The Role of Mathematics in Liberal Arts Education

Grabiner, Judith V.

doi:10.1007/978-94-007-7654-8_25

Judith V. Grabiner²

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Abstract

The history of the continuous inclusion of mathematics in liberal education in the West, from ancient times through the modern period, is sketched in the first two sections of this chapter. Next, the heart of this essay (Sects. 3, 4, 5, 6, and 7) delineates the central role mathematics has played throughout the history of Western civilization: not just a tool for science and technology, mathematics continually illuminates, interacts with, and sometimes challenges fields like art, music, literature, and philosophy – subjects now universally considered to be liberal arts. Section 8 adds an international perspective to the contemporary liberal arts story by describing some instructive mathematical achievements from many cultures and societies. Finally, Sect. 9 addresses how contemporary mathematics teaching can use the history of mathematics viewed as a liberal art to enhance the appreciation and understanding of mathematics for all students.

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History of Mathematics, Mathematics Education, and the Liberal Arts

Low-Attaining Secondary School Mathematics Students’ Perspectives on Recommended Teaching Strategies

Article Open access 12 October 2023

Lower secondary students’ encounters with mathematical literacy

Article Open access 16 July 2021

Notes

1.
In modern discourse about education, the term “liberal arts” has been defined and characterized in a variety of ways. It is of course built into the history of Western education. The influential Carnegie Foundation for the Advancement of Teaching has provided a list of contemporary “liberal arts” that goes beyond the traditional Western canon: English language and literature, foreign languages, letters, liberal and general studies, life sciences, mathematics, physical sciences, psychology, social sciences, visual and performing arts, area and ethnic studies, multi- and interdisciplinary studies, philosophy, and religion (Carnegie 1994, p. xx; Ferrall 2011, p. 9). In antiquity, as the next sections of this paper will describe, the list came to have seven items: arithmetic, geometry, astronomy, and music theory (the quadrivium), and grammar, rhetoric, and logic (the trivium) (Stahl 1977; Wagner 1983b). Those who compile such lists characterize liberal arts education as study undertaken for its own sake, as opposed to vocational education. As for the purpose of liberal arts education, it has been described as educating a free person, or as liberating the mind to pursue the truth, or as producing a cultivated person who can be both a good citizen and a leader of society. Liberal arts instruction has sometimes focused on the canonical texts of Western civilization as ways to build and reinforce the shared values of society. But such instruction has also been championed as suitable for the education of free individuals to be citizens in a democracy by developing the capacities for independent and critical thinking, logical analysis, effective communication, an understanding of the interrelations between different fields of learning, and imagination (Ferrall 2011; Kimball 1995; Nussbaum 2010; Sinaiko 1998). The present chapter recognizes and appreciates these disparate views. The Carnegie list is useful to illustrate the types of subjects that constitute a liberal arts education today, and the present essay shares the view that modern liberal education’s most important goal is to educate independent and thoughtful citizens.
2.
For Proclus, see Proclus (1970, p. 53). The term “mathematics” itself reveals a liberal arts origin; the Greek root “mathema” was first more general, connoting merely “something learned,” and the “mathematikoi” were the inner initiates of the Pythagorean school. For pre-Euclidean logically structured geometry, see Knorr (1975, esp. p. 7) and McKirahan (1992, pp. 16–18).
3.
See Chapple (1993a, p. 7), Cesareo (1993, p. 17), Cruz (1999, p. 250), and Taton (1964).
4.
Distinguishing experimental philosophy from reasoning from arbitrary hypotheses, Newton wrote, “In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction” (Newton 1934, p. 547). Once Newton had his general principles, his Principia could take the logical structure familiar from Euclid’s Elements.
5.
Khwārizmī’s title can be translated as “the book of restoring and balancing,” where the Arabic “al-jabr” or “restoring” was interpreted as adding the same thing to both sides of an equation and “al-muqabala” or “balancing” the subtraction of the same quantity from both sides of an equation (Berggren 1986, p. 7). The sense of “al-jabr” as “restoring” remains in Spanish, where, for instance, in Don Quixote, Part II, Chap. XV, a bonesetter is an “algebrista” (Merzbach and Boyer 2011, p. 207).
6.
The point will be clearer with a fuller quotation: “With the feeling…that he was setting forth an important axiom, he wrote: Freedom is the freedom to say that two plus two make four. If that is granted, all else follows” (Orwell 1949, p. 81; his italics).
7.
(Fauvel et al. 2003; Field 2003; Helmholtz 1954; Karp 1983; Jeans 1956; Newman 1956, pp. 2278–2309 Wardhaugh 2009; Wollenberg 2003)
8.
(Olson 2008, pp. 96–121; Richards 2002, pp. 11, 308–310). The Romantics would not admit that what epistemologists call “secondary qualities” like color, so constitutive of human experience, are mere epiphenomena reducible to “primary qualities” of matter in motion nor that greater understanding necessarily follows from mathematical description. As John Keats put it in criticizing “philosophy” (science) in his poem Lamia (part 2):

