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The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom - Uses of Historical Poems in the Mathematics Classroom

Author(s): 
Sarah Glaz (University of Connecticut)

The links between the history of mathematics, or mathematics-related art, and the teaching of mathematics are complex and difficult to classify. Generally acknowledged is the fact that both history and art can be used to enrich the teaching of mathematics by strengthening students' engagement with the material. The power of a historical anecdote or a poem to engage and charm may lie in the response to the contrast between the abstract and impersonal mathematical idea, and the emotional and aesthetic nature of a piece of related art or a historical tale associated with the idea. The power may also reside in the heightened interest generated by a presentation of mathematical ideas in a broad context, a context that includes historical, social, and artistic dimensions, in addition to the mathematical one. This pedagogical aspect of either history or art in the teaching of mathematics is present, even if the original reasons for their inclusion in the classroom may have been different. In addition to enrichment of pedagogy through engagement, both history and art are often used in the mathematics classroom to shape course content and to enhance learning, retention, and integration of material. The reference section includes a small, but representative, selection of sources: [5, 7, 21, 22, 23, 30, 31] contain discussions, ideas, and classroom resources for the use of history in teaching mathematics, while [2, 6, 8, 9, 18, 19, 20, 25, 29, 32] elaborate on parallel themes for the use of poetry.

The use of a combination of history and poetry in the high-school and college mathematics classroom is not as prevalent as using them separately. One reason may be that many of the historical sources of mathematics that were originally written in verse, such as, for example, much of the mathematics written in the Middle Ages in India, were translated as prose. Still, some historical poems are available in English, and poetry and history were also successfully combined in other ways to enhance the teaching of mathematics.

Poems, or rhymes, have been used for a long time as mnemonics for important numbers such as π and e, or to assist with the memorization of techniques or formulas, such as, for example, the formula for finding the roots of a quadratic equation. Some of these rhyming mnemonics are of historical vintage (see for example [18, 25, 32]).

An interesting use of the combination of history and poetry appears in [20], where history provides the motivation for the introduction of poetry in an algebra course. The motivating history is the flowering of algebra during the Middle Ages in India with its cultural tradition of recording mathematical results and problems in verse. Some of the most charming mathematical poems come from this tradition. For example, Bhaskara (1114-1185), the best known of medieval Indian mathematicians, wrote an algebra book some believe was intended for the education of his daughter, Lilavati. The book's title is also Lilavati (meaning "the beautiful"), and it was written entirely in verse. The translation of Lilavati into English in [10] is in prose, but luckily a few of the poems were translated as verses in other sources (see [18] for three such translations and their sources). Inspired by the mathematical poetry of medieval India, Barbara Jur [20] encouraged her algebra class to compose word-problems in poetry. The results have both mathematical and poetic merit. Jur's motivation was to enrich teaching by engagement, but in articles [2, 29] we find reports of such poetry writing experiments conducted in Pre-Calculus, Calculus, and Statistics classes that conclude that poetry writing in mathematics classes strengthens students’ understanding and integration of the subject matter.

The mathematical poetry of medieval India and the difficulties students have with word-problems in algebra feature in another article describing the use of poetry in a college algebra course [18]. Glaz and Liang [18] used poems from Lilavati and other historical sources to ease the difficulties students have with the transition between word-problems representing natural phenomena and the corresponding mathematical models—the equations representing the phenomena. The process yielded additional pedagogical benefits, such as the strengthening of students' number sense and mathematical intuition and the enhancement of retention and integration of the material [18].

In [18] the authors also introduce a poem by Glaz, "Calculus" [16, 17, 18], which, like "The Enigmatic Number e," combines biographical details about the mathematicians involved with the history of mathematical ideas. A poem of this kind may be used to enhance course content by acting as a springboard to class-wide or small group discussions, projects, or assignments on the topics presented in the poem. This kind of poetry-related classroom activities and their implications for the learning process are discussed in [9, 18, 25], while historical accounts in prose with suggestions for these kinds of classroom activities may be found in [5, 9, 31].

I will conclude this section with a few words about my own experience with the use of historical mathematical poetry. The poetry I use is either old poetry with mathematical content or contemporary poetry that combines mathematics and history in its content. In addition to class discussions, I assign group projects on, or related to, the mathematical content of the poem. As a rule, the more elementary the course, the more structured the project. The rationale behind my approach takes into account that students in higher level mathematics courses are academically and mathematically more mature and are therefore capable of extracting benefits from the inclusion of poetry and history in their class with less supervision. In particular, in a Problem Solving course for freshmen and in a remedial College Algebra course, the projects I assigned had step-by-step instructions to the mathematics involved along with an introduction describing some of the history of the poems. For samples of projects at this level, the reader may consult reference [18].

The poem "Calculus," mentioned above, was used in a Calculus class to enliven the review of the material at the end of the chapter on The Fundamental Theorem of Calculus. The project required students to "translate" from words into mathematical statements the lines of the poem describing mathematical concepts and theorems (such as, for example, the definition of the definite integral as a limit of Riemann sums and the Fundamental Theorem of Calculus itself). In a class for senior mathematics majors I offered Archimedes' "The Cattle Poem" [24] as a handout, with a recommendation that students explore any point that aroused their curiosity and an invitation to discuss their explorations with me. In general, students reacted to the poems and the projects with surprise and enjoyment—sometimes surprise at their unexpected enjoyment from a practice that is not common in the discipline.

Sarah Glaz (University of Connecticut), "The Enigmatic Number [i]e:[/i] A History in Verse and Its Uses in the Mathematics Classroom - Uses of Historical Poems in the Mathematics Classroom," Loci (November 2010), DOI:10.4169/loci003482

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