Sci & Educ DOI 10.1007/s11191-013-9638-x

Cultivating Parabolas in the Parlor Garden: Reconciling Mathematics Education and Feminine Ideals in Nineteenth-Century America Andrew Fiss

 Springer Science+Business Media Dordrecht 2013

Abstract This article introduces the justification problem for mathematics, which it explores through the case study of 1820s–1840s rationales for the teaching of mathematics to women in the United States. It argues that, while educators in the 1820s justified women’s studies through mental discipline (a common reason for men’s study), those of the 1830s–1840s increasingly relied on separate, gendered justifications, tied to emerging ideals of middle-class femininity. This article therefore emphasizes the contingency of the justification problem, which serves to break the present-day cycle of gender stereotypes regarding mathematics.

Mathematics faces a justification problem (Jensen et al. 1998). Why teach mathematics? What purposes, goals, methods, and values support its presence in varied curricula? How can current curricular choices be justified and future ones rationalized? Or, to go even further, as educational philosopher Paul Ernest does, ‘What should be the reason for teaching mathematics, if it is to be taught at all?’ (Ernest 1998, p. 33) Paul Ernest, a social constructivist, argues that no universal solution exists to the questions above and that the justification problem is eminently social, evolving, relative, and contingent (Ernest 1998). This article, following Ernest, explores the historical contingency of the justification problem through a specific case: when the teaching of algebra and geometry began to be extended to American women in the 1820s–1840s. It argues that, while educators in the 1820s justified women’s studies through mental discipline (a common reason for men’s study), those of the 1830s–1840s increasingly relied on separate, gendered rationales, tied to emerging ideals of middle-class femininity. The context of the burgeoning American middle class proves essential to our understanding of this change. The first half of the nineteenth century witnessed the economic and social transformation of the United States, resulting in the establishment of bifurcated

A. Fiss (&) Writing Program and History Department, Davidson College, Davidson, NC, USA e-mail: [email protected]

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gendered ideals located in the emerging middle class. While industrialization and economic growth shifted sites of production, increased trade, and fed urbanization, they also, in the words of historian Nancy Woloch, ‘opened new avenues of upward mobility’, especially in the Northeastern United States (Woloch 2000, p. 119). The new middle class opened new physical, literary, and educational spaces, including parlors, periodicals, common schools, and colleges, and these did allow both men and women access to scientific and intellectual discourse at a high level (Bledstein 1976; Warner 1978; Kohlstedt 1990). But men and women increasingly faced different ideals about their vocations. As American market economies developed, men increasingly left home for work in factories, trade, business, and the professions, and women gained authority over the home, where they would perform tasks of washing clothing, baking bread, and producing appropriately middle-class children (Woloch 2000, pp. 120–122). These vocational bifurcations led to idealized character traits: men as time-oriented, market-driven, aggressive, and individualistic and women as task-oriented, pure, pious, gentle, and altruistic (Woloch 2000, p. 125). Meanwhile, the ‘woman’s sphere’ of the home came to mean an escape from the crass, commercial, crowded world of market-minded men, and, though only a small percentage of American families had the means for women to escape work outside the home, many still supported the underlying ideals (Woloch 2000, p. 121). Gendered rationales for mathematics emerged in the 1820s–1840s, in response to the emerging ideals associated with the new middle class. Since this article is concerned with explicit statements of reasons for academic study, it primarily focuses on school advertisements and articles in the periodical press, as well as select educational monographs. In examining such sources, it asks: what rationales existed for mathematical study? How did they change, when, and why? In the past, how have institutional structures justified the extension of mathematics education? How have such reasons and goals affected students’ perception of the study? And how are answers to the justification problem interconnected with employment opportunities for mathematical adepts? Following the example set by educational historians Nerida Ellerton and Ken Clements, it uses ‘school’ as an umbrella term for a wide variety of educational institutions that were variously called ‘dame schools’, ‘common schools’, ‘public schools’, ‘subscription schools’, ‘evening schools’, ‘normal schools’, and ‘academies’. Such distinctions meant little at the time, and doing away with them does not confuse the historical record (Ellerton and Clements 2012, pp. 2–3). Furthermore, when gender enters the analyses, it follows Joan Scott’s definition of gender as both ‘a constitutive element of social relationships based on perceived differences between the sexes’ and ‘a primary way of signifying relationships of power.’ (Scott 1986, p. 1067) In this way, the historical analyses follow the justifications of mathematics education for both men and women, as well as the ways in which, for nineteenth-century Americans, mathematical knowledge equaled power. The following article begins with the reasons that adolescent boys learned mathematics: especially the rhetoric of ‘mental discipline’. It subsequently investigates the ways in which educators of the 1820s–1840s made mathematics an acceptable subject for young women to study. While in the 1820s proponents of women’s mathematics education extended ‘mental discipline’ to women, by the 1830s they developed new rationales stemming from ideals of middle-class femininity. By emphasizing the contingency of the emergence of gendered reasons for the study of mathematics in the United States, this article breaks the cycle of gendered stereotypes associated with mathematics, explored further in the final section.

