Does Math Make You Smarter?

By Manil Suri

Dr. Suri is a professor of mathematics at the University of Maryland, Baltimore County.

April 13, 2018

Does studying mathematics enhance your overall mental prowess?

Abraham Lincoln certainly believed so, embarking on the arduous task of mastering Euclid’s treatises on geometry to increase his cognitive capacities, in particular his linguistic and logical abilities. This idea — that mathematics strengthens your mind much as physical exercise strengthens your body, helping you negotiate a variety of mental challenges — goes all the way back to Plato. Alive and well in today’s world, it is one reason popularly given for why everyone should study mathematics.

So it can come as a surprise to learn that cognitive psychologists have a different take on the issue. Various studies point to the conclusion that subjecting the mind to formal discipline — as when studying geometry or Latin — does not, in general, engender a broad transfer of learning. There is no sweeping increase of a general capacity for tasks like writing a speech or balancing a checkbook.

But surely a narrower claim is true: that mathematics, so systematically built as it is on inference, must develop logical thinking. Right?

By “logical,” I mean the kind of thinking needed to solve the following problem:

Four cards are laid in front of you, each of which, it is explained, has a letter on one side and a number on the other. The sides that you see read E, 2, 5 and F. Your task is to turn over only those cards that could decisively prove the truth or falsity of the following rule: “If there is an E on one side, the number on the other side must be a 5.” Which ones do you turn over?

Clearly, the E should be turned over, since if the other side is not a 5, the rule is untrue. And the only other card that should be flipped is the 2, since an E on the other side would again disprove the rule. Turning over the 5 or the F doesn’t help, since anything on the other side would be consistent with the rule — but not prove it to be true.

This innocuous-looking puzzle, a variation of which was introduced by the British psychologist Peter Wason in 1966, has been called “the single most investigated paradigm in the psychology of reasoning.” If you answered E and 2, congratulations: You are among the roughly 10 percent of the public able to solve the puzzle. Many reasons have been advanced for this poor showing, including the lack of relevance of such an abstract exercise to people’s daily lives.

Most people reflexively eliminate the cards not explicitly specified in the rule (the F and the 2) and then continue with slower, more analytic processing only for the E and the 5. In this, they rely on an initial snap judgment about superficial similarity, a tendency that some scholars speculate evolved in humans because in most real-world contexts, quickly detecting such similarities is a good strategy for survival.

Interestingly, though, it turns out that if the puzzle’s abstract rule is translated into terms that are logically equivalent but grounded in real-world experience — as in, “If someone is drinking beer at a bar, she must be at least 21 years of age” — then the success rate jumps to 75 percent or more.

I learned about the Wason selection task and its intricacies from a fascinating recent book, “Does Mathematical Study Develop Logical Thinking?” by the education and cognition researchers Matthew Inglis and Nina Attridge. They conducted experiments that found that university students studying mathematics were just as likely as those studying history to quickly reject the F and the 2 cards. But differences emerged in the slower, more effortful cogitative phase that followed, leading to divergent success rates in the end: 18 percent for the mathematics students versus 6 percent for the history students.

Based on results from a slate of such reasoning tasks, Dr. Inglis and Dr. Attridge show that studying higher mathematics (at the advanced secondary and college levels) does lead to an increase in logical ability. In particular, mathematics students become more skeptical in their reasoning — they begin to think more critically.

But these gains, though significant enough to establish a causal relation between mathematics education and logical thinking, are too modest to settle the debate on how much mathematics should be prescribed as part of a general education, and for which students. (An 18 percent success rate is hardly compelling.) Moreover, there is the possibility of a self-selection effect: Students with the greatest potential to get a benefit in their logical reasoning might be disproportionately attracted to mathematics classes in the first place, so these gains might not apply to the entire population.

In any event, the most crucial finding of such research, in my view, is how much insight the psychological study of learning can contribute to the practical teaching of mathematics — two fields of endeavor that are too often pursued separately. It is sadly telling that while the Wason selection task is well known among psychologists, it is not familiar to most mathematicians and math teachers.