Philosophy will clip an angel’s wings,

Conquer all mysteries by rule and line,…

Unweave a rainbow.
9.
(Joseph 2011, p. 462). Sebokht, a Syrian bishop, was challenging the supposed universal superiority of Greek scientific thought by praising the superior methods of calculation using the base-10 place-value number system from India.
10.
There is now an extensive and reliable English-language literature on the mathematics of other cultures. See, for instance, Ascher (1998, 2002), Berggren (1986), (2007), Closs (1986), Dauben (2007), Gerdes (1999), Gillings (1972), Imhausen (2007), Katz (2000), (2007), Martzloff (1997), Plofker (2007), (2009), Robson (2007), (2008), Robson and Stedall (2009), Van Brummelen (2009), and Zaslavsky (1999).
11.
See Katz (2009, pp. 17–18, 84–88), Robson (2007, pp. 100, 140–141, 151), and Robson (2008, pp. 109–115, 218–219).
12.
The question is, if long or heavy syllables are two beats and short or light syllables are one beat, what is the number of different arrangements A(n) of long and short syllables for a line of n beats? For example, if there are two beats and if we use “S” for short and “L” for long, the arrangements are SS and L. If there are three beats, the arrangements are SSS, SL, and LS. If there are four beats, the arrangements are SSSS, SSL, and SLS (formed by placing an S in front of each of the arrangements for three beats), plus LSS and LL (formed by placing an L in front of each of the arrangements for two beats). Thus, A(4) = A(3) + A(2). Since A(2) = 2 and A(3) = 3 and since the method of forming A(n) from A(n-2) and A(n-1) must follow the same pattern, this gives the Fibonacci series (Singh 1985).
13.
Discussions about how this can and has been done, and how it has been assessed, may be consulted in Alternatives for Rebuilding Curricula Center (2003), Ball et al. (2005), Boaler and Staples (2008), Hill et al. (2005), and Tarr et al. (2008). A cross-cultural study involving Chinese and American teachers at the elementary-school level can be found in Ma (1999).
14.
Examples of books that might be suitable for such courses include Ascher (1998 and 2002), Frantz and Crannell (2011), Gerdes (1999), and Packel (1981). For details about the author’s courses, see Grabiner (2011). A superb online resource for liberal arts mathematics teaching is the Mathematical Association of America’s “magazine” of the history of mathematics and its uses in the classroom, Convergence (n.d.).
15.
On women in mathematics in general, see the online biographies maintained by Agnes Scott College (2012), the sourcebook Grinstein and Campbell (1987), and the Mathematical Association of America’s poster Women of Mathematics (MAA 2008). On important individual women in mathematics, see Arianrhod (2012), Brewer and Smith (1989), Dahan-Dalmédico (1991), Deakin (2007), Hagengruber (2012), Katz (2009, pp. 189–190, 616–617, 714–715, 787, 874, 896–898, 899), Koblitz (1983), Mazzotti (2007), Neeley (2001), Reid (1996), and Zinsser (2006).

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Acknowledgments

This essay is dedicated to the memory of Herman Sinaiko (1929–2011), peerless teacher and scholar of the liberal arts. I am grateful to the anonymous referees and to the editor, for their scholarly expertise, comments, and criticisms, which have materially improved this chapter. I also thank the Pitzer family, donors of the Flora Sanborn Pitzer professorship, for their generous support of my research.

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Mathematics, Pitzer College, Claremont, CA, USA
Judith V. Grabiner

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School of Education, University of New South Wales, Sydney, New South Wales, Australia
Michael R. Matthews

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Grabiner, J.V. (2014). The Role of Mathematics in Liberal Arts Education. In: Matthews, M. (eds) International Handbook of Research in History, Philosophy and Science Teaching. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7654-8_25

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