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1 Justifying Men’s Mathematics Education, 1820s What were common rationales for men’s mathematics education in early-nineteenth-century America? The importance of mathematics in the education of adolescent boys crystallized around the rhetoric of ‘mental discipline’. In this framework, mathematics represented the pursuit of mental perfection through frequent practice. Just as physical exercise allowed the body to acquire superior abilities, the mental exercise of mathematical problems allowed the mind to improve (Roberts 2001). Repeated mathematical exercise would confer skills that could be applied to any situation involving the presentation and interpretation of argument. The foremost mental disciplinarian, John Locke, argued that the ability to transform quotidian arguments into mathematical ones would not just improve the student’s abilities of communication, but would also raise him to the position of the judge of all other men. Locke had the autodidact, the primary figure in Of the Conduct of Understanding, study algebra, in order to gain the ability to weigh the assertions of others, and geometry, in order to separate distinct ideas and lay them out in logical order (Locke 1706). While he might not understand all of the content of the argument presented to him, the man proficient in mathematics could at least decide which claims to accept based on structure. When earlynineteenth-century American schoolteachers and college professors parroted such education rhetoric, they emphasized to their male students that mathematical study transformed them into leaders (Day 1814; Roberts 2001). Through the doctrine of ‘mental discipline’, mathematical knowledge was power. ‘Mental discipline’ provided one response to the question of the social utility of mathematics (Day and Kingsley 1829), and the typical sequence of mathematical subjects at school and college provided another. In the usual course of studies, a student learned reading, followed by writing and then arithmetic (Monaghan 2005). Studies in arithmetic took years, because of the framework for memorizing individual rules for specific types of problems, as well as extensive units about, for instance, the monetary systems of Europe and the United States (Cohen 1982; Ellerton and Clements 2012). If the student decided to continue pursuing mathematics, he would then take a year of algebra, especially devoted to the derivation of those rules that he had memorized in arithmetic classes (National Council of Teachers of Mathematics 1970, pp. 27–28). Translation and adaptations of Euclid’s Elements structured his proof-based year of geometry (Kidwell et al. 2008, p. 8). Finally, trigonometry was the third year past arithmetic, and it ended with units in mensuration, surveying, and navigation (Tolley 2003, pp. 82–83). Because of the presence of such ostensibly practical subjects at the end of the course of mathematics, students understood the usefulness of mathematics as its application to specific professional pursuits. The presence of navigation and surveying served to construct mathematics as an exciting, even dangerous, pursuit, and some educated men held on to the somewhat unfounded idea that arithmetic, algebra, and geometry were the beginning of the road to travel and adventure (Tolley 2003, pp. 83–84).