I propose we start to teach the Wason selection task in mathematics courses at the high-school level and higher. The puzzle captures so much that is essential to mathematics: the nuts and bolts of inference, the difficulty of absorbing abstract concepts when removed from the context of real-world experience, the importance of a slow, deliberative cogitative process and the pitfalls of instant intuitive judgments. I presented the puzzle to a recent college class of mathematics majors and they listened with rapt attention afterward — startled by their lowly 19 percent success rate.

Logical thinking may be promoted by mathematics, but it is a gradual and complex learning process. Psychological insight into learning, such as that offered by Wason’s puzzle, can give students a head start by educating them on the challenges they will face.https://www.nytimes.com/2018/04/13/opinion/sunday/math-logic-smarter.html

ゼロ除算の発見と重要性を指摘した：日本、再生核研究所

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\begin{document}

\title{\bf Announcement 409: Various Publication Projects on the Division by Zero\\

(2018.1.29.)}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

}

\date{\today}

\maketitle

The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristoteles (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).

In particular, Brahmagupta defined as $0/0=0$ in Brhmasphuasiddhnta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.

For the details, see the references and the site: http://okmr.yamatoblog.net/

We wrote two global book manuscripts \cite{s18} with 154 pages and \cite{so18} with many figures for some general people. Their main points are:

\begin{itemize}

\item The division by zero and division by zero calculus are new elementary and fundamental mathematics in the undergraduate level.

\item They introduce a new space since Aristoteles (BC384 - BC322) and Euclid (BC 3 Century - ) with many exciting new phenomena and properties with general interest, not specialized and difficult topics. However, their properties are mysterious and very attractive.

\item The contents are very elementary, however very exciting with general interest.

\item The contents give great impacts to our basic ideas on the universe and human beings.

\end{itemize}

Meanwhile, the representations of the contents are very important and delicate with delicate feelings to the division by zero with a long and mysterious history. Therefore, we hope the representations of the division by zero as follows:

\begin{itemize}

\item

Various book publications by many native languages and with the author's idea and feelings.

\item

Some publications are like arts and some comic style books with pictures.

\item

Some T shirts design, some pictures, monument design may be considered.

\end{itemize}

The authors above may be expected to contribute to our culture, education, common communications and enjoyments.

\medskip

For the people having the interest on the above projects, we will send our book sources with many figure files.

\medskip

How will be our project introducing our new world since Euclid?

\medskip

Of course, as mathematicians we have to publish new books on

\medskip

Calculus, Differential Equations and Complex Analysis, at least and soon, in order to {\bf correct them} in some complete and beautiful ways.

\medskip

Our topics will be interested in over 1000 millions people over the world on the world history.

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{kmsy}

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms16}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58

Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt

\\ http://dx.doi.org/10.4236/alamt.2016.62007.

\bibitem{ms18}

T. Matsuura and S. Saitoh,

Division by zero calculus and singular integrals. (Submitted for publication)

\bibitem{mms18}

T. Matsuura, H. Michiwaki and S. Saitoh,

$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.

\bibitem{msy}

H. Michiwaki, S. Saitoh and M.Yamada,

Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

\bibitem{mos}

H. Michiwaki, H. Okumura and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces,

International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 2017), 1-16.

\bibitem{osm}

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,

Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.

\bibitem{os}

H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).

\bibitem{o}

H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.

\bibitem{os18}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

(Submitted for publication).

\bibitem{ps18}

S. Pinelas and S. Saitoh,

Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.

\bibitem{romig}

H. G. Romig, Discussions: Early History of Division by Zero,

American Mathematical Monthly, Vol. {\bf 3}1, No. 8. (Oct., 1924), pp. 387-389.

\bibitem{s14}

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/

\bibitem{s16}

S. Saitoh, A reproducing kernel theory with some general applications,

Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .

\bibitem{s17}

S. Saitoh, Mysterious Properties of the Point at Infinity, arXiv:1712.09467 [math.GM](2017.12.17).

\bibitem{s18}

S. Saitoh, Division by zero calculus (154 pages: draft): http//okmr.yamatoblog.net/

\bibitem{so18}

S. Saitoh and H. Okumura, Division by Zero Calculus in Figures -- Our New Space --

\bibitem{ttk}

S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.

\end{thebibliography}

\end{document}

List of division by zero:

\bibitem{os18}

H. Okumura and S. Saitoh,

Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum.

Saburou Saitoh, Mysterious Properties of the Point at Infinity、
arXiv:1712.09467 [math.GM]

Hiroshi Okumura and Saburou Saitoh

The Descartes circles theorem and division by zero calculus.　２０１７．１１．１４

https://arxiv.org/abs/1711.04961

L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,　Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

T. Matsuura and S. Saitoh,

Matrices and division by zero　z/0=0,

Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58

Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt

\\ http://dx.doi.org/10.4236/alamt.2016.62007.

T. Matsuura and S. Saitoh,

Division by zero calculus and singular integrals. (Submitted for publication).

T. Matsuura, H. Michiwaki and S. Saitoh,

$\log 0= \log \infty =0$ and applications. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)

H. Michiwaki, S. Saitoh and M.Yamada,

Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

H. Michiwaki, H. Okumura and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces,

International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,

Journal of Technology and Social Science (JTSS), 1(2017), 70-77.

S. Pinelas and S. Saitoh,

Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/

S. Saitoh, A reproducing kernel theory with some general applications,

Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　https://sites.google.com/site/sandrapinelas/icddea-2017　報告

https://ameblo.jp/syoshinoris/entry-12287338180.html

1/0=0、0/0=0、z/0=0

http://ameblo.jp/syoshinoris/entry-12276045402.html

1/0=0、0/0=0、z/0=0

http://ameblo.jp/syoshinoris/entry-12263708422.html

1/0=0、0/0=0、z/0=0

http://ameblo.jp/syoshinoris/entry-12272721615.html

ソクラテス・プラトン・アリストテレス　その他

https://ameblo.jp/syoshinoris/entry-12328488611.html

アインシュタインも解決できなかった「ゼロで割る」問題

http://matome.naver.jp/odai/2135710882669605901

Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.

https://notevenpast.org/dividing-nothing/

私は数学を信じない。　アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。

ドキュメンタリー 2017: 神の数式第２回宇宙はなぜ生まれたのか

https://www.youtube.com/watch?v=iQld9cnDli4

〔NHKスペシャル〕神の数式完全版第3回宇宙はなぜ始まったのか

https://www.youtube.com/watch?v=DvyAB8yTSjs&t=3318s

〔NHKスペシャル〕神の数式完全版第1回この世は何からできているのか

https://www.youtube.com/watch?v=KjvFdzhn7Dc

NHKスペシャル神の数式完全版第4回異次元宇宙は存在するか

https://www.youtube.com/watch?v=fWVv9puoTSs

再生核研究所声明 411(2018.02.02): 　ゼロ除算発見4周年を迎えて

https://ameblo.jp/syoshinoris/entry-12348847166.html

ゼロ除算の論文

Mysterious Properties of the Point at Infinity

https://arxiv.org/abs/1712.09467

Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197

http://www.worldscientificnews.com/wp-content/uploads/2017/12/WSN-922-2018-171-197.pdf

2018.3.18．午前中　最後の講演：　日本数学会　東大駒場、函数方程式論分科会　講演書画カメラ用　原稿
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より

*057 Pinelas,S./Caraballo,T./Kloeden,P./Graef,J.(eds.): Differential and Difference Equations with Applications: ICDDEA, Amadora, 2017. (Springer Proceedings in Mathematics and Statistics, Vol. 230) May 2018 587 pp.