areas</a>, <a href="/facts/Rates_of_change/sMy2QGsK">rates of change</a> etc.</td></tr><tr><td>2</td><th scope="row">Intermediate</th><td>Rigorous</td><td>Students learn mathematical formalism in this phase, which typically includes <a href="/facts/Formal_logic/SaI7Y3wN">formal logic</a> and some <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>. This serves a foundation for learning <a href="/facts/Mathematical_proofs/7ECDrU80">mathematical proofs</a> 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 <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> or <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>. Students of <a href="/facts/Computer_science/lN3pEyPz">computer science</a> or <a href="/facts/Applied_mathematics/qyqklXXm">applied mathematics</a> may encounter this formalism during their underclassman undergraduate years via a course in <a href="/facts/Discrete_mathematics/XdRUnzUL">discrete mathematics</a>. Exposure to <a href="/facts/Counterexamples/dnThYpHW">counterexamples</a> is typical, and an ability to discern between <a href="/facts/Mathematical_proof/7ECDrU80">sound mathematical argumentation</a> and erroneous mathematical argumentation is developed.<p>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.</p></td><td>Upperclassman undergraduate years through early <a href="/facts/Graduate_school/e19zs58x">graduate</a> years</td><td>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.</td><td>The reintroduction of calculus in the context of rigorous <a href="/facts/Limit_of_a_function/HFbQ69e5">ε, δ notation</a></td></tr><tr><td>3</td><th scope="row">Advanced</th><td>Post-rigorous</td><td>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 <a href="/facts/Logical_intuition/stJIX3kK">instinctive feel</a> for <a href="/facts/Soundness/ndpamd2A">mathematical soundness</a>. 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 <a href="/facts/Problem-solving/9cuLz5pH">problem-solving</a> efficiency because now formal rigor is infused with a refined intuition. In this ideal <a href="/facts/Mental_state/sHsDj380">state</a>, every <a href="/facts/Heuristic/BcBEkv9c">heuristic argument</a> naturally suggests its rigorous counterpart, and vice versa. This readies the practitioner to tackle complex mathematical problems that otherwise would be inaccessible.<p>Students generally attain this phase towards their later graduate years, after they have begun reading <a href="/facts/List_of_mathematics_journals/WtnaW2YA">mathematical research papers</a>.</p></td><td>Later graduate years and beyond</td><td>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.</td><td>The improvised and provisional use of <a href="/facts/Infinitesimals/JKcTMIgD">infinitesimals</a> or <a href="/facts/Big-O_notation/weFFjSWg">big-O notation</a>; temporarily constructing informal, quasi-rigorous argumentation that can be later converted into a formal, rigorous argumentation</td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Logical_intuition/stJIX3kK">Logical intuition</a></li> <li><a href="/facts/Four_stages_of_competence/jsl3V8C4">Four stages of competence</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 <a href="/wiki/Lynn_Arthur_Steen" target="_blank">/wiki/Lynn_Arthur_Steen</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lew, Kristen. "How Do Mathematicians Describe Mathematical Maturity?" (PDF). Special Interest Groups of the Mathematical Association of America (SIGMAA). Retrieved 2019-12-07. <a href="http://sigmaa.maa.org/rume/crume2019/Papers/106.pdf" target="_blank">http://sigmaa.maa.org/rume/crume2019/Papers/106.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lew, Kristen. "How Do Mathematicians Describe Mathematical Maturity?" (PDF). Special Interest Groups of the Mathematical Association of America (SIGMAA). Retrieved 2019-12-07. <a href="http://sigmaa.maa.org/rume/crume2019/Papers/106.pdf" target="_blank">http://sigmaa.maa.org/rume/crume2019/Papers/106.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Math 22 Lecture A, Larry Denenberg <a href="http://www.larry.denenberg.com/math22/LectureA.pdf" target="_blank">http://www.larry.denenberg.com/math22/LectureA.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>LBS 119 Calculus II Course Goals, Lyman Briggs School of Science <a href="https://www.msu.edu/course/lbs/119/simpson/goals_content.html" target="_blank">https://www.msu.edu/course/lbs/119/simpson/goals_content.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>A Set of Mathematical Equivoques, Ken Suman, Department of Mathematics and Statistics, Winona State University <a href="http://course1.winona.edu/KSuman/Dictionary/Fill%20Ins/Mathematical%20Maturity.htm" target="_blank">http://course1.winona.edu/KSuman/Dictionary/Fill%20Ins/Mathematical%20Maturity.htm</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>A Set of Mathematical Equivoques, Ken Suman, Department of Mathematics and Statistics, Winona State University <a href="http://course1.winona.edu/KSuman/Dictionary/Fill%20Ins/Mathematical%20Maturity.htm" target="_blank">http://course1.winona.edu/KSuman/Dictionary/Fill%20Ins/Mathematical%20Maturity.htm</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Lew, K. (2019). How do mathematicians describe mathematical maturity? Cognition and Instruction, 37(2), 121-142. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>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/ <a href="https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/" target="_blank">https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Numberphile2. (2017, March 18). Terry Tao and “Cheating Strategically” (extra footage) - Numberphile [Video]. YouTube. https://www.youtube.com/watch?v=48Hr3CT5Tpk <a href="https://www.youtube.com/watch?v=48Hr3CT5Tpk" target="_blank">https://www.youtube.com/watch?v=48Hr3CT5Tpk</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>