2 Extending Mental Discipline to Women, 1820s Justifying mathematics education through its applications made mathematics unsuitable according to emerging ideals of middle-class femininity. While women were not as numerate or literate as their male counterparts in colonial America, new educational programs after the Revolution, particularly the presence of mental arithmetic at common

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schools, increasingly taught both men and women arithmetic (Cohen 1982, pp. 139–140; Cohen 2003). Educators, such as Benjamin Rush, broke new ground through arguing that women should be able to keep household accounts and help in a family business (Cohen 1982, p. 141). However, the emerging ideal of femininity as opposing market materialism generated myriad arguments against women learning about money and numbers (Cohen 1982, pp. 142–143; Monaghan 2005, p. 373). Furthermore, algebra, geometry, and trigonometry proved even more controversial. ‘GEOMETRY. The sound of this work in reference to females, is very terrific’, expounded James Fishburn, an educational writer, in 1828. ‘Parents startle at it as though it possessed some talismanic power of converting their delicate daughters into tempest-beaten rovers of the deep, and sun-burnt surveyors of the forest.’ (Fishburn 1828, p. 20) In focusing on the place of navigation and surveying at the end of a mathematical course of studies, Fishburn succinctly demonstrated the reasons why parents found mathematics unfit for their daughters: mathematics hardened young women’s ‘delicate’ bodies; in the guise of surveying and navigation, it exposed them to pain and disease in a distant wilderness. Educators therefore faced the justification problem writ large: how to rationalize the presence of mathematics in women’s curricula without some, albeit vague, connections to professional pursuits. In the early 1820s, New York teacher Emma Hart Willard publicized the problems inherent in offering mathematics for women. When Willard, an autodidact, took over the Waterford Academy near Albany, New York, her curricular choices drew intense public scrutiny (Lutz 1964, p. 76; Woody 1929, p. 309; Fenner and Fishburn 1944, p. 36). The school ended its academic year with a public examination when members of the community were invited to hear the pupils answer questions on their studies, and these events became a regional spectacle when the more advanced students started to answer questions in geometry (Lutz 1964, p. 77; DeBare 2004, p. 31). Mary Cramer, the daughter of a local politician, was the first, and the audience was variously suspicious and supportive. On the one hand, some found Cramer’s performance literally unbelievable. As educator Henry Fowler remembered, ‘Some said it was all a work of memory, for no woman ever did, or could, understand geometry’ (Fowler 1859, p. 147). In other words, some audience members thought that Cramer had been taught to mimic mathematical thought, as an animal might. On the other hand, others found such demonstrations inspiring. One member of the audience sent an anonymous letter to the Albany Gazette, later reprinted in the Saratoga Sentinel, that began, ‘On Tuesday, for the first time in my life, I had the pleasure of hearing classes of young ladies, from ten to eighteen years of age, demonstrate with correctness and promptitude the most abstruse propositions of Euclid.’ The correspondent continued, ‘the young ladies manifested proficiency, which would astonish those…who yet remain incredulous in regard to the powers of the female mind’ (Albanian 1821, p. 3). For this anonymous ‘Albanian’, Cramer and her classmates demonstrated the possibilities available in a new educational regime. As both of these responses show, in offering mathematics to women, Willard faced a divided public. After Willard became head of the school in nearby Troy, New York, she converted naysayers through soliciting powerful supporters. On visiting the Troy Female Seminary, the Revolutionary War General Gilbert du Motier, marquis de Lafayette encountered an astounding display. The students, arranged in ranks, greeted him outside the school, where they had hung patriotic decorations. Through this scene, a committee of local matrons approached him and read a formal letter, in which they called the female students ‘a living testimony to the blessing conferred by that Independence, which you, sir, so essentially contributed to establish, and in which our sex enjoy a prominent share’ (Foster 1824, p. 170). After he expressed his pleasure at the welcome, Lafayette ascended the steps of the

