A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart, Foreword by Keith Devlin NEW YORK: BELLEVUE LITERARY PRESS, 2009, PAPERBACK, 140 PP., US $12.95, ISBN: 978-1-934137-17-8 REVIEWED BY TIMO TOSSAVAINEN

aul Lockhart’s A Mathematician’s Lament was originally posted in 2002 as a 25-page essay, and it has been circulating in the mathematics education community ever since. Since Keith Devlin published it in his monthly online column on the MAA website in March 2008 [1], Lockhart’s lament has resonated around the world even more, and since 2009 it has also been available in printed form. It is difficult to review this passionate pamphlet in an objective manner. Lockhart describes the current state of affairs in mathematics education as a total failure or even as a nightmare, and he does so in an exceedingly provocative way. Even a reader who agrees in principle with Lockhart’s message may become annoyed with his excessive criticism that tends to overlook what is realistically possible in mathematics teachers’ education. What an accomplished professional research mathematician and a devoted teacher in one person such as Lockhart can exemplify goes far beyond what we may expect of the whole educational system or its employees, the ordinary teachers. A Mathematician’s Lament consists of two parts. In the first part, Lamentation, Lockhart argues, often using metaphors, that real mathematics has been removed from the school curriculum and replaced with meaningless mumbojumbo, i.e., rote memorization of mystical symbols and the rules for their manipulation. The main reason for the unfortunate situation is that both the reformers of K-12 mathematics education and the contemporary mathematics teachers themselves ignore what doing mathematics essentially is all about: an art of imagination done for pleasure, not a tool kit for surviving in work and society. Lockhart often exaggerates, but he usually has a point. Like Schmidt in his review [3], I agree that school mathematics indeed is typically taught without any reference to its historical and philosophical underpinnings, and raising this point is especially laudable. By focusing only on the end products of mathematical inquiry we effectively preclude our students from experiencing the creative side of mathematics, everything about genuine invention and discovery in it. In the second part, Exultation, Lockhart continues his criticism but also demonstrates what practicing the ‘‘art of

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imagination’’ could mean in the classroom. The first thing to do, in his opinion, is to ‘‘throw the stupid curriculum and textbooks out the window’’ and then ‘‘make up anything you want, so long as it isn’t boring’’! I read this radical advice as saying that the customary hierarchical arrangement of the prescribed topics of school mathematics is artificial to the point of preventing students from perceiving the true nature of mathematics as an organic whole. Interesting and beautiful problems arise from free and serendipitous play of thought, and one solved problem naturally leads to another. Lockhart’s examples of such fertile problems are illustrative, even if quite elementary. One may wonder, nonetheless, how far this kind of episodic approach can ultimately lead if we aim at understanding, for instance, the real numbers at a deeper level. Moreover, I doubt whether every student in an ordinary math class is even interested in becoming an artist of imagination—which is not to argue that the children were better served in the traditional approach either. It would be a waste of time to establish a detailed list of points on which I agree or disagree with Lockhart. In his Lament, Lockhart makes a strong case for the creativeness of a talented individual teacher but I find it more imperative to query what the education system should do. Lockhart is content to argue that every mathematics teacher should possess a personal relationship with the creative art of mathematics. One may wonder, though, how such a goal could be achieved without an increased and more constructive participation of professional research mathematicians in the mathematics teachers’ training and in the related pedagogical research [2]. One might, for instance, wish to establish new forums for mathematicians who are interested in pedagogical innovations and willing to report on their teaching experiments. Despite a wealth of journals devoted to research in mathematics education, there is a perceived lack of refereed journals that appreciate the professional mathematicians’ often very pragmatic point of view and somewhat condensed parlance. Tossavainen and Pehkonen provide a recent contribution to this debate [4]. A Mathematician’s Lament has hit some mathematics educators like a bolt from the blue. I suppose, however, that Lockhart did not set out to seek confrontation, neither did he pretend to appear as a universal problem-solver in the didactics of our science. His book is first of all a personal lamentation that has arisen from his seeing how too many basics have gone all wrong. Many of us who feel the same might have reacted differently, but his lively and persuasive pamphlet is a necessary reminder of how all is lost if the joy of doing mathematics and the students’ right to experience it are not at the heart of mathematics education. Keith Devlin in his foreword recommends this book as mandatory reading for every parent, educator, and government official with responsibilities toward the teaching of mathematics. Moreover, he would have loved to have had Paul Lockhart as his school math teacher. I definitely agree.

Ó 2014 Springer Science+Business Media New York DOI 10.1007/s00283-013-9438-9

REFERENCES

[1] K. Devlin, Lockhart’s Lament—The Sequel (Devlin’s Angle, May 2008). http://www.maa.org/external_archive/devlin/devlin_05_08. html. [2] A. Ralston, ‘‘Research Mathematicians and Mathematics Education: A Critique,’’ Notices Amer. Math. Soc. 51(4) (2004), 403–411. [3] W. Schmidt, ‘‘A Mathematician’s Lament—A Book Review,’’ Notices Amer. Math. Soc. 60(4) (2013), 461–462. [4] T. Tossavainen and E. Pehkonen, ‘‘Three Kinds of Mathematics: Scientific Mathematics, School Mathematics and Didactical Mathematics,’’ Far East J. Math. Educ. 11(1) (2013), 27–42.

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