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schools to meet Emma Willard, passing the inscription: ‘WE OWE OUR SCHOOLS TO FREEDOM; FREEDOM TO LA FAYETTE’ (Foster 1824, p. 171). The letter and inscribed decorations made the message of the meeting clear, in geometrical reasoning: that Lafayette’s actions led to the creation of schools that offered women novel academic opportunities, including mathematics. Few would dare criticize an institution endorsed (even implicitly) by such an influential Revolution War hero. Other schools noted the importance of justifying mathematics for women. ‘It may be asked of what use will the study of Algebra, Geometry, &c. &c. be to young Ladies?’ began the trustees of the new Female Department of the Providence High School (Dewitt and Kingsbury 1828, p. 428). In ruminating on this rhetorical question, the Providence trustees demonstrated the concerns with social utility inherent in the justification problem. New schools founded on the model of Willard’s innovative curricula faced similar anxieties. Catharine and Harriet Beecher consciously followed Willard’s lead and opened a school in Hartford, Connecticut that listed Jeremiah Day’s Algebra and Euclid’s Elements of Geometry among its studies (Tolley 2003, p. 81). But their educational program was short-lived, and Catharine Beecher’s advocacy of arithmetic faced scorn from the community (Cohen 1982, p. 146). Even the young women at such institutions needed to be convinced that mathematics was appropriate for their contemplation (Cohen 1982, p. 146). When faced with this conundrum, educators of the late 1820s relied on a modified rhetoric of mental discipline. The trustees of the new Female Department of the Providence High School answered the justification question through recourse to the well-regarded assertion that mathematics trained the mind (Dewitt and Kingsbury 1828, p. 428), and so too did Catharine Beecher, when she told her students that ‘the chief benefit to be derived from attending to both Arithmetic and the higher branches of Mathematics, is the beneficial influence they exert in calling into exercise, disciplining, and invigorating the powers of the mind’ (Cohen 1982, p. 146). Mental power, after all, was not assumed to be threatening on its own, particularly when directed toward oneself. Even girls, educators and parents agreed, needed to learn to govern their behavior. But extending mental discipline to the realm of women’s education was not such a simple matter. Since Locke’s arguments relied on the sense of raising the student to the pinnacle of judge of other men, they did not apply so neatly to women faced with nascent middle-class binaries of market/home. Rather, the trustees, Beecher, and others emphasized the importance of training the mind for further study in school instead. The Providence trustees asserted that ‘the mind must be prepared for the reception of…knowledge’ (Dewitt and Kingsbury 1828, p. 428), while the director of the South Carolina Female Institute argued that mathematics conferred ‘a habit of correct reasoning, and a method of pursuing knowledge’ important for further academic work (Marks 1828, p. 19). This rhetorical move did skirt concerns about social utility, but it also translated the justification problem for mathematics into a broader question: why cultivate the love of learning in students? John T. Irving, a founder of the curriculum of the New-York High-School for Females, provided a possible response: that women would pass on their academic enthusiasms to their children (pp. 20–24). So, when Irving included mathematical subjects in his curriculum, he did so for the explicit reason that the students could then pass on such lessons to future generations (Irving 1826, pp. 20–24). In the rhetoric of mental discipline, while Locke emphasized judgment and physical exertion (for young men), Irving and his contemporaries connected mathematics and mental power to academic achievement and intelligent progeny.

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3 Justifying Mathematics Through a Gendered Lens, 1830s–1840s While educators of the 1820s relied on a modified rhetoric of mental discipline, the 1830s and 1840s saw a marked divergence in justifications for men’s and women’s mathematical study amid a boom in schools offering the subject to pupils of both sexes. In her survey of ninety-one schools, historian Margaret Nash found that nineteen percent offered algebra in the 1820s, while sixty-seven percent did in the 1830s. Meanwhile, the percentages associated with geometry more than doubled, from thirty-four percent in the 1820s to seventyfour in the 1830s (2005, p. 87). Furthermore, at a time of drastic regional differences, New England schools showed the greatest opportunities for young women who wanted to learn mathematics, with ninety percent offering geometry, while Southern and Mid-Atlantic schools lagged far behind, with only half offering even algebra (Nash 2005, p. 87). But, with the growing popularity of middle-class gendered ideals, the rationale for such educational programs increasingly looked to the languages of cultivation and purity rather than mental discipline. In 1830–1831, Almira Hart Lincoln Phelps, Emma Willard’s sister, provided a new framework for discussing the virtues of mathematics for young women after she had become temporary head of the Troy Female Seminary during one of Willard’s lecturing tours. Phelps compared women’s minds to flowers or fruit trees, which benefited from careful attention and pruning. ‘We may now consider the human mind as a garden laid out before us; he who created this garden, planted in it seeds of various faculties; these do indeed spring up of themselves, but without education, they will be stinted in the their growth, choked with weeds, and never attain the strength and elevation of which they are susceptible,’ she professed in her lecture about the importance of mental discipline. ‘The skillful gardener knows that his roses require one mode of culture, his tulips another, and his geraniums another; and that attention to one of these, will not bring forward the other. So ought the mental cultivator to understand that the germs of the various faculties should be simultaneously brought forward.’ (Phelps 1836, pp. 98–99). With an original mixture of faculty psychology, educational philosophy, and horticultural metaphors, Phelps therefore justified pedagogical choices through recourse to mental cultivation, not mental discipline. Such rhetoric spread not just throughout the education literature of the 1830s but also to the periodical press, including the popular women’s journals The American Ladies’ Magazine and The Lady’s Book (SFW 1834; Hale 1838). This rhetorical shift allowed educators to argue against the common misconceptions concerning women’s mathematical studies. In 1840, Joseph Matthews, principal of the Oakland Female Seminary in Ohio, gave an address to the regional ‘College of Teachers’ that began with the complaints that he heard from parents about their daughters’ mathematical studies (1841, pp. 50–53). Assuming that young women could handle the presupposed mental strain associated with mathematics, parents worried that these studies would encourage daughters to revolt against their husbands and their God. Like Eve’s snake, mathematical knowledge could tempt them to overturn matrimonial and religious hierarchies (Matthews 1841, p. 51). Others questioned the fiscal sense of educating young women beyond their needs (Matthews 1841, p. 51). And still others complained that mathematics would destroy femininity, saying that ‘the rough and difficult studies of boys would destroy the peculiar charms of the female character by rendering it rough and masculine.’ (Matthews 1841, p. 51) While Matthews responded to the religious objection through chastising parents for using the Bible to justify their prejudices, and he argued with the financially minded through long speeches about the ways that education is far better than a mere monetary inheritance, his answer to the final objection relied on a modified

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sense of Phelps’s mental cultivation (Matthews 1841, pp. 51–52). Mathematical study does not harden or ‘roughen’, it ‘polish[es]’, ‘enlightens the intellect’, and ‘refines the whole soul’ (1841, p. 51). Matthews therefore proposed women’s minds, like precious metals, improved through processes of purification, and, in this case, mathematics provided the best tool to perform the task (1841, pp. 52–53). Thus, for Phelps and Matthews, as well as many of their contemporaries, the feminine, middle-class ideals of cultivation and purity provided persuasive answers to the justification problem (Woloch 2000, pp. 130–135). Such rhetorical strategies added purpose to the growing trend of offering algebra and geometry for young women but they did not support claims for separate but equal curricula. When Mark Hopkins, president of Williams College, spoke before the Mount Holyoke Seminary in 1840, he observed that no one had addressed the problems raised two decades earlier about geometry and trigonometry leading to navigation and surveying. ‘The great motive with men in studying languages and mathematics’, Hopkins said, ‘is not, generally to cultivate their faculties, but to prepare themselves for the attainment and practice of their professions. There evidently is not the same reason for teaching young ladies navigation, and engineering, and Hebrew, as if they were expected to take the command of our men-of-war, or lay our railroads, or expound the Old Testament’ (1847, p. 181). Like Phelps and Matthews, Hopkins too supported gendered justifications for mathematics education, but he noted the ways that such rationales would result in the establishment of separate, unequal curricula (Hopkins 1847, p. 81). Without men’s job prospects, women would necessarily need a shorter and more limited course of studies. Yet women did experience novel employment opportunities in the 1830s–1840s. At this time, expansion of the common school system created a shortage of willing teachers (Woloch 2000, p. 132). While New England seminaries had been supplying female teachers to that region for years, school boards elsewhere began to realize the fiscal sense of hiring a staff of female teaching assistants who could be paid half of men’s salaries or less (Melder 1972, pp. 19–32). While girls’ schools had been relying on female staffs for years, coeducational institutions increasingly hired women to teach all subjects, even mathematics, to girls as well as boys (Barnard 1841; Tolley 2003, pp. 86–87). For Catharine Beecher, supplying female teachers to states outside of New England became a personal mission (Woloch 2000, p. 134). Though women were assumed to have more familial obligations than men, and therefore less opportunity for travel, Beecher wrote articles, created societies, and founded schools, all to promote the idea of sending female schoolteachers to the West (Woody 1929, pp. 321–325). Beecher considered her teachers’ movement similar to a missionary movement: a chance to extend Northeastern ideals of middle-class femininity throughout the nation (Sklar 1973). The instructors at the Troy Female Seminary, noticing this atmosphere, increased their already trendsetting mathematics standards. The autodidact Emma Willard had taught her successors, but by the 1830s–1840s she had to leave Troy often for lecturing and teaching tours (Scott 1979). In her absence, her mathematics assistants took on more of her duties, and, in 1847, Mary Hastings took over as head mathematics teacher (Goodsell 1931, p. 38). Hastings, a Troy graduate who had assisted at a school in nearby Hudson, New York, had a passion for teaching mathematics and science, and she reworked the classes in geometry, trigonometry, applied mathematics, mechanics, optics, and astronomy, all the while adopting the same textbooks used at Yale (Goodsell 1931, p. 38). Thus, through Hasting’s influence, the Troy Female Seminary once again offered a radically progressive mathematics curriculum to young women. Students at similar institutions reveled in their educations, even when they found their studies difficult. A graduate of the Monticello (Illinois) Female Seminary remembered: ‘I

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had a natural distaste for mathematics, and my recollections of my struggles with trigonometry and conic sections are not altogether those of a conquering heroine. But my teacher told me that my mind had need of just that exact sort of discipline, and I think she was right’ (Larcom 1889, p. 266). Others, like an anonymous student from the Abbot Academy in Massachusetts, created games out of displaying their mathematical prowess against male relatives. ‘I became enamored of mental arithmetic’, she said, ‘and carried my Colburn’s Sequel [textbook] back and forth from school, trying to puzzle my father and brothers over the examples I had conquered.’ (McKeen and McKeen 1880, p. 18). Whether a personal battle or a way of competing with male relatives, mathematics therefore built the confidence of mid-century young women (DeBare 2004).

4 Conclusion The 1820s through the 1840s marked a key moment in the mathematical education of Americans. Programs that supported women’s mathematical study faced the objective, global version of the justification problem: rationalizing ‘the very existence of studies involving mathematics’ (Jensen et al. 1998, p. 10). Though the 1820s saw the extension of mental discipline (the common rationale for men’s mathematics classes) to women, the 1830s–1840s witnessed an increasing reliance on divergent, gendered ideals. Thus, responses to the justification problem changed in response to the emerging middle-class conceptions of the market-minded, crass male and the pure, cultivated female. This emphasis on contingency is especially necessary when dealing with the case of gender and American mathematics, since it is often treated without sufficient historical reflection. Educational commentator Bridget Murray and the American Association for the Advancement of Science generalize that boys trans-historically outperformed girls in mathematics (Murray 1995, p. 43; 1990, p. xviii). Even the historian Stanley Guralnick confirms these views, relying on the sense that women trans-historically have received less exposure to mathematics than men (1975, pp. 54–55). The propagation of these impressions becomes a self-fulfilling prophecy: views of mathematics as trans-historically gendered lead to unequal educational opportunities, unequal participation rates in mathematics careers, and an ultimate confirmation of the initial gender stereotyping (Ernest 1998, p. 43). Through presenting a more complicated view of the history of women’s mathematics education, this article attempts to break the cycle. While educational historian Kim Tolley has elsewhere demonstrated that nineteenth-century American girls did sometimes exceed boys in their mathematical training and achievements (Tolley 2003), this article argues that separate, gendered rationales associated with mathematics emerged at a particular historical time and under specific circumstances. This suggests unanswered research questions about other situations in which students are taken to be representatives of social groups instead of as individuals. In cases of race, class, and national origin, as well as gender, it is important to understand if trans-historical generalizations and self-fulfilling prophecies exist in the teaching of mathematics and science, leading to the establishment of what the National Science Foundation and its offshoots call URMs: ‘underrepresented minorities’ (Poirier et al. 2009, p. 1; Smith 2014). To complicate such separate rationales, other studies can explore the contingency of the justification problem associated with various splits in pedagogical rhetoric. An international comparison would be particularly valuable in this research program, and some scholars, such as Bent Jørgensen, have already started compiling comparisons on a nation-

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by-nation basis (Jørgensen 1998). More work can and should be done to create multifaceted histories of mathematics education.

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